Wikipedia:Reference desk/Archives/Science/2019 August 24

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August 24

Thomas Edison

There are many claims being made that Edison stole works from others. These are true or conspiracy theories?

https://www.historicmysteries.com/did-thomas-edison-steal-inventions/

http://newsreeling.com/about-thomas-edisons-lies-and-19-stolen-inventions — Preceding unsigned comment added by 2409:4061:2106:1A01:3483:A09F:5A25:16CD (talk) 05:13, 24 August 2019 (UTC)

The OP cites two examples of muck raking journalism.

Ref. 1 puts malicious slants on Edison's legal patenting in USA of X-ray fluoroscopy and the phonograph where we are told that Edison "padded his bank account at Rontgen's expense". Note that Röntgen expressly chose not to patent his discovery of X-rays for which he received multiple rewards including the first Nobel Prize in Physics, and which would inevitably have been acknowledged as Prior art in Edison's patent application. The smear piece continues with a "sinister story" seeking to connect Edison with an unsolved murder that he "may have nothing to do with" and accuses Edison of driving French cinematographer Georges Méliès into bankrupcy. A little research finds that unauthorized displays in the US of Méliès' film "A Trip to the Moon" brought him fame that continued with production of more films, later as a partner in Edison's Motion Picture Patents Company (1908). Méliès' decline can be attributed to an unlucky Pacific film expedition in 1913, death of his wife and the onset of WW1 during which the French army confiscated film prints in order to recover silver and celluloid, the latter for shoe heels.

Ref. 2 alleges in superficial manner "19 stolen inventions" of which 16 have nothing to do with Edison. Its writer scolds "naughty" Edison for co-inventing the light bulb with Joseph Swan when the reality is that thousands of light bulbs were manufactured with the trademark EDISWAN, reflecting the agreeable merger of their business interests in 1883. She tells the nonsense that "During the 1980’s Edison developed his “Edison Vitascope”. Again she scolds Edison as "naughty" for developing the principle of recording and reproducing sound starting in 1877. How could Edison dare usurp the glory of Édouard-Léon Scott de Martinville who earlier invented a Phonautograph that could only scratch a wriggly line on a cylinder with no pretense at that time of being able to play it back (sarcasm)?

"the men with the muck rakes are often indispensable to the well being of society; but only if they know when to stop raking the muck..." said Theodore Roosevelt. DroneB (talk) 14:10, 24 August 2019 (UTC)

Did you mean 1890's? I don't think Edison was all that active in the 1980's. :) ←Baseball Bugs ^{What's up, Doc?} carrots→ 15:17, 24 August 2019 (UTC)

No. Sic erat scriptum. DroneB (talk) 22:38, 24 August 2019 (UTC)

Indeed, Edison credits Rontgen with the discovery of X-rays in his patent, claiming as his invention not the x-ray itself, but a particular means of using them to induce fluorescence. The implicit argument that Rontgen deserves credit for anything to do with x-rays reminds me of Morse's attempt to patent the very concept of using electromagnetism to communicate (see Samuel_Morse#Litigation_over_telegraph_patent). Someguy1221 (talk) 03:34, 27 August 2019 (UTC)

I think the above is sufficient to dismiss the claims, but I would also note that a false claim that Edison stole something is not a "conspiracy theory". A conspiracy theory alleges that two or more people contributed to some malicious action. If Edison had stolen someone's intellectual property, he could have done it all on his own. --76.69.116.4 (talk) 04:37, 25 August 2019 (UTC)

It has some elements of a conspiracy theory, though. ←Baseball Bugs ^{What's up, Doc?} carrots→ 05:06, 25 August 2019 (UTC)

Well, it has the "theory" element but not the "conspiracy" element. I don't think 1 counts as "some". --76.69.116.4 (talk) 20:23, 26 August 2019 (UTC)

Conspiracy makes it easier to explain why the alleged thief got away with it and the alleged victim could not prevail. Officials would be involved in the cover-up, bullies hired to keep others quiet, people knowing the real owner would disregard him, the thief in turn would had help those helping him in their own evil schemes, the whole ring of villains would be Master of the universe, untouchables, etc. Makes the scenario more appealing and harder to disprove Gem fr (talk) 07:31, 27 August 2019 (UTC)

why JFET

hello, in AGC circuits, they mostly have an amplifier with a JFET as the control element (voltage-controlled resistor) in the feedback path or the input voltage divider. why? specifically, why not BJT (BC547) or MOSFETs (such as the 2N7001/BS170), moreso as JFETs seem to become more and more exotic with each passing year (do they?)... Is there a way to "emulate" JFET behaviour using a BJT? Aecho6Ee (talk) 10:06, 24 August 2019 (UTC)

A JFET is better suited to use as a variable resistance to AC than a BJT because:

• The JFET's effective resistance between source and drain is a gradual function of the control voltage applied to the gate while the BJT responds abruptly to a change in the control voltage above or below a nominal 0.65V that depends on temperature applied to the base-emitter diode.
• The reverse-biased JGET gate draws virtually no current.
• JFETs are generally symmetrical which allows applying AC voltage between source and gate. A BJT must run at a finite collector current that is stabilised against temperature variation by additional components.

The impression that JFETs seem to become more exotic can arise from continual improvements in available component's power and frequency limits. However for applications such as gain control in a compander, the inexpensive 2N3819 N-Channel JFET still serves well. DroneB (talk) 12:27, 24 August 2019 (UTC)

Missing keys

They say that there is no such thing as a stupid question (only stupid answers). The following might be the exception that proves the rule:

On a keyboard (the musical kind) why are there "missing" black keys? In other words, why are there some pairs of white keys that don't have a black key between them? If you took a frequency that was exactly half-way between the those two white keys, and made a black key for that frequency, would the universe implode or something? 2606:A000:1126:28D:8DFA:9F88:DA44:2E31 (talk) 18:01, 24 August 2019 (UTC)

The names of the notes in the popular key of C major. DroneB (talk) 19:58, 24 August 2019 (UTC)

The gap between any two adjacent keys of any colour is exactly a semitone (about a 6% frequency difference). See Equal temperament. The colour of the notes is just a keyboard convention. There are no "missing" notes. Putting an extra note between any two adjacent notes would give a quarter tone which is seldom used in Western music, but perhaps can be heard in Indian and Arabic music. Dbfirs 18:27, 24 August 2019 (UTC)

A piece of music is normally composed using only or mainly notes from a preselected key or group of 7 notes sometimes taught as Do, Re, Mi, Fa, Sol, La, and Si. The starting note Do may be any one of 11 consecutive piano keys. When the tonic note Do is on the leftmost key in the illustration, the key of the music is said to be "C major" and it is playable using white keys only. There are 11 other possible major keys corresponding to the other 11 possible tonic (starting) keys. The grouping of black and white piano keys allows immediate recognition of their fixed letter values. The particular set of black and white piano keys employed in any given major key (or in the 12 minor keys that differ from major keys in how the Solfège is spaced) will be deduced by the performer from the Key signature in the musical notation. DroneB (talk) 19:58, 24 August 2019 (UTC)

That 'Si' instead of 'Ti' might come as a shock to Rodgers and Hammerstein. ←Baseball Bugs ^{What's up, Doc?} carrots→ 20:59, 24 August 2019 (UTC)

Decomposers since 1979 and 1960 respectively. R.I.P. DroneB (talk) 22:34, 24 August 2019 (UTC)

Blame Sarah Ann Glover who changed 'Si' to' Ti'. For that matter, 'Do' was originally 'Ut'. {The poster formerly known as 87.81.230.195} 2.122.61.224 (talk) 02:26, 25 August 2019 (UTC)

«Ut queant laxi, Resonare fibris, Mira gestorum, Famuli tuorum, Solve polluti, Labii reatum, Sancte Iohannes.»Gem fr (talk) 07:38, 26 August 2019 (UTC)

Mention of the Solfège is an inevitable reminder of The Nairobi Trio. ←Baseball Bugs ^{What's up, Doc?} carrots→ 22:53, 24 August 2019 (UTC)

Thanks for the smart answers to my stupid question. I suspect the answer to my follow-up question will be found in the links provided (which I will actually read) -- which is: Since the interval between adjacent keys (regardless of color) is the same, how come improvising (aka: "noodling"} with only the black keys sounds distinct from doing the same with only the white keys? --Or, is it just my imagination?-- I'll probably figure this out on my own from the links provided (and re-reading the helpful answers until they make sense to me), but any shortcuts would be appreciated. Thanks again. 2606:A000:1126:28D:8DFA:9F88:DA44:2E31 (talk) 04:32, 25 August 2019 (UTC) ... P.s.: no, Baseball Bugs, I don't catch catfish with my bare hands

Playing only the black keys (starting from F#) gives a pentatonic scale, which is associated with some oriental music or fol styles, whereas the white keys (starting from C) give the "classical" "western" major scale. AndrewWTaylor (talk) 06:52, 25 August 2019 (UTC)

and a pentatonic scale can be played on white notes (with optionally just the odd black) and it will sound the same on an instrument with Equal temperament, just higher or lower. It's easier to play on all black notes. Dbfirs 20:44, 25 August 2019 (UTC)

Thanks to all for the music theory primer!  But now I have to get that damn song out of my head.  —2606:A000:1126:28D:7931:7AE7:6D26:288 (talk) 17:05, 25 August 2019 (UTC)

• The thing with the black and white keys is that it is not much different in purpose than the dots on the neck of a guitar. It provides reference for the player to orient themselves on the instrument. Like a guitar, a piano player's fingers do not stay stationary on the same keys (as they might on other instruments like a flute or a trumpet) so they need either visual or tactile means to orient themselves. The key pattern allows that. They keys also match the western note pattern in the sense that the white keys represent the specific lettered notes (A, B, C, D, E, F, and G) while the black keys represent the sharps between those notes (A#, C#, D#, E#, G#). B# is enharmonic with C in equal temperament and E# is enharmonic with F, so those sharps don't need separate keys, which is why there are two less black keys. In equal temperament, flat notes are enharmonic with all existing notes already (Ab = G#, Bb = A#, Cb = B, Db = C#, Eb = D#, Fb = E, Gb = F#) so they don't need other keys. (It should be noted, if you really want to get in the weeds on this, that other tuning systems such as Pythagorean tuning and just intonation do not have such enharmonic notes, so strictly speaking you would need more keys in those systems. Those systems would work fine in singing and instruments like a violin, but become unwieldy on a piano or guitar unless you do something weird like microtonal music) --Jayron32 12:43, 26 August 2019 (UTC)

Thx again. I've also found useful information at Five black-key pentatonic scales of the piano and Isomorphic keyboard. My background in physics seems to be more of a hindrance than a help in understanding music theory, however; musicology and physics overlap, but there is a definite language barrier. 2606:A000:1126:28D:7931:7AE7:6D26:288 (talk) 15:16, 26 August 2019 (UTC)

Actually, music theory and physics overlap quite nicely. There are a few basic bits to understand the music theory we are talking about here.

1. The western system is built on a series of 12 intervals. Those intervals define all notes in relation to the base note of the key called the root note. The most important interval is the octave, which in all tuning systems is defined as a 2:1 ratio of the root note's frequency. Thus, if a note is defined as 440 Hz (concert A is the name of that, but that isn't important), then the octave of that note has a fundamental frequency of 880 Hz (and is also a A). The other intervals are all given various names like major third and perfect fifth and minor sixth and so on. "third" because you would apply 3 times to get an octave --perfect ratio 2^1/3:1 --, fifth because you apply 5 times, etc. Gem fr (talk) 07:18, 27 August 2019 (UTC)

I just want to clarify that Gem fr. edited my post to add some clarifications. I also want to point out that there's some disagreement between his clarifications, and information that can be found in other sources, especially the Wikipedia article titled Interval. It also disagrees with my experience in this. My prior understanding, which is similar to what the Wikipedia article notes, is that the name for an interval indicates which number interval (in order) it represents in one of the various diatonic scale. Thus, a second interval in the key of C is a D, the third is an E and so on. The difference between major and minor (or augmented and diminished) is that the interval is dropped or raised a semitone. Thus, a minor second in the key of C is a Db, while an augmented fourth is a F#, from the fourth note after C (CDEF, then RAISE the F). This also explains why some keys have ideas like double sharp or double flat; in a key like Eb major, the fifth is Bb, so the diminished fifth has to be Bbb (Eb, F, G, Ab, Bb). You can't call this an A, because the fourth is already notated as Ab; you only get one of each letter. The names don't actually have much to do with the ratios or multiples; just which place in the seven notes of the diatonic scale it falls. Also, they say that taking the perfect fifth 5 times gets an octave; except you don't. Because 3:2 times 5 is 15:2, and that's not any octave I know of, which will always be some whole-number multiple of the root (2:1, 3:1, etc.) If you follow the circle of fifths around 5 steps, you get a minor seventh interval, which is a whole tone off from an octave.--Jayron32 14:13, 27 August 2019 (UTC)

hum. Sorry I actually messed up matter instead of clarifying as intended. The only good point is that it gave you a chance to elaborate, otherwise I would just delete Gem fr (talk) 23:41, 27 August 2019 (UTC)

1. When two notes are played together, their soundwaves undergo wave interference. If the interval between the notes is resonant (that is, where the ratio of the frequencies is a small, whole number) humans tend to perceive such sounds as pleasant (what is called in music consonance, while intervals that do not come in small whole-number ratios have dissonance. A composer can evoke various visceral emotional responses in the listener by using consonance and dissonance in creative ways.
2. It isn't possible to build a fully consonant system of intervals AND also have the intervals keep the correct ratios for all possible key centers. That is, if we build a system of intervals where every interval is the most possible consonant ratios, and then change the keys, all of the intervals become messed up. A perfectly tuned system of intervals is called just intonation. If you create such a perfect system of intervals for one key (say "C major") and then try to play music on the same instrument in a different key (say "B minor"), you find all of your intervals are now messed up. In order to make this easier, you want to have an instrument that can play in all possible keys equally, so what you do is take all of these various errors and spread them out more-or-less evenly over each interval, so that every interval is a little bit wrong, but not wrong enough to sound bad. There are lots of ways to do this, but modern music is generally tuned to 12-tone equal temperament, which seems to work well for most instruments.

Common tones between chords are the same pitch, with the other notes tuned in pure intervals to the common tones.

@Jayron32: Strictly speaking, it is possible if you allow an unlimited number of notes (but then you often end up accumulating tiny little intervals called commas, until where you end up is probably not where you started, because there are no more enharmonic equivalents) – and also strictly speaking, if you have only 12 notes you cannot even have the intervals keep the correct ratio in one key centre, because if you have a C major scale, you cannot simultaneously have a just F-major triad (F = 4/3, A = 5/3, C = 2) and a just D-minor triad (D = 9/8, F = 4/3, A = 27/16) unless you accept having two different A's which differ by the syntonic comma 81:80. This syntonic comma provides a long-known example of the first situation: in the example given at right by Giovanni Benedetti in 1563, the A in the soprano must be a perfect fifth over the D in the bass and alto and a major sixth above the C in the bass; since the A comes first and the C can match it later, the C ends up being raised up by one syntonic comma (symbolised as a " " accidental), and then we repeat the cycle four times until the G we end up with has gone up almost one equal-tempered semitone. (I copied the picture and some of the caption from Comma (music).) Double sharp (talk) 06:48, 28 August 2019 (UTC)

1. In terms of labeling these notes, the letters assigned to the notes are set up to match the C major diatonic scale. The reason for this is long and convoluted and took many hundreds of years of history, so is probably outside of the scope of this discussion. Since there are 7 notes in the diatonic scale, you need 5 more notes to get a full 12 notes, so the system of sharps and flats is used to indicate the missing notes from that scale. To change keys, you keep the same intervals but you change where you start counting from. --Jayron32 16:19, 26 August 2019 (UTC)

Hope that all helps! --Jayron32 16:19, 26 August 2019 (UTC)

2606:A000:1126:28D:7931:7AE7:6D26:288 (talk) 16:32, 26 August 2019 (UTC)

The scale is traditionally built up from powers of 3, dividing by enough powers of 2 to make all the notes fit into the interval 1:2. The first five notes thus obtained are

• 1/1, 9/8, 81/64, 3/2, 27/16, (2/1)

or equivalently (by rotation)

• 1/1, 9/8, 4/3, 3/2, 16/9, (2/1)

The smallest interval among these is 8:9. One more step gives

• 1/1, 9/8, 4/3, 3/2, 27/16, 16/9, (2/1)

Now the smallest interval is 243:256 (approx 19:20), less than half (measured by logarithms) of the previously smallest interval. That may be why many traditions stop at a pentatonic scale.

Go two more steps, to eight notes, and the smallest interval doesn't get smaller but now you have two adjacent small steps, which add up to the biggest step (8:9) of the seven-tone scale. I guess that last note was considered melodically superfluous, a good reason to stop at seven.

And if you continue to thirteen notes, the newest note creates an interval of 524288:531441 (approx 73:74), which is less than a third of the previous smallest interval, and also only slightly more than the amount by which 81/64 is flatted to make the harmonious 5/4, so that last note is really useless.

Thus there are good reasons for scales of five, seven and twelve tones, unequally spaced – and a legitimate motive, stated elsewhere, to bend the twelve for equal spacing. —Tamfang (talk) 20:45, 26 August 2019 (UTC)

Further to fretted instruments, as mentioned by Jayron above, there are such things as true temperament guitars, which incorporate really odd-looking frets. We don't have an article (yet), but examples can be found on YouTube, as for example in the demonstration video here. {The poster formerly known as 87.8`1.230.195} 2.122.61.224 (talk) 01:01, 27 August 2019 (UTC)

I wish the video said something about the theory of that fretboard; what does it optimize? — If I could play guitar I'd definitely want a custom fretboard made to my theoretical improvement on meantone. — Recently I went looking for electric baroque videos and was disappointed to see no guitar with non-factory frets! —Tamfang (talk) 02:59, 27 August 2019 (UTC)

Tamfang, If you search on YouTube for "True temperament guitar" you should see quite a number of other videos on the subject, some of which do go into the theory more. One such came up as a suggestion to me a week or so ago, before which I'd never heard of them: I linked the one above merely because it was the first on the list when i searched again because of the query here. {The poster formerly known as 87.81.230.195} 2.122.61.224 (talk) 15:20, 27 August 2019 (UTC)

...which lead me to "xenharmonic" (as if things weren't complicated enough). 2606:A000:1126:28D:7931:7AE7:6D26:288 (talk) 03:34, 27 August 2019 (UTC)

My guess with that guitar is that it is optimized for playing barre chords. He mentions several times about the major third interval being better. In standard tuning, when playing an "E" chord and on barre chords built on the "E", the major third is on the "G" string, which you'll notice is where the fret is bent on all of the strings. The deal is (and remember, he's trying to sell you on this guitar) the fact that the third occurs on that string is basically unique to the open "E" chord and on "E"-shaped barre chords. On other chords, the major third would not necessarily appear on that string. That string may have a fifth or an octave or something like that; or what about extended chords with sevenths and ninths, or suspended chords with seconds and fourths? That bent fret would not necessarily have the "correct" spacing for THOSE intervals. So now what? Someone has built a guitar better optimized for a very narrow set of chords. But you've probably made it worse for other applications. The other issue with an electric guitar is that many styles of music don't use thirds in that way. Jazz famously uses lots of chord extensions and chord suspensions. Also, any time you're playing music with heavy Distortion or overdrive, as in hard rock or heavy metal, you'll usually want to play with power chords because the distortion contains extra frequencies that clash in horrible ways with the third. This is NOT a symptom of the intonation of the guitar; even a perfectly tuned major third would have these problems. A power chord has no third interval. So you aren't even playing that interval on many types of music that one would use that instrument for. Basically, it feels to me like someone fixed a minor problem, and in doing so created bigger problems. --Jayron32 12:36, 27 August 2019 (UTC)

What would a 3:2:1 hard rock or metal chord sound like? I suppose you could have a not bass guitar play what the bass plays but at 6:4:2? Sagittarian Milky Way (talk) 00:27, 28 August 2019 (UTC)

What do you mean by 3:2:1 chord? Do you mean a root, a fifth and an octave? That's basically what every heavy metal rhythm guitar has played, like, ever. That's what a power chord is. Tune to drop-d tuning and you can play that with one finger. 6:4:2 is the exact same ratio. Bass guitarists don't usually play triads other than Lemmy, so if you want to know what a bass player sounds like playing power chords, listen to some Motorhead. If that isn't what you mean, can you use the correct terminology so we all can respond to your question? --Jayron32 01:17, 28 August 2019 (UTC)

No, root, octave, and 3 times the hertz of the root. And I suppose you could have a higher-pitched guitar play 2 times the bass root's hertz, 4 times and 6 times but I didn't know that bass guitars didn't usually play triads. What chords do they play? 4 notes? 2? Sagittarian Milky Way (talk) 02:05, 28 August 2019 (UTC)

Because of octave equivalence (notes a whole number of octaves apart sound equivalent), your root-octave-twelfth chord of 1:2:3 is essentially identical to a power chord. (It is not really a chord in the traditional sense but an interval/dyad, since if you discard octave doublings it is just a perfect fifth, but within the context of the musical language it is found in it acts enough like a chord to justify the name.) Double sharp (talk) 04:52, 28 August 2019 (UTC)

Bass guitarists usually play melody lines known as basslines. The role of a bass player is to 1) provide harmonic accompaniment to the upper register instruments and 2) provide rhythmic structure along with the drummer. Basically, the bass is the bridge between the drummer and the more midrange instruments such as guitar and vocals. Bass players will sometimes play double stops, but really they mostly play single-note melody lines. That's because of the sonic properties of the bass guitar: the attack and sonic spectrum of the notes of a bass don't really play chords well. Full triads played on bass guitar tend to sound muddy and indistinct. Bass guitars changed a bit in the mid-1960s with the advent of new pickup, effects pedal, and amplification techniques that added significant midrange to bass guitars, and allowed the advent of the power trio where the bass takes up some of the role of the rhythm guitarist in rock ensembles. You can hear this in the bass guitar tones in bands like Cream and Motorhead, where the bass has a fuller tone and where the bassist plays more double stops and even triads. --Jayron32 11:58, 28 August 2019 (UTC)

Western classical music, the tradition in which the piano evolved as an instrument, is based on triads. There are two kinds of triads, called major and minor. Both triads are approximations of the chord formed by harmonics 1, 3, and 5 of a note (you need three to make a triad; the even harmonics are discovered because of something called octave equivalence: multiplying a frequency by any power of two makes it sound like an equivalent, even if non-identical note). The major triad is a closer approximation. The first harmonic is called the tonic: it is the basis of the triad. The third harmonic is called the "dominant" and the fifth harmonic is called the "mediant". Confusingly, the interval the third harmonic forms from the tonic is called the "fifth", and the interval the fifth harmonic forms from the tonic is called a "third". The number names are reversed because of their positions in the major scale.

As Tamfang has explained, the major scale is theoretically made from a stack of third harmonics up and down from the tonic (i.e. multiplying the frequency by 3/2), and has a major triad on the note it is based on (called the tonic). If you stack four 3/2 fifths, you get to 81/64 (octave equivalence lets you insert as many factors of 2 as you wish), which is close to 80/64 = 5/4; therefore we "fudge" the intervals away from an exact 3/2 and 5/4 to make this exact. (This is called tempering out the syntonic comma of 81/80, i.e. setting the difference between the 80th and 81st harmonics to zero, and distorting the factors of 3 and 5 to match.) And if you keep stacking ideal 3/2 fifths indefinitely, at the twelfth fifth you get something very close to the original note, and so we traditionally distort every fifth very slightly to make this correspondence exact, and declare that our musical universe consists only of these twelve notes (and of course, any power of two times their frequencies, so really these are twelve types of notes). (This is called tempering out the Pythagorean comma of 531441/524288.) So a major triad is not really formed from frequencies k, 3k, and 5k, but rather of the close approximations k, 2^19/12k ≈ 2.9966k, and 2^28/12k ≈ 5.0397k. These subtitutes are close enough for the ear to accept as standing for 3k and 5k, but allow for a richer musical language; it also helps that twelve notes is a good number for categorical perception (fewer types of notes and there's not enough richness; many more types of notes and it becomes difficult to tell which is which if they go by quickly).

Why then is the major scale made from five third harmonics up from the tonic and only one down? Because the "upward", "dominant", or "sharp" direction (e.g. 3/2 as a fifth over a note) is stronger than the "downward", "subdominant", or "flat" direction (4/3 as a fifth under a note, equivalent to a fourth over it). This is expected because harmonics rise from a note. Since the major scale is so fundamental, the piano keyboard gives the notes of one particular major scale a privileged position on the white keys, while the other five notes are black keys. (Other major scales of course have to mix white and black keys.) The notes inside a scale are called diatonic; the notes outside a scale are called chromatic. Chromatic notes are dissonant, which means that they have to resolve (traditionally stepwise, meaning to an adjacent note) to a diatonic note. But even diatonic notes can be dissonant and have to resolve stepwise. In effect, everything in a Western classical piece of music is theoretically dissonant to the tonic triad, and to make grammatical sense the piece must end there. (See the disclaimer at the bottom about this applying only in its strictest form to the so-called Classical period of about the 1770s to the 1820s, but that is in a way the theoretically most fundamental use case.)

In music, we traditionally single out one major scale and call its tonic "C". We then use the letters A through G to name the rest of its notes in such a way that when we force all seven notes in this stack of fifths within one range of frequencies from k to 2k (where k is the frequency of C), they go alphabetically from C to C (due to tradition; they wrap around from G back to A). So the notes are C, D, E, F, G, A, and B; they are generated by stacking fifths on top of and below C. Stacking fifths above C leads to G, D, A, E, and finally B; one fifth below C leads to F. This is why we name the intervals this way: G is the fifth note in a scale starting on C. (So the intervals C-D, C-E, C-F, C-G, C-A, and C-B are respectively a second, third, fourth, fifth, sixth, and seventh; C to the next C, a frequency doubling is called an octave, from the Latin root "octo" for eight. You can keep going with the number names; two octaves make a fifteenth, an octave and a fifth make a twelfth, and so on – yes, there is basically a fencepost error here due to inclusive counting. But normally past the fifteenth you use constructions like "two octaves plus a third" rather than force everyone to count how much a seventeenth is.) You can build a triad on any one of these seven notes simply by picking alternate members of the scale: the triad on C is C-E-G, that on D is D-F-A, and so on. (But note that these are two different kinds of triad; the first is a major triad and the second is a minor triad. More on this below. As you wrap around, for example, the triad on G is G-B-D.)

In the pentatonic scale we stop at C, G, D, A, and E, because that is the maximum number of fifths we can stack and end up with no adjacent notes, a condition called anhemitonia. In the major scale we have seven notes, because that is the maximum number of fifths we can stack and end up with no three consecutive notes from our universe of twelve, a condition called ancohemitonia. The major scale stops stacking fifths when it gets to B because this note happens to be directly adjacent to and below the tonic C, which gives it a powerful pull towards the tonic; this is called a leading note and is a dissonance essential for a strong harmonic feeling in Western classical music. (It also happens to be the fifth harmonic of G, the dominant, which strengthens it.) As we can see, harmonic feeling and movement in music is conditioned both by triads and by the scale; these sometimes work together and sometimes don't, creating tension. The other important point about this note B is that since it is the highest note in our stack of fifths, it doesn't have a real (technically called perfect) fifth above it; B to F is one note short of being a perfect fifth. This is called a diminished fifth or tritone and is a very dissonant interval indeed. Often this dissonant triad on B is united with the dominant triad on G to create what is called the dominant seventh chord G-B-D-F (it's a seventh because its outermost notes are a seventh apart), a powerful dissonance that heightens the pull towards resolution to the tonic. (This resolution is stepwise, as G is common to both chords and can be left out as implied, and then B, D, and F respectively go to C, C, and E.) Stacking a further third creates the even more powerful dominant ninth chord G-B-D-F-A and its curtailed half-diminished seventh chord B-D-F-A.

If we continue on beyond B and beyond F we get onto the missing "black notes" of the scale. We name these by analogy. If a fifth above A is E, then a fifth above B must be some kind of F. But it's not actually F – that's one step too low. So we call it F-sharp, symbolised F♯. And then we can continue stacking fifths to F♯, C♯, G♯, D♯, A♯, E♯, and B♯, at which point we have traversed all twelve notes and arrived back at C = B♯ (we already knew this, of course, because B was adjacent to C). In fact F = E♯ as well. This is why you don't have black keys between E/F and B/C on a piano (your original question): they are already adjacent in our twelve-tone universe, so there is no need for anything in between. (If we were to continue stacking fifths beyond B♯, we would have to arrive at F-double-sharp for similar reasons, which we symbolise F, and then continue to B. Normally musicians do not go far enough to necessitate going another round into triple sharps. ^_^ These are important because you may want to use any note as a basis for a major scale, and you want the seven-different-letters pattern to work right. So if you started a major scale on G♯, it would need to stack five fifths above it for the sharp notes, which would have to take you to F for the leading note. You don't usually need triple sharps because you'd much rather for easy reading begin a major scale on A than on G. ^_^)

Similarly, if a fifth below G is C, then a fifth below F must be some kind of B. But it's not actually B – that's one step too high. So we call it B-flat, symbolised B♭. And then we can continue stacking fifths to E♭, A♭, D♭, G♭, C♭, F♭ – and now we need double flats to continue, which are symbolised in the obvious way. So we go on to B, E, A, and D = C (because we have again gone through all twelve notes). (And by filling in equivalents upwards we also see A = G, E = D, and so on until we again get C♭ = B and F♭ = E as expected, providing another demonstration of why there aren't any black keys between E/F and B/C. Again, normally musicians do not go far enough to necessitate going another round into triple flats.) This closed chain of fifths is called the circle of fifths. We use both sharp and flat names for notes, even though they refer to the same pitch, because their semantic meaning is different: E-G♯ is a kind of third, but E-A♭ is a kind of fourth, and from the context you can distinguish them by meaning even without any audible difference in pitch (like homophones).

Now I mentioned that there are two kinds of triads: major (like C-E-G) and minor (like D-F-A). In a major scale you will find two kinds of intervals that differ in size but are approximately identified as within the same category: for example, C-E and E-G are both three-note intervals, or "thirds", but the first one is bigger; these are called a major third and a minor third respectively). Because the ear can identify these as roughly equivalent, we can exchange the places of the major and the minor third in the major triad to make a minor triad. A major triad has a major third below a minor third; a minor triad has a minor third below a major third. A minor triad on C is hence C-E♭-G (by the rule of having all seven letters used, the mediant must be some form of E; E is one note too high, so it's E-flat). We can then fill in a natural minor scale that theoretically goes two fifths above the tonic and four fifths below it. Starting on C this is C-D-E♭-F-G-A♭-B♭-C; on A it nicely is on the white keys as A-B-C-D-E-F-G-A. But the minor triad is a poorer approximation of the harmonic series than the major one (its approximation to 5 is 2^27/12 ≈ 4.7568, which is close but sounds audibly one semitone different; it is frankly more of an approximation to 4 * 6/5 = 4.8, the minor-third distance between the 5th and 6th harmonics as it should be). To make it a functional scale, you must make many notes variable and hence chromatic. To add a leading note you must raise B♭ to B; but that creates an awkwardly large gap to A♭, which is sometimes raised to A as well; and sometimes you want a fifth below A♭ to get a strong upper leading tone and have D♭ instead of D. (But sometimes you do want that B-A♭ gap as a powerful dissonance, as part of the diminished seventh chord B-D-F-A♭!) The first two are standard alterations that create what are called the harmonic minor scale C-D-E♭-F-G-A♭-B-C and a melodic minor scale C-D-E♭-F-G-A-B-C; the last alteration is generally considered chromatic, this flattened second note being called the Neapolitan scale degree. In general the harmonic minor scale is the closest fit to actual practice in terms of what is used for pieces in minor, but this poorer approximation and chromatic variability (not to mention the fact that the altered notes in the minor scale are all in the weaker "downward" flat direction) means that the minor mode is more unstable and expressive than the major. Indeed, pieces in minor often end in major, which acts like a resolution (in the simplest form, ending with a major triad after a lot of minor-key music, it is called the Picardy third); theoretically speaking, they are different modes, but they have the same tonic and can loosely be considered the same key.

Note that what I have given above is an account of tonality (the technical name for this musical system based on triads) as it was conceived in about the 1770s through 1820s (the so-called Classical period of classical music), where keys and harmonic forces are the most clearly defined. What happened beforehand and afterwards is less clearly hierarchical, but I think the basics are where to start here. (You can find a more advanced look at the axiomatic principles of tonality and equal temperament in Charles Rosen's famous book The Classical Style on pp. 23–29, which is a source for many of the things I have stated in the last few paragraphs.) Double sharp (talk) 06:36, 28 August 2019 (UTC)