Talk:Perfect fifth

Sound files?

Aren't there some sound files someone can upload so that we can hear what each diatonic interval sounds like? Artichoke84 10:57, 13 March 2007 (UTC)

Ratio

In this article a just fifth "corresponds to a pitch ratio of 2:3".

From what I know pitch ratio, or interval is the ratio of higher frequency over lower frequency. Hence it should be 3:2 instead. What do you think?

I agree and I've changed it. (24 October '05)

I've added to the ratios conversation, by reordering the wording of a sentence. I also stated that the P5 is harmonically significant because it is the half-point of the octave (E.g., 2^7/12 = appr. 1.5) See Normalizing the Musical Scale for info to back up that claim. (GaulArmstrong)

It's true that normalising the frequency scale enables the 3/2 ratio of the perfect fifth to appear as 1.5 and therefore the halfway point of the octave. But that's not why it's harmonically significant. Its harmonic significance lies in the fact that it corresponds (in just intonation) with the very important 3rd harmonic generated by the fundamental. I think the more traditional view of having the midpoint of the octave represented by the note that corresponds with the ratio equal to the square root of 2 (1.414) i.e., the 6 semitone augmented 4th/ diminished 5th tritone identifies that interval's harmonic significance nicely and accurately. (Mark 10 February 2007)

I still contend the point, although your generally correct. I agree that the 3rd harmonic is the important subject, and not the 7th step of whetever-tet system we're using. No contest. But, what is central to my point is that the 3rd harmonic is the half-point of the octave. Take the 2nd harmonic and the 4th one (both U1's). They span exactly one octave. Now the 3rd harmonic (P5), you might guess is the midpoint of this octave because (2 4)/2=3. And surprisingly, it is this simple. But because frequency is an exponential scale based on 2^x, the midpoint is right-skewed to log₂(1.5). It doesn't seem to be the midpoint of the octave, because it falls on the 1.582...-point of the linear frequency scale. In linear terms-- no, the half-point is not the P5, but the tritone. So, if I pick up my guitar and play the note half-way between E and E, I'm going to play A#. But in terms of 'the way your and my ears work', the P5 is the half-point of the octave -- because we hear a linear pitch scale and not the exponential frequency scale. The question is which takes precedence, our ears or our tablature? I say our ears, and that's why the P5 is the half-point. (GaulArmstrong Feb 11 2007)

Well, no. Our ears say that the intervals E-A# and A#-E are identical. Then, A# is the midpoint between E and E, at least if we are going to accept the usual meaning of "midpoint". Musicians recognize a tritone whenever they hear it, even if they don't have a perfect pitch (relative pitch is of course necessary). And our ears say clearly that the intervals E-B and B-E are different. That's how we can tell the difference between a 5th and a 4th by ear. No need of playing, not to say looking at any instrument. Also, I am unable to see how a P5 could be considered the midpoint of the octave in any other way. Old Palimpsest 21:27, 23 February 2007 (UTC)

Merge with dominant

It has been suggested that this article be merged with Dominant. I strongly disagree. They are significantly different concepts.

What the article does now need is some repair work regarding the following points:

"It is the most important interval for chord structure, song development, and western tuning systems."

No - the 'third' is the most important interval in western (tonal) harmony - That's why it's called tertian harmony. As for tuning systems, the musical tuning article lists only Pythagorean tuning as being based on perfect fifths.

"Gregorian chant was the first formal composition"

Perhaps in Europe, but Indian, Persian and Arabic formal composition all predate Gregorian chant. And maybe not even in Europe, if we include music based on Greek modal systems.

(Heading) "Use in ....tonal systems" - This term needs needs explaining or a link to a relevant article.

I've made a couple of minor fixes for now, mainly rewording, and removed an irrelevant sentence concerning thirds, but would appreciate some comments on the above points before making any further changes - Thanks (Mark - 28 March 06)

• hello mark, glad for a speedy response. i made some changes to reflect your points.

"tonal systems" could be a better term, i agree, something that includes keys and scales and whatever.

"most important interval" - changed, silly to argue this point. with the impact on tuning systems, i was thinking of like gamelan music tuning, which sounds funny mostly because of its sharp 5th.

gregorian has been softened to "a very early formal music composition" or something

merge, lets keep the tag for now, to get other opinions maybe

thanks Spencerk 20:54, 28 March 2006 (UTC)

Hi Spencer. That's much better now. My only remaining concern is about it being merged with 'dominant'. Do you feel they should share an article? I can't see that they have that much in common, except that the dominant is a perfect fifth above the tonic. Anyway, thanks for your positive response to my comments (Mark 28March 06)

Which title would be kept? In other words, which article should be merged into which? Hyacinth 08:56, 29 March 2006 (UTC)

I'm a professed mergist, yet I don't think "Perfect fifth" and "Dominant" should be merged. They're totally different concepts. The interval of a perfect fifth is used in music that totally lacks the concept of tonic and dominant. —Keenan Pepper 15:30, 29 March 2006 (UTC)

I second Keenan's view that they should not be merged as they are completely different concepts. I can see no advantage in merging them. Fretsource (29-03-06)

Don't merge with dominant. The perfect fifth an interval. The dominant refers to a change of harmonic centre (which happens to be at the interval of a fifth). They're really very different. Rainwarrior 21:16, 4 April 2006 (UTC)

• i removed the merge suggestion and put dominant (music) in a 'see also'. There is a section called "popular wang ba dans", i dont know what this means and have found nothing on in google or wikipedia. Am beginning to think that it is vandalism because "wang ba dan" is a chinese swear word. Also, i'd like to see "music with a perfect fifth" clarified too, like does that mean contains a perfect fifth harmony? cause that would be crazy. Would be awesome to have this clarified on the page. any thoughts?Spencerk 21:53, 5 April 2006 (UTC)

I don't know where this "wang ba dan" thing is you're talking about, but it sounds like vandalism. It probably doesn't mean anything. Rainwarrior 22:37, 5 April 2006 (UTC)

I think the whole contribution of 70.24.220.81 should be reverted. Apart from the mysterious 'wang ba dans' the content is far from encyclopedic, and seems to be more about military marches (with inappropriate and unsuccessful attempts at humour), and the list of music containing perfect fifths is obviously nonsense - especially the 'obscure' Chopin concerto. Yet, I'm not sure it's vandalism as he (or she) seems to have gone to a lot of effort. I vote revert, but let's leave it for a bit to let others have a look or for the author to respond. (Mark 6 April 06)

I was right - He's not a vandal, he's a nut, as confirmed by another (now reverted) contribution to Chopin, which reported that 'wall-mines' were put in place to deter Chopin from damaging the walls by kicking them while playing... (wait for it)... perfect fifths. (Mark)

Diatonic Fifths

About diatonic fifths, it is said that the perfect fifth is

"one of three musical intervals that span five diatonic scale degrees; the others being the diminished fifth, which is one chromatic semitone smaller, and the augmented fifth, which is one chromatic semitone larger."

As far as I know, there is no augmented fifth in a diatonic scale; there are six perfect fifths and one diminished fifth. I think there is a confusion here with the diatonic minor sixths (E-C, A-F and B-G in the diatonic natural scale). Although they are indeed enharmonic to augmented fifths, they span six diatonic degrees, not five.

If you all agree, I suggest a correction here. Old Palimpsest 18:26, 23 February 2007 (UTC)

Eb to B in C harmonic minor? Hyacinth (talk) 00:13, 10 January 2008 (UTC)

Contradiction?

Isn't it contradictory to speak of a fifth as being "perfect" when equal temperament is used? Equality of temperament necessarily means that the fifth is not quite perfect and will have a slightly dissonant sound. I hoped to hear an actually perfect and fully consonant fifth with a 3-to-2 frequency ratio. Where's the sound file for that? Michael Hardy (talk) 05:07, 12 March 2008 (UTC)

---

This article begins as follows:

The perfect fifth or diapente (sometimes abbreviated P5) is a musical interval which is responsible for the most consonant, or stable, of the unison and octave.

I don't quite understand that. Could it have been intended to say that it is more consonant than any interval except the unison and the octave?

Later in the same paragraph it says the article says:

The prefix perfect identifies it as belonging to the group of perfect intervals (perfect fourth, perfect octave) so called because of their extremely simple pitch relationships resulting in a high degree of consonance.

With all this talk of "high degree of consonance", it seems very (pardon the pun) unjust to use as an example, for the reader to listen to, the approximation that occurs in the equally tempered scale. That interval is obviously NOT so highly consonant. If I had suitable software and knew how to create these sound files, I'd replace the one that's here. Michael Hardy (talk) 23:47, 12 March 2008 (UTC)

Michael, are you telling us that you can not only hear the difference between a just perfect fifth and an equal tempered one, but can also hear that the equally tempered one is less consonant? The difference is only 2 cents!! As far as I'm aware the limit of human pitch discrimination is around 5 cents, 3 or 4 if you're super sensitive - but 2 cents? Isn't that superhuman? (Mark - 28/3/08) —Preceding unsigned comment added by 84.68.179.71 (talk) 22:13, 28 March 2008 (UTC)

I don't know whether I can hear the difference between the two notes, the "G" and the slightly sharp "G", if you play then in succession, but I believe I can hear the difference between the two intervals---the C with the justly tuned G, and the C with the equally tempered G. I think the fact that it's played with another note at the same time may make it perceptible where it otherwise would not be. And I ask you: have you tried it, to see whether you can hear the difference? I can certainly hear the slight dissonance when that pair of notes is played on a piano. In about 2001 I encountered a musician who had an electronic instrument with a keyboard and with the ability to change from equal temperament to a scale on which the ratio of G to C was 3:2. I could hear the dissonance on the equally tempered scale and not on the one with the 3:2 ratio. Michael Hardy (talk) 23:19, 28 March 2008 (UTC)

I've just done as you suggested. I mixed a 5 second, 440Hz sine wave with a 660Hz one to get a "just" perfect fifth and then mixed the 440Hz one with one 700 cents sharper to get the equally tempered fifth. The only difference I could hear is what you'd expect - beats. There are about 8 noticeable pulses within the 5 second duration of the tempered mix. Obviously that beat frequency is too low to cause dissonance. Dissonance only occurs when the beats are too rapid to be perceived individually but are perceived instead as a roughness of sound. Remember, I'm using sine waves though. Complex wave mixes such as from certain musical instrument (especially synthesizers) may introduce dissonances via their various overtones, which are absent from pure sine waves. That's irrelevent to this article though, which should focus purely on the fifth, regardless of the harmonic series of the constituent tones. (Mark - 29 March 08) —Preceding unsigned comment added by 84.68.6.30 (talk) 20:28, 29 March 2008 (UTC)

Can you post those sound files hear so that I, and others discussion this, can hear them? Then the discussion might be able to advance beyond this impasse. Michael Hardy (talk) 17:09, 30 March 2008 (UTC)

There's an impasse? There's no need to post any files. The difference between a 'just' perfect fifth and an equally tempered perfect fifth is well known. The only difference is that the tempered one causes about 2 (faintly) audible beats per second. This is NOT dissonance.

Both are mathematically precise - one is precisely 3/2, the other is precisely 700 cents.

Both are equally consonant. There's nothing in the tempered one that can cause dissonance. A 2 beats per second variation in amplitude can't be perceived as dissonance by any stretch of the imagination. If you can hear dissonance in any properly tuned tempered perfect fifth, then it's probably caused by an interaction of the harmonics produced by the instrument, speaker cabinet, ear wax or whatever, not the perfect fifth itself.

Last but not least, the term perfect fifth is a musical term, describing a musical interval and its function within the music system that named it. It's not a physics or mathematical term describing frequency ratios. It's not concerned with the mathematics of any chosen tuning system. Whether its just or tempered, musically, it's still a perfect fifth. So I think you're missing the point in criticising the article's use of a tempered perfect fifth rather than a just perfect fifth. Musically, both are perfect fifths. (Mark - 30 March 2008) —Preceding unsigned comment added by 84.64.78.152 (talk) 19:20, 30 March 2008 (UTC)

OK, so you're UNWILLING to post those things here.

There's also a question of more complicated things than sine waves. The Fourier series of a seemingly simple square wave has higher-frequency components---overtones. The thing in the sound file here doesn't sound like a simple sine wave. Your remarks may apply to sine waves, but that's not what we've got here. Michael Hardy (talk) 21:47, 30 March 2008 (UTC)

That's right. I'm UNWILLING to post them here as this article is about a musical term. I might have considered it if the article was relating to acoustics or physics. But it's not, and such discussions are irrelevant here. The article mentions the difference between both types of perfect fifth and that's fine. If the reader wants to delve deeper into the differences from a mathematical or any other non-musical perspective, they will go to articles on equal temperament, acoustics, wave theory, etc, where such concepts are presumably discussed in detail.

Whether or not dissonance is introduced by any particular tuning system, or by the complex waveform of a particular instrument, in no way affects the status of an interval's quality. An equally tempered major third is noticeably less consonant than a just one - yet it's no less a major third because of it. (Mark - 30 March 2008) —Preceding unsigned comment added by 62.136.27.173 (talk) 22:45, 30 March 2008 (UTC)

How can the question of whether they're dissonant or consonant be irrelevant to the article when the article begins with a statement that this interval is more consonant than any other except the unison and the octave? Besides, the definition of the term is not all that matters in an article about the term. There's also the question of whether, or why, the concept is important. That depends in part on these issues of dissonance or consonance.

Also, you seem to assume that any comments I could post after hearing the sound files you're talking about could be of interest only if YOU are able to anticipate, BEFORE posting them, what I would say that would be of interest. Michael Hardy (talk) 23:32, 30 March 2008 (UTC)

OK, no knowlegdeable person is working on this...

It's been more than two weeks since I pointed out problems here, and I don't see people knowledgeable in music coming forward to say anything about my concerns, either to tell me I've got it wrong or otherwise, or to edit the article. I'm going to alter the mangled opening sentence of the article so that it says what I'm guessing it meant. Then I'll see if I can find some musically knowledgeable Wikipedians to assist further. Michael Hardy (talk) 21:38, 28 March 2008 (UTC)

It seems you misunderstand the distinction between intervals and tuning. There are serval types of fifths, viz dimished fifth, perfect fifth, and augmented fifth. How these intervals are implemented is the tuning. Does that answer your question? Brettr (talk) 10:03, 30 March 2008 (UTC)

How would it either answer any question I had or address any other concern I had? Michael Hardy (talk) 17:06, 30 March 2008 (UTC)

 — Preceding unsigned comment added by 95.113.10.112 (talk) 23:51, 4 April 2015 (UTC) 

"perfect"?

Perfect!

A perfect fifth is an exact (math relation) harmony, of 2 frequencies (pitches), where the 2nd pitch is exactly 3x the first, and divided by 2 (an octave). The frequency difference between any pair of frequencies is called an musical interval and that pair of pitches has a frequency ratio of 3/2 (meaning multiply by 3, & divided by 2). "Perfect" means "no error" (of harmonic (multiples)), thus exact math (=without tolerance).

(Wrongly stated below as, "or very nearly so".
A fifth is either perfect, or it is not.)

In music theory, a perfect fifth is the musical interval corresponding to a pair of pitches with a frequency ratio of 3:2, or very nearly so.

(That statement above is (almost) correct, if we delete

", or very near so",
The colon ":" also does not explain the math well enough
(for most people).

Using concert pitch at A 440 Hz

440 Hz x 3= 1320 Hz)/2=660 Hz Perfect fifth.

As can be seen

440 Hz x 3= 1320 Hz):(440 Hz x 2=880 Hz) is in the next octave above, unnecessarily so.

660 Hz/3=220 Hz):(440 Hz/2=220 Hz) refers to the octave below concert pitch 440 Hz.)

An equal tempered fifth is not harmonic, at 659.26 Hz, thus "beats" with 0.74 Hz (error)!

A good ear can hear that.

That error is a form of modulation.

Such a small error (1.955 cent) is tolerable,

but if it gets large enough, like a Pythagorean comma (~24 cent=1.955 cent x 12)

it becomes very noticeable, & that (large error) is called a wolf

(because it sounds so terrible, it growls).

We use the word fifth to say, the first and "fifth" finger are used to play this interval ("pair" of notes) on the keyboard.

Meaning we (ignore &) don't bother mentioning the 1st note, which is there (being played).

Peter Sheedy

Email [email protected]

"perfect"?

The article now says this:

The term perfect identifies it as belonging to the group of perfect intervals (perfect fourth, perfect octave) so called because of their simple pitch relationships and their high degree of consonance.

Several people who have posted on this page have told me that the fifth on an equally tempered scale should be included as one example of a "perfect fifth". If that is true, then the statement that it has this "simple pitch relationship" and "high degree of consonanace" is false. Michael Hardy (talk) 03:25, 8 April 2008 (UTC)

Hello MH, I took out your recent change, but wouldn't mind it going back in provided you can find a legitimate reference source to back it up. Along this lines, I looked at Piston's widely-used harmony text, and I found that it covers harmony without a single mention of tuning and temperaments; i.e. Piston evidently considered them to be separate matters. So if you really want to make the claim that an equally-tempered fifth doesn't count as perfect, I think you should find a published reference source that says this. Sincerely, Opus33 (talk) 16:01, 8 April 2008 (UTC)

Well, evidently this Wikipedia article fails to treat them as "separate matters" when it speaks of consonance between notes on a scale. Michael Hardy (talk) 16:19, 8 April 2008 (UTC)

Actually, "perfect" probably relates to the fact that the difference of the tempered and harmonic fifths is smallest of any note in the scale. For example a harmonic or Just major 3rd (e.g. 5/4) is 13.69 cents smaller than a tempered major 3rd. That is 7 times as large as the difference between perfect harmonic and tempered perfect fifths. A harmonic or Just minor 3rd (e.g 6/5) is 15.64 cents larger than a tempered minor 3rd or 8 times as large the difference between a perfect harmonic and tempered fifth. The difference between a harmonic and tempered perfect fifth is 1.955 cents. 99.56.137.94 (talk) 18:35, 16 February 2011 (UTC)

An interesting theory, but how does this account for the designation of "perfect" consonances (and their differentiation from "imperfect" ones) as far back as the 13th century?—Jerome Kohl (talk) 19:59, 16 February 2011 (UTC)

As I take it, "perfect" relates to those intervals on a chromatic scale that cannot be discriminated in a "minor" or "major" version and thus appear only once. The attribute is hence redundant, as the term "fifth" inherently bears this property. Compare the German language, where both "Quinte" and "reine Quinte" exist -- the latter is mostly used to emphasize the fact that a fifth is a perfect interval (meaning there is only one such interval to a given base tone), but otherwise left out. I can only quote German sources though, so I will not edit the article. --Doubaer (talk) 13:10, 29 April 2013 (UTC)

When the terms "fifth", "fourth", or "octave" are used alone (in English or in German), the assumption is that the "perfect" form is meant. However, both languages have qualifiers other than "perfect"/"reine", so that, in German, one may also speak of a "verminderte Quinte" or "übermäßige Quinte". For this reason, the terms "perfect fifth" ("reine Quinte"), "perfect fourth" ("reine Quarte"), etc. are redundant only in casual usage. Still, perhaps this informal practice should be made more explicit in this article (it is presently hinted at in the first sentence).—Jerome Kohl (talk) 16:27, 29 April 2013 (UTC)

I agree, and there is a further discrimination between major/minor (e.g. große Terz, kleine Terz) and augmented/diminished (e.g. übermäßige/verminderte Terz), the latter of which are enharmonics of major second and perfect fourth, respectively. I personally regard all augmented/diminished intervals as modifications of diatonic intervals through use sharp or flat signs, so as you state, I take it that "perfect" relates to the non-modified (not aug./dim) interval, that also cannot be discriminated in major or minor. Is that correct? --Doubaer (talk) 14:11, 12 June 2013 (UTC)

That is the way I understand the terminology as well, yes, with just one small caveat: while the characterizations "augmented" and "diminished" really only make sense in a diatonic context, so that distinction between the diminished fifth and augmented fourth depend upon notation based on a diatonic scale, naming the interval of a tritone remains a bit of a conundrum unless, in an undifferentiated twelve-tone context, an alternative notation is employed (such as the numeric measurement of interval size in semitones). In such a case, however, the notion of "third", "fourth", "fifth", etc. also becomes meaningless.—Jerome Kohl (talk) 16:13, 12 June 2013 (UTC)

Imperfect article?

I think I agree with Michael Hardy in some of the comments above. As I read the old literature, a "perfect fifth" is always contrasted with an imperfect fifth based on "temperament". The perfect fifth is the 3:2 ratio; other ratios are not perfect. And yes, you can hear the difference, if not a degree of consonance, then in the beats. For example, this guy defines "perfect harmony" as producing no noticable beats or roughness. The easy audibility of beats, via Tartini's tones, is a tool widely used to tune a fifth just imperfect enough to be equally tempered, in tuning a piano for example.

The articles just intonation, equal temperament, etc. share this flaw. Or has the usage changed and I need to read newer sources? Dicklyon (talk) 06:22, 4 September 2008 (UTC)

Both you and MH seem have two different but related concepts confused. If I sit at my piano and play a

fifth

PS: I've deleted your adjective "perfect"

because if you are playing on a (standard) piano, it is equal tempered;

(=not purely harmonic (German: unrein)).

If you use that adjective (in the wrong place) there,

then you have confused everything from the very start.

on C, I play C and G. That doesn't matter how my piano is tuned, conventionally in twelve-tone equal temperament or a just temperament. As I mentioned above I could also play an augmented fifth and a diminished fifth. The 3:2 ratio is a historical point and a perfect fifth is now defined as 7

pure

PS: I've added the adjective "pure",

to help you distinguish.

Please see my comment above in "Perfect?" (Perfect!).

semitones and a semitone as 12 logarithic (=equal tempered) intervals in an octave. Inspite of the tuning used a perfect fifth is a lot more consonant than any other interval save the octave.

All of this has been answered above but it's seems that MH doesn't accept them and considers the rest of us ignorant. As 84.64.78.152 said above:

Last but not least, the term perfect fifth is a musical term, describing a musical interval and its function within the music system that named it. It's not a physics or mathematical term describing frequency ratios. It's not concerned with the mathematics of any chosen tuning system. Whether its just or tempered, musically, it's still a fifth (adjective deleted).

Brettr (talk) 09:24, 4 September 2008 (UTC)

Well, yes, I saw that point of view above. But what about the issue of references that I brought up? Should both definitions be included in the article if the references differ? Can you provide a link to book with your definition? Dicklyon (talk) 14:39, 4 September 2008 (UTC)

Is there a quote at your reference? I only see a map.

If you read the previous discussions then why didn't you refer to them instead of ignoring them? It's very frustrating to having repeat the same arguments every time someone else comes along who may or may not understand the subject. In spite of the continued arguments against this you are still claiming a perfect fifth can only be 3:2, confusing interval with tuning systems. Looking at your user page it's obvious you are very mathematically orientated, maybe you need to let go of that. A perfect fifth is perfectly consonant whether it's 3:2 or 3.14159:2.

PS: Please don't make me laugh. That (3.14..) is a terrible wolf!

The article Consonance_and_dissonance might help explain this.

I would like to see a quote of an "imperfect fifth"

PS: Every fifth that is not (exactly) 3/2 is imperfect.

The best (= most common) imperfect fifth is an equal tempered fifth. Tempered is imperfect!

Perfection is difficult to maintain.

Guitars go out of tune while playing them a while,

but they are tuned with perfect fifths.

Pianos are NOT!

because as far as I know there is no such thing (except perhaps a wolf fifth), you have a diminished fifth, perfect fifth and augmented fifth. Perhaps you could refer to a pure and impure perfect fifth". Brettr (talk) 15:22, 4 September 2008 (UTC)

PS: It shows how confused you are,

you mentioned a wolf fifth above (3.14),

& then try to sell an impure as perfect.

That (confusion) might be a perfect deal for you,

but not for the receivers (=us).

I did refer to the discussion by saying that I supported Michael Hardy's points. I got here via the other articles when researching the meaning of "perfect fifth" based on this same argument in real life, with my son, who had taken a music theory course. So I realize there are multiple points of view on it. The thing is, I can't find a book source for the definition in the article, only for the 3:2 ratio definition. Many sources (especially older ones) also contrast specifically with "imperfect" or "tempered" fifths.

PS: There you go. It should be obvious, by now. Tempered are NOT perfect!

For example, Helmholtz.  But even in voice training books I see 3:2 as defining a perfect fifth.  What does "perfect" mean if not tuned to the integer ratio?  In my "this guy" above ref, see Art. 4 on p.294. Besides Helmholtz, Lord Rayleight also speaks of imperfect fifth, as do many ohters.  And there's no need to divert the content discussion by talking about me; see WP:NPA.  Dicklyon (talk) 15:56, 4 September 2008 (UTC)

By the way, the example you provide, 3.14159:2, is actually quite dissonant, being much closer to an augmented fifth. Dicklyon (talk) 16:01, 4 September 2008 (UTC)

Here is a good one by Kepler, with commentary by Stephen Hawking. Dicklyon (talk) 16:43, 4 September 2008 (UTC)

You think that pointing out that your mathematical background might be influencing your view is a personal attack? I'd apologise if I could see any misunderstanding or even merit in that, as it is I'm offended. All through this page we have been saying "perfect" is not a mathematical term.

PS: I'm sorry, but perfect is no error (& can be proved mathematically).

I'm doing my best to explain this and you get offended! But thanks for the link WP:NPA

The appropriate response to such statements is to address the issues of content rather than to accuse the other person of violating this policy. Accusing someone without justification of making personal attacks is also considered a form of personal attack.

Music developed without mathematics.

PS: That is not true, Pythagoras developed the math with harp strings lengths,

& gave us the circle of fifths.

The confusion in these webpages is unacceptable.

It was developed by people doing such things as plucking strings and blowing into bits of bamboo. It was only in modern times that it was understood mathematically - at least in full - and by some.

PS: It was only in modern times that we understood the math better.

What does "perfect" mean if not tuned to the integer ratio?

PS: Exactly! Perfect means integer.

All the points you have raised repeatedly have been answered above especially this exact one. Perfect does not refer to the 3:2 ratio or 'exactly' the 3:2 ratio.

--- PS: Both of those are identical. ---

It is that the most consonant of all (12) intervals (except octave and unison). Did you read the references I gave to Consonance_and_dissonance or Equal temperament ? All the intervals are based on simple ratios (sort of) but I've never heard anyone refer to a perfect major sixth even if it is exactly tuned to exactly a 5:3 ratio. Sure with more understanding we know why 3:2 is more consonant than the others but no one labelled a 3:2 ratio as perfect. 

PS: (Why not?) They didn't have to! Equal temperament was invented (much later) after the mid ages.

It took organ builders 300 years to solve the problems of polyphonic instruments.

Pythagoras had fewer problems with monophonic strings & flutes.

The problems in math don't happen until you push to the limits.

The international society for organ builders (ISO) has some interesting articles on these problems,

but they are not online. (Including the nomenclature.) The math is quite precise (=exact).

---

It's sound and it's role in music NOT numbers. But woe be me and the WP:NPA if I mention mathematics again. Until the Baroque era the major third was not considered consonant, only the perfect fourth, perfect fifth, octave where consonant.

---

PS: You may be surprised(? I don't know), but (at least) 24 (pure) notes exist (not just 12),

all producible using the circle of fifths.

Tuning up (the scale) with 12 fifths (Pythagorean tuning),

& tuning down with 12 fifths (Anti_pythagorean tuning, =Silbermann tuning)

(from A 440 Hz).

Flats are not sharpes (& visa versa); & each (of the 24) has its own frequency.

We threw out all of them (=more than half) with equal temperament.

So actually we're talking about (ruffly) 36 notes (or frequencies, intervals) per octave.

But more than that (36) are possible. Clearly laid out in ISO, with the math.

---

Helmholtz: While I can't wade through 500 odd pages of terse music theory on a screen, it is discussing the development of the diatonic system and the term imperfect fifth is being used to describe pre or non diatonic music.

Books refer to the 3:2 ratio because not many musicians can handle base 12 logarithms. 3/2 = 1.5, 2^7/12 = 1.498307

--- PS: Well done!

But isn't it called "base 2", with 1/12th exponents;

or the "12th root of 2", to the exponent of the semitone number? ---

Rayleigh: uses the term imperfect fifth to refer to a diminished fifth. G->Db and D -> Ab

Amongst your list of otters: I C&P

• "The interval of the imperfect fifth is a minor semitone less than the perfect fifth. " ie diminished fifth.
• "Thus in mode I from D to A is a perfect fifth, and from A to the upper D, or final, ... for from В to F is an imperfect fifth ".
• I don't know what Kepler was on about as I can't find the text.

Of course you could have read those yourself and saved me the trouble of researching your own citations. Thanks for wasting my time. Quite obviously you can't just google for the answers you want without understanding the subject and providing citations for texts that are 500 years old is not relevant. Listen to your son, he's done a music course so he knows what he's talking about.

ps thanks for the wiki links to Kepler and Dawking and pointing out that pi is not particularly useful in music theory Brettr (talk) 17:51, 5 September 2008 (UTC)

It's clear that the usage and definitions vary. I'll work on adding sourced info to back up BOTH interpretations. As for "perfect major sixth", here are 36 books that include that phrase. Dicklyon (talk) 17:59, 5 September 2008 (UTC)

I made a start at it. Improvements are welcome, of course. Dicklyon (talk) 20:00, 6 September 2008 (UTC)

PS: I hope (& I really do) that I have cleared some of the confusion.

Have I?

Misleading audio file

I removed this audio file, which was meant to show the beat in a perfect fifth tuned in equal temperament:

The idea of inserting an audio file was great, but in this example the "beat" was actually a slight tremolo which was clearly present in each separate note, and was not not due to their interference. This was highly misleading. The beat in a 700 cent interval is not so easy to hear. We should find a better example, similar to these (from just intonation):

An A-major scale, followed by three major triads, and then a progression of fifths in just intonation.

An A-major scale, followed by three major triads, and then a progression of fifths in equal temperament. If you listen to the above file, and then listen to this one, you might be able to hear a slight buzzing in this file.

Unfortunately, I was not able to locate or produce an audio file containing only a perfect fifth. Some of you might be able to cut from the above two examples the A-E fifth that you can hear for a moment when the A scale is played together with a continuous A. Some very interesting examples of beat are given in the page beat (acoustics), but they are not limited to the perfect fifth. Paolo.dL (talk) 20:19, 26 June 2010 (UTC)

Starwars

the intro in the starwars theme is a perfect 4th not a 5th! —Preceding unsigned comment added by 80.216.26.171 (talk) 19:10, 8 March 2011 (UTC)

hemiola

According to the New Harvard Dictionary of Music, the pitch ration 3:2 is known as a hemiola. I put this fact in the article yesterday and it has been removed. Why?Dr clave (talk) 14:46, 3 February 2012 (UTC)

According to the edit summary by User:Justlettersandnumbers, "where & when is it referred to as hemiola? please add reference to sources". The link to the Wikipedia article Hemiola is insufficient, since Wikipedia cannot be a reference for itself. Of course, if all readers were fluent in Ancient Greek, this would be superfluous. Sadly, today's education system has let us down badly.—Jerome Kohl (talk) 17:06, 3 February 2012 (UTC)

Thanks, JK, sorry, Dr. C.! If it's in that dictionary, we should probably put it in the article somewhere. It is, however, a most unusual application of the term, and I'd be interested to know, out of personal curiosity, where and when it was so used. Julian Rushton mentions the usage in Grove, but doesn't apparently give date, place or indeed any reference (or have I misread?). It doesn't appear to be a common enough usage to deserve much prominence, though I'm very open to correction. Sesquialtera also appears to have been used for pitch ratios, btw. Justlettersandnumbers (talk) 18:34, 3 February 2012 (UTC)

Yes, the Harvard Dictionary citation would probably be sufficient. The reason both terms are used is that they are mutual translations of one another (Greek and Latin). Latin music theory treatises often sought to upgrade their status by using Greek terms, or simply didn't bother to translate well-established Greek forms. Later, when educated people were no longer expected to communicate exclusively in Latin, vernacular treatises often did the same thing: peppering English, French, German, etc. prose with learned Latin and Greek terms. I have added a note in the Hemiola article about an apparent OR claim regarding which of these two terms is "obviously" the more frequent. Personally, I find "hemiola" unfamiliar in pitch-interval naming (but "sesquialtera" fairly common), but both words quite familiar in rhythmic theory. However, the Greek word (hemiola) I associate more with modern music theory (17th century to the present), while the Latin (sesquialtera) I think is more common in good old-fashioned, traditional theory (16th century and before).—Jerome Kohl (talk) 19:30, 3 February 2012 (UTC)

I find diapente to be the common name for the interval of a fifth, at least from Zarlino to Bach or so. Hmm, I'm about to climb on to a hobby-horse. Can you cite any written source that mentions hemiola in its modern sense before about 1934? (I ask because I'd really like to know). The ratio of 2:3 described in the hemiola article is subsesquialtera, not sesquialtera. But maybe we should move this there? Justlettersandnumbers (talk) 19:43, 3 February 2012 (UTC)

The word diapente (Greek for "through five [strings]") refers to the interval of a fifth, but not specifically to the one tuned in a ratio of 3:2 (I hadn't noticed that the article got the numbers of the ratio backward but I shouldn't be surprised, considering the many other lapses). As far as I am aware, the earliest use in English music-theory sources of the word "hemiola" to describe rhythm is Thomas Morley's Plaine & Easie Introduction to Practicall Musicke (1597). It is found in Latin treatises as early as 1517 to describe rhythmic proportions, in Andreas Ornithoparchus's Musice active micrologus, liber secundus of 1517 (hardly surprising to discover this, since Morley relied heavily on Ornithoparchus), as well as in, e.g., Nicolaus Listenius (Musica , 1549), Adrian Petit Coclico (Compendium musices, 1552), Lampadius (Compendium musices, 1554), Hermann Finck (Practica musica, 1556), and in Luca Lossio's tantalizingly named Erotemata musicae practica (1563). In reference to pitch ratios, the sources go back much earlier, as far as the anonymous, late-9th or early 10th-century Alia musica ("et hoc est quod ait: musicos esse numeros, qui per ter et quater dividuntur, comparati eidem duodenario. Nam VIII. qui per quater, et VIIII. qui per ter dividuntur, ad eundem sub epitrita vel hemiola proportione comparantur …").—Jerome Kohl (talk) 20:24, 3 February 2012 (UTC)

Does an English encyclopedia article on pitch need to use a sesquipedalian term that may have been tossed into an early treatise as a fancy foreign-sounding way of saying "one and a half?" I will leave it to more learned heads than mine, but this is the first time I've ever seen "hemiola" refer to a pitch ratio. Rhythmic hemiola is another matter, and a lot of fun. __ Just plain Bill (talk) 21:02, 3 February 2012 (UTC)

To be perfectly honest, when I said I found the use of hemiola "unfamiliar in pitch-interval naming", what I really meant was that I had never before today encountered this usage in English. Perhaps readers in other parts of the anglophone (to use a terser word than "English-speaking") world have different experiences. If so, I hope they will come forward and set the record straight. Until then, I could not possibly disagree with you less.—Jerome Kohl (talk) 21:32, 3 February 2012 (UTC)

Re-inserted hemiola reference, but this time, with dictionary citation. Thank you for the guidence.Dr clave (talk) 02:47, 4 February 2012 (UTC)

No probs. The fuss kicked up was disproportionate, but nevertheless raised some productive issues. I have again changed the word "idealized", since all theoretical interval constructions are "ideals", not merely the ones defined by whole-number ratios.—Jerome Kohl (talk) 03:54, 4 February 2012 (UTC)

Clarification

I must agree that this is written for the (very) advanced reader. (;-) I'm an Engineer and amateur musician and wanted to to clear up some frequency relationships so I could talk about them to another Engineer/amateur-musician and decided to review the fifth. The term "interval" seems to be a poor choice when it says: “The perfect fifth may be derived from the harmonic series as the interval between the second and third harmonics.”. As I read the phrase “the fifth may be derived as”, it equates to the simple “is”. The word “interval” implies the space between, or the difference. The *difference* in frequency between any two adjacent harmonics is the fundamental. Now, clearly, the "ratio" of the third harmonic to the second is 3/2, which "gives" the ratio of the fifth to the tonic. [I had that reversed - too many numbers on the spreadsheet] . I have a hard time calling a “ratio” an “interval”. In the Pitch Ratio section the article describes this much better. So I'm questioning the purpose of that sentence. Is it simply a reference pointing to *some* relationship, or is it trying to be a precise definition? If is only a refere3nce, then I suggest something lmore ike: "The perfect fifth may be derived from a relationship between the second and third harmonic frequencies." Or leave it out and rely on the later section to be precise. P.S. I am not talking about whether it is precisely 3:2 for good consonance, just the term "interval". -- Steve -- (talk) 23:40, 5 November 2012 (UTC)

There is a link to Interval (music) in the first sentence of this article; it is a commonly used term of art in this musical context. Perceived musical pitch is pretty close to a logarithmic function of frequency; the arithmetic difference in frequency between adjacent harmonics has very little relevance here.

The business about "may be derived” provides motivation for the worldwide prevalence of this commonly-heard interval. Any string player can easily play the second and third harmonics on an open string. Similarly, a brass player can easily play the second and third harmonics on any fixed bore length. Such simple practical demonstrations were possible long before humanity understood the concept of cycles per second. __ Just plain Bill (talk) 00:44, 6 November 2012 (UTC)

The lead section, as currently worded, is absolutely meaningless to 95% of readers. This means to say that nearly everyone who reads it will close the tab with absolutely no increase in their knowledge of perfect fifths. The first paragraph, at least, need to provide some kind of information about what a perfect fifth is.

I've put a large amount of careful work in to some kind of cogent presentation of the information. If there is some kind of minor semantic error, please correct it, and don't revert the article unless you have some intention of addressing the fundamental issue of the lead section comprising uniformly of technical jargon that is utterly meaninless to nearly every reader.

InternetMeme (talk) 15:07, 9 March 2014 (UTC)

While I appreciate the editor's intent to make the article easier to read by a lay audience, I'm not sure this helps. In particular, the fifth is not "defined by" the fifth scale degree of a major or minor scale. Every note of every diatonic scale has at least one other note, and in most cases two notes, a fifth away from it, within that scale or its surrounding octaves. Just plain Bill (talk) 15:44, 9 March 2014 (UTC)

Right. The wording was just intended to point out why the term is "perfect fifth". I've altered the wording to better explain this. InternetMeme (talk) 15:58, 9 March 2014 (UTC)

If you want, you could check MOS:INTRO to get a better idea of my motives here. The main thing I want to achieve is a lead paragraph that an average high-school level person could easily understand, and within 20 seconds, come away with a basic understanding of what a perfect fifth is. In particular, I want this to be possible without requiring the reading of other wikilinked music articles. InternetMeme (talk) 16:08, 9 March 2014 (UTC)

It still claims the interval is "named after its relationship to the fifth note in a ... scale." Got a source for that claim?

It also mentions a "symmetrical pitch ratio". That makes little or no sense.

Right now you are at 4rr which counts as edit warring. I don't intend to make any further issue out of that, but kindly recognize that you don't yet have consensus here. Just plain Bill (talk) 16:10, 9 March 2014 (UTC)

My last edit summary read "please reach consensus on the talkpage before adding this again"; how was it, InternetMeme, that you failed to read that when you were busy reverting it? Or did you just choose to re-insert the current illiterate gibberish in the article without discussion or consideration of the views of others? Just plain Bill has generously described your incompetent edits as good faith; but they are increasingly looking like intentional disruption. Please discuss any proposed changes here, as you have already twice (at least) been asked. Justlettersandnumbers (talk) 16:38, 9 March 2014 (UTC)

That MoS page mentions links in the lead as being desirable. In general, hyperlinks make for better flow in writing— with them, readers somewhat familiar with the context do not need to wade through wordy explanations. InternetMeme, what parts of the lead do you think are confusing or unclear? General assertions such as "unreadable to 95% of people" will be difficult to support. Specifics, please. Just plain Bill (talk) 17:14, 9 March 2014 (UTC)

While I agree that there are some problems with the change, the current "a musical interval encompassing five staff positions" is also unsourcable and strange, supported only by a link to another article with an unsourced definition of what that means. Dicklyon (talk) 17:31, 9 March 2014 (UTC)

That is a problem, I agree. Perhaps it could be cast in terms of steps in a diatonic scale. With one exception (e.g. B up to F making a tritone) every instance of taking five steps up or down from any note in any given diatonic scale makes the interval of a fifth. The interval (music) article could use clarification about this as well. Just plain Bill (talk) 17:44, 9 March 2014 (UTC)

Right, some discussion at last (thank you, Dicklyon and Just plain Bill). I readily agree that the current wording is far from perfect, and I think InternetMeme is right to draw attention to that (but not to edit-war over it); the proposed changes have however to date made things worse not better, being apparently based on a less-than-perfect grasp of the topic and unsupported by any reference. In reply to Dicklyon: the ordinal numbers that describe musical intervals derive from the number of staff positions or note names/letters that compose them – B to Fb is a fifth, B to E is a fourth. This is regardless of the quality of the interval, which is determined by the number of semitones between the two notes. I think this is so well known as not to need a reference (sort of Paris-is-the-capital-of-France), but is covered in manuals of music basics such as the AB Guide to Music Theory, p. 46. A perfect fifth spans five note-letters and seven semitones. One difficulty I see in presenting that here is that we don't have an article on Fifth (music) (it's a redirect), which I suppose is why this article starts with an explanation of what a fifth is. I tentatively suggest the following for the first paragraph:

The perfect fifth is a musical interval. In Western music theory it spans five staff positions or note letters, and includes seven semitones. For example, the rising interval from C to G is a perfect fifth, as the note G lies seven semitones above C, and there are five staff positions or note names from C to G. Diminished and augmented fifths span the same number of staff positions, but consist of a different number of semitones.

Any good? Justlettersandnumbers (talk) 18:38, 9 March 2014 (UTC)

Given the convention that scales don't re-use letters, "note names" could be a good choice. Just plain Bill (talk) 18:48, 9 March 2014 (UTC)

Well I think the problem there is that some scales (diminished and bebop scales, for example) do re-use letters, while others (pentatonic, Hexatonic scales) don't use them all; which is why any definition of intervals in terms of scales is risky in my view. Note names are unambiguously assigned to staff positions – G# (and any other alteration of G) is on the same line as the G clef, Ab is in the space above like any other alteration of A. But if "note names" makes people happier then let's go with that. The AB Guide that I cited above discusses intervals in terms of note names. Justlettersandnumbers (talk) 19:11, 9 March 2014 (UTC)

I'm not able to see the AB Guide online, and I had trouble finding anything about encompassing staff positions. I'd like to see us follow the terminology of a couple of good sources that we can agree do a good job of it. Can someone point to some (online links, preferably)? Dicklyon (talk) 01:51, 10 March 2014 (UTC)

Okay guys, you've gone right back to a Music Theory 301 definition. The problem with this is that 95% of people haven't spent a couple of years studying music, and the first paragraph will have about as much meaning to them as the instruction manual to a 747. I have no particular preference for the logical flow of information, but what is important to me is that we use words and principles that an average lay-person can understand.

The proper way to teach is to start from first principles, and build upwards: It's a very unsuccessful method to start with a fully formed theory and listing all of the parts while simultaneously defining them.

The perfect fifth is a concept so fundamental that it precedes musical theory (although it is named after concepts from it). In fact, the perfect fifth is probably one of the most fundamental building blocks of music, and it can be explained using primitive principles, and without confusing the reader with the introduction of advanced musical theory. If they reach that level of interest, that's what the body of the article is for. Not the introductory paragraph.

InternetMeme (talk) 02:43, 10 March 2014 (UTC)

I'd just like to add that I can see that both of the Just guys clearly have a good advanced understanding of musical theory, and they'd both surely be way better than I at writing the bulk of this article. However, I'm not certain they have the required skills to break down a concept in to very simple principles suitable for an introductory paragraph. The two things I'd like you to consider are:

1. How very little the average Wikipedia reader knows about music theory.
2. The basic concept behind a perfect fifth seems very fundamental to human hearing, and probably precedes the invention of music theory. It can therefore probably be described independently of music theory (however, an explanation of the 'name' "perfect fifth" obviously requires music theory)

Also, one of the Just guys incorrectly stated that my edit displayed a "less-than-perfect grasp of the topic and unsupported by any reference." I make no claims as to the grasp, but I supported it with the reference I have repeated above. It's derived from basic maths, and is probably a very good starting point for an understanding of the topic at hand.

It was also incorrectly stated that I was edit warring: If you were to have looked at my edits, you would see that each time I was amending my edit with respect to the specific objections given: Each edit I made was iteratively corrected.

InternetMeme (talk) 03:23, 10 March 2014 (UTC)

I don't think it makes sense to give much space in the lead to the 3:2 ratio of vibrations. I'm not sure it needs any mention in the lead at all— that pitch ratio is mentioned prominently in the body of the article. In a realistic example from music teaching, learning about pitch often starts with solfeggio, with steps easily countable between do and sol to make a fifth.

I might be persuaded to include something linking the interval to the fifth scale degree, since that is how Ottó Károlyi describes it in his Introducing Music, but I'd rather not, since that could imply that the fifth needs to be rooted in a scale's tonic, adding to the confusion. How about this:

The perfect fifth is a musical interval. In Western music theory it spans five steps in a diatonic scale, and includes seven semitones. For example, the rising interval from C to G is a perfect fifth, as the note G lies seven semitones above C, and there are five steps from C to G. (see revised wording below)

That leaves out the bit about augmented and diminished fifths, which are mentioned later in the article. Does that work for you all? Just plain Bill (talk) 15:18, 10 March 2014 (UTC)

To avoid a narrow focus on Western theory, I like this better:

The perfect fifth is a musical interval used in music cultures worldwide. In Western music theory it is described as spanning five steps in a diatonic scale, and including seven semitones. For example, the rising interval from C to G is a perfect fifth, as the note G lies seven semitones above C, and there are five steps from C to G.

To support "used worldwide" see Pentatonic scale#Pervasiveness. To keep the article's completeness more or less intact, the part about diatonic and chromatic semitones in meantone temperament would move into the "Alternative definitions" section. Just plain Bill (talk) 13:07, 11 March 2014 (UTC)

Hmm. The concepts "perfect" and "fifth" are surely entirely Western; I readily agree that fifths are found in the musics of many cultures, even though they aren't called that; but are they perfect? I think not; would they sound any different if we called them diminished sixths? I'm also still unhappy with using steps of diatonic scales rather than note names or staff positions in the definition as diatonic scales contain diminished and augmented fifths. Moving the stuff about different qualities of semitone to later in the article seems an excellent plan.

I'm sorry the AB Guide is not available online. It's the basic guide I refer to when I want to know how things are taught to children; I could quote a couple of sentences here if my interpretation of it is in doubt. An earlier Associated Board publication, Rudiments and Theory of Music, also defines intervals in terms of note names (p. 27). Roger Sessions, Harmonic Practice p. 11, defines them in terms of the number of degrees. Justlettersandnumbers (talk) 18:42, 11 March 2014 (UTC)

Mentioning "five (diatonic) steps and seven semitones" was intentional. (I am unaware of any five-step augmented fifths in the diatonic scale, but I had to count on my fingers to see.) Leaving off the "worldwide" part is OK by me, for the reasons you mention (even though the nameless interval determined by 2nd and 3rd harmonics appears worldwide, whatever a particular musical culture may call it, and however it is used there. Not sure the "what does 'perfect' mean?" topic needs opening again...)

For now I'll wait for others to chime in. Just plain Bill (talk) 20:01, 11 March 2014 (UTC)

Yes, I was wondering about that (what perfect means), something I'd like to know. Does it derive like other kinds of musical perfection (e.g., in mensural notation) from the Trinity? Eb to B in a C melodic minor ascending scale is an augmented fifth, btw. Justlettersandnumbers (talk) 20:55, 11 March 2014 (UTC)

Musical theology is above my pay grade here. I only know enough to say that an equal-tempered fifth may still be "perfect" even though it ain't the same as tuning the strings of my devil's box to beatless Pythagorean fifths. I believe the natural minor scale is diatonic, with the melodic minor and that B♮ being a chromatic alteration of it. Just plain Bill (talk) 22:25, 11 March 2014 (UTC)

Once more, with feeling

Interesting stuff. I think I've worked something out here: We have two separate interests in regards to the basic definition of a perfect fifth: You guys are interested in defining why a perfect fifth is called a perfect fifth, and also how it relates to music theory. On the other hand, I'm more interested in defining a more fundamental issue that predates music theory, which what causes a perfect fifth to have any significance to human hearing (which is that it is the interval between two tones that produce a highly symmetrical waveform). Both of these things are important to get across in the lead. Part of what made me realize the disparity in our interests is that one of you guys wasn't even certain about how symmetry is involved. But you guys are clearly better equipped to deal with the music theory aspect.

Here's a sort of thought experiment that I don't think you've read to explain the simplest implementation of the concept of the article that I can think of:

Imagine it's the year 4000 BC, long before the invention of modern musical theory or the naming of notes, etc'. Now imagine you're some ancient human of the day, and that you get hold of some kind of prehistoric thread and string two lengths of it between the two beams of a forked stick. Assuming you figure out some way of winding them to make them tighter, imagine you tune the first string to any random note near the middle of human hearing. Now, while plucking the first string, you tune the second string until you reach a note that's above the first, and which has a pleasing sonorousness. Although you have no way of knowing it, that note is vibrating three times for every two times the first note vibrates.

http://mathforum.org/library/drmath/view/52470.html

The important thing is there is something going on that predates music theory, and in fact, that is itself one of the building blocks of music theory. And it can be explained and understood in very simple terms completely independently of music theory, though explaining how it relates to music theory is very important to anyone wanting to gain a fuller understanding of how music works.

InternetMeme (talk) 11:25, 13 March 2014 (UTC)

Yes, I read that the first time, before you cut it from the earlier discussion and pasted it here. A better thought experiment would start with a bowstring. Playing around with harmonics on something like a guitar string is fun; I recommend it.

Couple of points:

• The bit about what "perfect" means is a side issue in this discussion. I see the present focus as defining how wide this interval is (five steps, seven semitones, or something equally unambiguous) in terms that are both correct and understandable.
• You keep using the word "symmetry." I don't think it means what you think it means. You got any particular axis of reflection to point out? Very few of us keep oscilloscopes in our back pockets, so talking in terms of waveforms is not suited to a general music theory audience. In simple fact, it is just the kind of thing I believe we should stay away from: getting all technical in an article's introduction.

If you believe that the article should be accessible to a general audience, I agree with that. You seem to be saying that we should use this article for teaching music theory, or one aspect of it, from the ground up. That isn't going to fly: "Wikipedia is an encyclopedic reference, not a textbook. The purpose of Wikipedia is to present facts, not to teach subject matter." This article is not meant to be a basic course in music theory. __Just plain Bill (talk) 13:57, 13 March 2014 (UTC)

p.s. Discussing "what causes a perfect fifth to have any significance to human hearing" might deserve a new topic on this talk page. __ Just plain Bill (talk) 14:49, 13 March 2014 (UTC)

Yes, what Bill said. I apologise for introducing digressions about the meaning of "perfect" and the scope of "diatonic". What causes harmonic consonance to have meaning for humans probably falls under psychology of music. The perfect fifth is, like other such intervals in traditional Western music theory, a functional musical interval; it doesn't sound any different (in modern tunings) from a diminished sixth or a doubly-augmented fourth, but its harmonic and melodic function is totally different from that of those little-used constructs. Justlettersandnumbers (talk) 15:34, 13 March 2014 (UTC)

I've just noticed that this talk page topic was originally created on 5 November 2012 by user "-- Steve --". He was pointing out that the lead is very difficult to understand, and he's apparently an engineer. You guys have both been working on this article since at least that time, and neither of began to address the issue until I stumbled across this article, saw all of the technical jargon in the lead, and tried to help.

In fact, the lead as written today is word-for-word identical to the lead as it was back in 2012. Why haven't you guys done anything about it?

InternetMeme (talk) 03:30, 14 March 2014 (UTC)

Hey guys, this is a serious question. I'm interested in improving the lead of this article. The fact that the lead is very confusing was brought to everyone's attention over a year ago. A year has gone by, and nobody has done anything about it. I've tried to help, and you guys repeatedly reverted and now have gone silent, leaving the lead once again in limbo.

InternetMeme (talk) 05:29, 18 March 2014 (UTC)

Over a year ago, a retired engineer came along, complaining that "interval" is the wrong word to use for a jump between musical pitches. (It isn't the wrong word. It is exactly what such jumps are called.) I don't see how that called for all hands on deck to rewrite an introduction that has been stable since the middle of 2010, and which has had the attention of experts since then, including at least one music professor.

What you call "technical jargon" I call "terms of art." To discuss any specialized topic, it makes sense to learn the rudiments of its vocabulary. With that in mind, if you want to suggest specific changes to the introduction, this is the place to do it. Just plain Bill (talk) 17:48, 19 March 2014 (UTC)

Hey there. Thanks for your reply. You're talking about a different part of what EngineerSteve said. I'm not debating the usage of the term interval, and I'm referring to his sentence "I must agree that this is written for the (very) advanced reader." Rather than I describing what I think could be improved, it might be nice to get in contact with EngineerSteve to see if he can elaborate on what he's interested in improving.

Also, just for reference, I'd like to point out that I have no problem with any of the other parts of the article. My concern is entirely with the lead section.

InternetMeme (talk) 15:11, 22 March 2014 (UTC)

You are certainly welcome to try contacting him. He has been off the air here for a while. When he disputes the use of "interval" to describe, well, an interval, in my eyes that contaminates the credibility of whatever else he said. That includes his assessment of what's appropriate for a (very) advanced reader. What parts of the lead section do you think are too difficult to understand? Just plain Bill (talk) 16:01, 22 March 2014 (UTC)

Hey there. Conversely, I think the fact that he misunderstood the meaning of the term "interval" puts him in a good position to comment on the level of complexity of the lead section: The lead section is aimed at people that have no specialist knowledge in a subject (people who probably don't know what the term "interval" means). His misunderstanding of the term interval qualifies him to comment as an average joe who has a minor curiosity about the meaning of the term "Perfect fifth"; exactly the target demographic of the lead section.

For example of the target demographic: Maybe a reader is enjoying a fantasy novel, and reads a phrase such as "The dawn bells chimed for breakfast: First, a lone note, followed by an ascending perfect fifth*". The reader might wonder "What is a perfect fifth? Why would someone choose to use a note with that relationship over any other note? Knowing how it relates to any musical scale won't particularly help them, but if they learn that two notes with a perfect fifth interval have a 2:3 ratio, and that symmetrical ratios sound nice, they get some kind of resolution to the question.

• It's worth noting that in common parlance, the term "perfect fifth" is used to refer to a note in relation to another, rather than the interval itself, which may bear mentioning in the lead.

InternetMeme (talk) 07:01, 23 March 2014 (UTC)

Yes, an interval is two notes in relation to each other. Sometimes they come one after another, and sometimes they sound at the same time. The intro has a link to Interval (music) in its first sentence, so those who don't know what that is can find it, and those who do know don't need to waste their time reading (or skipping over) an elaborate word picture. What is this "symmetry" you keep mentioning? Just plain Bill (talk) 12:52, 23 March 2014 (UTC)

It's not a good idea to require readers to consult other articles in the very first sentence, as they're already trying to learn a new term. Having to learn another term simultaneously (and then another, and yet another) pretty much halts the learning process (this applies only to the lead section, of course). A brief description of a term is a sort of compromise between that and not introducing new terms at all; putting the brief description in brackets provides a visual cue to allow educated readers to skip over it easily.

It's hard to say for sure, but I'd say the upper limit for introducing new terms in a lead section is probably one per paragraph. Any more than this and the flow of the learning process is disrupted too much, and an average reader will either switch off or just stop reading.

I apologize for not providing a source for my motivation, so here is the part I'm interested in:

In general, introduce useful abbreviations, but avoid difficult to understand terminology and symbols. Mathematical equations and formulas should be avoided when they conflict with the goal of making the lead section accessible to as broad an audience as possible. Where uncommon terms are essential, they should be placed in context, linked and briefly defined. The subject should be placed in a context familiar to a normal reader. For example, it is better to describe the location of a town with reference to an area or larger place than with coordinates. Readers should not be dropped into the middle of the subject from the first word; they should be eased into it.

(Wikipedia:Lead_section#Provide_an_accessible_overview)

InternetMeme (talk) 16:17, 23 March 2014 (UTC)

Two direct questions yet unanswered:

• What parts of the lead section do you think are too difficult to understand?
• What is this "symmetry" you keep mentioning?

I see no point in continuing this river of text until those are answered, directly and concisely. Just plain Bill (talk)

You are the one who keeps mentioning symmetry. InternetMeme mentioned it once. There are various musical theories involving symmetries, such as that harmonies involve time-translation symmetries of waveforms (in a perfect fifth, translation by two periods of one tone, which is three periods of the other, leaves the waveform unchanged; short translation symmetries are generally more harmonious than longer ones). I'm not sure if this is the symmetry he has in mind, or whether he means the up-doing interval in relation to the down-going interval or something like that. In terms of the lead section being hard to understand, I think he said that already. For one thing, it needs to mention the 3:2 ratio prominently and early, rather than just the more abstract music theory language, which will be meaningless to many readers. Dicklyon (talk) 17:04, 23 March 2014 (UTC)

I have added a new lead sentence at the start that says what a perfect fifth is without reference to classical and Western music theory or staff positions. This will be easier to understand for some readers, and more informative for many. I'm sure it can be improved or better integrated, but something like this is needed. Dicklyon (talk) 17:37, 23 March 2014 (UTC)

[a few edit conflicts later] Actually, he's mentioned it a couple of times on this talk page, along with its inclusion in the (reverted) change he made, without clarifying what he meant after repeated inquiry. An exposition of translation symmetry, and how it relates to perceived consonance, might fit somewhere in the article, if reliably sourced, but I do not think it is appropriate for a general Wikipedia audience, especially not in the intro of an article such as this one. I believe InternetMeme's expectation, that a high school student without much music education could get it in 20 seconds after reading an introduction without wikilinks, is not a realistic one.

How about this to replace the first two paragraphs:

In classical music from Western culture, a fifth is a musical interval spanning five steps in a diatonic scale, for example do re mi fa sol in solfeggio. The perfect fifth (often abbreviated P5) is a fifth spanning seven semitones. The interval from C to G is one example of a perfect fifth, as the note G lies seven semitones above C, and there are five staff positions from C to G. Diminished and augmented fifths span the same number of staff positions, but consist of a different number of semitones (six and eight, respectively). In a diatonic scale, the dominant note is a perfect fifth above the tonic note. The perfect fifth may be derived from the harmonic series as the interval between third and second harmonics, with a frequency ratio of 3:2.

That doesn't mention the types of semitone in meantone, but that can come later in the article. They are still semitones. When Karolyi op. cit. first defines an interval, he does so in terms of scale degrees in a C major scale, so steps in a scale seems appropriate and supportable. The term "perfect fifth" is, as far as I can tell, particular to western theory. Just plain Bill (talk) 18:20, 23 March 2014 (UTC)

The perfect fifth is also well known in engineering, acoustics, psychophysics, and such, where it is often defined without reference to staff positions or semitones. For example in this book and this book and this book. Also in non-Western music as in this book. In all cases, perfect fifth means a 3:2 pitch ratio. Mentioning "do re mi fa sol in solfeggio" is just noise to most people outside of Western music theory. And defining a fifth before a perfect fifth may make sense within the context of Western musical theory, but it completely confuses the issue for everyone else. Start with the perfect fifth, and after that go into how it relates to the general fifth of Western music theory. Dicklyon (talk) 18:32, 23 March 2014 (UTC)

As for the "symmetry" thing, I think it's a red herring. I searched and found the "symmetrical" in the same one place, where he said "the interval between two tones that produce a highly symmetrical waveform", which suggests that he did mean the translation symmetry of waveform that I was referring to. I agree we can leave that alone and stop worrying about it. Dicklyon (talk) 18:46, 23 March 2014 (UTC)

Fair enough, as long as "symmetry" or "symmetrical" doesn't show up in the lead.

The various technical disciplines you mention took the terminology from pre-existing music theory, or I'm a monkey's uncle. Including the solfeggio is a nod in the direction of accessibility to a general audience, namely anyone who has seen Julie Andrews singing in that Rodgers and Hammerstein show. I do not believe we need to include sa ri ga ma pa along with do..sol. __ Just plain Bill (talk) 18:51, 23 March 2014 (UTC)

Yes, we understand that the terminology came from Western music theory; that doesn't mean that the introductory definition needs to be made accessible only to people who know Western music theory. And while I could happily sing the Julie Andrews tune from memory, the words and harmonies tell me nothing useful about how to interpret these sentences, since I'm not a musician. Dicklyon (talk) 18:55, 23 March 2014 (UTC)

(edit conflict)Unfortunately that isn't what a perfect fifth is. Of course it's true that a perfect fifth is in most modern tunings tuned close to a pure fifth, and exactly to one in Pythagorean and just tunings; but that doesn't make those things identical. The way a fifth is tuned does not affect its definition or function as a musical interval. However, it might be a good idea to include that similarity in the lead section if that would satisfy the anxieties for the supposedly fantasy-novel-reading target audience of this page expressed above. I believe that the lead should include something about the perfect fifth being the most consonant interval after the octave, the only two intervals to have been regarded as consonant through all the history of Western music. And possibly something about its fundamental importance in harmony. Justlettersandnumbers (talk) 18:56, 23 March 2014 (UTC)

The solfeggio is simply another way of counting steps, accessible to those who don't read staff notation. One, two, three, four, five, caught a hare alive. The last paragraph of the current rev's intro mentions the consonance of this interval. __ Just plain Bill (talk) 19:20, 23 March 2014 (UTC)

I know that; but most readers outside of music theory will find it unfamiliar, and even if they understand it, it doesn't get to the point that the perfect fifth is the 3:2 pitch ratio (or nearly so). The lead sentence that Justlettersandnumbers reverted allowed for other tunings: "In music theory, a perfect fifth is the musical interval corresponding to a pair of note pitchs with a frequency ratio of 3:2, or very nearly so." Dicklyon (talk) 19:44, 23 March 2014 (UTC)

Suggested wording

In music theory, a perfect fifth is the musical interval corresponding to a pair of note pitches with a frequency ratio of 3:2, or very nearly so.

In classical music from Western culture, a fifth is a musical interval encompassing five steps in a diatonic scale, and the perfect fifth (often abbreviated P5) is a fifth spanning seven semitones. For example, the interval from C to G is a perfect fifth, as the note G lies seven semitones above C, and there are five staff positions from C to G. Diminished and augmented fifths span the same number of staff positions, but consist of a different number of semitones (six and eight, respectively).

The perfect fifth may be derived from the harmonic series as the interval between the second and third harmonics. In a diatonic scale, the dominant note is a perfect fifth above the tonic note.

The perfect fifth is more consonant, or stable, than any other interval except the unison and the octave. It occurs above the root of all major and minor chords (triads) and their extensions. Until the late 19th century, it was often referred to by one of its Greek names, diapente.<ref>{{cite book | title = A Dictionary of Christian Antiquities | author = William Smith and Samuel Cheetham | publisher = London: John Murray | year = 1875 | page = 550 | url = http://books.google.com/books?id=1LIPFk6oFVkC&pg=PA550&dq=diatessaron diapason diapente fourth fifth }}</ref> Its inversion is the perfect fourth. The octave of the fifth is the twelfth.

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That includes Dick's suggestion for a first sentence, along with some trimming by yours truly. Will that work? __ Just plain Bill (talk) 19:42, 23 March 2014 (UTC)

Looks good to me. Dicklyon (talk) 19:46, 23 March 2014 (UTC)

Thanks. I'm still not completely comfortable with "encompassing five steps" since that leaves open the possibility of a fence-post miscount. Karolyi is careful to point out that "Both first and last notes are counted." [op. cit. p.18] Still thinking... __ Just plain Bill (talk) 20:07, 23 March 2014 (UTC)

Not clear that "steps" is the right word, either, since some of them are what are called "half steps" by some. Just need to be clear what you're counting. Maybe "musical interval representing the relationship between the first and fifth notes in a diatonic scale"? Dicklyon (talk) 20:10, 23 March 2014 (UTC)

Pipe a link to Steps and skips and that's taken care of. Or, it could be "musical interval equivalent to the relationship between the first and fifth notes in a diatonic scale", but that is getting close to the kind of circuitous locution I like to avoid. __ Just plain Bill (talk) 20:35, 23 March 2014 (UTC)

Yes, it says a step is 1 or 2 half-steps. Go for it. Dicklyon (talk) 20:38, 23 March 2014 (UTC)

On further thought, that still leaves the fence-post problem.

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In music theory, a perfect fifth is the musical interval corresponding to a pair of note pitches with a frequency ratio of 3:2, or very nearly so.

In classical music from Western culture, a fifth is a musical interval encompassing five consecutive notes in a diatonic scale. The perfect fifth (often abbreviated P5) is a fifth spanning seven semitones. For example, the interval from C to G is a perfect fifth, as the note G lies seven semitones above C, and there are five staff positions from C to G. Diminished and augmented fifths span the same number of staff positions, but consist of a different number of semitones (six and eight, respectively).

The perfect fifth may be derived from the harmonic series as the interval between the second and third harmonics. In a diatonic scale, the dominant note is a perfect fifth above the tonic note.

The perfect fifth is more consonant, or stable, than any other interval except the unison and the octave. It occurs above the root of all major and minor chords (triads) and their extensions. Until the late 19th century, it was often referred to by one of its Greek names, diapente. Its inversion is the perfect fourth. The octave of the fifth is the twelfth.

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I will go quiet for a while, leaving that for others to comment. __ Just plain Bill (talk) 20:56, 23 March 2014 (UTC)

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EngineerSteve Returns

Yikes! What did I start... As requested, I have returned. You can jump to the end for my wording suggestions. This'll take way too many words, but since this Talk is progressing maturely, I give this my full and complete attention. Here goes...

I am not trying to argue anything, merely present my thinking, as a "stopped by to learn something" reader in hopes it helps bring consensus to those trying to help future readers. But...BACKGROUND first.

I am NOT a highly experienced Wiki editor, so this is going to require much previewing to get it to look correct, but give it the full attention I feel it deserves in light of the above discussion.

I do enjoy helping people understand things that I do. See my profile. I also do not read music, except, if needed, with great pain. I am a native American English speaker.

I had questions, but now, I don't recall everything I wanted to learn. I did feel it was time, after having developed some sense of how music is assembled while playing around completely self-taught with guitar for 50 years, that I should learn some more music theory in order to discuss it with a friend, and turned to Wiki and YouTube. Now that I know, it seems like an easy thing to explain, so why all the discussion...

Re-reading my original post, I see that I was stumbling over many of the original intro words, that now, have been changed AND that the Intro is better now...

With all due respect to those of you who are much more advanced that I, I believe this is an issue of being too familiar with the subject; something I constantly wrestle with as a VERY broadly experienced Electrical Engineer teaching beginners and Technicians.

I started reading the above l-o-n-g Talk, but scanned much of it and seeing me mentioned again, pretty much skipped to the end. I also saw some good suggestions, but did scan it.

As a side note, IF there are differences in western vs. other "fifths", I recommend alluding to this in the intro (as one post did), then have a section explaining the differences/definitions.

While I see that much of what Meme said aligns with my thoughts, I was not interested in understanding why our ears/brain makes the Perfect Fifth "work" in music and BE harmonious (when the two are played together), though I have heard an explanation on radio which I yet have to "do the math on" to see if it makes sense. (it has to do with the harmonics of the two notes at the ends of the P5 interval somehow being related in the way they excite sensor-hairs in the cochlea - say no more).

I just wanted a good understanding of it.

One more preface item addressing things I saw while scanning this Talk. I was involved, but gave up in another wiki article which had what I will call a "battle of early, absolute correctness". As a teacher, I work very hard to get students to understand the fundamentals FULLY because this allows easily understanding the more complex aspects. An intro which has MANY aspects that are absolutely correct, may indeed be correct, but easily confuses the beginner and looses them. AFTER fundamentals are understood, really understood; then, walk them cogently into all those other correct details. Also, every field has "terms of art", but jargon is easier to type and shouldn't be offensive -- and starting in an intro, right-off with too much, is also confusing - even if they are linked to definitions; for those disrupt the flow.

Near the end of the above Talk (growing as I type this) I see that one possible basis for consensus may be an issue with the easily understood "fence-post" issue.

Keep in mind that this is all now that I understand more about P5.

Here's the beef In hopes it helps find words to meet everyone's criteria, yet inform the "lay reader", here is the current intro followed by my stream of consciousness:

"In classical music from Western culture, a fifth is a musical interval encompassing five staff positions (see Interval number for more details), and the perfect fifth (often abbreviated P5) is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones."

First, hitting "Encompassing" slows me a bit in hopes this will be cleared up. For me, this is not a precise word at this point (though the word makes sense, knowing what P5 is now - hint).

Next, not classically trained, "Five staff positions" is marginally informative/confusing and I ignore it since I know accidentals make this part more confusing and actually this can be interpreted as making this mystical P5 a differing number of half-steps depending on which note you start counting. My simplistic view is that the scale is 12 frequencies/tones with adjacent ratios of the 12th root of two with some of them being called accidentals. Therefore, I'm not there yet, but anticipating clarification soon...

Continuing:

"...and the perfect fifth (often abbreviated P5) is a fifth interval ...)

The last "fifth" is redundant. (yes there are other fifths, but that should come later)

Continuing:

" spanning seven semitones,..."

Euricka!. For me, this is pay dirt! Spamming includes the ends of the interval.

"For example, the interval from C to G is a perfect fifth, as the note G lies seven semitones above C,"

THEN, this IS concrete confirmation for me - seven half-steps...C to G...got it!. If the intro had ONLY that small explanation, I'd be home free in a much shorter sentence...as long the accidentals don’t mess things up later - and, as I understand the Perfect Fifth, it is always seven semi-tones come hell or you know what....

Continuing:

"and there are five staff positions from C to G. "

Five "staff positions" is redundant with the same words earlier, except there are only four positions "from" C to G -- the fence-post problem pops up - semantics can be a pain. However, I can ignore this issue (and initially did) until I start counting instead of just saying five...ok...whatever. "From" C is then: D-E-F-G is four "from "C"...oops - ence-posts, better keep reading because those accidentals are a continuing source of confucion for me, and I must assume many newbies... Nit picking, perhaps, but confusing none-the-less.

Continuing:

"Diminished and augmented fifths span the same number of staff positions, but consist of a different number of semitones (six and eight, respectively)."

NOW, I think I can guess that since there is a dim and aug fifth, that the word "perfect" is used to distinguish it from the other two, no? If not, then something about why this fifth is called perfect is in order. It may be later, but I think it belongs in the intro.

For this change of subject to more advanced concepts, I think a bit of introductory, change of pace is needed. See below.

HOWEVER, Wait a minute here. If a good "definition" earlier of the PERFECT Fifth is ""five staff positions", why are these "the same number of...", yet not the seven half-steps. Now THAT can't be anything but confusing. One might be tempted to say that this article is titled "Perfect Fifth" not "Fifths".

For me, the word "distance" works a bit better than "interval", but I think this is because the initial confusion put up a block to this word as seeming more like jargon to be understood later. In driving, the interval is the space, or distance between cars. Now that I understand it, "interval" is ok. Also, Interval is independent from timing of the two notes, or whether or not both are even present near each other.

Continuing:

"..., or in meantone, four diatonic semitones and three chromatic semitones...."

This stops me dead! I'm looking for the "fifth" and this is going into other parts of music theory, again, unnecessary for the initial P5 understanding. While this may very well be 100% correct and for those of you who are classically trained, it may eat away at your sense of correct/complete-ness, but I feel it adds nothing to the fundamental understanding of the P5 and belongs further down.

MY EXAMPLE WORDING: As a "lay person"...Complete definitions of the Perfect Fifth that works for me, as I understand it, are as follows. If incorrect, hopefully this helps find the words:

The Perfect Fifth is the distance between notes seven half steps apart. OR

The Perfect Fifth is the interval between notes separated by seven half steps.

THEN with an obvious transition... The Fifth has two modified (or additional) forms. The Augmented Fifth is one more semitone, or an interval of eight and the Diminished Fifth is one less semi-tone, or an interval of six.

NOW go ahead and add all that other fifth information... I'm not going there.

Agree or disagree, but I hope this helps find words that help others in the future.

P.S. RE: Just Plain Bills very recent post (others I like):

RE: "is a fifth spanning seven semitones". Here, again the word "fifth" seems redundant. The word "spanning" seems to fall into the same category as the "encompassing" earlier - a bit too vague for a concise definition.

The phrase "...may be derived from the harmonic series..." seems unnecessarily wordy to me. The word "harmonics", while at the, end clearly defined the domain we are in.

It seems sufficient to me to just say: "The perfect fifth IS the interval between the second and third harmonics." While true (2:3, or 3:2 - whatever), HOWEVER, while mathematically true, I don’t think the "fifth" is concerned with harmonics - that would be timbre. We are addressing the relation between two (pure) notes, no? We have a note and another note that differ by a fifth. Harmonics of either note are another subject, so I feel this addition is also unnecessary.

While the terms dominant and tonic are familiar to me, this can be a bit later addition to the definition in other terms: "In a diatonic scale, the dominant note is a perfect fifth above the tonic note.

Also talking about the consonance of the pair is indeed good here since it gives at least SOME "reason" for the existence of the fifth in the first place.

RE: "For example, the interval from C to G is a perfect fifth, as the note G lies seven semitones above C."

Looks really good because the "lies seven semitones above" clearly defines it,

but:

"...and there are five staff positions from C to G. " sure seems complicating for the initial intro - especially since the Aug and Dim fifths are "the same, but different".

I better stop now. Regards -- Steve -- (talk) 02:36, 24 March 2014 (UTC)

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But as InternetMeme points out, this most important of consonant musical intervals long predates things like semitones and staff positions. It can be first explained for what it is, the pitch ratio 3:2, the simplest ratio of small integers after 1:1 and 2:1. Dicklyon (talk) 03:48, 24 March 2014 (UTC)

As for "interval", I can see no reason to switch to any less standard word for the pitch relationship between notes. We explain it, of course, by link or otherwise. Dicklyon (talk) 03:50, 24 March 2014 (UTC)

When we say the P5 may be derived from the second and third harmonics, it is not so much about the timbre created by a mix of overtones, but about individual harmonics sounding by themselves. Steve may be familiar with what guitarists call harmonics. For example, touching an open string lightly at the twelfth fret and picking close to the bridge gives a sound that is mostly made up of the second harmonic, an octave above the pitch of the open string. The seventh fret's (third) harmonic sounds a fifth and an octave above the open string, the fifth fret's (fourth) harmonic sounds two octaves above, and so on... Kindly forgive me if I'm repeating stuff you've known for a long time.

Violinists can do similar things, and buglers play nothing but overtones. They aren't spurious emissions to be avoided, but are intentionally excited loops and nodes in the vibrating medium, in the way one might take a signal from an oscillator, put it through a nonlinear process if need be, and use a narrow bandpass filter to select a particular overtone. Musicians can often adjust the overtone mix to get a particular tone color, doing things like picking closer to the bridge or further from it, but that's beyond the present scope.

See if this works for you:

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In music theory, a perfect fifth is the musical interval corresponding to a pair of pitches with a frequency ratio of 3:2, or very nearly so.

In classical music from Western culture, a fifth is an interval spanning five consecutive notes in a diatonic scale. The perfect fifth (often abbreviated P5) spans seven semitones. For example, the interval from C to G is a perfect fifth, as the note G lies seven semitones above C. The perfect fifth may be derived from the harmonic series as the interval between the second and third harmonics. In a diatonic scale, the dominant note is a perfect fifth above the tonic note.

The perfect fifth is more consonant, or stable, than any other interval except the unison and the octave. It occurs above the root of all major and minor chords (triads) and their extensions. Until the late 19th century, it was often referred to by one of its Greek names, diapente. Its inversion is the perfect fourth. The octave of the fifth is the twelfth.

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I've cut out the bit about diminished and augmented fifths. There may be a slick way to work them back in, if someone feels strongly that they belong in the lead. Just plain Bill (talk) 04:05, 24 March 2014 (UTC)

Wow. This discussion has started to become epic. And hi there, Steve! I'm very impressed by your skill at breaking down large ideas into their components for individual analysis and criticism.

Firstly, I'd like to say that much to my surprise, I'm 80% happy with JustPlainBill's prototype lead above. I'd like to tweak it somewhat, but the basic flow, and restrained use of jargon is reaching a point that I feel is going to inform most readers before they close the tab in bewilderment.

Secondly, I should aplologize for not defining my use of the word "symmetrical". I thought that there was no possible way that JustPlainBill couldn't see the symmetry in a 2:3 ratio, but I now realize that it may not be the proper way to describe this kind of thing. So wherever I've used the term "symmetrical", please substitute whatever word you'd use to describe very basic ratios: "even"? "Consonant"? "Stable"? I'm not sure which one is best; all I desire is that we use a word that is going to be comprehended by average readers.

Back to Bill's prototype intro, I'd like to expand the first paragraph slightly. Taking a page out of EngineerSteve's book, I'll try to describe a thought process as I read it:

In music theory, a perfect fifth is the musical interval...

1) "What's an interval? I only just started reading this page two seconds ago, I don't want to start reading another already."

...corresponding to a pair of pitches with a frequency ratio of 3:2...

2) "Why 3:2? Why not 4:5? Or 2.983:17? What makes that significant?

...or very nearly so.

3) "Very nearly so? I could substitute 3:2.75 and it would still work? So this 3:2 thing doesn't even matter?

To address point one, I'd like to add a very bried definition of the word "interval" in brackets "(gap between notes)"? "(Span)"? You can choose, but given that the word "interval" can be used to refer to a chronological gap, I think it's important to clarify.

To address point two, I'd like to go back to the "morning bells at dawn" scenario that Justlettersandnumbers summarily dismissed earlier. Let's go back to what a perfect fifth is really all about: Two notes, sounded either in unison or in succession. Imagine an untrained listener were to hear those two notes, what might they think about them that they might not think about any two other randomly chosen notes?

1. That sounds nice! (Sounds pleasing to the ear? Sonorous?)

2. Why does it sound nice? (Pairs of notes with symm... err, consonant relationships sound nice)

3. What kind of relationship do these two notes have? (These two notes have a frequency ratio of 2:3)

So, keeping in mind that pretty much the whole point of music is to evoke feelings in humans via their ears, my modified version of the first paragraph would go something like this (and the order of the statements above is malleable):

In music theory, a perfect fifth is the musical interval corresponding to a pair of notes with a frequency ratio of 3:2 (or a close approximation). Musical intervals with even ratios such as 3:2 produce pairs of notes that sounds pleasant together.

In my mind, that gets the fundamental significance of what a perfect fifth out of the way, and leaves the reader with a basis on which to build their understanding as they delve further into the article. I assume that "pitch" is a more accurate term than "note", but I think in this context it's okay to generalize the terms a bit to cater for lay readers: Everyone knows what a note is. Not everyone knows what a pitch is.

My other main concern the reference to "seven semitones". Although this makes perfect sense to me, I think a lay reader might ask "Why seven? It's a fifth, not a seventh!". Mentioning that intervals are named after major scales and not chromatic scales would help clear this up.

Overall, though: Thanks JustPlainBill for your work on this, and for coming up with a pretty good version of the lead section that takes in to consideration most of the myriad concerns we've enumerated here!

InternetMeme (talk) 05:49, 24 March 2014 (UTC)

I've just realized there's a big problem with my prototype lead paragraph: Most people don't understand what frequencies (and therefore frequency ratios) are. I think it could be important to re-add the bit "This means that one note makes three vibrations for every two made by the other". I think this kind of clarification is appropriate in a lead section.

InternetMeme (talk) 06:07, 24 March 2014 (UTC)

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In music theory, a perfect fifth is the musical interval corresponding to a pair of pitches with a frequency ratio usually given as 3:2.

In classical music from Western culture, a fifth is an interval spanning five consecutive notes in a diatonic scale. The perfect fifth (often abbreviated P5) spans seven semitones. For example, the interval from C to G is a perfect fifth, as the note G lies seven semitones above C. The perfect fifth may be derived from the harmonic series as the interval between the second and third harmonics. In a diatonic scale, the dominant note is a perfect fifth above the tonic note.

Formed by a ratio of small integers (3:2 being the simplest such ratio after 1:1 and 2:1) the perfect fifth is more consonant, or stable, than any other interval except the unison and the octave. It occurs above the root of all major and minor chords (triads) and their extensions. Until the late 19th century, it was often referred to by one of its Greek names, diapente. Its inversion is the perfect fourth. The octave of the fifth is the twelfth.

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There is no getting around the simple fact that words have meanings. "Note" has several meanings, even when limited to a musical context. "Pitch" is less ambiguous and more accurate. If the reader doesn't understand, there is the link for further exploration. Click the link, or don't click and read on, to see if it starts to make sense from the context. That's the beauty of hypertext.

I am having a really hard time seeing 3:2 as an "even" ratio. Talking about numbers, even and odd have pretty well-known meanings. Simple integer ratios do produce smoother sound.

The reason for the "nearly so" is that modern tuning practice (equal temperament) puts the perfect fifth at about 2.996614154 to 2. That is too much information for the first sentence of the intro, in my opinion. __ Just plain Bill (talk) 12:35, 24 March 2014 (UTC)

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Upon awakening AND before reading posts after my last (which undoubtedly correct me), I realized that I have committed the exact error I hoped to avoid and used words that tell the wrong story. I must re-think this because the tonic is clearly the "first" and one must count the first fence-post in order to get to the "fifth", or dominant, and, therefore have the basis for the term "fifth" in describing this concept. Though my initial rationale was correct, I appologize for the bad concluding remarks. Returning after a year, I wasn't up to speed enough to comment as I did. Regards -- Steve -- (talk) 13:34, 24 March 2014 (UTC)

Hi guys. I'm interested in the prototype first paragraph:

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In music theory, a perfect fifth is the musical interval corresponding to a pair of pitches with a frequency ratio usually given as 3:2.

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Keep in mind that we're writing a lead here, not the main article. We need to keep things as accessible as possible, and avoid requiring the navigation of wikilinks. There are a few issues:

1. Most readers don't know what a musical interval is. It needs a brief (2-3 word) explanation in brackets.
2. It uses the word "pitch", which isn't as universally understood as the word "note". The word "note" is precise enough for a lead, and the body of the article can be used to elaborate.
3. The frequency ratio is not merely usually given as 3:2. The ideal frequency ratio is exactly 3:2, and any other ratio used is an approximation of that ratio.
4. It doesn't explain why the ratio is 3:2. The reader is left to wonder why the ratio shouldn't be something like 13.5:17. It needs to indicate that 3:2 ratios are symmetrical/even/whateveryoulike, and that these ratios sound pleasant/consonant/whateveryoulikeagain.
5. The ratio might be better written as 2:3, as that lists the tonic frequency first, followed by the fifth.

InternetMeme (talk) 14:00, 24 March 2014 (UTC)

I think it would be futile to try to make the content immediately understandable to everyone, without them needing to follow any wikilinks. This is what links are for. What I'd like to see, however, is a lead that is accessible to most readers who are likely to care. That means both the ones from music theory background and the ones from engineering/acoustics background. For some, talking about staffs and scales will make more sense, and for others ratios of frequencies will make more sense. Most in both camps will know basically what is meant by pitch, and musical interval, and those who don't can follow the links. That is, I reject your premise that "Most readers don't know what a musical interval is." Most will probably know it, though in different ways, and those who don't can click and find out; if they don't know, why are they reading about a perfect fifth? Same with "pitch"; what use is a perfect fifth to someone unfamiliar with pitch? As for the "usually given as", I agree that misses the mark; but "is exactly" also misses the mark. That's why I had proposed the "very nearly" wording; an equal tempered perfect fifth is still a perfect fifth, not an approximation to one. And the explaning of "why" is a huge tangent, certainly not suitable for the lead, though we could certainly say that it is a most pleasing or consonant interval, or has been regarded as such since ancient times, or whatever along those lines. As for 2:3 vs 3:2, these are about equally common in books; there is no standard convention, but it is certainly not unusual to express ratios in a to:from form, or note:tonic. As for the potential fencepost error misinterpretation, I might change "the interval spanning five consecutive notes in a diatonic scale" to "the interval from the first to the last of five consecutive notes in a diatonic scale". Dicklyon (talk) 00:00, 25 March 2014 (UTC)

"the interval from the first to the last of five consecutive notes in a diatonic scale" is clear, unambiguous, and reasonably compact. I like it.

The bit about "or very nearly so" poses a problem or two. As InternetMeme pointed out, it may cause some readers to wonder why it needs to be said. It does need to be said, but in-line wordy explanation of the motivation would be tangential to the gist of that first single-sentence paragraph of about two dozen words.

One solution would be an explanatory footnote. There aren't any of those in the article as it now stands.

Another solution would be linking "or very nearly so" to the section on Use in tuning and tonal systems, where just and tempered perfect fifths are further explained.

__ Just plain Bill (talk) 02:08, 25 March 2014 (UTC)

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EngineerSteve's more carefully thought-out comments.

Since some above comments refer to past revisions it is hard to know why some comments are made.

Yes. I am doing my best to approach this like some non-engineer stopping by for a little dose of music theory and not a full course, just as I was

I'm puzzled by the desire to include harmonic talk and I don't think talking about second and third harmonics in the lead is appropriate - I believe it will certainly confuse and chase away. While that ratio is the same as the ratio of the Fifth to the Tonic, I am pretty sure that 'fifth' is only used to define a relationship of two notes, not any harmonics. I really believe having the word harmonics in the lead is inappropriate for that reason.

Since I started this, allow me to identify items that can be addressed separately. I see several points that seem possibly appropriate for a lead:

1 - Western vs. non-western fifths (probably just a passing mention)

2 - The relationship of the two notes to each other (in pitch, frequency, diatonic steps, chromatic steps, OR staff positions). Is it a difference, or a span including the end points...

3 - Human consonance, harmony, or pleasure in hearing a fifth. Intervals too small are to be avoided as they sound dissonant)

4 - The source of (or reason for) the word "fifth".

5 - The source of (or the reason for) the word "perfect".  ? Is this word used only to distinguish it from the Augmented and Diminished fifths??

6 - Precise meaning of words like interval, span, or encompass.

7 - Fifths that are "very nearly so". I need clarification. DOES this refer to the Aug and Dim fifths???

8 - Jargon usage.

RE: #1 (western). It was hard following the comments. It seems to me that something like this is ok: "The [?perfect?] fifth is a musical interval used in music cultures worldwide. In Western music theory it is ...". Then, if appropriate address non-western fifths in the article.

RE #2 & #6 THE interval. Poking around the net I see some other definitions and even they seem to mix-up counting the steps "in between" - with - including (spanning) the first, which is counting Tonic.

This: http://piano.about.com/od/musicaltermsa1/g/GL_fifth.htm uses (diatonic) "spanning Five scale degrees", but then says "equals seven half steps".

I also see some inconsistencies (above) in the way you authors define the interval, or span; so the fence-post-count error is an easy trap to fall into. I think it was Just__Bill who wrote: "...spanning five consecutive notes in a diatonic scale. The perfect fifth ... spans seven semitones." Should be "spans eight semi-tones".

If we "span" five diatonic notes we better "span" eight chromatic notes.

If we are seven chromatic steps away, then we should be four diatonic steps away.

(One of the YouTube videos that helped me mentioned the 5 vs 4 thing and even said that he could consider it a fourth (or four steps), only for understanding purposes).

I'm coming to the conclusion that "span" includes the Tonic and is ok as long as the "diatonic C to G" example immediately follows to confirm the meaning. Then, I'm comfortable with something like: "In the chromatic scale, this is identical to a span of eight notes/semi-tones/steps (separated by seven steps).

I also point out the the current image easily shows the five staff positions as including the tonic/root.

It seems reasonable to me that it is also ok to add the "Staff positions" at this point because some readers will most certainly have more training and this will add to their experience. Dummies like me can easily ignore that as being for the more advanced. It doesn't confuse me UNLESS it is used as the primary definition.

RE: #3 Human perception For me, a mention here would have been appropriate. Perhaps: "The fifth is found to be a pleasing-to-the-ear combination of notes / tones. This is referred to as consonance." and Wililink to Consonance and dissonance.

# 4 Why Fifth? The Fifth is the fifth note in the Diatonic scale and THAT is where it gets its name - This must be. I think this can be noted in the lead.

#5 Why Perfect? I'll leave this to yous guys... I seems that is it used simply to differentiate this from the augmented and diminished fifths. This is out of my pay-grade as bill says.

#6 Span. Covered in #2 above

#7 Nearly so fifths I think it's been covered...

#7 Jargon usage I believe jargon should not be very heavy in the lead. It must be there. However, things like the follow-up example of the "C to G" is a good way to clear up any confusion caused by the use of "span" right before it.

Question Ignoring the Aug and Dim fifths, I do not know if the term fifth is only used to talk about it in absolute/fixed terms of "the dominant G in the key of C" way, or would you also use it more relatively and talk about, say: " a fifth above the E" when still in the key of C...?

So, for your consideration; assembling my sample lead... (parens) means I think it may or may not improve it. Also, Wikilinks and bolding where appropriate.

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The fifth is an (important]) musical interval used in music cultures worldwide. In Western music theory the Perfect Fifth is an interval that spans five diatonic notes. For example, the interval from C to G is a perfect fifth and, as shown in the figure, it also spans five staff positions. Note that it also spans eight chromatic notes (or semitones), or one is seven semitones higher (than the other). This is called the Perfect fifth, often abbreviated P5, and sometimes just called the fifth.

In a diatonic scale, the dominant note is a perfect fifth above the tonic note, is the fifth note in the scale and thus derives the name of fifth.

The perfect fifth is more consonant, stable, or pleasing to the ear, than any other interval except the unison and the octave. It occurs above the root of all major and minor chords (triads) and their extensions.

There are also Augmented and Diminished fifths that are useful. These are one more and one less semitone respectively and also span five staff positions even though the semitone spans differ.

(It is interesting to note that the Perfect Fifth is a frequency ratio of 3:2.)

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Notes' I intentionally left out the "meantone" 4 3 thing (obviously an intermediate or advanced thing), any reference to harmonics and the Greek thing, though I have no objection to adding a little Greek culture....(;-)

I also wasn't able to see any trend in the other comments about What, or Why it is "perfect" - #5.

It also seems appropriate to leave the "or nearly so" talk to the section already addressing it. The beginner doesn't need to know that, will probably not go too deep into the article, and the more advanced reader will read more and see it.

Regards, -- Steve -- (talk) 05:45, 25 March 2014 (UTC)

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Hi there,

I believe there's a small problem here: People are being misdirected a little by the fact that the term "perfect fifth" has two completely separate (though partly related) meanings. It'd probably be better if there was another term, but as far as I'm aware, there isn't:

1. The first concept Perfect Fifth describes is the well known one we're discussing, relating to scales, Western music, etc'.
2. The second concept it describes is the one I'm more interested in, and it's a concept that's existed many years before any kind of music theory had been invented. In fact, more than just existing independently of music theory, the concept itself is one of the cornerstones of music theory: It goes back to the thing I wrote earlier, which I'll paste here in case EngineerSteve missed it:

Imagine it's the year 4000 BC, long before the invention of modern musical theory or the naming of notes, etc'. Now imagine you're some ancient human of the day, and that you get hold of some kind of prehistoric thread and string two lengths of it between the two beams of a forked stick. Assuming you figure out some way of winding them to make them tighter, imagine you tune the first string to any random note near the middle of human hearing. Now, while plucking the first string, you tune the second string until you reach a note that's above the first, and which has a pleasing sonorousness. Although you have no way of knowing it, that note is vibrating three times for every two times the first note vibrates.

http://mathforum.org/library/drmath/view/52470.html

The unfortunate thing is that the only term we have to describe that concept is Perfect Fifth, which misleadingly implies that it is dependent on scales and music theory. In addition, the concept didn't get a name until Western music theory had been invented. The name (though not the concept) is dependent on Western music theory, and as a consequence, the concept itself is often viewed through that lens, and is seen as dependent upon it, when in fact it is completely independent of it, and in fact predates it significantly.

Now, possibly I'm missing something, and there is in fact a separate term to describe that concept, but as far as I know, there isn't. The two possible solutions are:

1. Make a new article (or locate the existing one) to describe the concept of a pair of harmonious notes with a 2:3 ratio.

1. Keep the two concepts together in this article, and describe them both consecutively. I think it'd make sense to describe this concept first, as it predates the other concept, is more fundamental to the basis of music, and is simpler and easier to understand as it can be described independently of any complex music frameworks.

InternetMeme (talk) 14:07, 25 March 2014 (UTC)

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EngineerSteve here...

I read about Cave Man, have often wondered the same thing and I include it in my Number #3 Human perception a.k.a. pleasing-ness.

Short story...

Months ago I came into the middle of a radio program that was saying something to the effect that...

Tones form standing waves in the cochlea. The loops and nodes of the two standing waves will line up in some (easily calculated) places. This appears to possibly be the basis for consonance, or dissonance when they don't allign.

Try Searching on "coclea standing waves". I don't have time to read and pick one appropriate, sorry.... The brain is a marvelous thing...

My motivation in defining the subtopics that I see was to allow easier discussions on each. Change if you wish, but I have expresssed my view the best I can and will back away for a while - ugh! other responsibilities. I appreciate everyone's efforts on this.

Regards, -- Steve -- (talk) 15:48, 25 March 2014 (UTC)

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Thanks for your input! It's great to have you back here to elaborate on your original observations. Since you're busy now, I guess it's up to us to consider everything that's been said here and come up with a lead that benefits from our collective ideas. I particularly look forward to Dicklyon's ideas about the two rather different meanings of the term. It's a tricky one. I kind of wish there was another word to define the sound made by two tones with a 2:3 ratio, as it really is a fundamental thing. I'll have to look in to your research regarding the wave effects in the cochlea. Maybe we'll eventually get to the heart of the phenomenon ; )

InternetMeme (talk) 16:27, 25 March 2014 (UTC)

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OK. you're after the why, rather than just the what (sounds "good"). Fifths (and Thirds, I guess)... Try Googling "Why does Harmony sound good?" ... Good luck, -- Steve -- (talk) 17:15, 25 March 2014 (UTC)

I gotta' put this down. Remember ... Google is your Friend! I Googled: "Why does Harmony sound good to the ear?". Third hit:

http://news.sciencemag.org/math/2011/09/how-ear-distinguishes-sweet-sounds-sour-notes

-- Steve -- (talk) 17:26, 25 March 2014 (UTC)

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Simple! That's the word I've been looking for! Simple ratios sound good!

Thanks very much for that link. The last paragraph reminds me a little of my cave man analogy ; ) I think the big question for me is: Where do we draw the line? I think that getting in to information such as how the cochlea works is going too far. I think the reader will be satisfied with the logic:

A fifth is based on a 2:3 ratio --> A 2:3 ratio is simple --> Simple ratios sound good.

In addition, this logic would mirror a reader's experience if they were to listen to a pair of perfect fifth pitches. They would hear it, think to themselves "That sounds good!", and they could find the explanation here: "Oh, so it's a simple ratio, and simple ratios sound good."

InternetMeme (talk) 06:59, 26 March 2014 (UTC)

I just undid an unexplained removal of a genious method of auralizing a perfect fifth: Humming the first two works of "Twinkle, Twinkle little star". The last time it was seen before being butchered then removed from the lead without reasonable explanation is in this revision:

https://en.wikipedia.org/w/index.php?title=Perfect_fifth&oldid=204011700

InternetMeme (talk) 19:08, 28 March 2014 (UTC)

References

Self-referential definition

The article says

"Perfect intervals are also defined as those natural intervals whose inversions are also perfect, ...".

But this is a circular definition since it uses the concept of perfect to define perfect. Should it be rewritten like this:

"Perfect intervals are also defined as those natural intervals whose inversions are also natural, ..."? 81.175.221.202 (talk) 14:33, 7 January 2024 (UTC)