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Hindemith's 5-limit just intonation derivation
of 12-tone equal temperament

[Joe Monzo]

Paul Hindemith, in his book The Craft of Musical Composition, volume 1, describes a derivation of 12-et as a tempering of a 12-tone just intonation scale, which in turn is generated as subharmonics of higher partials.

In Genesis of a Music (p 420 in the 2nd edition), Harry Partch provides an informative diagram which succinctly illustrates Hindemith's explanation.

Below is a 3-dimensional Musica lattice in 2,3,5-prime-space, showing the same data as Partch's diagram.

Hindemith 5-limit JI derivation of 12-et: Musica lattice

Below is the data for the above lattice, showing each pair of notes group together: the higher partial (white lattice cube) comes first, then below it is the generated scale note (orange cube).

  2,3,5-monzo      ratio   partial

[ 1   0,   0 >     2 / 1    2
[ 0   0,   0 >     1 / 1    1

[ 0   1,   0 >     3 / 1    2
[-1   1,   0 >     3 / 2    1

[ 2   0,   0 >     4 / 1    3
[ 2  -1,   0 >     4 / 3    1

[ 0   0,   1 >     5 / 1    3
[ 0  -1,   1 >     5 / 3    1

[ 0   0,   1 >     5 / 1    4
[-2   0,   1 >     5 / 4    1

[ 1   1,   0 >     6 / 1    5
[ 1   1,  -1 >     6 / 5    1

[ 2   0,   0 >     4 / 1    5
[ 3   0,  -1 >     8 / 5    2    NOTE: involves 2 prime-factors, so blue line is angled

[-1   2,   0 >     9 / 2    4
[-3   2,   0 >     9 / 8    1

[ 4  -1,   0 >    16 / 3    3
[ 4  -2,   0 >    16 / 9    1

[ 4  -1,   0 >    16 / 3    5
[ 4  -1,  -1 >    16 / 15   1

[-2   1,   1 >    15 / 4    2
[-3   1,   1 >    15 / 8    1

[ 6  -1,  -1 >    64 / 15   3
[ 6  -2,  -1 >    64 / 45   1

[-3   2,   1 >    45 / 8    4
[-5   2,   1 >    45 / 32   1
		

Partch notes that Hindemith confounds the last two scale-notes into one.

It can be seen that the scale shown in orange on the lattice is exactly the same as the "closest to 1/1" 12-et bingo-card tiling derived by Joe Monzo.

. . . . . . . . .
[Gene Ward Smith, Yahoo tuning group message 55422 (Wed Aug 11, 2004 11:39 am)]

It seems to me all that is being said here is that

  1. 12-et can be described as a tempering of the Malcolm Monochord or its inverse, New Albion.
  2. The fundamental theorem of arithmetic -- that any positive rational number can be uniquely factored into primes -- is really true.

There's also a suggestion that Hindemith is willing, Peter Sault style, to switch between Malcolm and New Albion.

Point number one is of course true for any epimorphic 12-scale, and point number two removes any necessity of showing that a rational number is the product of primes in any particular case, since this is always true.

. . . . . . . . .

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