Talk:Just intonation

Key of examples

Not that there's anything wrong with it, but is there any reason for the examples being changed from C major to F major? Just curious. --Camembert (22 August 2003)

Outline

My proposed outline:

1. introduction: Just intonation is any musical tuning in which the frequencies of notes are related by whole number ratios. Another way of considering just intonation is as being based on lower members of the harmonic series. Any interval tuned in this way is called a just interval. Intervals used are then capable of greater consonance and greater dissonance, however ratios of extrodinarily large numbers, such as 1024:927, are rarely purposefully included just tunings.
2. Why JI, Why ET
   1. JI is good
   2. 1. "A fifth isn't a fifth unless its just"-Lou Harrison
   3. Why isn't just intonation used much?
      1. Circle of fifths: Loking at the Circle of fifths, it appears that if one where to stack enough perfect fifths, one would eventually (after twelve fifths) reach an octave of the original pitch, and this is true of equal tempered fifths. However, no matter how just perfect fifths are stacked, one never repeats a pitch, and modulation through the circle of fifths is impossible. The distance between the seventh octave and the twelfth fifth is called a pythagorean comma.
      2. Wolf tone: When one composes music, of course, one rarely uses an infinite set of pitches, in what Lou Harrison calls the Free Style or extended just intonation. Rather one selects a finite set of pitches or a scale with a finite number, such as the diatonic scale below. Even if one creates a just "chromatic" scale with all the usual twelve tones, one is not able to modulate because of wolf intervals. The diatonic scale below allows a minor tone to occur next to a semitone which produces the awkward ratio 32/27 for Bb/G.
3. Just tunings
   1. Limit: Composers often impose a limit on how complex the ratios used are: for example, a composer may write in "7-limit JI", meaning that no prime number larger than 7 features in the ratios they use. Under this scheme, the ratio 10:7, for example, would be permitted, but 11:7 would not be, as all non-prime numbers are octaves of, or mathematically and tonally related to, lower primes (example: 12 is an octave of 6, while 9 is a multiple of 3).
   2. Diatonic Scale: It is possible to tune the familiar diatonic scale or chromatic scale in just intonation but many other justly tuned scales have also been used.
4. JI Composers: include Glenn Branca, Arnold Dreyblatt, Kyle Gann, Lou Harrison, Ben Johnston, Harry Partch, Terry Riley, LaMonte Young, James Tenney, Pauline Oliveros, Stuart Dempster, and Elodie Lauten.
5. conclusion

http://www.musicmavericks.org/features/essay_justintonation.html

Hyacinth (30 January 2004)

Problems of this article

There are many problems in this article, as appears from the warnings on top. Some of these problems already are obvious in the lead, which says:

In music, just intonation or pure intonation is the tuning of musical intervals as whole number ratios (such as 3:2 or 4:3) of frequencies. An interval tuned in this way is said to be pure, and is called a just interval. Just intervals (and chords created by combining them) consist of tones from a single harmonic series of an implied fundamental. For example, in the diagram, if the notes G3 and C4 (labelled 3 and 4) are tuned as members of the harmonic series of the lowest C, their frequencies will be 3 and 4 times the fundamental frequency. The interval ratio between C4 and G3 is therefore 4:3, a just fourth

• Not all intervals can be said "just" or "pure", I think that the expression only concerns the octave, pure fifths and fourths, and pure major and minor thirds. (This should be checked in Sauveur's writings where the expression "pure" originates, but I have been unable to do so just now.)
• It is not true that "just intonation [...] is the tuning of musical intervals as whole numbers ratios" – the Pythagorean comma, for instance, 531441⁄524288, also is an interval formed of a ratio of whole numbers.
• Also, it is not true that "just intervals consist of tones from a single harmonic series of an implied fundamental": the just minor third in the perfect minor chord cannot be found in the harmonic series of the fundamental of that chord.

There is a somewhat better article on Intonation juste in the French Wikipedia, which could inspire the revision of the article in English. — Hucbald.SaintAmand (talk) 15:26, 11 January 2023 (UTC)

Sauveur, in Methode Générale Pour former les Systêmes temperés de Musique (available here), describes the "just diatonic system" as formed of degrees corresponding to the numbers 24 27 30 32 36 40 45 48 for C D E F G A B C. He adds that in this system all octaves are equal [48:24 = 2], as are the minor seconds [48:45 = 32:30 = 16:15]; the major seconds are inequal [being either 27:24 = 9:8 or 30:27 = 10:9]. He then says that some minor thirds are "just", but not all, and that the major thirds are all the same, and continues saying that most fourths and fifths are "just", but not all. As one sees, which intervals are called "just" is not entirely clear. This, so far as I can tell, is the first usage of the term "just" to describe intervals. — Hucbald.SaintAmand (talk) 15:51, 11 January 2023 (UTC)

Taking your comments in order:

1. Musicians (and music theorists) do sometimes refer to these other intervals as just or pure, even if Sauveur doesn't. It seems pedantic to insist that all vocabulary derive from a single source in 1707. All music terminology is applied inconsistently.

2. How about we just say "ratios of small whole numbers"? That would eliminate the Pythagorean comma problem, although it would leave it unexplained why some small ratios, such as 7:5 or 13:11, aren't considered part of this meaning. I'm comfortable with the vagueness of "small," since there are different opinions about some intervals. The French article breaks it down to primes, so that what I would consider a just-tuned second, 9:8, is broken into its prime factors, but that makes the explanation more complicated.

3. The minor third E-G can be derived from the fundamental C, even if you're using it in an E minor triad. The sentence you cite doesn't say the implied fundamental is the root of the chord. Even the Pythagorean comma could theoretically be derived from a single, impossibly-low fundamental frequency.

I agree, in general, though, that there's room for improvement, and the French article seems like it might be helpful. —Wahoofive (talk) 18:21, 11 January 2023 (UTC)

The French article says "Until the 20th century, just intonation aimed particularly at just consonances, fifths and fourths, thirds and sixths". No source is given, but this appears to make sense. It continues discussing Euler's extension to harmonic 7, which suggests that just intonation was limited before to numbers 1 to 6 (the Renaissance senario) until the development of the 7-limit diatonic scale.

Note that saying this does not imply that these numbers correspond to sounds being harmonic partials of a single fundamental. There is no reason to claim that "the minor third E–G can be derived from the fundamental C", because nobody would do that. It makes more sense to say that it can be derived from numbers 5 and 6. Describing the 5-limit C major scale as derived from the fundamental F, or the 5-limit A minor scale as derived from the fundamental B♭, as done here really makes little sense. Removing these mentions already would improve the article. — Hucbald.SaintAmand (talk) 09:32, 13 January 2023 (UTC)

I don't understand how the ratios are derived. You say "For example, in the diagram, if the notes G3 and C4 (labelled 3 and 4) are tuned as members of the harmonic series of the lowest C, their frequencies will be 3 and 4 times the fundamental frequency. The interval ratio between C4 and G3 is therefore 4:3, a just fourth." From this, I would assume I could also say that "the interval ratio between G3 and C4 is 3:4, a just fourth." Is this true? This example is confusing. How about saying what all the ratios are, starting at the low C and working your way up? It then goes on to discuss "the numbers 2 and 3 and their powers, such as 3:2, a perfect fifth, and 9:4, a major ninth." - I don't see from the first example how these ratios are derived. Could you please explain this in a more clear and thorough way? For example, what does the first number in the ratio refer to and what does the second number refer to? Some clarification of how these numbers are derived would be helpful. — Preceding unsigned comment added by 24.103.122.30 (talk) 12:52, 25 April 2023 (UTC)

There is indeed something confusing in this description of the interval ratios, particularly in the mention of the harmonic series. It should be corrected, but let's leave that for later. The intervals considered in Just Intonation are consonances, which were shown by the Pythagoreans in Greece about 27 centuries ago, on the consideration of string lengths, to correspond to simple ratios of whole numbers: 2:1 for the octave, 3:2 for the pure fifth, 4:3 for the pure fourth. Modern explanations often are based on concordances between the partials of the sounds considered (see Consonance and dissonance), which may explain why what you quote refers to harmonic partials. But once again, let's leave that for now.

Whether you describe the intervals as 2:1, 3:2 and 4:3, or 1:2, 2:3 and 3:4 depends (a) on whether you consider the intervals in ascending or in descending order and (b) whether you consider them to concern string lengths or frequencies. A pure fourth, for instance, is pure in either direction. The ratios are merely that, ratios. The numbers themselves describe a proportionality. 4:3, for instance, may refer to string lengths of 40 and 30 cm, or of 60 and 45, or whatever; or to frequencies of 400 and 300 Hz, or of 500 and 375 Hz, etc., as you want. The ratio is 4:3 in all cases. If 4 and 3 correspond to string lenths, then 4 denotes the lower sound (the longer string); if they correspond to frequencies, the reverse. Is that clearer? I hope so, but tell me if something remains unclear. — Hucbald.SaintAmand (talk) 19:38, 25 April 2023 (UTC)

Pure or Just tuning: Fundamental considerations.

What is the relationship of "simple" ratios with "music"? The argument is, thirds, fifths and octaves "looks like" their counterparts in the harmonic series. Is this enough justification to say, music intervals must be same like harmonics? There is one consideration completely neglected: The ear can identify individual "instruments", e.g. a flute vs an oboe. The reason the ear can do so is because each natural sound consists harmonics each with unique "deviation" from being exact multiples of the fundaments. The Reasons are "inhamonicity" in strings, and "mouth effect" in pipes. In order to prove this, a program is made able to play music in the "stretched octave". Instead of the octave being 2:1, or 1200 cents, ET, the octave is gradually increased in steps to 1220 cents. At 1200 cents, dense music chords are heard to be much confused. Individual instruments cannot be discerned. The whole chord sounds "muddy". https://www.youtube.com/watch?v=UB9P7VCJmhk As the octave is gradually "stretched", this phenomenon decreases, until it completely disappears at 1220 cents. It can be explained that the ear will generate via OAE accurate frequencies at 2x and 3x to "check" the deviation of these harmonics. The slight deviation can be measured by the BM tightening or relaxing the muscles, the degree of such action will reflect the magnitude of each unique deviation. We should also remind us the fact the eye can recognize peoples' faces, with only very slight differences in curvature. If the ear actually uses this method to identify "sounds", then music should AVOID using intervals comparable to harmonics! There are multiple studies of listeners' preference of music octaves. The result is already established as fact that the music octave is indeed stretched at around 1220 cents. Also, another study confirms "beats" does not contribute to the judgement of intonation. Rather, there is evidence that "beats" is a factor that contributes to the "aesthetic value" of the music. https://www.youtube.com/watch?v=CAN0HjKPLXI There is also the missing attribute how music carries emotion. We know that ET lacks variety, and most 4,000 proposals in "scales.zip" how a scale can be formed lacks uniformity. This attribute is easily remedied by using two 6-note per octave "queues", with each whole tone being 9/8, octave is (9/8)^6 = 2.0273 or 1223.5 cents, the octave identified in many scientific studies. By putting two 6-note queues in parallel, with an offset of the two different sizes of the half-tone, 2187/2048, and 256/243, we get the 12-note music scale. Violin teachers told us the leading note should be sharper. This is true, this way, notes in the lower tetrachord are on the "flatter" queue, while the upper tetrachord notes are on the "sharper" queue. This way, music melodies have the mechanism to carry emotion, even without western style harmony. Also, in key change, a new queue may need be created, another way to create emotion. This presently proposed scale is 3-limit. Any pair of notes will make a interval 2^m * 3^n, meeting the "definition", if it can be so called, of "Pure" or "Just" intonation. With OAE, the ear can create distortion waves as reference to check harmonic deviations, it is likely the ear can voluntarily generate odd or even harmonics waves, since these are different by the distortion being symmetric or not. It might be difficult for the ear to identify if odd harmonics are 3rd, 5th or 7th. Hence, it may not be useful to consider higher harmonics than 3 or 4. If this is true, the violin should tune the highest string, E-string, higher. It should be 669Hz instead of 660Hz. The performance will be much more musical, emotional, and solve all problems with intonation. This is the opposite of the practice of many violinists tuning the G-string sharper. Vssimo (talk) 11:34, 4 September 2024 (UTC)

The relationship between "simple" ratios and music can be described as this : consonances between complex sounds (i.e. stable sounds with harmonic partials) arise from a blending of harmonic partials. When two sounds are an octave apart, all the partials of the higher are in unison with one partial on two of the lower. When they are a fifth apart, the concordance is between one partial on two of the higher, one on three of the lower – and you can see the "simple ratio", 1:2 in the first case, 2:3 in the second. This is a simple way of viewing things. It would be more exact to consider dissonances; which arise when the concordance between partials is not exact and produces beats. In other words, it certainly is not that the octave correponds to partial 2, the 5th to partial 3, etc. These intervals are between two sounds, each of which with its own partials.

In the case of inharmonic partials (i.e. less stable sounds; but in musical sounds the inharmonicity remains limited), the concordance is not perfect and beats are produced, but consonance is then defined by a minimum of beating. The article Consonance and dissonance tries to explain all that. Feel free to modify it, but you will need better references than YouTube presentations. — Hucbald.SaintAmand (talk) 18:00, 18 November 2024 (UTC)