Table of rank-3 temperament lattice bases: Difference between revisions

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Below are tabulated bases (temperamental Minkowski bases) of minimal temperamental complexity for the pitch classes of various regular temperaments.
Below are tabulated bases (temperamental Minkowski bases) of minimal temperamental complexity for the pitch classes of various regular temperaments.


=Rank three=
== Rank-3 ==
 
=== 7-limit temperaments ===
==Seven limit==
{| class="wikitable center-3 center-5 right-7"
 
{| class="wikitable"
|-
|-
! | Comma
! Comma
! | Name
! Name
! | Gen 1
! Gen 1
! | Length
! Length
! | Gen 2
! Gen 2
! | Length
! Length
! | Angle
! Angle
|-
|-
| | 686/675
| 686/675
| | [[senga]]
| [[Sengic]]
| | 15/14
| 15/14
| | 0.6977
| 0.6977
| | 3/2
| 3/2
| | 1.4208
| 1.4208
| | 103.732
| 103.732
|-
|-
| | 64/63
| 64/63
| | [[Archytas]]'
| [[Archytas]]
| | 3/2
| 3/2
| | 1.0508
| 1.0508
| | 5/4
| 5/4
| | 2.3219
| 2.3219
| | 90.000
| 90.000
|-
|-
| | 875/864
| 875/864
| | [[keema]]
| [[Keemic]]
| | 6/5
| 6/5
| | 0.8879
| 0.8879
| | 3/2
| 3/2
| | 1.3391
| 1.3391
| | 77.834
| 77.834
|-
|-
| | 81/80
| 81/80
| | [[Didymus]]'
| [[Didymus]]
| | 3/2
| 3/2
| | 0.5451
| 0.5451
| | 8/7
| 8/7
| | 2.8074
| 2.8074
| | 90.000
| 90.000
|-
|-
| | 3125/3087
| 3125/3087
| | [[gariboh]]
| [[Gariboh]]
| | 25/21
| 25/21
| | 0.5928
| 0.5928
| | 21/20
| 21/20
| | 1.2018
| 1.2018
| | 80.810
| 80.810
|-
|-
| | 2430/2401
| 2430/2401
| | [[nuwell]]
| [[Nuwell]]
| | 7/6
| 7/6
| | 0.6247
| 0.6247
| | 9/7
| 9/7
| | 1.1894
| 1.1894
| | 94.044
| 94.044
|-
|-
| | 50421/50000
| 50421/50000
| | [[trimyna]]
| [[Trimyna]]
| | 7/5
| 7/5
| | 0.3158
| 0.3158
| | 5/4
| 5/4
| | 1.7938
| 1.7938
| | 94.101
| 94.101
|-
|-
| | 245/243
| 245/243
| | [[sensamagic]]
| [[Sensamagic]]
| | 3/2
| 3/2
| | 0.9644
| 0.9644
| | 9/7
| 9/7
| | 1.0807
| 1.0807
| | 96.914
| 96.914
|-
|-
| | 126/125
| 126/125
| | [[starling]]
| [[Starling]]
| | 6/5
| 6/5
| | 1.068
| 1.068
| | 5/4
| 5/4
| | 1.206
| 1.206
| | 100.364
| 100.364
|-
|-
| | 4000/3969
| 4000/3969
| | [[octagar]]
| [[Octagar]]
| | 63/50
| 63/50
| | 0.8966
| 0.8966
| | 21/20
| 21/20
| | 1.0605
| 1.0605
| | 97.743
| 97.743
|-
|-
| | 1728/1715
| 1728/1715
| | [[orwellisma]]
| [[Orwellismic]]
| | 7/6
| 7/6
| | 0.7526
| 0.7526
| | 3/2
| 3/2
| | 1.3931
| 1.3931
| | 97.508
| 97.508
|-
|-
| | 1029/1024
| 1029/1024
| | [[gamelisma]]
| [[Gamelismic]]
| | 8/7
| 8/7
| | 0.5192
| 0.5192
| | 5/4
| 5/4
| | 2.3219
| 2.3219
| | 90.000
| 90.000
|-
|-
| | 225/224
| 225/224
| | [[marvel]]
| [[Marvel]]
| | 16/15
| 16/15
| | 1.256
| 1.256
| | 3/2
| 3/2
| | 1.369
| 1.369
| | 106.958
| 106.958
|-
|-
| | 3136/3125
| 3136/3125
| | [[hemimean]]
| [[Hemimean]]
| | 28/25
| 28/25
| | 0.5055
| 0.5055
| | 3/2
| 3/2
| | 1.5849
| 1.5849
| | 90.000
| 90.000
|-
|-
| | 5120/5103
| 5120/5103
| | [[hemifamity]]
| [[Aberschismic]]
| | 3/2
| 3/2
| | 0.5670
| 0.5670
| | 10/9
| 10/9
| | 1.8063
| 1.8063
| | 82.112
| 82.112
|-
|-
| | 6144/6125
| 6144/6125
| | [[porwell]]
| [[Porwell]]
| | 35/32
| 35/32
| | 0.9584
| 0.9584
| | 35/24
| 35/24
| | 1.1877
| 1.1877
| | 92.681
| 92.681
|-
|-
| | 65625/65536
| 65625/65536
| | [[horwell]]
| [[Horwell]]
| | 5/4
| 5/4
| | 0.6213
| 0.6213
| | 16/15
| 16/15
| | 1.3863
| 1.3863
| | 64.638
| 64.638
|-
|-
| | 703125/702464
| 703125/702464
| | [[meter]]
| [[Metric]]
| | 75/56
| 75/56
| | 0.7305
| 0.7305
| | 375/224
| 375/224
| | 0.7675
| 0.7675
| | 97.289
| 97.289
|-
|-
| | 420175/419904
| 420175/419904
| | [[wizma]]
| [[Wizmic]]
| | 245/216
| 245/216
| | 0.5541
| 0.5541
| | 324/245
| 324/245
| | 0.9731
| 0.9731
| | 79.663
| 79.663
|-
|-
| | 2401/2400
| 2401/2400
| | [[breedsma]]
| [[Breedsmic]]
| | 49/40
| 49/40
| | 0.7858
| 0.7858
| | 8/7
| 8/7
| | 1.1241
| 1.1241
| | 107.367
| 107.367
|-
|-
| | 4375/4374
| 4375/4374
| | [[ragisma]]
| [[Ragismic]]
| | 10/9
| 10/9
| | 0.7890
| 0.7890
| | 6/5
| 6/5
| | 0.9208
| 0.9208
| | 105.299
| 105.299
|}
|}


==Eleven limit==
=== 11-limit temperaments ===
 
{| class="wikitable center-3 center-5 right-7"
{| class="wikitable"
|-
|-
! | Commas
! Commas
! | Name
! Name
! | Gen 1
! Gen 1
! | Length
! Length
! | Gen 2
! Gen 2
! | Length
! Length
! | Angle
! Angle
|-
|-
| | 225/224, 385/384
| 225/224, 385/384
| | [[marvel]]
| [[Marvel]]
| | 16/15
| 16/15
| | 1.0364
| 1.0364
| | 5/4
| 5/4
| | 1.0759
| 1.0759
| | 104.028
| 104.028
|-
|-
| | 225/224, 441/440 <!-- was 441/400, presumed incorrect -->
| 225/224, 441/440 <!-- was 441/400, presumed incorrect -->
| | [[prodigy]]
| [[Prodigy]]
| | 16/15
| 16/15
| | 0.9111
| 0.9111
| | 3/2
| 3/2
| | 0.9477
| 0.9477
| | 65.933
| 65.933
|-
|-
| | 99/98, 176/175
| 99/98, 176/175
| | [[minerva]]
| [[Minerva]]
| | 16/15
| 16/15
| | 0.8997
| 0.8997
| | 5/4
| 5/4
| | 1.0457
| 1.0457
| | 98.604
| 98.604
|-
|-
| | 126/125, 176/175
| 126/125, 176/175
| | [[thrush]]
| [[Thrush]]
| | 5/4
| 5/4
| | 0.8576
| 0.8576
| | 6/5
| 6/5
| | 0.9314
| 0.9314
| | 74.624
| 74.624
|-
|-
| | 56/55, 100/99
| 56/55, 100/99
| | [[thrasher]]
| [[Thrasher]]
| | 6/5
| 6/5
| | 0.9089
| 0.9089
| | 5/4
| 5/4
| | 1.2007
| 1.2007
| | 98.845
| 98.845
|}
|}


Revision as of 07:01, 7 June 2026

Todo: improve synopsis, clarify, link

Below are tabulated bases (temperamental Minkowski bases) of minimal temperamental complexity for the pitch classes of various regular temperaments.

Rank-3

7-limit temperaments

Comma Name Gen 1 Length Gen 2 Length Angle
686/675 Sengic 15/14 0.6977 3/2 1.4208 103.732
64/63 Archytas 3/2 1.0508 5/4 2.3219 90.000
875/864 Keemic 6/5 0.8879 3/2 1.3391 77.834
81/80 Didymus 3/2 0.5451 8/7 2.8074 90.000
3125/3087 Gariboh 25/21 0.5928 21/20 1.2018 80.810
2430/2401 Nuwell 7/6 0.6247 9/7 1.1894 94.044
50421/50000 Trimyna 7/5 0.3158 5/4 1.7938 94.101
245/243 Sensamagic 3/2 0.9644 9/7 1.0807 96.914
126/125 Starling 6/5 1.068 5/4 1.206 100.364
4000/3969 Octagar 63/50 0.8966 21/20 1.0605 97.743
1728/1715 Orwellismic 7/6 0.7526 3/2 1.3931 97.508
1029/1024 Gamelismic 8/7 0.5192 5/4 2.3219 90.000
225/224 Marvel 16/15 1.256 3/2 1.369 106.958
3136/3125 Hemimean 28/25 0.5055 3/2 1.5849 90.000
5120/5103 Aberschismic 3/2 0.5670 10/9 1.8063 82.112
6144/6125 Porwell 35/32 0.9584 35/24 1.1877 92.681
65625/65536 Horwell 5/4 0.6213 16/15 1.3863 64.638
703125/702464 Metric 75/56 0.7305 375/224 0.7675 97.289
420175/419904 Wizmic 245/216 0.5541 324/245 0.9731 79.663
2401/2400 Breedsmic 49/40 0.7858 8/7 1.1241 107.367
4375/4374 Ragismic 10/9 0.7890 6/5 0.9208 105.299

11-limit temperaments

Commas Name Gen 1 Length Gen 2 Length Angle
225/224, 385/384 Marvel 16/15 1.0364 5/4 1.0759 104.028
225/224, 441/440 Prodigy 16/15 0.9111 3/2 0.9477 65.933
99/98, 176/175 Minerva 16/15 0.8997 5/4 1.0457 98.604
126/125, 176/175 Thrush 5/4 0.8576 6/5 0.9314 74.624
56/55, 100/99 Thrasher 6/5 0.9089 5/4 1.2007 98.845
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