Xenharmonic Wiki

354edo: Difference between revisions

← Older edit
VisualWikitext
Cmloegcmluin (talk | contribs)
autopatrolled, emailconfirmed
5,171 edits
I've asked for the clutter of pages of different forms for the words defactor and enfactor to be deleted, so now pages that linked to them need to be updated to use the remaining working link
→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
 
(15 intermediate revisions by 5 users not shown)
Line 1: Line 1:
{{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]]
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
Line 55: Line 57:
| 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)
| 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>(200/189)
| 18/17
| [[Term]] / terminator
| [[Term (temperament)|Term]] / terminator
|-
|-
| 6
| 6
| 64\354<br>(5\354)
| 5\354
| 216.95<br>(16.95)
| 16.95
| 567/500<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>(200/189)
| 25/24
| [[Trivish]]
|-
| 6
| 29\354
| 98.31
| 18/17
| [[Semiterm]]
| [[Semiterm]]
|-
| 118
| 2\354
| 6.78
| ?
| [[Oganesson]]
|}
|}
<nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]]


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