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[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 | 0.0 | -0 | 968.83 | ||
| +1 | 701.04 | -1 | 267.78 | ||
| +2 | 433.26 | -2 | 535.57 | ||
| +3 | 165.47 | -3 | 803.35 | ||
| +4 | 866.52 | -4 | 102.31 | ||
| +5 | 598.73 | -5 | 370.09 | ||
| +6 | 330.95 | -6 | 637.88 | ||
| +7 | 63.16 | -7 | 905.66 | ||
| +8 | 764.21 | -8 | 204.62 | ||
| +9 | 496.42 | -9 | 472.40 | ||
| +10 | 228.64 | -10 | 740.19 | ||
| +11 | 929.68 | -11 | 39.15 | ||
| +12 | 661.90 | -12 | 306.93 | ||
| +13 | 394.11 | -13 | 574.71 | ||
| +14 | 126.33 | -14 | 842.50 | ||
| +15 | 827.37 | -15 | 141.46 | ||
| +16 | 559.59 | -16 | 409.24 | ||
| +17 | 291.80 | -17 | 677.02 | ||
| +18 | 24.02 | -18 | 944.81 | ||
| +19 | 725.06 | -19 | 243.77 | ||
| +20 | 457.28 | -20 | 511.55 |
Bolded ratios are 7/4-reduced harmonics up to 21.
- ↑ 1.0 1.1 1.2 1.3 Optimal generator from the Sevish Scale Workshop