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Mathematical explanation of music and white/black notes in a piano

math.stackexchange.com

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Re: Mathematical explanation of music and white/black notes in a piano

#21

Post by nix »

manlon wrote: post #10
nix wrote: post #7

Earlier quoted context omitted.

Supposedly there is a mathematical reason for the 2-2-1-2-2-2-1 spacing of the major scale. If you take all possible pairs of notes in the diatonic scale, you get a richer distribution of intervals than you can produce with any other seven-note selection from the twelve note scale. Similarly, the classic pentatonic scale provides the best set of intervals for any five-note selection from the twelve. A better set of i…

If you start with the 12 chromatic tones and start addding notes to a scale going up a circle of fifths, there are two natural stopping points where you have spanned the octave with a complete-sounding set of notes with relatively equal spacing and no gaps: five notes, which gives whole-step and minor-third intervals; and seven notes, which gives whole-step and half-step intervals. These two scales correspond to the…

The drawback to this explanation is that the diatonic scale is 10,000 years older than the "circle of fifths". So it presumably had some appeal to musicians as well as to music theorists.

Re: Mathematical explanation of music and white/black notes in a piano

#22

Post by levesque »

alextgordon wrote: post #16
extension wrote: post #12

The top answer on stackexchange is a superb explanation of musical science that most of the finest musicians in history could not give you. They just don't teach you these things in music class. Musical culture seems to resist illumination, perhaps fearing that the "magic" will somehow be broken. The irony is that music is , in a sense, a mathematical illusion, but revealing the trick only makes it more fascinating.…

Musical culture seems to resist illumination, perhaps fearing that the "magic" will somehow be broken. The irony is that music is, in a sense, a mathematical illusion, but revealing the trick only makes it more fascinating. This is true, but I think the answer is far more mundane. Musicians operate at a couple of levels of abstraction above that SO answer, so such details become irrelevant to them. Imagine if the sta…

Yeah, and now we argue about whether or not knowing the basic mechanics of your typical language (be it programming or music) can help you achieve a greater global understanding and thus allow you to be a better musician/programmer overall.

Re: Mathematical explanation of music and white/black notes in a piano

#23

Post by JeremyBanks »

alextgordon wrote: post #16
extension wrote: post #12

The top answer on stackexchange is a superb explanation of musical science that most of the finest musicians in history could not give you. They just don't teach you these things in music class. Musical culture seems to resist illumination, perhaps fearing that the "magic" will somehow be broken. The irony is that music is , in a sense, a mathematical illusion, but revealing the trick only makes it more fascinating.…

Musical culture seems to resist illumination, perhaps fearing that the "magic" will somehow be broken. The irony is that music is, in a sense, a mathematical illusion, but revealing the trick only makes it more fascinating. This is true, but I think the answer is far more mundane. Musicians operate at a couple of levels of abstraction above that SO answer, so such details become irrelevant to them. Imagine if the sta…

I'm not sure I understand. Your argument for why musicians should start at a higher level of abstraction is that starting programmers at a higher level of abstract works out poorly?

Re: Mathematical explanation of music and white/black notes in a piano

#24

Post by JonnieCache »

I can highly recommend this lecture, Notes and Neurons, from the 2009 World Science Festival. It features a panel of neuroscientists discussing the possible physiological encodings of the various mathematical structures discussed here. It also includes some amazing participative musical performances from Bobby McFerrin.

http://www.worldsciencefestival.com/video/notes-neurons-full

Re: Mathematical explanation of music and white/black notes in a piano

#25

Post by sampo »

nix wrote: post #7
hugh3 wrote: post #3

I think there are different answers at different levels. On one level, the white notes are the notes of the C major scale, and the black notes are the semitones which are left over. Why not put all the semitones in one row? That would be much harder to play. Why split it into "C major" and "leftovers"? A bunch of semi-arbitrary decisions made by early harpsichord manufacturers, I guess, which happen to make the instr…

Supposedly there is a mathematical reason for the 2-2-1-2-2-2-1 spacing of the major scale. If you take all possible pairs of notes in the diatonic scale, you get a richer distribution of intervals than you can produce with any other seven-note selection from the twelve note scale. Similarly, the classic pentatonic scale provides the best set of intervals for any five-note selection from the twelve. A better set of i…

So if we stard from C, and take steps of fifths, we get the notes in the order: C, G, D, A, E, B, F#, C#, G#, D#, A#, F (and then C again).

It is true that if you take the first 5, you get a scale with pattern of intervals C--D--E--G--A--C, and then it you'd add the sixth note (B), that would be only 1 step from C. So if we aim for a "nice" distribution of intervals, 5-note scale is a good stopping point.

But your explanation doesn't give any light on why the 7-note scale would be the next stopping point.

Re: Mathematical explanation of music and white/black notes in a piano

#26

Post by seertaak »

extension wrote: post #12

The top answer on stackexchange is a superb explanation of musical science that most of the finest musicians in history could not give you. They just don't teach you these things in music class. Musical culture seems to resist illumination, perhaps fearing that the "magic" will somehow be broken. The irony is that music is , in a sense, a mathematical illusion, but revealing the trick only makes it more fascinating.…

> They just don't teach you these things in music class.

Go read "How Equal Temperament Destroyed Harmony". By a musician, for musicians. I would add there are a number of conductors and musicians that insist on playing compositions in their original tunings.

> Musical culture seems to resist illumination

And I'm sure elsewhere in cyberspace at this precise moment some musician is lamenting the ability of scientists to understand their craft. Plus ca change...

Re: Mathematical explanation of music and white/black notes in a piano

#27

Post by russellallen »

extension wrote: post #12

The top answer on stackexchange is a superb explanation of musical science that most of the finest musicians in history could not give you. They just don't teach you these things in music class. Musical culture seems to resist illumination, perhaps fearing that the "magic" will somehow be broken. The irony is that music is , in a sense, a mathematical illusion, but revealing the trick only makes it more fascinating.…

Of course they teach those things in "music class". They teach them in music theory class at university level for a start. You're creating a strawman.

Just look at the Wikipedia entry on tuning, for example: http://en.wikipedia.org/wiki/Musical_tuning

Do you think that was all created by non-musicians while 'most of the finest musicians in history' cowered in the corner afraid of the magic being broken?

Re: Mathematical explanation of music and white/black notes in a piano

#28

Post by extension »

seertaak wrote: post #26
extension wrote: post #12

The top answer on stackexchange is a superb explanation of musical science that most of the finest musicians in history could not give you. They just don't teach you these things in music class. Musical culture seems to resist illumination, perhaps fearing that the "magic" will somehow be broken. The irony is that music is , in a sense, a mathematical illusion, but revealing the trick only makes it more fascinating.…

> They just don't teach you these things in music class. Go read "How Equal Temperament Destroyed Harmony". By a musician, for musicians. I would add there are a number of conductors and musicians that insist on playing compositions in their original tunings. > Musical culture seems to resist illumination And I'm sure elsewhere in cyberspace at this precise moment some musician is lamenting the ability of scientists…

I realize that some people know this stuff and some places teach it as part of an advanced music education, it's just remarkably uncommon considering how fundamental it is. I would have meticulously qualified my earlier generalization but I foolishly assumed that my hyperbole would not be taken literally.

I'm speaking as a musician when I say that we resist illumination. I've raised this topic with many teachers and peers and there is a sort of disbelief in the connection between the artistic experience of music and the underlying mathematics.

I don't think this is because musicians are ignorant luddites, I think it's because the emotional response to music is primal and not at all intuitive. Why do we feel so strongly about harmonically related tones? It doesn't sound like anything, in the way a painting or a sculpture looks like something. Abstract art is a relatively recent invention, yet music has been almost purely abstract for as long as anyone remembers.

Re: Mathematical explanation of music and white/black notes in a piano

#29

Post by dmoney »

nix wrote: post #21
manlon wrote: post #10

Earlier quoted context omitted.

If you start with the 12 chromatic tones and start addding notes to a scale going up a circle of fifths, there are two natural stopping points where you have spanned the octave with a complete-sounding set of notes with relatively equal spacing and no gaps: five notes, which gives whole-step and minor-third intervals; and seven notes, which gives whole-step and half-step intervals. These two scales correspond to the…

The drawback to this explanation is that the diatonic scale is 10,000 years older than the "circle of fifths". So it presumably had some appeal to musicians as well as to music theorists.

Where do you find 10,000 year old music?

Re: Mathematical explanation of music and white/black notes in a piano

#30

Post by nix »

dmoney wrote: post #29
nix wrote: post #21

Earlier quoted context omitted.

The drawback to this explanation is that the diatonic scale is 10,000 years older than the "circle of fifths". So it presumably had some appeal to musicians as well as to music theorists.

Where do you find 10,000 year old music?

You infer it from the existence of 10,000 year old musical instruments.

See http://en.wikipedia.org/wiki/Diatonic_scale#Prehistory - which also says that the circle of fifths was described much earlier than I thought.

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