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das monde:
My understanding from your diaries is that major keys are more harmonic (their intervals represent "smaller" fractions), aren't they?

It's interesting that you can split the scale into equal steps without breaking anything too badly, so it could just be that there's a tendency to pick the first X ratios which can be distinguished by a typical listener and can be made practically distinct on an instrument.

If you try to work with a finer split you get more notes than people can hear, can play, can write, and can be bothered to remember.

So scales will tend to have a small number of notes - 5, 7, 12 - modified by local rules about ordering, ornaments and intonation, which will give each scale its flavour. Scales based on harmonics seem to be the most natural, but others are possible, and the exact ratios will always be a little fuzzy, and not quite perfect physics.

by ThatBritGuy (thatbritguy (at) googlemail.com) on Wed May 14th, 2008 at 05:06:42 AM EST
[ Parent ]
It appears that the world is very lucky with the 12 tones.

Why do we not use a ten-tone or twenty-tone equal-tempered scale? Is there something special about twelve?

The answer is: Yes, the twelve-tone equal-tempered scale is remarkable. The nearly perfect intervals seen in the table above are not typical of other equal-tempered scales. Consider the six basic consonant intervals less than an octave (described above): 3/2, 4/3, 5/4, 6/5, 5/3, 8/5. The twelve-tone equal-tempered scale is the smallest equal-tempered scale that contains all six of these pure intervals to a good approximation - within one percent.

Let's compare the twelve-tone equal-tempered scale to some other scales.

    * All equal-tempered scales with 14 notes or fewer (except the twelve-tone equal-tempered scale) contain at most only two of the six basic intervals within one percent.
    * Several equal-tempered scales with between 15 and 30 notes (notably the 19-tone and 24-tone scales) contain all six basic intervals, but in none of these scales are the intervals more nearly pure than in the twelve-tone equal-tempered scale.
    * The 31-tone equal-tempered scale has all six basic intervals to a good approximation, some with better accuracy than the twelve-tone scale, but the most important fifth (3/2) interval is less accurate than in the twelve-tone scale (218/31=1.495). Some Indonesian music actually uses a 31-tone equal-tempered scale.
    * The 41-tone equal-tempered scale is the first with a better fifth (3/2) interval than the twelve-tone scale (224/41=1.5004).
    * The 53-tone equal-tempered scale has all six basic intervals with a better accuracy than the twelve-tone scale (231/53=1.49994).

by das monde on Wed May 14th, 2008 at 05:38:42 AM EST
[ Parent ]
That has probably something to do with continued fractions.

When the capital development of a country becomes a by-product of the activities of a casino, the job is likely to be ill-done. — John M. Keynes
by Migeru (migeru at eurotrib dot com) on Wed May 14th, 2008 at 05:46:01 AM EST
[ Parent ]
To approximate the major fifth 3:2 ratio, approximations of log[2](3/2) are needed. The partial quotients of the continuous fraction are

[0, 1, 1, 2, 2, 3, 1, 5, 2, 23, 2, 2, 1, 1, 55, 1, ... ]

the first few approximants are

0, 1, 1/2, 3/5, 7/12, 24/41, 31/53, 179/306, 389/665, 9126/15601, ...

so the 5, 12, 41, 53, 306, 665 tone scales as good for the 3:2, 4:3, 8:3... harmonics as you can cope for. Especially 12, 53 and 665 tone scales are good, because the next partial quotients 3, 5 or 23 are large. (The devil must be using the 666-scale.)

What about other important ratios?

The continuous fraction for 5:4 is [0, 3, 9, 2, 2, 4, 6, 2, 1, 1, 3, 1, 18, 1, ...]
with the first two approximants 0, 1/3, 9/28, 19/59, 47/146, 207/643, 1289/4004...
Here we have 1/3=4/12 (and there is nothing better until the denominator 28). This luck is a bit of coincidence.

The continuous fraction for 5:3 is [0, 1, 2, 1, 4, 22, 4, 1, 1, 13, 137, 1, 1, ...]
with the first two approximants 0, 1, 2/3, 3/4, 14/19, 311/422, 1258/1707....
Here again, 2/3=8/12 and 3/4=9/12 (and there is nothing better until the denominator 19), which must be very convenient for the major and minor subtonalities...

So the 12 is helped by the fact that it is richly divisible.

by das monde on Wed May 14th, 2008 at 06:44:07 AM EST
[ Parent ]
Thanks, that's exactly what I had in mind.

Pity the pythagoreans didn't know about logarithms.

When the capital development of a country becomes a by-product of the activities of a casino, the job is likely to be ill-done. — John M. Keynes

by Migeru (migeru at eurotrib dot com) on Wed May 14th, 2008 at 06:50:27 AM EST
[ Parent ]
i never saw those words put together before...

amazing, thanks.

12 is a number that has such mythic resonance.

'The history of public debt is full of irony. It rarely follows our ideas of order and justice.' Thomas Piketty

by melo (melometa4(at)gmail.com) on Wed May 14th, 2008 at 07:05:00 AM EST
[ Parent ]
I was reminded of Backgammon--I had a few games last night--I think there's something very ancient about those two cubes (six sides)--it seems that to play Backgammon well you have to understand percentages--statistics--and twelve is a large enough number to create adequate subtlety--

Don't fight forces, use them R. Buckminster Fuller.
by rg (leopold dot lepster at google mail dot com) on Wed May 14th, 2008 at 07:19:06 AM EST
[ Parent ]
Taking the diary way OT (I can't post videos where I am, otherwise I'd be--posting music videos!)

Dice were probably originally made from the ankle bones (specifically the talus or "astragalus") of hoofed animals (such as oxen), colloquially known as "knucklebones", which are approximately tetrahedral. Modern Mongolians still use such bones, known as shagai, for games and fortunetelling. In addition to bone, ivory, wood, metal, and stone materials have been commonly used. Recently, the use of plastics, including cellulose acetate and bakelite, is nearly universal. It is almost impossible to trace clearly the development of dice as distinguished from knucklebones, because ancient writers confused the two. It is certain, however, that both were used in prehistoric times.

Dice have been used throughout Asia since before recorded history.

The oldest known dice were excavated as part of a 5000-year-old backgammon set, at the Burnt City archeological site in south-eastern Iran. Excavations from ancient tombs in the Harappan civilization,[4] seem to further indicate a South Asian origin. Dicing is mentioned as an Indian game in the Rig Veda, Atharva Veda[5] and Buddha games list. It is also mentioned in the great Hindu epic, the Mahabharata, where Yudhisthira plays a game of dice against the Kauravas for the northern kingdom of Hastinapura.

http://en.wikipedia.org/wiki/Dice#History



Don't fight forces, use them R. Buckminster Fuller.
by rg (leopold dot lepster at google mail dot com) on Wed May 14th, 2008 at 07:30:02 AM EST
[ Parent ]
I wish I knew a couple of friends nearby who played backgammon, or better still a cafe where you could play. I find playing it relaxing and stimulting at the same time

You can't be me, I'm taken
by Sven Triloqvist on Sat May 17th, 2008 at 12:45:03 PM EST
[ Parent ]
That's because you're not a mathematician :-P

There is a comment somewhere else in this thread quoting an article with the question of why we don't use 10 regular intervals. The Babylonians knew that 12, 24, 30, 60 and 360 were richly divisible, that's where their number system comes from and note the only place it survives is in astronomy and timekeeping.

There is nothing new under the Sun, etc.

When the capital development of a country becomes a by-product of the activities of a casino, the job is likely to be ill-done. — John M. Keynes

by Migeru (migeru at eurotrib dot com) on Wed May 14th, 2008 at 07:20:43 AM EST
[ Parent ]
i love how you give our Great Solar Father His Own Capital, lol...

'The history of public debt is full of irony. It rarely follows our ideas of order and justice.' Thomas Piketty
by melo (melometa4(at)gmail.com) on Wed May 14th, 2008 at 11:15:54 AM EST
[ Parent ]

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