354edo: Difference between revisions

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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.
{{EDO intro|354}}


== 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 [[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]]); 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]].  


=== Prime harmonics ===
=== Prime harmonics ===
{{Primes in edo|354}}
{{Harmonics in equal|354}}
 
=== Subsets and supersets ===
Since 354 factors into 2 × 3 × 59, 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|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]] (¢)
Line 61: Line 64:
|+Table of rank-2 temperaments by generator
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Periods<br>per 8ve
! Generator<br>(reduced)
! Generator<br>(Reduced)
! Cents<br>(reduced)
! Cents<br>(Reduced)
! Associated<br>ratio
! Associated<br>Ratio
! Temperaments
! Temperaments
|-
|-
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| 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]] / terminator
|-
|-
Line 87: Line 90:
| 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
| 118
|167\354<br>(2\354)
| 167\354<br>(2\354)
|566.101<br>(6.78)
| 566.101<br>(6.78)
|165/119<br>(?)
| 165/119<br>(?)
|[[Oganesson]]
| [[Oganesson]]
|}
|}
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->

Revision as of 05:57, 9 June 2023

← 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

Template:EDO intro

Theory

354edo is 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), 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. It provides the optimal patent val for stearnscape.

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
(Reduced) 		Cents
(Reduced) 		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) 		567/500
(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
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