51edo: Difference between revisions

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<b>51-EDO</b> divides the [[Octave|octave]] into 51 equal parts of 23.529 [[cent|cent]]s each, which is about the size of the [http://en.wikipedia.org/wiki/Pythagorean_comma Pythagorean comma] (even though this comma itself is mapped to a different interval). It tempers out [[250/243|250/243]] in the [[5-limit|5-limit]], [[225/224|225/224]] and [[2401/2400|2401/2400]] in the [[7-limit|7-limit]], and [[55/54|55/54]] and [[100/99|100/99]] in the [[11-limit|11-limit]]. It is the [[Optimal_patent_val|optimal patent val]] for [[Porcupine_rank_three_family|sonic]], the rank three temperament tempering out 250/243, 55/54 and 100/99, and also for the rank four temperament tempering out 55/54. It provides an alternative tuning to [[22edo|22edo]] for [[Porcupine_family|porcupine temperament]], with a nice fifth but a rather flat major third, and the optimal patent val for 7 and 11-limit [[Porcupine_family#Porky|porky temperament]], which is sonic plus 225/224.
{{Infobox ET
| Prime factorization = 3 × 17
| Step size = 23.52941¢
| Fifth = 30\51 ≈ 706¢
| Major 2nd = 9\51 ≈ 212¢
| Minor 2nd = 3\51 ≈ 71¢
| Augmented 1sn = 6\51 ≈ 141¢
}}
 
== Theory ==
 
{{primes in edo|51}}
'''51-EDO''' divides the [[Octave|octave]] into 51 equal parts of 23.529 [[cent|cent]]s each, which is about the size of the [http://en.wikipedia.org/wiki/Pythagorean_comma Pythagorean comma] (even though this comma itself is mapped to a different interval). It tempers out [[250/243|250/243]] in the [[5-limit|5-limit]], [[225/224|225/224]] and [[2401/2400|2401/2400]] in the [[7-limit|7-limit]], and [[55/54|55/54]] and [[100/99|100/99]] in the [[11-limit|11-limit]]. It is the [[Optimal_patent_val|optimal patent val]] for [[Porcupine_rank_three_family|sonic]], the rank three temperament tempering out 250/243, 55/54 and 100/99, and also for the rank four temperament tempering out 55/54. It provides an alternative tuning to [[22edo|22edo]] for [[Porcupine_family|porcupine temperament]], with a nice fifth but a rather flat major third, and the optimal patent val for 7 and 11-limit [[Porcupine_family#Porky|porky temperament]], which is sonic plus 225/224.
 
== Intervals ==
 
{| class="wikitable center-all right-2 left-3"
|-
! Degrees
! [[Cents|Cents]]
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0.000
| Perfect 1sn
| P1
| D
|-
| 1
| 23.529
| Up 1sn
| ^1
| ^D
|-
| 2
| 47.059
| Downminor 2nd
| vm2
| vEb
|-
| 3
| 70.588
| Minor 2nd
| m2
| Eb
|-
| 4
| 94.118
| Upminor 2nd
| ^m2
| ^Eb
|-
| 5
| 117.647
| Downmid 2nd
| v~2
| ^^Eb
|-
| 6
| 141.176
| Mid 2nd
| ~2
| vvvE, ^^^Eb
|-
| 7
| 164.706
| Upmid 2nd
| ^~2
| vvE
|-
| 8
| 188.235
| Downmajor 2nd
| vM2
| vE
|-
| 9
| 211.765
| Major 2nd
| M2
| E
|-
| 10
| 235.294
| Upmajor 2nd
| ^M2
| ^E
|-
| 11
| 258.824
| Downminor 3rd
| vm3
| vF
|-
| 12
| 282.353
| Minor 3rd
| m3
| F
|-
| 13
| 305.882
| Upminor 3rd
| ^m3
| ^F
|-
| 14
| 329.412
| Downmid 3rd
| v~3
| ^^F
|-
| 15
| 352.941
| Mid 3rd
| ~3
| ^^^F, vvvF#
|-
| 16
| 376.471
| Upmid 3rd
| ^~3
| vvF#
|-
| 17
| 400.000
| Downmajor 3rd
| vM3
| vF#
|-
| 18
| 423.529
| Major 3rd
| M3
| F#
|-
| 19
| 447.509
| Upmajor 3rd
| ^M3
| ^F#
|-
| 20
| 470.588
| Down 4th
| v4
| vG
|-
| 21
| 494.118
| Perfect 4th
| P4
| G
|-
| 22
| 517.647
| Up 4th
| ^1
| ^G
|-
| 23
| 541.176
| Downdim 5th
| vd5
| vAb
|-
| 24
| 564.706
| Dim 5th
| d5
| Ab
|-
| 25
| 588.235
| Updim 5th
| ^d5
| ^Ab
|-
| 26
| 611.765
| Downaug 4th
| vA4
| vG#
|-
| 27
| 635.294
| Aug 4th
| A4
| G#
|-
| 28
| 658.824
| Upaug 4th
| ^A4
| ^G#
|-
| 29
| 682.353
| Down 5th
| v5
| vA
|-
| 30
| 705.882
| Perfect 5th
| P5
| A
|-
| 31
| 729.412
| Up 5th
| ^5
| ^A
|-
| 32
| 752.941
| Downminor 6th
| vm6
| vBb
|-
| 33
| 776.471
| Minor 6th
| m6
| Bb
|-
| 34
| 800.000
| Upminor 6th
| ^m6
| ^Bb
|-
| 35
| 823.529
| Downmid 6th
| v~6
| ^^Bb
|-
| 36
| 847.059
| Mid 6th
| ~6
| vvvB, ^^^Bb
|-
| 37
| 870.588
| Upmid 6th
| ^~6
| vvB
|-
| 38
| 894.118
| Downmajor 6th
| vM6
| vB
|-
| 39
| 917.647
| Major 6th
| M6
| B
|-
| 40
| 941.176
| Upmajor 6th
| ^M6
| ^B
|-
| 41
| 964.706
| Downminor 7th
| vm7
| vC
|-
| 42
| 988.235
| Minor 7th
| m7
| C
|-
| 43
| 1011.765
| Upminor 7th
| ^m7
| ^C
|-
| 44
| 1035.294
| Downmid 7th
| v~7
| ^^C
|-
| 45
| 1058.824
| Mid 7th
| ~7
| ^^^C, vvvC#
|-
| 46
| 1082.353
| Upmid 7th
| ^~7
| vvC#
|-
| 47
| 1105.882
| Downmajor 7th
| vM7
| vC#
|-
| 48
| 1129.412
| Major 7th
| M7
| C#
|-
| 49
| 1152.941
| Upmajor 7th
| ^M7
| ^C#
|-
| 50
| 1176.471
| Down 8ve
| v8
| vD
|-
| 51
| 1200.000
| Perfect 8ve
| P8
| D
|}
 
[[Category:51edo]]
[[Category:51edo]]
[[Category:Equal divisions of the octave]]
[[Category:Equal divisions of the octave]]
[[Category:theory]]
[[Category:theory]]

Revision as of 15:50, 8 June 2021

← 50edo 51edo 52edo →
Prime factorization 3 × 17
Step size 23.5294 ¢ 
Fifth 30\51 (705.882 ¢) (→ 10\17)
Semitones (A1:m2) 6:3 (141.2 ¢ : 70.59 ¢)
Consistency limit 3
Distinct consistency limit 3

Theory

Script error: No such module "primes_in_edo". 51-EDO divides the octave into 51 equal parts of 23.529 cents each, which is about the size of the Pythagorean comma (even though this comma itself is mapped to a different interval). It tempers out 250/243 in the 5-limit, 225/224 and 2401/2400 in the 7-limit, and 55/54 and 100/99 in the 11-limit. It is the optimal patent val for sonic, the rank three temperament tempering out 250/243, 55/54 and 100/99, and also for the rank four temperament tempering out 55/54. It provides an alternative tuning to 22edo for porcupine temperament, with a nice fifth but a rather flat major third, and the optimal patent val for 7 and 11-limit porky temperament, which is sonic plus 225/224.

Intervals

Degrees Cents Ups and Downs Notation
0 0.000 Perfect 1sn P1 D
1 23.529 Up 1sn ^1 ^D
2 47.059 Downminor 2nd vm2 vEb
3 70.588 Minor 2nd m2 Eb
4 94.118 Upminor 2nd ^m2 ^Eb
5 117.647 Downmid 2nd v~2 ^^Eb
6 141.176 Mid 2nd ~2 vvvE, ^^^Eb
7 164.706 Upmid 2nd ^~2 vvE
8 188.235 Downmajor 2nd vM2 vE
9 211.765 Major 2nd M2 E
10 235.294 Upmajor 2nd ^M2 ^E
11 258.824 Downminor 3rd vm3 vF
12 282.353 Minor 3rd m3 F
13 305.882 Upminor 3rd ^m3 ^F
14 329.412 Downmid 3rd v~3 ^^F
15 352.941 Mid 3rd ~3 ^^^F, vvvF#
16 376.471 Upmid 3rd ^~3 vvF#
17 400.000 Downmajor 3rd vM3 vF#
18 423.529 Major 3rd M3 F#
19 447.509 Upmajor 3rd ^M3 ^F#
20 470.588 Down 4th v4 vG
21 494.118 Perfect 4th P4 G
22 517.647 Up 4th ^1 ^G
23 541.176 Downdim 5th vd5 vAb
24 564.706 Dim 5th d5 Ab
25 588.235 Updim 5th ^d5 ^Ab
26 611.765 Downaug 4th vA4 vG#
27 635.294 Aug 4th A4 G#
28 658.824 Upaug 4th ^A4 ^G#
29 682.353 Down 5th v5 vA
30 705.882 Perfect 5th P5 A
31 729.412 Up 5th ^5 ^A
32 752.941 Downminor 6th vm6 vBb
33 776.471 Minor 6th m6 Bb
34 800.000 Upminor 6th ^m6 ^Bb
35 823.529 Downmid 6th v~6 ^^Bb
36 847.059 Mid 6th ~6 vvvB, ^^^Bb
37 870.588 Upmid 6th ^~6 vvB
38 894.118 Downmajor 6th vM6 vB
39 917.647 Major 6th M6 B
40 941.176 Upmajor 6th ^M6 ^B
41 964.706 Downminor 7th vm7 vC
42 988.235 Minor 7th m7 C
43 1011.765 Upminor 7th ^m7 ^C
44 1035.294 Downmid 7th v~7 ^^C
45 1058.824 Mid 7th ~7 ^^^C, vvvC#
46 1082.353 Upmid 7th ^~7 vvC#
47 1105.882 Downmajor 7th vM7 vC#
48 1129.412 Major 7th M7 C#
49 1152.941 Upmajor 7th ^M7 ^C#
50 1176.471 Down 8ve v8 vD
51 1200.000 Perfect 8ve P8 D
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