User:2^67-1/Ed12

Disclaimer: written a la MMTM

The equal division of 12/1 (ed12/1) is a tuning obtained by dividing the twelfth harmonic (12/1) in a certain number of equal steps.

Properties

Division of 12 into equal parts does not necessarily imply directly using this interval as an equivalence. The question of equivalence has not even been posed yet. The utility of this interval as an equivalence, despite being irrational, is that it serves as two times the upper bound of the range of most peoples' voices, which the author takes to be about √12. The twelfth harmonic is pretty far as much as equivalences go, so here pure 3-smooth scales are required, leading the ratios to be square roots of 3-smooth integers or fractions. Taking √12 as the period gives us 2, 3, 4, 7, 11, and 18 note MOS-scales within the √12, or 4, 6, 8, 14, 22, or 36 note MOS-scales within the 12/1. This is the pochhammeroid temperament, named by Cole.

Proposed names for 12/1-equivalent temperaments

Explanation of notation: the notation a&b refers to the temperament shared by ED12s a and b, which is similar to the x31eq notation, and thus the MOS scales aL bs or bL as. A temperament collection thus refers to all the MOS scales with a certain number of notes.

Cole's names

Proposed names for 12/1-equivalent MOS scales

Cole's names

• 14L 22s - Pochhammeroid