User:UnbihexiumFan/Temperaments: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
extension
No edit summary
 
(11 intermediate revisions by 2 users not shown)
Line 1: Line 1:
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.
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 [[User talk:UnbihexiumFan/Temperaments|talk page]].


== Stearnsmic 7/4-period temperaments ==
== Stearnsmic 7/4-period temperaments ==
Line 17: Line 17:
| +0
| +0
| 0.00
| 0.00
| [[1/1]]
| '''[[1/1]]'''
| -0
| -0
| 968.83
| 968.83
| [[7/4]]
| '''[[7/4]]'''
|-
|-
| +1
| +1
Line 102: Line 102:
'''Bolded''' ratios are 7/4-reduced harmonics up to 21.
'''Bolded''' ratios are 7/4-reduced harmonics up to 21.


=== High-accuracy 7/4.2.3.11/5.13.17 extension ===
=== 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:
 
{| class="wikitable"
! # Gens
! Cents<ref name="SSW2">Optimal generator from the [https://sevish.com/scaleworkshop Sevish Scale Workshop], subgroup given as 7/4.2.3.5/4.11/10</ref>
! Approximate ratios
! # Gens
! Cents<ref name="SSW2" />
! 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.
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:


{| class="wikitable"
{| class="wikitable"
! # Gens
! # Gens
! Cents<ref name="SSW">Optimal generator from the [https://sevish.com/scaleworkshop Sevish Scale Workshop]</ref>
! Cents<ref name="SSW" />
! Approximate ratios
! Approximate ratios
! # Gens
! # Gens
Line 115: Line 220:
|-
|-
| +0
| +0
| 0.0
| 0.00
|  
| '''[[1/1]]'''
| -0
| -0
| 968.83
| 968.83
|  
| '''[[7/4]]'''
|-
|-
| +1
| +1
| 701.04
| 701.04
|  
| [[3/2]]
| -1
| -1
| 267.78
| 267.78
|  
| [[7/6]]
|-
|-
| +2
| +2
| 433.26
| 433.26
|  
| [[9/7]]
| -2
| -2
| 535.57
| 535.57
|  
| [[49/36]], [[15/11]]
|-
|-
| +3
| +3
| 165.47
| 165.47
|  
| [[11/10]]
| -3
| -3
| 803.35
| 803.35
|  
| [[35/22]], [[27/17]]
|-
|-
| +4
| +4
| 866.52
| 866.52
|  
| [[33/20]]
| -4
| -4
| 102.31
| 102.31
|  
| [[17/16]], [[18/17]], [[35/33]]
|-
|-
| +5
| +5
| 598.73
| 598.73
|  
| [[17/12]], [[24/17]]
| -5
| -5
| 370.09
| 370.09
|  
| [[26/21]], [[21/17]]
|-
|-
| +6
| +6
| 330.95
| 330.95
|  
| [[17/14]], [[39/32]], [[40/33]]
| -6
| -6
| 637.88
| 637.88
|  
| [[13/9]], [[49/34]]
|-
|-
| +7
| +7
| 63.16
| 63.16
|  
| [[28/27]], [[27/26]]
| -7
| -7
| 905.66
| 905.66
|  
| [[27/16]]
|-
|-
| +8
| +8
| 764.21
| 764.21
|  
| [[14/9]]
| -8
| -8
| 204.62
| 204.62
|  
| [[9/8]]
|-
|-
| +9
| +9
| 496.42
| 496.42
|  
| [[4/3]]
| -9
| -9
| 472.40
| 472.40
|  
| [[21/16]]
|-
|-
| +10
| +10
| 228.64
| 228.64
|  
| '''[[8/7]]'''
| -10
| -10
| 740.19
| 740.19
|  
| [[49/32]], [[26/17]]
|-
|-
| +11
| +11
| 929.68
| 929.68
|  
| '''[[12/7]]'''
| -11
| -11
| 39.15
| 39.15
|  
| [[49/48]], [[45/44]], [[52/51]]
|-
|-
| +12
| +12
| 661.90
| 661.90
|  
| [[22/15]]
| -12
| -12
| 306.93
| 306.93
|  
| [[105/88]]
|-
|-
| +13
| +13
| 394.11
| 394.11
|  
| [[44/35]], [[34/27]]
| -13
| -13
| 574.71
| 574.71
|  
| [[39/28]]
|-
|-
| +14
| +14
| 126.33
| 126.33
|  
| [[14/13]]
| -14
| -14
| 842.50
| 842.50
|  
| [[13/8]]
|-
|-
| +15
| +15
| 827.37
| 827.37
|  
| [[34/21]], [[21/13]]
| -15
| -15
| 141.46
| 141.46
|  
| [[13/12]]
|-
|-
| +16
| +16
| 559.59
| 559.59
|  
| '''[[3328/2401]]'''
| -16
| -16
| 409.24
| 409.24
|  
| [[91/72]]
|-
|-
| +17
| +17
| 291.80
| 291.80
|  
| [[77/65]]
| -17
| -17
| 677.02
| 677.02
|  
| [[65/44]]
|-
|-
| +18
| +18
| 24.02
| 24.02
|  
| [[64/63]]
| -18
| -18
| 944.81
| 944.81
Line 249: Line 354:
| +19
| +19
| 725.06
| 725.06
|  
| [[32/21]]
| -19
| -19
| 243.77
| 243.77
|  
| [[39/34]]
|-
|-
| +20
| +20
| 457.28
| 457.28
|  
| '''[[64/49]]'''
| -20
| -20
| 511.55
| 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:
 
{| class="wikitable"
! 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{{demiflat}}
| +1
| 349.08
| [[11/9]]~[[27/22]]
| -1
| A{{demiflat}}
| 850.92
| [[18/11]]~[[44/27]]
|-
| G
| +2
| 698.15
| '''[[3/2]]'''
| -2
| F
| 501.85
| [[4/3]]
|-
| B{{demiflat}}
| +3
| 1047.23
| [[11/6]]
| -3
| D{{demiflat}}
| 152.77
| [[12/11]]
|-
| D
| +4
| 196.30
| '''[[9/8]]'''
| -4
| B{{flat}}
| 1003.7
| [[16/9]]
|-
| F{{demisharp}}
| +5
| 545.38
| '''[[11/8]]'''
| -5
| G{{demiflat}}
| 654.62
| [[16/11]]
|-
| A
| +6
| 894.46
| [[27/16]]
| -6
| E{{flat}}
| 305.54
| [[32/27]]
|-
| C{{demisharp}}
| +7
| 43.53
| [[33/32]]~[[64/63]]
| -7
| C{{demiflat}}
| 1156.47
|
|-
| E
| +8
| 392.61
| [[81/64]]
| -8
| A{{flat}}
| 807.39
|
|-
| G{{demisharp}}
| +9
| 741.68
| [[32/21]]
| -9
| F{{demiflat}}
| 458.32
| '''[[21/16]]'''
|-
| B
| +10
| 1090.76
|
| -10
| D{{flat}}
| 109.24
|
|-
| D{{demisharp}}
| +11
| 239.84
| [[8/7]]
| -11
| B{{sesquiflat}}
| 960.16
| '''[[7/4]]'''
|-
| F{{sharp}}
| +12
| 588.91
|
| -12
| G{{flat}}
| 611.09
|
|-
| A{{demisharp}}
| +13
| 937.99
| [[12/7]]
| -13
| E{{sesquiflat}}
| 262.01
| [[7/6]]
|-
| C{{sharp}}
| +14
| 87.06
| [[22/21]]
| -14
| C{{flat}}
| 1112.94
| [[21/11]]
|-
| E{{demisharp}}
| +15
| 436.14
| [[9/7]]
| -15
| A{{sesquiflat}}
| 763.86
| [[14/9]]
|-
| G{{sharp}}
| +16
| 785.22
| [[11/7]]
| -16
| F{{flat}}
| 414.78
| [[14/11]]
|-
| B{{demisharp}}
| +17
| 1134.29
| [[27/14]]
| -17
| D{{sesquiflat}}
| 65.71
| [[28/27]]
|-
| D{{sharp}}
| +18
| 283.37
| [[33/28]]
| -18
| B{{flat2}}
| 916.63
|
|}
|}


'''Bolded''' ratios are 7/4-reduced harmonics up to 21.
'''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{{flat}} on C) and the 19th harmonic is mapped to the minor third (-6 generators, E{{flat}} 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{{sesquisharp}} on C).

Latest revision as of 19:21, 8 February 2026

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[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.

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[2] Approximate ratios # Gens Cents[2] 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[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.

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).

  1. ↑ 1.0 1.1 1.2 1.3 Optimal generator from the Sevish Scale Workshop
  2. ↑ 2.0 2.1 Optimal generator from the Sevish Scale Workshop, subgroup given as 7/4.2.3.5/4.11/10
Retrieved from "https://en.xen.wiki/index.php?title=User:UnbihexiumFan/Temperaments&oldid=223600"