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User:UnbihexiumFan/Temperaments: Difference between revisions

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'''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 263: Line 368:


'''Bolded''' ratios are 7/4-reduced harmonics up to 21. The 7/4-reduced 17th harmonic, [[17408/16807]], is found at +36 generators.
'''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.
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