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 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 three=


==Seven limit==
==Seven limit==
Line 7: Line 9:
{| class="wikitable"
{| class="wikitable"
|-
|-
| | Comma
! | Comma
| | Name
! | Name
| | Gen 1
! | Gen 1
| | Length
! | Length
| | Gen 2
! | Gen 2
| | Length
! | Length
| | Angle
! | Angle
|-
|-
| | 686/675
| | 686/675
| | Keema
| | [[senga]]
| | 15/14
| | 15/14
| | 0.6977
| | 0.6977
Line 24: Line 26:
|-
|-
| | 64/63
| | 64/63
| | Archytas
| | [[Archytas]]'
| | 3/2
| | 3/2
| | 1.0508
| | 1.0508
Line 32: Line 34:
|-
|-
| | 875/864
| | 875/864
| | Keema
| | [[keema]]
| | 6/5
| | 6/5
| | 0.8879
| | 0.8879
Line 40: Line 42:
|-
|-
| | 81/80
| | 81/80
| | Didymus
| | [[Didymus]]'
| | 3/2
| | 3/2
| | 0.5451
| | 0.5451
Line 48: Line 50:
|-
|-
| | 3125/3087
| | 3125/3087
| | Gariboh
| | [[gariboh]]
| | 25/21
| | 25/21
| | 0.5928
| | 0.5928
Line 56: Line 58:
|-
|-
| | 2430/2401
| | 2430/2401
| | Nuwell
| | [[nuwell]]
| | 7/6
| | 7/6
| | 0.6247
| | 0.6247
Line 64: Line 66:
|-
|-
| | 50421/50000
| | 50421/50000
| | Trimyna
| | [[trimyna]]
| | 7/5
| | 7/5
| | 0.3158
| | 0.3158
Line 72: Line 74:
|-
|-
| | 245/243
| | 245/243
| | Sensamagic
| | [[sensamagic]]
| | 3/2
| | 3/2
| | 0.9644
| | 0.9644
Line 80: Line 82:
|-
|-
| | 126/125
| | 126/125
| | Starling
| | [[starling]]
| | 6/5
| | 6/5
| | 1.068
| | 1.068
Line 88: Line 90:
|-
|-
| | 4000/3969
| | 4000/3969
| | Octagar
| | [[octagar]]
| | 63/50
| | 63/50
| | 0.8966
| | 0.8966
Line 96: Line 98:
|-
|-
| | 1728/1715
| | 1728/1715
| | Orwellisma
| | [[orwellisma]]
| | 7/6
| | 7/6
| | 0.7526
| | 0.7526
Line 104: Line 106:
|-
|-
| | 1029/1024
| | 1029/1024
| | Gamelisma
| | [[gamelisma]]
| | 8/7
| | 8/7
| | 0.5192
| | 0.5192
Line 112: Line 114:
|-
|-
| | 225/224
| | 225/224
| | Marvel
| | [[marvel]]
| | 16/15
| | 16/15
| | 1.256
| | 1.256
Line 120: Line 122:
|-
|-
| | 3136/3125
| | 3136/3125
| | Hemimean
| | [[hemimean]]
| | 28/25
| | 28/25
| | 0.5055
| | 0.5055
Line 128: Line 130:
|-
|-
| | 5120/5103
| | 5120/5103
| | Hemifamity
| | [[hemifamity]]
| | 3/2
| | 3/2
| | 0.5670
| | 0.5670
Line 136: Line 138:
|-
|-
| | 6144/6125
| | 6144/6125
| | Porwell
| | [[porwell]]
| | 35/32
| | 35/32
| | 0.9584
| | 0.9584
Line 144: Line 146:
|-
|-
| | 65625/65536
| | 65625/65536
| | Horwell
| | [[horwell]]
| | 5/4
| | 5/4
| | 0.6213
| | 0.6213
Line 152: Line 154:
|-
|-
| | 703125/702464
| | 703125/702464
| | Meter
| | [[meter]]
| | 75/56
| | 75/56
| | 0.7305
| | 0.7305
Line 159: Line 161:
| | 97.289
| | 97.289
|-
|-
| | 420175 / 419904
| | 420175/419904
| | Wizma
| | [[wizma]]
| | 245/216
| | 245/216
| | 0.5541
| | 0.5541
Line 168: Line 170:
|-
|-
| | 2401/2400
| | 2401/2400
| | Breedsma
| | [[breedsma]]
| | 49/40
| | 49/40
| | 0.7858
| | 0.7858
Line 176: Line 178:
|-
|-
| | 4375/4374
| | 4375/4374
| | Ragisma
| | [[ragisma]]
| | 10/9
| | 10/9
| | 0.7890
| | 0.7890
Line 188: Line 190:
{| class="wikitable"
{| 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
Line 204: Line 206:
| | 104.028
| | 104.028
|-
|-
| | 225/224&441/400
| | 225/224, 441/440 <!-- was 441/400, presumed incorrect -->
| | Prodigy
| | [[prodigy]]
| | 16/15
| | 16/15
| | 0.9111
| | 0.9111
Line 212: Line 214:
| | 65.933
| | 65.933
|-
|-
| | 99/98&amp;176/175
| | 99/98, 176/175
| | Minerva
| | [[minerva]]
| | 16/15
| | 16/15
| | 0.8997
| | 0.8997
Line 220: Line 222:
| | 98.604
| | 98.604
|-
|-
| | 126/125&amp;176/175
| | 126/125, 176/175
| | Thrush
| | [[thrush]]
| | 5/4
| | 5/4
| | 0.8576
| | 0.8576
Line 228: Line 230:
| | 74.624
| | 74.624
|-
|-
| | 56/55&amp;100/99
| | 56/55, 100/99
| | Thrasher
| | [[thrasher]]
| | 6/5
| | 6/5
| | 0.9089
| | 0.9089
Line 236: Line 238:
| | 98.845
| | 98.845
|}
|}
[[Category:lattice]]
 
[[Category:overview]]
[[Category:Lattice]]
[[Category:table]]
[[Category:Lists of temperaments]]
[[Category:temperaments]]
[[Category:Tables]]
[[Category:todo:link]]

Revision as of 18:21, 16 December 2024

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 three

Seven limit

Comma Name Gen 1 Length Gen 2 Length Angle
686/675 senga 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 keema 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 orwellisma 7/6 0.7526 3/2 1.3931 97.508
1029/1024 gamelisma 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 hemifamity 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 meter 75/56 0.7305 375/224 0.7675 97.289
420175/419904 wizma 245/216 0.5541 324/245 0.9731 79.663
2401/2400 breedsma 49/40 0.7858 8/7 1.1241 107.367
4375/4374 ragisma 10/9 0.7890 6/5 0.9208 105.299

Eleven limit

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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