354edo: Difference between revisions

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== 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]], but the approximation to higher [[harmonic]]s are much improved.   
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, the equal temperament 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.   
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.  
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 ===
Line 30: Line 30:
| 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
| 4.23
| 4.23
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| 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
| 5.36
| 5.36
Line 44: Line 44:
| 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
| 4.93
| 4.93
Line 51: Line 51:
| 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
| 4.60
| 4.60
Line 58: Line 58:
| 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
| 4.43
| 4.43
Line 105: Line 105:
| [[Oganesson]]
| [[Oganesson]]
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct


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

Revision as of 14:10, 16 January 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

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, 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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