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354edo: Difference between revisions

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→Regular temperament properties: + trivish. Also convert the table of rank-2 temps to minimal form since we're already reducing it to the first semi-octave
 
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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|354}}
{{ED intro}}


== Theory ==
== Theory ==
354edo is [[enfactoring|enfactored]] in the 5-limit, with the same tuning as [[118edo]], defined by [[tempering out]] the [[schisma]] and the [[parakleisma]], but the approximation to higher [[harmonic]]s are much improved.   
354edo is [[enfactored]] in the 5-limit, with the same tuning as [[118edo]], defined by [[tempering out]] the [[schisma]] and the [[parakleisma]], but the approximation to higher [[harmonic]]s are much improved.   


In the 7-limit, the equal temperament tempers out 118098/117649 (stearnsma), 250047/250000 ([[landscape comma]]), and 703125/702464 ([[meter]]); in the 11-limit, [[540/539]], and [[4000/3993]]; in the 13-limit, [[729/728]], [[1575/1573]], [[1716/1715]], [[2080/2079]], [[4096/4095]], and [[4225/4224]]. In the 13-limit, particularly 2.3.5.13 subgroup, one should consider [[peithoian]], as it preserves 5-limit tuning of 118edo while also improving the first harmonic 118edo tunes inconsistently.   
In the 7-limit, it tempers out 118098/117649 (stearnsma), 250047/250000 ([[landscape comma]]), and 703125/702464 ([[meter]]); in the 11-limit, [[540/539]], and [[4000/3993]]; in the 13-limit, [[729/728]], [[1575/1573]], [[1716/1715]], [[2080/2079]], [[4096/4095]], and [[4225/4224]]. In the 13-limit, particularly 2.3.5.13 subgroup, one should consider [[peithoian]], as it preserves 5-limit tuning of 118edo while also improving the first harmonic 118edo tunes inconsistently.   


354edo provides the [[optimal patent val]] for [[stearnscape]], the 72 & 282 temperament, and 13- and 17-limit [[terminator]], the 171 & 183 temperament.  
354edo provides the [[optimal patent val]] for [[stearnscape]], the {{nowrap|72 & 282}} temperament, and 13- and 17-limit [[terminator]], the {{nowrap|171 & 183}} temperament.  


=== Prime harmonics ===
=== Prime harmonics ===
Line 17: Line 17:
== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning Error
! colspan="2" | Tuning error
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
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| 2.3.5.7
| 2.3.5.7
| 32805/32768, 118098/117649, 250047/250000
| 32805/32768, 118098/117649, 250047/250000
| {{mapping| 354 561 822 994 }}
| {{Mapping| 354 561 822 994 }}
| -0.0319
| −0.0319
| 0.1432
| 0.1432
| 4.23
| 4.23
Line 35: Line 36:
| 2.3.5.7.11
| 2.3.5.7.11
| 540/539, 4000/3993, 32805/32768, 137781/137500
| 540/539, 4000/3993, 32805/32768, 137781/137500
| {{mapping| 354 561 822 994 1225 }}
| {{Mapping| 354 561 822 994 1225 }}
| -0.0963
| −0.0963
| 0.1817
| 0.1817
| 5.36
| 5.36
Line 42: Line 43:
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 540/539, 729/728, 1575/1573, 4096/4095, 31250/31213
| 540/539, 729/728, 1575/1573, 4096/4095, 31250/31213
| {{mapping| 354 561 822 994 1225 1310 }}
| {{Mapping| 354 561 822 994 1225 1310 }}
| -0.0871
| −0.0871
| 0.1671
| 0.1671
| 4.93
| 4.93
Line 49: Line 50:
| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 540/539, 729/728, 936/935, 1156/1155, 1575/1573, 4096/4095
| 540/539, 729/728, 936/935, 1156/1155, 1575/1573, 4096/4095
| {{mapping| 354 561 822 994 1225 1310 1447 }}
| {{Mapping| 354 561 822 994 1225 1310 1447 }}
| -0.0791
| −0.0791
| 0.1559
| 0.1559
| 4.60
| 4.60
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| 2.3.5.7.11.13.17.19
| 2.3.5.7.11.13.17.19
| 540/539, 729/728, 936/935, 969/968, 1156/1155, 1445/1444, 1521/1520
| 540/539, 729/728, 936/935, 969/968, 1156/1155, 1445/1444, 1521/1520
| {{mapping| 354 561 822 994 1225 1310 1447 1504 }}
| {{Mapping| 354 561 822 994 1225 1310 1447 1504 }}
| -0.0926
| −0.0926
| 0.1509
| 0.1509
| 4.43
| 4.43
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{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Periods<br>per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br>Ratio*
! Associated<br>ratio*
! Temperaments
! Temperaments
|-
|-
| 2
| 2
| 128\354<br>(49\354)
| 49\354
| 433.90<br>(166.10)
| 166.10
| 9/7<br>(11/10)
| 11/10
| [[Pogo]]
| [[Pogo]]
|-
|-
| 3
| 3
| 147\354<br>(29\354)
| 29\354
| 498.31<br>(98.31)
| 98.31
| 4/3<br>(18/17)
| 18/17
| [[Term (temperament)|Term]] / terminator
| [[Term (temperament)|Term]] / terminator
|-
|-
| 6
| 6
| 64\354<br>(5\354)
| 5\354
| 216.95<br>(16.95)
| 16.95
| 17/15<br>(245/243)
| 245/243
| [[Stearnscape]]
| [[Stearnscape]]
|-
|-
| 6
| 6
| 147\354<br>(29\354)
| 21\354
| 498.31<br>(98.31)
| 71.19
| 4/3<br>(18/17)
| 25/24
| [[Trivish]]
|-
| 6
| 29\354
| 98.31
| 18/17
| [[Semiterm]]
| [[Semiterm]]
|-
|-
| 118
| 118
| 167\354<br>(2\354)
| 2\354
| 566.101<br>(6.78)
| 6.78
| 165/119<br>(?)
| ?
| [[Oganesson]]
| [[Oganesson]]
|}
|}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
<nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]]


[[Category:Stearnscape]]
[[Category:Stearnscape]]
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