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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).
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).
[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.
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
amazing, thanks.
12 is a number that has such mythic resonance. "These days, there's nothing more ridiculous than the truth." Leonard Pitts Jr
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
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
