User:UnbihexiumFan/Temperaments

A collection of temperaments that I have found that may or may not have yet been discovered. A lot of these are the same as already-known temperaments but with non-octave periods. I am not very good with technical details so even though they are included as info on most temperaments I will not be putting it here.

Stearnsmic 7/4-period temperaments

While searching for temperaments with period 7/4 and generator 3/2 I found that -8 generators (117649/104976) provides a close approximation of 9/8. The difference between these intervals is 118098/117649, which has apparently already been named the stearnsma. Tempering this comma given mapping generators ~7/4 and ~3/2 gives a pretty nice temperament which is essentially the same as no-five stearnsmic with different generators, but gives easier access to the perfect fifth and to septimal thirds.

Interval chain for the 7/4.2.3 temperament tempering the stearnsma:

# Gens	Cents	Approximate ratios	# Gens	Cents	Approximate ratios
+0	0.00	1/1	-0	968.83	7/4
+1	701.32	3/2	-1	267.51	7/6
+2	433.80	9/7	-2	535.02	49/36
+3	166.29	54/49	-3	802.53	343/216
+4	867.61	81/49	-4	101.22	343/324
+5	600.10	486/343, 343/243	-5	368.73	2401/1944, 81/49
+6	332.59	98/81	-6	636.24	81/56
+7	65.08	28/27	-7	903.75	27/16
+8	766.39	14/9	-8	202.44	9/8
+9	498.88	4/3	-9	469.95	21/16
+10	231.37	8/7	-10	737.46	49/32
+11	932.68	12/7	-11	36.14	49/48

Each half-period can be taken to represent

Bolded ratios are 7/4-reduced harmonics up to 21.

Extensions

The 17th harmonic can be added by equating 17/12 and 24/17 with the half-octave, tempering 442/441 and the 13th harmonic can be added by equating 27/26 and 28/27, tempering 729/728. This provides a comma basis of 442/441, 729/728, and 289/288.