ellis@flairvax.UUCP (Michael Ellis) (07/14/83)
Hmmm... a hexadecimally tempered musical scale... I don't think it
would work, even in an alien culture from outer space. Here's why.
Our tone scale evolved from obvious resonances based on integer
relationships, like 1:2 (C:C+), 2:3 (C:G), 3:4 (C:F), 4:5 (C:E) and
so on. In integers, the notes follow this series:
C D E F G A B C+
24 27 30 32 36 40 45 48
You don't have to be from our culture to hear the beauty of this
sequence, just as foreign music based on different ratios sounds
naturally beautiful to us (to me, anyway). Note that even foreign
music shares the most basic ratios (1:2, 2:3, 3:4) with our scale.
Now here's where tempering comes in. Suppose you want to play a song
starting a note higher or lower, but preserve the mathematical
intervals between the notes in the song. The solution was to try
adding notes in between the big gaps (like F & G). Mathematically,
this means selecting an integer N>8 such that the Nth roots of 2
contain a subset having the above integer relationships.
Doing this exactly would mean a keyboard with zillions of keys, and I
heard that such instruments were actually attempted. But, one of the
happier mathematical bonanzas of history are the 12th roots of two.
Selected values lie remarkably close to the desired integer ratios,
ESPECIALLY the important ones (2:3 & 3:4). If these were not close,
a piano would sound horrible. For your amusement, here are the
values:
2^(0/12) = 1 C 24 1
2^(1/12) = 1.05946 C#
2^(2/12) = 1.12246 D 27 1.125
2^(3/12) = 1.18921 D#
2^(4/12) = 1.25992 E 30 1.25
2^(5/12) = 1.33484 F 32 1.333.. (amazingly close)
2^(6/12) = 1.41421 F#
2^(7/12) = 1.49831 G 36 1.5 (amazingly close)
2^(8/12) = 1.5874 G#
2^(9/12) = 1.68179 A 40 1.666..
2^(10/12) = 1.7818 A#
2^(11/12) = 1.88775 B 45 1.875
2^(12/12) = 2 C 48 2.0
So you might say, a smaller interval should get closer. Try using the
16th roots of 2 rather than the 12th roots. Check out hexadecimal
tempering yourself:
2^(0/16) = 1 C 24 1
2^(1/16) = 1.04427
2^(2/16) = 1.09051
2^(3/16) = 1.13879 D 27 1.125
2^(4/16) = 1.18921
2^(5/16) = 1.24186 E 30 1.25 (only one that's better)
2^(6/16) = 1.29684
2^(7/16) = 1.35426 F 32 1.333.. (horrible)
2^(8/16) = 1.41421
2^(9/16) = 1.47683 G 36 1.5 (horrible)
2^(10/16) = 1.54221
2^(11/16) = 1.61049
2^(12/16) = 1.68179 A 40 1.666..
2^(13/16) = 1.75625
2^(14/16) = 1.83401
2^(15/16) = 1.91521 B 45 1.875
2^(16/16) = 2 C 48 2.0
These values are generally worse than the 12th roots, except for E
(only marginally better) and A (the same). Not to mention the extra
construction problems, and more complex fingering required, and so
on. I bet it'd only sound good for punk.
Michael Ellis - Fairchild AI Lab - Palo Alto CA - (415) 858-4270ellis@flairvax.UUCP (Michael Ellis) (07/14/83)
Correction on my incorrect mathematical language in the "Hexadecimal
Tempering" article. Please substitute:
"integer powers of the Nth root of 2"
...wherever I said:
"Nth roots of 2"
Sorry - Michaelthomas@utah-gr.UUCP (Spencer W. Thomas) (07/18/83)
I once heard some music made on a 19 note scale. The guy who did it claimed that the 19th roots of 2 come even closer to the ideal ratios. Sounded (naturally) weird. =Spencer