User:UnbihexiumFan/Temperaments

A collection of temperaments that I have found that may or may not have yet been discovered. If you find any inaccuracies feel free to point them out on the talk page.

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.

7/4.2.3.5 extension

Each half-octave can be equated with 7/5~10/7, tempering out 50/49. While the resulting temperament is not very accurate, it gives a fairly simple mapping of pental thirds. It has a comma basis of 50/49 and 245/243. This temperament is equivalent to hedgehog but with a 7/4 period. The 11th harmonic can be added by equating 10/9 with 11/10, tempering out 100/99. The resulting temperament has subgroup 7/4.2.3.5.11 and comma basis 50/49, 100/99, and 55/54.

Interval chain:

# Gens	Cents	Approximate ratios	# Gens	Cents	Approximate ratios
+0	0.0	1/1	-0	968.83	7/4
+1	700.10	3/2	-1	268.73	7/6, 25/21, 33/28
+2	431.37	9/7, 14/11	-2	537.46	49/36, 11/8, 15/11, 25/18, 27/20
+3	162.64	10/9, 11/10, 12/11	-3	806.18	35/22
+4	862.74	5/3	-4	106.09	21/20, 15/14, 35/33
+5	594.01	10/7, 7/5	-5	374.81	5/4, 27/22
+6	325.28	6/5, 11/9, 40/33	-6	643.54	35/24
+7	56.56	36/35, 22/21, 28/27, 56/55	-7	912.27	27/16, 55/32
+8	756.65	11/7, 14/9, 54/35	-8	212.17	9/8
+9	487.93	4/3, 33/25	-9	480.90	21/16
+10	219.20	8/7	-10	749.63	49/32, 25/16
+11	919.30	12/7	-11	49.53	49/48, 25/24, 33/32

Bolded ratios are 7/4-reduced harmonics up to 21. The 7/4-reduced 5th harmonic, 80/49, is found at +15 generators, and the 7/4-reduced 11th harmonic, 2816/2401, is found at +28 generators.

18ed7/4 provides a good tuning for this temperament.

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.

Interval chain:

# 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.

29ed7/4 provides a good tuning for this temperament.

243/242+2079/2048-based temperaments

These temperaments temper out the rastma, 243/242, and an unnamed comma 2079/2048. They are similar to mohajira, but they can be tuned sharper to provide a better perfect fifth. In fact, mohajira is one possible full 11-limit extension, though this will focus on tunings sharper than mohajira.

Interval chain in the 2.3.7.11-limit:

Note name	# Gens	Cents	Approximate ratios	Note name	# Gens	Cents	Approximate ratios
C	+0	0.00	1/1	-0	C	1200.00	2/1
E	+1	349.08	11/9~27/22	-1	A	850.92	18/11~44/27
G	+2	698.15	3/2	-2	F	501.85	4/3
B	+3	1047.23	11/6	-3	D	152.77	12/11
D	+4	196.30	9/8	-4	B	1003.7	16/9
F	+5	545.38	11/8	-5	G	654.62	16/11
A	+6	894.46	27/16	-6	E	305.54	32/27
C	+7	43.53	33/32~64/63	-7	C	1156.47
E	+8	392.61	81/64	-8	A	807.39
G	+9	741.68	32/21	-9	F	458.32	21/16
B	+10	1090.76		-10	D	109.24
D	+11	239.84	8/7	-11	B	960.16	7/4
F	+12	588.91		-12	G	611.09
A	+13	937.99	12/7	-13	E	262.01	7/6
C	+14	87.06	22/21	-14	C	1112.94	21/11
E	+15	436.14	9/7	-15	A	763.86	14/9
G	+16	785.22	11/7	-16	F	414.78	14/11
B	+17	1134.29	27/14	-17	D	65.71	28/27
D	+18	283.37	33/28	-18	B	916.63

Bolded ratios are octave-reduced harmonics up to 21.

No-5's 19-limit extension

While the major third is too sharp to be seen as 5/4, it can be seen as 24/19 or 64/51. Treating it equal to both tempers out 513/512 and 4131/4096, providing a comma basis of 2057/2052, 513/512, 154/153, and 243/242. The 17th harmonic is mapped to the minor second (-10 generators, D on C) and the 19th harmonic is mapped to the minor third (-6 generators, E on C). The 13th harmonic can be added by setting 28/27 equal to 27/26, tempering out 729/728. This maps the 13th harmonic, 13/8, to the sesqui-augmented fifth (+23 generators, G on C).