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

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

High-accuracy 7/4.2.3.11/5.13.17 extension

The 17th harmonic can be added by equating 17/12 and 24/17 with the half-octave, tempering 442/441, the 13th harmonic can be added by equating 27/26 and 28/27, tempering 729/728, and the interval 11/5 can be added by equating 54/49 with 11/10, tempering out 540/539. This provides a high-accuracy temperament with a comma basis of 442/441, 729/728, 289/288, and 540/539.

# Gens	Cents	Approximate ratios	# Gens	Cents	Approximate ratios
+0	0.00	1/1	-0	968.83	7/4
+1	701.04	3/2	-1	267.78	7/6
+2	433.26	9/7	-2	535.57	49/36, 15/11
+3	165.47	11/10	-3	803.35	35/22, 27/17
+4	866.52	33/20	-4	102.31	17/16, 18/17, 35/33
+5	598.73	17/12, 24/17	-5	370.09	26/21, 21/17
+6	330.95	17/14, 39/32, 40/33	-6	637.88	13/9, 49/34
+7	63.16	28/27, 27/26	-7	905.66	27/16
+8	764.21	14/9	-8	204.62	9/8
+9	496.42	4/3	-9	472.40	21/16
+10	228.64	8/7	-10	740.19	49/32, 26/17
+11	929.68	12/7	-11	39.15	49/48, 45/44, 52/51
+12	661.90	22/15	-12	306.93	105/88
+13	394.11	44/35, 34/27	-13	574.71	39/28
+14	126.33	14/13	-14	842.50	13/8
+15	827.37	34/21, 21/13	-15	141.46	13/12
+16	559.59	3328/2401	-16	409.24	91/72
+17	291.80	77/65	-17	677.02	65/44
+18	24.02	64/63	-18	944.81
+19	725.06	32/21	-19	243.77	39/34
+20	457.28	64/49	-20	511.55	91/68

Bolded ratios are 7/4-reduced harmonics up to 21. The 7/4-reduced 17th harmonic, 17408/16807, is found at +36 generators.