354edo: Difference between revisions

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m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"
→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
 
Line 21: Line 21:
! 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 29: Line 29:
| 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
Line 36: 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
Line 43: 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
Line 50: 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
Line 57: 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
| {{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
Line 69: Line 69:
|+ style="font-size: 105%;" | 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 />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
<nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]]


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

Latest revision as of 15:21, 20 June 2026

← 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 49\354 166.10 11/10 Pogo
3 29\354 98.31 18/17 Term / terminator
6 5\354 16.95 245/243 Stearnscape
6 21\354 71.19 25/24 Trivish
6 29\354 98.31 18/17 Semiterm
118 2\354 6.78 ? Oganesson

* In minimal-generator form

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