Xenharmonic Wiki

106edo: Difference between revisions

← Older edit
VisualWikitext
Fredg999 category edits (talk | contribs)
autopatrolled, Bots, emailconfirmed
4,397 edits
m Sort key
Overthink (talk | contribs)
autopatrolled
2,697 edits
→Theory: Removed confusing MMTM sentence from years ago
 
(12 intermediate revisions by 8 users not shown)
Line 1: Line 1:
The '''106 equal division of the octave''' ('''106edo''') divides the [[octave]] into 106 [[equal]] parts of 11.321 [[cent]]s each.
{{Infobox ET}}
{{ED intro}}


Since 106 = 2 × 53, 106edo is closely related to [[53edo]], and is [[contorted]] through the [[7-limit]], tempering out the same commas ([[32805/32768]], [[15625/15552]], [[1600000/1594323]], 2109375/2097152 in the [[5-limit]], 3125/3087, [[225/224]], 4000/3969, [[1728/1715]], [[2430/2401]], [[4375/4374]] in the 7-limit) as the [[patent val]] for 53edo. In the 11-limit it also tempers out [[243/242]], [[3025/3024]] and [[9801/9800]], so that it [[support]]s [[spectacle]] temperament and [[borwell]] temperament.
== Theory ==
Since 106 = 2 × 53, 106edo is closely related to [[53edo]], and is [[contorted]] through the [[7-limit]], tempering out the same commas ([[32805/32768]], [[15625/15552]], [[1600000/1594323]], [[2109375/2097152]] in the [[5-limit]], 3125/3087, [[225/224]], 4000/3969, [[1728/1715]], [[2430/2401]], [[4375/4374]] in the 7-limit) as the [[patent val]] for 53edo. In the 11-limit it also tempers out [[243/242]], [[3025/3024]] and [[9801/9800]], so that it [[support]]s [[spectacle]] temperament and [[borwell]] temperament. Unfortunately, it is now only consistent to the [[5-odd-limit]] due to 7/5 being closer to 51 steps rather than its patent val mapping of 52 steps. A superset of 53edo with much a higher consistency limit is [[159edo]].


The division is notable for the fact that it is related to the [[turkish cent]], or türk sent, which divides 106edo into 100 parts just as ordinary cents divides 12edo into 100 parts, thereby making it the [[relative cent]] division for 106edo. Conversely, it makes the Pythagorean [[relative cent]] (or pion, symbol π<sup>¢</sup>, π<sup>r¢</sup>), which most closely approximates equally dividing an exact [[3/2]], if you care about such a thing.
The division is notable for the fact that it is related to the [[turkish cent]], or türk sent, which divides 106edo into 100 parts just as ordinary cents divides 12edo into 100 parts, thereby making it the [[relative cent]] division for 106edo.


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|106|columns=16}}
{{Harmonics in equal|106|columns=16}}


== Scales ==
53edo for comparison:
Since 106edo has a step of 11.321 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}.
{{Harmonics in equal|53|collapsed=1|columns=16}}
{| class="wikitable"
 
|+Circulating temperaments in 106edo
== Intervals ==
!Tones
{{Interval table}}
!Pattern
 
!L:s
== Instruments ==
|-
* [[Lumatone mapping for 106edo]]
|5
|[[1L 4s]]
|22:21
|-
|6
|[[4L 2s]]
|18:17
|-
|7
|[[1L 6s]]
|16:15
|-
|8
|[[2L 6s]]
| 14:13
|-
|9
|[[7L 2s]]
|12:11
|-
|10
|[[6L 4s]]
| 11:10
|-
|11
|[[7L 4s]]
|10:9
|-
|12
|[[10L 2s]]
| rowspan="2" |9:8
|-
|13
|[[2L 11s]]
|-
|14
|[[8L 6s]]
| rowspan="2" |8:7
|-
|15
|[[1L 14s]]
|-
|16
|[[10L 6s]]
| rowspan="2" |7:6
|-
|17
|4L 13s
|-
|18
|16L 2s
| rowspan="4" |6:5
|-
|19
|[[11L 8s]]
|-
|20
|6L 14s
|-
|21
|1L 20s
|-
|22
|18L 4s
| rowspan="5" |5:4
|-
|23
|14L 9s
|-
|24
|10L 14s
|-
|25
|[[6L 19s]]
|-
|26
|2L 24s
|-
|27
|25L 2s
| rowspan="9" |4:3
|-
|28
|22L 6s
|-
|29
|19L 10s
|-
|30
|16L 14s
|-
|31
|13L 18s
|-
|32
|10L 22s
|-
|33
|7L 26s
|-
|34
|4L 30s
|-
|35
|1L 34s
|-
|36
|34L 2s
| rowspan="17" |3:2
|-
|37
|32L 5s
|-
|38
|30L 8s
|-
|39
|28L 11s
|-
|40
|26L 14s
|-
|41
|24L 17s
|-
|42
|22L 20s
|-
|43
|20L 23s
|-
|44
|18L 26s
|-
|45
|16L 29s
|-
|46
|14L 32s
|-
|47
|12L 35s
|-
|48
|10L 38s
|-
|49
|8L 41s
|-
|50
|6L 44s
|-
|51
|4L 47s
|-
|52
|2L 50s
|-
|53
|[[53edo]]
|equal
|-
|54
|52L 2s
| rowspan="31" |2:1
|-
|55
|51L 4s
|-
|56
|50L 6s
|-
|57
|49L 8s
|-
|58
|48L 10s
|-
|59
|47L 12s
|-
|60
|46L 14s
|-
|61
|45L 16s
|-
|62
|44L 18s
|-
|63
|43L 20s
|-
|64
|42L 22s
|-
|65
|41L 24s
|-
|66
|40L 26s
|-
|67
|39L 28s
|-
|68
|38L 30s
|-
|69
|37L 32s
|-
|70
|36L 34s
|-
|71
|35L 36s
|-
|72
|34L 38s
|-
|73
|33L 40s
|-
|74
|32L 42s
|-
|75
|31L 44s
|-
|76
|30L 46s
|-
|77
|29L 48s
|-
|78
|28L 50s
|-
|79
|27L 52s
|-
|80
|26L 54s
|-
|81
|25L 56s
|-
|82
|24L 58s
|-
|83
|23L 60s
|-
|84
|22L 62s
|}


== See also ==
== See also ==
Retrieved from "https://en.xen.wiki/w/106edo"