354edo: Difference between revisions

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== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{{comma basis begin}}
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
|-
| 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
| &minus;0.0319
| 0.1432
| 0.1432
| 4.23
| 4.23
Line 36: Line 28:
| 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
| &minus;0.0963
| 0.1817
| 0.1817
| 5.36
| 5.36
Line 43: Line 35:
| 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
| &minus;0.0871
| 0.1671
| 0.1671
| 4.93
| 4.93
Line 50: Line 42:
| 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
| &minus;0.0791
| 0.1559
| 0.1559
| 4.60
| 4.60
Line 57: Line 49:
| 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
| &minus;0.0926
| 0.1509
| 0.1509
| 4.43
| 4.43
|}
{{comma basis end}}


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
Note: 5-limit temperaments supported by [[118edo|118et]] are not included.  
Note: 5-limit temperaments supported by [[118edo|118et]] are not included.  


{| class="wikitable center-all left-5"
{{rank-2 begin}}
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>Ratio*
! 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>(18/17)
| 4/3<br />(18/17)
| [[Term (temperament)|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)
| 17/15<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>(18/17)
| 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]]
|}
{{rank-2 end}}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
{{orf}}


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

Revision as of 04:40, 16 November 2024

← 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, 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.

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

Template:Comma basis begin |- | 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 Template:Comma basis end

Rank-2 temperaments

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

Template:Rank-2 begin |- | 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 Template:Rank-2 end Template:Orf

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