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{{Infobox ET
{{Infobox ET}}
| Prime factorization = 2 × 3 × 59
{{ED intro}}
| Step size = 3.38983¢
| Fifth = 207\354 (701.69¢) (→ [[118edo|69\118]])
| Semitones = 33:27 (111.86¢ : 91.53¢)
| Consistency = 9
}}
The '''354 equal divisions of the octave''' ('''354edo'''), or the '''354(-tone) equal temperament''' ('''354tet''', '''354et''') when viewed from a [[regular temperament]] perspective, is the [[EDO|equal division of the octave]] into 354 parts of about 3.39 [[cent]]s each.


== 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]]. In the 7-limit, it tempers out 118098/117649 (stearnsma), 250047/250000 ([[landscape comma|landscape]]), and 703125/702464 ([[meter comma|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]]. It provides the [[optimal patent val]] for [[stearnscape]].  
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, 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 {{nowrap|72 & 282}} temperament, and 13- and 17-limit [[terminator]], the {{nowrap|171 & 183}} temperament.  


=== Prime harmonics ===
=== Prime harmonics ===
{{Primes in edo|354}}
{{Harmonics in equal|354}}
 
=== Subsets and supersets ===
Since 354 factors into {{factorization|354}}, 354edo has subset edos {{EDOs| 2, 3, 6, 59, 118, and 177 }}.


== 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]]
! 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
|-
|-
Line 27: Line 29:
| 2.3.5.7
| 2.3.5.7
| 32805/32768, 118098/117649, 250047/250000
| 32805/32768, 118098/117649, 250047/250000
| [{{val| 354 561 822 994 }}]
| {{mapping| 354 561 822 994 }}
| -0.0319
| −0.0319
| 0.1432
| 0.1432
| 4.23
| 4.23
Line 34: 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
| [{{val| 354 561 822 994 1225 }}]
| {{mapping| 354 561 822 994 1225 }}
| -0.0963
| −0.0963
| 0.1817
| 0.1817
| 5.36
| 5.36
Line 41: 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
| [{{val| 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 48: 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
| [{{val| 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
| [{{val| 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
Line 62: Line 64:


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
Note: 5-limit temperaments supported by [[118edo|118et]] are not included.
{| 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 octave
|-
! Generator<br>(reduced)
! Periods<br />per 8ve
! Cents<br>(reduced)
! Generator*
! Associated<br>ratio
! Cents*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
| 2
| 2
| 128\354<br>(49\354)
| 128\354<br />(49\354)
| 433.90<br>(166.10)
| 433.90<br />(166.10)
| 9/7<br>(11/10)
| 9/7<br />(11/10)
| [[Pogo]]
| [[Pogo]]
|-
|-
| 3
| 3
| 147\354<br>(29\354)
| 147\354<br />(29\354)
| 498.31<br>(98.31)
| 498.31<br />(98.31)
| 4/3<br>(200/189)
| 4/3<br />(18/17)
| [[Term]] / terminator
| [[Term (temperament)|Term]] / terminator
|-
|-
| 6
| 6
| 64\354<br>(5\354)
| 64\354<br />(5\354)
| 216.95<br>(16.95)
| 216.95<br />(16.95)
| 567/500<br>(245/243)
| 17/15<br />(245/243)
| [[Stearnscape]]
| [[Stearnscape]]
|-
|-
| 6
| 6
| 147\354<br>(29\354)
| 147\354<br />(29\354)
| 498.31<br>(98.31)
| 498.31<br />(98.31)
| 4/3<br>(200/189)
| 4/3<br />(18/17)
| [[Semiterm]]
| [[Semiterm]]
|-
| 118
| 167\354<br />(2\354)
| 566.101<br />(6.78)
| 165/119<br />(?)
| [[Oganesson]]
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct


[[Category:Equal divisions of the octave]]
[[Category:Stearnscape]]
Retrieved from "https://en.xen.wiki/w/354edo"