User:UnbihexiumFan/Temperaments: Difference between revisions
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== Stearnsmic 7/4-period temperaments == | == 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 [[Stearnsmic clan|no-five stearnsmic]] with different generators, | 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 [[Stearnsmic clan|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: | Interval chain for the 7/4.2.3 temperament tempering the stearnsma: | ||
| Line 35: | Line 35: | ||
| 535.02 | | 535.02 | ||
| [[49/36]] | | [[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. | |||
Revision as of 21:38, 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 |
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.
- ↑ 1.0 1.1 Optimal generator from the Sevish Scale Workshop