476edo: 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]] (%)
|-
|-
| 2.9
| 2.9
Line 44: Line 36:
| 0.1586
| 0.1586
| 6.29
| 6.29
|}
{{comma basis end}}


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| 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
|-
|-
| 1
| 1
Line 72: Line 58:
| 4/3<br>(105/104)
| 4/3<br>(105/104)
| [[Oquatonic]] (5-limit)
| [[Oquatonic]] (5-limit)
|}
{{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}}

Revision as of 03:04, 16 November 2024

← 475edo 476edo 477edo →
Prime factorization 22 × 7 × 17
Step size 2.52101 ¢ 
Fifth 278\476 (700.84 ¢) (→ 139\238)
Semitones (A1:m2) 42:38 (105.9 ¢ : 95.8 ¢)
Dual sharp fifth 279\476 (703.361 ¢)
Dual flat fifth 278\476 (700.84 ¢) (→ 139\238)
Dual major 2nd 81\476 (204.202 ¢)
Consistency limit 7
Distinct consistency limit 7

Template:EDO intro

Theory

476edo is consistent to the 7-odd-limit, but the harmonic 3 is about halfway its steps, while its 5 and 7 are both tuned flat. To start with, consider the 2.3.5.7 patent val, as well as 2.9.15.21 and 2.9.5.7 subgroups.

Using the patent val, the equal temperament tempers out 2401/2400 and 19683/19600 in the 7-limit, supporting harry. The 11-limit 476e val tempers out 3025/3024 and 41503/41472, whereas the patent val tempers out 243/242, 441/440, 540/539, 4000/3993, 8019/8000, and 9801/9800, supporting 11-limit harry.

Odd harmonics

Approximation of odd harmonics in 476edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -1.11 -0.60 -0.76 +0.29 +0.78 -1.03 +0.81 +0.93 -0.03 +0.65 -0.54
Relative (%) -44.2 -23.8 -30.1 +11.6 +31.1 -40.9 +32.0 +36.8 -1.3 +25.7 -21.5
Steps
(reduced) 		754
(278) 		1105
(153) 		1336
(384) 		1509
(81) 		1647
(219) 		1761
(333) 		1860
(432) 		1946
(42) 		2022
(118) 		2091
(187) 		2153
(249)

Subsets and supersets

476 factors into 22 × 7 × 17, with subset edos 2, 4, 7, 14, 17, 28, 34, 68, 119, and 238. 952edo, which doubles it, gives a good correction to the harmonic 3, but unfortunately it is inconsistent in the 5-odd-limit.

Regular temperament properties

Template:Comma basis begin |- | 2.9 | [1509 -476⟩ | [⟨476 1509]] | -0.0460 | 0.0460 | 1.82 |- | 2.9.5 | [33 -17 9⟩, [-65 0 28⟩ | [⟨476 1509 1105]] | +0.0554 | 0.1482 | 5.88 |- | 2.9.5.7 | 703125/702464, 4802000/4782969, [25 3 -3 8⟩ | [⟨476 1509 1105 1336]] | +0.1091 | 0.1586 | 6.29 Template:Comma basis end

Rank-2 temperaments

Template:Rank-2 begin |- | 1 | 205\476 | 516.81 | 27/20 | Larry (476) |- | 2 | 205\476
(33\476) | 516.81
(83.19) | 27/20
(21/20) | Harry (11-limit, 476) |- | 28 | 197\476
(6\476) | 496.64
(15.13) | 4/3
(105/104) | Oquatonic (5-limit) Template:Rank-2 end Template:Orf

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