Approximation to Pierce-Bohlen

This is a "wet finger" approximation to the Pierce-Bohlen scale.
The second column shows the difference in cents.

1: 12/11 4.332829 3/4-tone
2: 33/28 -8.161428
3: 9/7 -3.828599 septimal major third
4: 7/5 -2.704731 septimal diminished fifth
5: 32/21 -2.302062 wide fifth
6: 5/3 6.533331 major sixth
7: 9/5 -6.533331 just minor seventh
8: 63/32 2.302062 octave - septimal comma
9: 15/7 2.704731 major diatonic semitone + 1 octave
10: 7/3 3.828599 minimal 10th
11: 28/11 8.161428
12: 11/4 -4.332829 harmonic augmented fourth + 1 octave
13: 3/1 -0.000000 perfect 12th

So a 3:5:7:9 chord involves pitch classes 0-6-10-13.


Marcus writes:
> I realize that we can do NonJust Equal Temperament, Just Non Equal Temperament
> (what others are there?).

All combinations are possible, Just Equal Temperament is rarely used,
an example is:

1: 9/8 203.9100 major whole tone
2: 81/64 407.8201 Pythagorean major third
3: 729/512 611.7302 Pythagorean tritone
4: 6561/4096 815.6403 Pythagorean augmented fifth
5: 59049/32768 1019.550 Pythagorean augmented sixth
6: 531441/262144 1223.460 Pythagorean comma + 1 octave

And non-just non-equal-temperament is what Brian has been writing about a lot.

> Would you (or anybody on the list) mind elaborating scale construction base on
> partial frequency formulas? i.e., Given the really neat partial freq ratios
> that mclaren described, how do I construct a scale from them?

You can just take the same frequencies, or a subset of them, or divide the
larger ones by a power of 2 to octave reduce them.

> For 12TET, did someone merely lay out the harmonic series to some order,
> octave transpose them, and then make a best fit to a 2^(n/12) scale
> (discarding partials 7,11,13 in this case)?

It has been known for a very long time and there are different ways to
structure 12-tET. It can be viewed as a cycle of tempered fifths, as the
completion of the diatonic scale, which on its turn can be viewed as three
triads a fifth apart, or as a shorter cycle of fifths, etc. So the
third harmonic alone is enough to get 12-tET, by ignoring the Pythagorean
comma.

Manuel Op de Coul coul@ezh.nl

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