User:UnbihexiumFan/Temperaments: Difference between revisions

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Each half-period can be taken to represent
'''Bolded''' ratios are 7/4-reduced harmonics up to 21.
 
=== High-accuracy 7/4.2.3.11/5.13.17 extension ===


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


=== Extensions ===
{| class="wikitable"
! # Gens
! Cents<ref name="SSW">Optimal generator from the [https://sevish.com/scaleworkshop Sevish Scale Workshop]</ref>
! Approximate ratios
! # Gens
! Cents<ref name="SSW" />
! 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
|
|}


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

Revision as of 23:05, 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 0.0 -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. ↑ 1.0 1.1 1.2 1.3 Optimal generator from the Sevish Scale Workshop
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