400edo: Difference between revisions

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== Theory ==
== Theory ==
400edo is consistent in the [[21-odd-limit]]. It tempers out the unidecma, {{monzo| -7 22 -12 }}, and the qintosec comma, {{monzo| 47 -15 -10 }}, in the 5-limit; [[2401/2400]], 1959552/1953125, and 14348907/14336000 in the 7-limit; 5632/5625, [[9801/9800]], 117649/117612, and [[131072/130977]] in the 11-limit; [[676/675]], [[1001/1000]], [[1716/1715]], [[2080/2079]], [[4096/4095]], [[4225/4224]] and 39366/39325 in the 13-limit, supporting the [[decoid]] temperament and the [[quinmite]] temperament. It tempers out 4914/4913 and [[24576/24565]] in the 17-limit, and [[1729/1728]] with 93347/93312 in the 19-limit.  
400edo is [[consistent]] in the [[21-odd-limit]]. It tempers out the unidecma, {{monzo| -7 22 -12 }}, and the qintosec comma, {{monzo| 47 -15 -10 }}, in the 5-limit; [[2401/2400]], 1959552/1953125, and 14348907/14336000 in the 7-limit; 5632/5625, [[9801/9800]], 117649/117612, and [[131072/130977]] in the 11-limit; [[676/675]], [[1001/1000]], [[1716/1715]], [[2080/2079]], [[4096/4095]], [[4225/4224]] and 39366/39325 in the 13-limit, supporting the [[decoid]] temperament and the [[quinmite]] temperament. It tempers out [[936/935]], [[1156/1155]], 2058/2057, [[2601/2600]], 4914/4913 and [[24576/24565]] in the 17-limit, and 969/968, [[1216/1215]], [[1521/1520]], and [[1729/1728]] in the 19-limit.  


400edo doubles [[200edo]], which holds a record for the best 3/2 fifth approximation. 400 is also the number of years in the Gregorian calendar's leap cycle. 400edo supports the Sym454 calendar scale with 231\400 as the generator, which can be treated as 5/12 syntonic comma meantone, which is the first meantone in the continued fraction that offers good precision. Other items like 1/3 and 2/5 eventually become inconsistent with the edo.  
400edo doubles [[200edo]], which holds a record for the best 3/2 fifth approximation.  


The leap week scale offers an interest in that 1/7th of its generator, 33\400, is associated to [[18/17]], making it an interpretation of [[18/17s equal temperament]]. Since it tempers out the 93347/93312, a stack of three 18/17s is equated with 19/16.  
400 is also the number of years in the Gregorian calendar's leap cycle. 400edo supports the Sym454 calendar scale with 231\400 as the generator, which is close to 5/12 syntonic comma meantone. The leap week scale offers an interest in that 1/7th of its generator, 33\400, is associated to [[18/17]], making it an approximation of [[18/17 equal-step tuning]]. Since it tempers out the 93347/93312, a stack of three 18/17's is equated with 19/16.  


=== Prime harmonics ===
=== Prime harmonics ===
{{Primes in edo|400|columns=15}}
{{Primes in edo|400|columns=15}}


=== Table of intervals ===
== Selected intervals ==
{| class="wikitable"
{| class="wikitable center-1"
|+
|+
!Step
! Step
!Name
! Eliora's Naming System
!Associated ratio
! Associated ratio
!Notes
|-
|-
|0
| 0
|unison
| unison
|1/1 exact
| 1/1
|
|-
|-
|28
| 28
|5/12-meantone semitone
| 5/12-meantone semitone
|6561/6250
| 6561/6250
|
|-
|-
|33
| 33
|small septendecimal semitone
| small septendecimal semitone
|[[18/17]]
| [[18/17]]
|
|-
|-
|35
| 35
|septendecimal semitone
| septendecimal semitone
|[[17/16]]
| [[17/16]]
|
|-
|-
|37
| 37
|diatonic semitone
| diatonic semitone
|[[16/15]]
| [[16/15]]
|
|-
|-
|99
| 99
|undevicesimal minor third
| undevicesimal minor third
|[[19/16]]
| [[19/16]]
|
|-
|-
|100
| 100
|symmetric minor third
| symmetric minor third
|
|  
|
|-
|-
|200
| 200
|symmetric tritone
| symmetric tritone
|[[99/70]], [[140/99]]
| [[99/70]], [[140/99]]
|
|-
|-
|231
| 231
|Gregorian leap week fifth
| Gregorian leap week fifth
|118/79, twelfth root of 800000/6561
| 118/79
|
|-
|-
|234
| 234
|perfect fifth
| perfect fifth
|[[3/2]]
| [[3/2]]
|
|-
|-
|323
| 323
|harmonic seventh
| harmonic seventh
|[[7/4]]
| [[7/4]]
|
|-
|-
|372
| 372
|5/12-meantone seventh
| 5/12-meantone seventh
|12500/6561
| 12500/6561
|
|-
|-
|400
| 400
|octave
| octave
|2/1 exact
| 2/1
|
|}
|}


Revision as of 20:00, 10 January 2022

The 400 equal divisions of the octave (400edo) is the equal division of the octave into 400 parts of exact 3 cents each.

Theory

400edo is consistent in the 21-odd-limit. It tempers out the unidecma, [-7 22 -12⟩, and the qintosec comma, [47 -15 -10⟩, in the 5-limit; 2401/2400, 1959552/1953125, and 14348907/14336000 in the 7-limit; 5632/5625, 9801/9800, 117649/117612, and 131072/130977 in the 11-limit; 676/675, 1001/1000, 1716/1715, 2080/2079, 4096/4095, 4225/4224 and 39366/39325 in the 13-limit, supporting the decoid temperament and the quinmite temperament. It tempers out 936/935, 1156/1155, 2058/2057, 2601/2600, 4914/4913 and 24576/24565 in the 17-limit, and 969/968, 1216/1215, 1521/1520, and 1729/1728 in the 19-limit.

400edo doubles 200edo, which holds a record for the best 3/2 fifth approximation.

400 is also the number of years in the Gregorian calendar's leap cycle. 400edo supports the Sym454 calendar scale with 231\400 as the generator, which is close to 5/12 syntonic comma meantone. The leap week scale offers an interest in that 1/7th of its generator, 33\400, is associated to 18/17, making it an approximation of 18/17 equal-step tuning. Since it tempers out the 93347/93312, a stack of three 18/17's is equated with 19/16.

Prime harmonics

Script error: No such module "primes_in_edo".

Selected intervals

Step Eliora's Naming System Associated ratio
0 unison 1/1
28 5/12-meantone semitone 6561/6250
33 small septendecimal semitone 18/17
35 septendecimal semitone 17/16
37 diatonic semitone 16/15
99 undevicesimal minor third 19/16
100 symmetric minor third
200 symmetric tritone 99/70, 140/99
231 Gregorian leap week fifth 118/79
234 perfect fifth 3/2
323 harmonic seventh 7/4
372 5/12-meantone seventh 12500/6561
400 octave 2/1

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3.5 [-7 22 -12⟩, [47 -15 -10⟩ [⟨400 634 929]] -0.1080 0.1331 4.44
2.3.5.7 2401/2400, 1959552/1953125, 14348907/14336000 [⟨400 634 929 1123]] -0.0965 0.1170 3.90
2.3.5.7.11 2401/2400, 5632/5625, 9801/9800, 46656/46585 [⟨400 634 929 1123 1384]] -0.1166 0.1121 3.74
2.3.5.7.11.13 676/675, 1001/1000, 1716/1715, 4096/4095, 39366/39325 [⟨400 634 929 1123 1384 1480]] -0.0734 0.1407 4.69
2.3.5.7.11.13.17 676/675, 936/935, 1001/1000, 1156/1155, 1716/1715, 4096/4095 [⟨400 634 929 1123 1384 1480 1635]] -0.0645 0.1321 4.40
2.3.5.7.11.13.17.19 676/675, 936/935, 969/968, 1001/1000, 1156/1155, 1216/1215, 1716/1715 [⟨400 634 929 1123 1384 1480 1635 1699]] -0.0413 0.1380 4.60

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per octave 		Generator
(reduced) 		Cents
(reduced) 		Associated
ratio 		Temperaments
1 83\270 249.00 [-26 18 -1⟩ Monzismic
1 101\400 303.00 25/21 Quinmite
1 153\400 459.00 125/96 Majvam
2 61\400 183.00 10/9 Unidecmic
5 123\400
(37\400) 		369.00
(111.00) 		10125/8192
(16/15) 		Qintosec (5-limit)
10 83\400
(3\400) 		249.00
(9.00) 		15/13
(176/175) 		Decoid

Scales

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