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{{Infobox ET}}
{{Infobox ET}}
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.
{{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]]. 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 21: 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 28: 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 35: 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 42: 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 49: 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 59: Line 67:


{| 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|###]] <!-- 3-digit number -->
[[Category:Stearnscape]]

Latest revision as of 22:57, 20 February 2025

← 353edo 354edo 355edo →
Prime factorization 2 × 3 × 59
Step size 3.38983 ¢ 
Fifth 207\354 (701.695 ¢) (→ 69\118)
Semitones (A1:m2) 33:27 (111.9 ¢ : 91.53 ¢)
Consistency limit 9
Distinct consistency limit 9

354 equal divisions of the octave (abbreviated 354edo or 354ed2), also called 354-tone equal temperament (354tet) or 354 equal temperament (354et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 354 equal parts of about 3.39 ¢ each. Each step represents a frequency ratio of 21/354, or the 354th root of 2.

Theory

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 harmonics 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 72 & 282 temperament, and 13- and 17-limit terminator, the 171 & 183 temperament.

Prime harmonics

Approximation of prime harmonics in 354edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -0.26 +0.13 +0.67 +1.22 +0.15 +0.13 +0.79 -1.16 +0.93 +0.73
Relative (%) +0.0 -7.7 +3.7 +19.6 +36.1 +4.4 +3.8 +23.4 -34.1 +27.5 +21.5
Steps
(reduced) 		354
(0) 		561
(207) 		822
(114) 		994
(286) 		1225
(163) 		1310
(248) 		1447
(31) 		1504
(88) 		1601
(185) 		1720
(304) 		1754
(338)

Subsets and supersets

Since 354 factors into 2 × 3 × 59, 354edo has subset edos 2, 3, 6, 59, 118, and 177.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3.5.7 32805/32768, 118098/117649, 250047/250000 [⟨354 561 822 994]] −0.0319 0.1432 4.23
2.3.5.7.11 540/539, 4000/3993, 32805/32768, 137781/137500 [⟨354 561 822 994 1225]] −0.0963 0.1817 5.36
2.3.5.7.11.13 540/539, 729/728, 1575/1573, 4096/4095, 31250/31213 [⟨354 561 822 994 1225 1310]] −0.0871 0.1671 4.93
2.3.5.7.11.13.17 540/539, 729/728, 936/935, 1156/1155, 1575/1573, 4096/4095 [⟨354 561 822 994 1225 1310 1447]] −0.0791 0.1559 4.60
2.3.5.7.11.13.17.19 540/539, 729/728, 936/935, 969/968, 1156/1155, 1445/1444, 1521/1520 [⟨354 561 822 994 1225 1310 1447 1504]] −0.0926 0.1509 4.43

Rank-2 temperaments

Note: 5-limit temperaments supported by 118et are not included.

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
2 128\354
(49\354) 		433.90
(166.10) 		9/7
(11/10) 		Pogo
3 147\354
(29\354) 		498.31
(98.31) 		4/3
(18/17) 		Term / terminator
6 64\354
(5\354) 		216.95
(16.95) 		17/15
(245/243) 		Stearnscape
6 147\354
(29\354) 		498.31
(98.31) 		4/3
(18/17) 		Semiterm
118 167\354
(2\354) 		566.101
(6.78) 		165/119
(?) 		Oganesson

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

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