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Why 12 notes in Western music?

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Re: Why 12 notes in Western music?

#111

Post by rrrrrrrrrrrryan »

tshaddox wrote: post #106
rrrrrrrrrrrryan wrote: post #104

Earlier quoted context omitted.

I don't think you need to imagine "creatures" - there are humans on earth whose culture's music doesn't have the concept of octaves, let alone fifths. All the real action in their music is in its rhythmic complexity. But my comment was in reply to this statement specifically: > Somewhere there is a perfect universe where 12 fifths form an octave. For this to be true, I think you'd indeed need a new universe where 3^1…

I'm not talking about explicit notions of octaves and octave equivalence, although those do exist in very many musical traditions and appear to be extremely widespread in musical traditions where tones have names (if not ubiquitous—I'm not aware of any exceptions). I was referring to the claims that octave equivalence is in some sense hard-wired in the human brain, or sometimes claimed to be all or many mammalian bra…

I'm much less qualified: I took a "World Music 101" class in college that described some Native American tribes whose songs had drums, and cries that fell in pitch from high to low to call down the spirits from above, but their music had no concept of octaves.

And I agree with you. While I can imagine a culture that hadn't "discovered" the octave, it's difficult to imagine people who can't even perceive octaves when presented with them, and quite easy to imagine other creatures that cannot perceive them.

Re: Why 12 notes in Western music?

#113

Post by kazinator »

aidenn0 wrote: post #103
kazinator wrote: post #93

Earlier quoted context omitted.

That's just from modulo math. A fifth is 7 semitones, which is relatively prime to 12. Thus 7x (mod 12) hits all the elements of the modulo 12 congruence for x in 0..11. We cover all notes in the first twelve steps. But say we are not assuming a twelve note system in the first place; how do we get twelve notes? Going up a fifth and then down a fourth is very close to a tone. We can do that five times before we approx…

> That's just from modulo math. A fifth is 7 semitones, which is relatively prime to 12. Thus 7x (mod 12) hits all the elements of the modulo 12 congruence for x in 0..11. We cover all notes in the first twelve steps. > But say we are not assuming a twelve note system in the first place; how do we get twelve notes? My diagram showed an (approximate) 12 note cycle assuming only a 3:2 ratio for a fifth. There are lots…

Yes, the near cycle is a coincidence; it just comes from 1.5 ^ 12 ~= 129.746, which is close to the power-of-two 128.

It's because 3^12 is close to 2^19, to about 1.36 percent.

This is all abstract arithmetic; I don't believe it's exclusively how musicians discovered chromaticity. That likely has multiple origins, one of which is likely about filling in the five "missing" half step "slots" in a diatonic mode.

The 3:2 perfect fifth being an important interval isn't a coincidence; that's rooted in how the frequencies blend together without any beats being heard. The second harmonic of a fundamental is a fifth above the octave, and all that.

Speaking of which, the progression of harmonics, which is just multiples of a frequency rather than a geometric series like stacked fifths, can also derive diatonic scale notes.

Re: Why 12 notes in Western music?

#114

Post by ssalazar »

gnulinux wrote: post #71
CobrastanJorji wrote: post #65

Earlier quoted context omitted.

You don't need a new universe. You just need a species that hears sounds a little differently. We won't like listening to their music, but it'll be really great for them.

This may or may not be true. In fact, it seems unlikely to me your claim is true. In nature, sounds produce harmonics i.e. when two objects collide they usually create waves of frequency f, 2f, 3f, 4f... in various (usually exponentially decreasing) weights. It's very rare to find pure sounds (i.e. only f frequency) in nature. The interval between f and 2f is an octave apart (1:2 ratio); the interval between 2f and 3…

> This means there might be something universal about octave and fifth

There is! At least for the kinds of instruments that are conventionally used in Western music. The harmonic series arises naturally from the physical properties of a string or wind instrument (e.g. violins, guitars, pianos, flutes, brass, organs, etc). As a very rough description of the physical phenomena, the tones we hear arise from a full spectrum, atonal excitation (like a pluck or a reed flapping) bouncing back and forth along the length of string or tube, which is basically a one-dimensional "waveguide". Frequencies that are aligned with the harmonic series naturally reinforce themselves, in the same way that putting energy at the top of the arc of a playground swing has more of an effect than in the middle.

Notably, musical instruments that are not strings or tubes, or more general sound-producing bodies, have more complicated patterns of sound waves dispersing through them, and don't typically follow the harmonic series. Pitched percussion, drum heads, or bells have more complicated harmonic spectra than the standard harmonic series (as they are generally thought of as 2D or 3D waveguides where cancellation/reinforcement patterns are less straightforward), as do less musically conventional sounds like knocking two rocks together or striking an arbitrary surface.

Re: Why 12 notes in Western music?

#115

Post by maxdamantus »

tshaddox wrote: post #106
rrrrrrrrrrrryan wrote: post #104

Earlier quoted context omitted.

I don't think you need to imagine "creatures" - there are humans on earth whose culture's music doesn't have the concept of octaves, let alone fifths. All the real action in their music is in its rhythmic complexity. But my comment was in reply to this statement specifically: > Somewhere there is a perfect universe where 12 fifths form an octave. For this to be true, I think you'd indeed need a new universe where 3^1…

I'm not talking about explicit notions of octaves and octave equivalence, although those do exist in very many musical traditions and appear to be extremely widespread in musical traditions where tones have names (if not ubiquitous—I'm not aware of any exceptions). I was referring to the claims that octave equivalence is in some sense hard-wired in the human brain, or sometimes claimed to be all or many mammalian bra…

The basic reason for octave equivalence is not from human subjectivity. It's from the fact that something that repeats "x" times is also something that repeats "x/2" times, eg a sequence such as:

  ABCABCABCABCABCABC
Can be thought of as "ABC" repeated 6 times or "ABCABC" repeated 3 times. If you replace the letters with numbers, you can treat the numbers as samples of a sound wave.

Just to be clear: the magical number here is 2 (not 12 or 8 or 7) since "octave equivalence" refers to the fact that you can multiply a frequency by 2 and get the same note.

Unless the sound is a perfect sine wave, there isn't a particular frequency associated with it due to these alternative interpretations.

Re: Why 12 notes in Western music?

#116

Post by aidenn0 »

kazinator wrote: post #113
aidenn0 wrote: post #103

Earlier quoted context omitted.

> That's just from modulo math. A fifth is 7 semitones, which is relatively prime to 12. Thus 7x (mod 12) hits all the elements of the modulo 12 congruence for x in 0..11. We cover all notes in the first twelve steps. > But say we are not assuming a twelve note system in the first place; how do we get twelve notes? My diagram showed an (approximate) 12 note cycle assuming only a 3:2 ratio for a fifth. There are lots…

Yes, the near cycle is a coincidence; it just comes from 1.5 ^ 12 ~= 129.746, which is close to the power-of-two 128. It's because 3^12 is close to 2^19, to about 1.36 percent. This is all abstract arithmetic; I don't believe it's exclusively how musicians discovered chromaticity. That likely has multiple origins, one of which is likely about filling in the five "missing" half step "slots" in a diatonic mode. The 3:2…

I'm not certain of this, but my understanding is that Pythagoras generated these notes (actually more than 12 because there was a distinction between flats and the matching sharps) circa 500 BCE, and this predates chromaticity in music in the west.

Re: Why 12 notes in Western music?

#117

Post by joshka »

tshaddox wrote: post #89
rrrrrrrrrrrryan wrote: post #80

Earlier quoted context omitted.

Eh, I think you'd need a new universe, as it's a pretty basic principle of math: 3^12 ~= 2^19 You can take two long strings of equal length (A & B), and pluck them, and they'll make the same sound. Then you can take scissors cut string A in half, and it'll sound different. (This is an octave.) Then you can cut string B into thirds, and it too will sound different. If you pluck both of your new strings at the same tim…

But who says the creatures need to perceive sound in such a way that the harmonic series has sensory significance? To be honest I’ve never seen a compelling evolutionary explanation for why “hearing the harmonic series” developed in the first place. It obviously seems useful to be able to perceive sounds generated by (roughly) harmonic oscillators, since those occur naturally for various reasons, but why octave equiv…

Look at it from a frequency perspective instead of genetic.

When you overlay two waveforms that are related x:1, the zero points of the waveforms align. The wave resets at the same instant. If instead the waveforms are related not exactly, then you get a change in where the reset point is that drifts and causes the wave to exhibit beating (changes in volume).

If you have a 2:1 (octave) relationship between 2 waveforms, then you won't hear the beating as the beat frequency exactly overlaps the frequency of the lower frequency.

If you have say two notes that are not quite in the octave ratio, then you hear beating at the difference in frequency. E.g. say 440Hz and 888Hz (instead of 880Hz), you have beating occurring at 448Hz, which you'll hear as an 8Hz (448Hz-440Hz) wobble in the sound volume of the combined wave.

Re: Why 12 notes in Western music?

#118

Post by navi0 »

inopinatus wrote: post #101
jacquesm wrote: post #61

Earlier quoted context omitted.

Somewhere there is a perfect universe where 12 fifths form an octave.

I've heard there is a secret chord, that prophets play to please their lords. But if you care for music less than algebra, just know it goes like this; the fourth, the fifth: the minor falls, the major lifts.

Well played

Re: Why 12 notes in Western music?

#119

Post by lynndotpy »

mcbrit wrote: post #105

Let's talk about splitting things up in useful ways. 12=2 * 2 * 3. Splitting something in half is useful; splitting it half again remains useful. Splitting in half a third time is arguably less useful than splitting it into a third. So 12 is the made-to-order number that lets you split it in half, twice, and in thirds, once. Which naturally leads to seconds and minutes, or 60: 60=2 * 2 * 3 * 5 Because dividing the wh…

No: Having 12 notes is a neat but accidental outcome of the musical scale.

On the derivation of the scale:

Two pure tones go good together when they have "ratios" of frequency. So, 440Hz and 660Hz would interfere in a pleasing way. This is the same way that it's "nice" when tiles on a floor match your gait in a way you can follow a pattern, or when two blinkers sync up.

So, it's nice when tones are fractions of one another like 3/2, 4/3, 5/3, etc. A ratio like pi/2 would sound weird, and very close frequencies (like 440Hz and 440.5Hz) would interfere to make a beat of 0.5Hz. (I'm sure we can all agree that a ratio of 4/3 is a much nicer fraction 440/441. In practice, this doesn't matter much, because we rarely use pure tones. This is why equal temperament scales don't sound abysmal.)

A natural ratio is just "2/1". This division gives you your octave. One octave up from 440 is 880. One octave down from 440 is 220.

This plays nicely into the second problem: Human perception is logarithmic in many things, tone included. For a musical scale to be perceived to have equal differences in pitch between notes, it needs to be roughly evenly spaced on the logarithmic scale.

So, we need to select a set of frequencies in [440Hz, 880Hz) (where 440Hz is arbitrary) that (1) arenice fractions of one another, but are also (2) evenly spaced on the logarithmic scale.

By nice mathematical luck, the 12-tone chromatic scale fulfills that!

---

On the questionable qualities of "12":

I don't think the nice divisibility of the number 12 matters here, in a scale where only 7 of the notes (with the most pleasing ratio) get the title of "major". Notation is written on a musical staff where the other 5 notes are folded away.

Furthermore, if you swap out the base of 2 for a base of 3 and re-derive the scale, you get other scales. One is the Bohlen-Pierce scale, which has 13 notes. I derived a similar scale one time, and I remember finding a 7 or 11 note scale. I forget which exactly, but the point being that these are prime numbers.

So, I'm wondering, how would 12 being divisible matter for music? I don't compose much, so I mean this question genuinely.

---

On the main point:

> I'm not sure that the linked article, or the current top comment (Circle of Fifths) meaningfully extends beyond this "useful parts" hypothesis.

I ask this in good faith: Did you read the linked article before commenting?

If you didn't, then :\

If you did, then I'm curious:

The scale derivation I described here is described in the article, with nice visualizations. If you did read the article, how do you now have this take? How could a 10-tone scale fit into this, and how would having a less nicely-divisible number matter?

Re: Why 12 notes in Western music?

#120

Post by jacquesm »

joshka wrote: post #117
tshaddox wrote: post #89

Earlier quoted context omitted.

But who says the creatures need to perceive sound in such a way that the harmonic series has sensory significance? To be honest I’ve never seen a compelling evolutionary explanation for why “hearing the harmonic series” developed in the first place. It obviously seems useful to be able to perceive sounds generated by (roughly) harmonic oscillators, since those occur naturally for various reasons, but why octave equiv…

Look at it from a frequency perspective instead of genetic. When you overlay two waveforms that are related x:1, the zero points of the waveforms align. The wave resets at the same instant. If instead the waveforms are related not exactly, then you get a change in where the reset point is that drifts and causes the wave to exhibit beating (changes in volume). If you have a 2:1 (octave) relationship between 2 waveform…

There are some assumptions about phase in there.
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