User:TromboneBoi9/sandbox

13 equal temperament

13 equal temperament or 13-ET is the tuning system splitting the octave into thirteen equal parts, with each part about 92.37 cents large.

From the perspective of 12-ET, 13-ET is extremely unorthodox since it misses the perfect fifth 3:2 by more than 30 cents. This puts any music in syntonic temperaments extremely out of tune. Its intervals exhibit a “phasing out” effect, in which intervals get more and more out of tune (from 12-ET) toward the half-octave and then get more and more in tune toward the octave.

Major seconds and major thirds are approximated well, with the major third as close to 5:4 ratio as that of 12-ET. Besides those, it also approximates some odd harmonics that 12-ET misses, particularly the 11^th, 13^th, and 21^st.

13-ET Approximations of Odd Harmonics

	3	5	7	9	11	13	15	17	19	21	23
Error (cents)	+36.5	-17.1	-45.7	-19.3	+2.5	-9.8	+19.4	-12.6	-20.6	-9.2	+17.9
Steps	21	30	36	41	45	48	51	53	55	57	59
Reduced	8	4	10	2	6	9	12	1	3	5	7

Scale systems and notation

Since diatonic systems are out of tune in 13-ET, we can instead use 13-ET's major second to generate the scale instead. Doing so can create to different scale systems which can be used for harmony and notation.

Archaeotonic

Archaeotonic is a scale system with 6 “whole steps” and 1 “half step” rather than 5 whole steps and 2 half steps. Archaeotonic can be notated in the same way as diatonic 12-ET, except that E♯ is its own note between E and F.

The circle of major seconds goes:

…

G♭

A♭

B♭

C

D

E

F

G

A

B

C♯

D♯

E♯

…