User:UnbihexiumFan/Temperaments: Difference between revisions

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| '''[[7/4]]'''
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| 433.26
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| [[9/7]]
| -2
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| 535.57
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| [[49/36]], [[15/11]]
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|-
| +3
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| 165.47
| 165.47
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| [[11/10]]
| -3
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| 803.35
| 803.35
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| [[35/22]], [[27/17]]
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| +4
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| 866.52
| 866.52
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| [[33/20]]
| -4
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| 102.31
| 102.31
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| [[17/16]], [[18/17]], [[35/33]]
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|-
| +5
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| 598.73
| 598.73
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| [[17/12]], [[24/17]]
| -5
| -5
| 370.09
| 370.09
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| [[26/21]], [[21/17]]
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| 330.95
| 330.95
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| [[17/14]], [[39/32]], [[40/33]]
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| [[13/9]], [[49/34]]
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| 63.16
| 63.16
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| [[28/27]], [[27/26]]
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| [[27/16]]
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| 764.21
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| [[14/9]]
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| 204.62
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| [[9/8]]
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|-
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| 496.42
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| [[4/3]]
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| -9
| 472.40
| 472.40
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| [[21/16]]
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|-
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| 228.64
| 228.64
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| '''[[8/7]]'''
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| -10
| 740.19
| 740.19
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| [[49/32]], [[26/17]]
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|-
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| 929.68
| 929.68
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| '''[[12/7]]'''
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| 39.15
| 39.15
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| [[49/48]], [[45/44]], [[52/51]]
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| 661.90
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| [[22/15]]
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| 306.93
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| [[105/88]]
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| 394.11
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| [[44/35]], [[34/27]]
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| 574.71
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| [[39/28]]
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|-
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| 126.33
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| [[13/8]]
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| [[34/21]], [[21/13]]
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| [[13/12]]
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| '''[[3328/2401]]'''
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| [[91/72]]
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| [[77/65]]
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| [[65/44]]
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| [[39/34]]
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| '''[[64/49]]'''
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'''Bolded''' ratios are 7/4-reduced harmonics up to 21.
'''Bolded''' ratios are 7/4-reduced harmonics up to 21. The 7/4-reduced 17th harmonic, [[17408/16807]], is found at +36 generators.

Revision as of 23:28, 10 January 2026

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[1] Approximate ratios # Gens Cents[1] 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[1] Approximate ratios # Gens Cents[1] 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.

  1. ↑ 1.0 1.1 1.2 1.3 Optimal generator from the Sevish Scale Workshop
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