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

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{{Todo|inline=1|improve synopsis|clarify|link|comments=Expand the intro to more thoroughly explain what this is. Link to here from other relevant pages.}}
Below are tabulated [[Rank-3 temperament #Euclidean metric on the lattice|temperamental Minkowski bases]] of minimal [[Tenney–Euclidean metrics #TE temperamental norm|temperamental complexity]] for the pitch classes of various [[regular temperament]]s.


Below are tabulated bases (temperamental Minkowski bases) of minimal temperamental complexity for the pitch classes of various regular temperaments.
== Rank-3 ==
 
=== 7-limit temperaments ===
=Rank three=
{| class="wikitable center-3 center-5 right-7"
 
==Seven limit==
 
{| 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
|}
|}


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