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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 ==


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, which gives easier access to the perfect fifth and to septimal thirds.
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 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 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]]
|}
|}


(not completed yet... will finish soon)
'''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:
 
{| 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.
 
Interval chain:
 
{| class="wikitable"
! # Gens
! Cents<ref name="SSW" />
! Approximate ratios
! # Gens
! Cents<ref name="SSW" />
! 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:
 
{| 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 octave-reduced harmonics up to 21.
 
=== No-5's 19-limit extension ===


has a good 7/4.2.3.13.17 extension that tempers 442/441, 729/728, 289/288
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).
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