Maths vs. Music - Tuning and harmony

[Thumpalumpacus]

As an interesting side note into the study of vibration, the naturally arising mathematic order in music gives us the 8 note octave with 12 semitones.  
http://www.math.uwaterloo.ca/~mrubinst/tuning/12.html

An octave has twelve, not eight, agreed-upon notes. An octave has twelve chromatic notes, of which eight are diatonic, in Western European music, but can have as little as four or as many as all twelve agreed-upon notes, in Western nomenclature, depending on scale and idiom. A pentatonic octave has (you guessed it!) five notes. There are also things like hexatonics, gypsy scales, and so on. The Western major scale (non-musicians, think do-re-mi-fa-so-la-ti-do) of eight notes is only one of many, many scales. It just so happens that it's the one you're acculturated to hearing.

However, those aren't the only notes one can play. There are notes in between the notes, which is why the Indians invented the sitar (with its scalloped fretboard for bending to those notes) and why the Africans invented slides for stringed instruments. You could also look up the "blue notes" used in American blues, located between the m3 and M3, the b5 and the perfect 5, and between the submediant 6 and the leading tone.

They are notes too. They just don't have formalized names.

Go to 3:18 in this clip, listen for about twelve seconds, and name all the notes that he plays.





I guarantee you can't, because a good 50% (at least!) of them are well between the "Western notes". Ain't no names for them ... and there ain't no magic theory sprouting from your head explaining them, either.

There's no "mathematical order" to music. Octaves subdivide on resonances, no doubt -- but that doesn't mean that western tempered tuning, which is your obvious referent, is the only physical expression of music. Mathematics is a language we use in order to speak with each other, but the mathematics doesn't define the music.

The finger pointing at the moon is not the moon. Music theory does not prescribe music, it describes it ... as any musician would know.

[Arkilogue]

There's no "mathematical order" to music. Octaves subdivide on resonances, no doubt -- but that doesn't mean that western tempered tuning, which is your obvious referent, is the only physical expression of music. Mathematics is a language we use in order to speak with each other, but the mathematics doesn't define the music.

The finger pointing at the moon is not the moon. Music theory does not prescribe music, it describes it ... as any musician would know.

None?

[Thumpalumpacus]

Cherry-pick much? Look at the sentence immediately following the one you bolded, and quit being so obstreperous.

No one, myself included, said resonances don't happen. Perhaps you should read my entire post, for content rather than argumentation, before you deign to reply again.

[Alex K]

There's no "mathematical order" to music. Octaves subdivide on resonances, no doubt -- but that doesn't mean that western tempered tuning, which is your obvious referent, is the only physical expression of music. Mathematics is a language we use in order to speak with each other, but the mathematics doesn't define the music.

The finger pointing at the moon is not the moon. Music theory does not prescribe music, it describes it ... as any musician would know.

None?

Looks deeper before you realize that you don't end up in the same place after going round once if you use pure intervals due to the pythagorean comma. A pity, really

[Arkilogue]

I guarantee you can't, because a good 50% (at least!) of them are well between the "Western notes". Ain't no names for them ... and there ain't no magic theory sprouting from your head explaining them, either.

So then why does the sand suddenly jump to ordered patterns rather than a gradual transition?

[bennyboy]

Looks deeper before you realize that you don't end up in the same place after going round once if you use pure intervals due to the pythagorean comma. A pity, really

Yep, and the math should be almost painfully obvious. How many iterations of (3/2)^n (a circle of perfect fifths) are ever going to arrive at an even multiple of 2? When you then start taking each of those notes and their overtones (f/2, f/3, f/4 etc.), you quickly realize that any amount of modulation among keys will almost instantly lead to chaos.

Tuning systems require two things: (1) fudge frequencies to make the math work; (2) have people's tolerances built to accept the impure sounds.

[Thumpalumpacus]

I guarantee you can't, because a good 50% (at least!) of them are well between the "Western notes". Ain't no names for them ... and there ain't no magic theory sprouting from your head explaining them, either.

So then why does the sand suddenly jump to ordered patterns rather than a gradual transition?

We were talking about music.

Step away from the bong, and answer my points without discursion. It's apparent when you make jumps like these that you're at a loss for words. Ain't my horse, I'm not gonna ride it ... but if you're not willing to actually discuss things (rather than propound your agenda), I'm not willing to give you a sounding-board. The choice is yours; you can either follow the discussion, or lose audience.

[Thumpalumpacus]

Looks deeper before you realize that you don't end up in the same place after going round once if you use pure intervals due to the pythagorean comma. A pity, really

Yep, and the math should be almost painfully obvious.  How many iterations of (3/2)^n (a circle of perfect fifths) are every going to arrive at an even multiple of 2?  When you then start taking each of those notes and their overtones (f/2, f/3, f/4 etc.), you quickly realize that any amount of modulation among keys will almost instantly lead to chaos.

Tuning systems require two things: (1) fudge frequencies to make the math work; (2) have people's tolerances built to accept the impure sounds.

There's the reason why tempered tuning was invented.

As much as Arky wants us to think that mathematically-perfect relationships are sonorous to the human brain, the fact is, they aren't -- which was the point of my mentioning blue notes. Mathematically imperfect, but a hell of a lot better than equal-temperament.

These are things a musician knows.

[bennyboy]

I guarantee you can't, because a good 50% (at least!) of them are well between the "Western notes". Ain't no names for them ... and there ain't no magic theory sprouting from your head explaining them, either.

So then why does the sand suddenly jump to ordered patterns rather than a gradual transition?

It's constructive and destructive interference patterns, I think.  The order isn't just from the frequencies, but from the placement of the sound sources.  You could probably measure the frequencies and use them to figure out for yourself how the sound sources were arranged, actually, and their distances apart in centimeters. Specifically, I think that those parts of the sounding surface which are vibrating most will "throw away" the sand over them, while those which are vibrating least will not, leading the sand to collect in the low-vibration regions.

You could probably do a super-simple version of this experiment by covering the lowest string of a guitar with flour, and then strongly plucking it. I would guess that you'd see a kind of pattern of flour remaining on the string after some of the dust got thrown off, and that those points with the least dust would be even divisions of the string: the exact middle (i.e. the octave point) which would be vibrating a lot, then the third and 2/3rd points, then the 1/4 and 3/4. Not sure if this would actually work, due to finger oils on the strings etc., but you can understand the idea I think.

[Thumpalumpacus]

That's really thoughtful, Benny. Hadn't considered that angle.