Xenharmonic Wiki

107edo: Difference between revisions

← Older edit
VisualWikitext
Francium (talk | contribs)
4,816 edits
No edit summary
BudjarnLambeth (talk | contribs)
autopatrolled, emailconfirmed
21,262 edits
 
(11 intermediate revisions by 4 users not shown)
Line 1: Line 1:
{{Infobox ET}}
{{Infobox ET}}
'''107edo''' is the [[EDO|equal division of the octave]] into 107 parts of 11.214953271 cents each. It is inconsistent to the 5-limit and higher limit, with four mappings possible for the 7-limit: {{val|107 170 248 300}} (patent val), {{val|107 169 248 300}} (107b), {{val|107 170 249 300}} (107c), and {{val|107 170 249 301}} (107cd). Using the patent val, it tempers out the [[Magic family|small diesis]], [[3125/3072]] and 33554432000/31381059609 in the 5-limit; [[1029/1024]], 2240/2187, and 3125/3087 in the 7-limit; [[100/99]], 1232/1215, and 1331/1323 in the 11-limit. Using the 107b val, it tempers out the [[syntonic comma]], [[81/80]] and {{monzo|-61 -1 27}}; in the 5-limit; [[2401/2400]], [[2430/2401]], and 234375/229376 in the 7-limit; [[385/384]], 1350/1331, 1375/1372, and 1944/1925 in the 11-limit. Using the 107c val, it tempers out the immunity comma, 1638400/1594323 and the valentine comma, 1990656/1953125 in the 5-limit; [[126/125]], [[1029/1024]], and 307328/295245 in the 7-limit; [[121/120]], [[176/175]], [[441/440]], and 184877/177147 in the 11-limit. Using the 107cd val, it tempers out [[1728/1715]], 4000/3969, and 28672/28125 in the 7-limit; 121/120, [[896/891]], 1375/1372, and 3168/3125 in the 11-limit.
{{ED intro}}


It is the 28th [[prime edo]].
== Theory ==
107edo is in[[consistent]] to the [[5-odd-limit]] and higher limits, and [[harmonic]]s [[3/1|3]], [[5/1|5]], and [[7/1|7]] are about halfway between its steps. Either the 2.9.5.7 or 2.9.15.21 [[subgroup]] can be used. For the full 7-limit, it has four possible [[mapping]]s: {{val| 107 170 248 300 }} ([[patent val]]), {{val| 107 '''169''' 248 300 }} (107b), {{val| 107 170 '''249''' 300 }} (107c), and {{val| 107 170 '''249''' '''301''' }} (107cd).  


==Theory==
Using the patent val, it tempers out 3125/3072 ([[magic comma]]) and {{monzo| 28 -22 3 }} in the 5-limit; [[1029/1024]], [[2240/2187]], and [[3125/3087]] in the 7-limit; [[100/99]], 1232/1215, and 1331/1323 in the 11-limit.
Since 107edo has a step of 11.214953271 cents, it also allows one to use its MOS scales as circulating temperaments.
 
{| class="wikitable"
Using the 107cd val, it tempers out [[1728/1715]], [[4000/3969]], and 28672/28125 in the 7-limit; [[121/120]], [[896/891]], [[1375/1372]], and 3168/3125 in the 11-limit.
|+Circulating temperaments in 107edo
 
!Tones
Using the 107c val, it tempers out 1638400/1594323 ([[immunity comma]]) and 1990656/1953125 ([[valentine comma]]) in the 5-limit; [[126/125]], 1029/1024, and 307328/295245 in the 7-limit; 121/120, [[176/175]], [[441/440]], and 184877/177147 in the 11-limit.
!Pattern
 
!L:s
Using the 107b val, it tempers out 81/80 ([[syntonic comma]]) and {{monzo| -61 -1 27 }}; in the 5-limit; [[2401/2400]], [[2430/2401]], and 234375/229376 in the 7-limit; [[385/384]], 1350/1331, 1375/1372, and 1944/1925 in the 11-limit.
|-
|5
|[[2L 3s]]
|22:21
|-
|6
|[[5L 1s]]
|18:17
|-
|7
|[[2L 5s]]
|16:15
|-
|8
|[[3L 5s]]
|14:13
|-
|9
|[[8L 1s]]
|12:11
|-
|10
|[[7L 3s]]
|11:10
|-
|11
|[[8L 3s]]
|10:9
|-
|12
|[[11L 1s]]
| rowspan="2" |9:8
|-
|13
|[[3L 10s]]
|-
|14
|[[9L 5s]]
| rowspan="2" |8:7
|-
|15
|[[2L 13s]]
|-
|16
|[[11L 5s]]
| rowspan="2" |7:6
|-
|17
|[[5L 12s]]
|-
|18
|17L 1s
| rowspan="4" |6:5
|-
|19
|[[12L 7s]]
|-
|20
|[[7L 13s]]
|-
|21
|[[2L 19s]]
|-
|22
|19L 3s
| rowspan="5" |5:4
|-
|23
|[[15L 8s]]
|-
|24
|11L 13s
|-
|25
|7L 18s
|-
|26
|3L 23s
|-
|27
|26L 1s
| rowspan="9" |4:3
|-
|28
|23L 5s
|-
|29
|20L 9s
|-
|30
|17L 13s
|-
|31
|14L 17s
|-
|32
|11L 21s
|-
|33
|8L 25s
|-
|34
|5L 29s
|-
|35
|2L 33s
|-
|36
|35L 1s
| rowspan="18" |3:2
|-
|37
|33L 4s
|-
|38
|31L 7s
|-
|39
|29L 10s
|-
|40
|27L 13s
|-
|41
|25L 16s
|-
|42
|23L 19s
|-
|43
|21L 22s
|-
|44
|19L 25s
|-
|45
|17L 28s
|-
|46
|15L 31s
|-
|47
|13L 34s
|-
|48
|11L 37s
|-
|49
|9L 40s
|-
|50
|7L 43s
|-
|51
|5L 46s
|-
|52
|3L 49s
|-
|53
|1L 52s
|-
|54
|53L 1s
| rowspan="32" |2:1
|-
|55
|52L 3s
|-
|56
|51L 5s
|-
|57
|50L 7s
|-
|58
|49L 9s
|-
|59
|48L 11s
|-
|60
|47L 13s
|-
|61
|46L 15s
|-
|62
|45L 17s
|-
|63
|44L 19s
|-
|64
|43L 21s
|-
|65
|42L 23s
|-
|66
|41L 25s
|-
|67
|40L 27s
|-
|68
|39L 29s
|-
|69
|38L 31s
|-
|70
|37L 33s
|-
|71
|36L 35s
|-
|72
|35L 37s
|-
|73
|34L 39s
|-
|74
|33L 41s
|-
|75
|32L 43s
|-
|76
|31L 45s
|-
|77
|30L 47s
|-
|78
|29L 49s
|-
|79
|28L 51s
|-
|80
|27L 53s
|-
|81
|26L 55s
|-
|82
|25L 57s
|-
|83
|24L 59s
|-
|84
|23L 61s
|-
|85
|22L 63s
|}


=== Odd harmonics ===
{{Harmonics in equal|107}}
{{Harmonics in equal|107}}


==Regular temperament properties==
=== Octave stretch ===
107edo’s approximations of 3/1, 5/1, 7/1, 13/1, 17/1 and 19/1 are all improved by [[AS|1ed175/174]], a [[Octave stretch|stretched-octave]] version of 107edo. The trade-off is a slightly worse 2/1 and 11/1.
 
There are also several nearby [[Zeta peak index]] (ZPI) tunings which can be used to improve some harmonics at the expense of others: 622zpi, 623zpi, 624zpi, 625zpi, 626zpi, 627zpi, 628zpi and 629zpi.
 
The details of each of those ZPI tunings are visible in [[User:Contribution]]’s gallery of [[User:Contribution/Gallery of Zeta Peak Indexes (1 - 10 000)|Zeta Peak Indexes (1 - 10 000)]]. Warning: due to its length, that page may slow down your device while it is open. The effect will go away after you close the page.
 
=== Subsets and supersets ===
107edo is the 28th [[prime edo]], following [[103edo]] and before [[109edo]]. [[214edo]], which doubles it, provides correction for the approximation to harmonics 3, 5, and 7.
 
== Intervals ==
{{Interval table}}
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" |[[Subgroup]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" |[[Comma list|Comma List]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" |Optimal<br>8ve Stretch (¢)
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! colspan="2" |Tuning Error
! colspan="2" | Tuning Error
|-
|-
![[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
! [[TE simple badness|Relative]] (%)
|-
|-
|2.3
| 2.9
|{{monzo|170 -107}}
| {{monzo| 339 -107 }}
|{{val|107 170}}
| {{mapping| 107 339 }}
| -1.4471
| +0.322
| 1.4453
| 0.322
| 12.89
| 2.87
|-
|-
|2.3.5
| 2.9.5
|3125/3072, {{monzo|18 -23 8}}
| 9765625/9565938, {{monzo| -34 10 1 }}
|{{val|107 170 248}}
| {{mapping| 107 339 248 }}
| -0.2497
| +0.933
| 2.0685
| 0.904
| 18.44
| 8.06
|-
|-
|2.3.5.7
| 2.9.5.7
|2240/2187, 1029/1024, 3125/3072
| 225/224, 84035/82944, {{monzo| 14 -6 7 -4 }}
|{{val|107 170 248 300}}
| {{mapping| 107 339 248 300 }}
| +0.1987
| +1.087
| 1.9529
| 0.827
| 17.41
| 7.37
|-
|-
|2.3.5.7.11
| 2.9.5.7.11
|100/99, 1232/1215, 1375/1372, 1029/1024
| 225/224, 441/440, 26411/26244, 161280/161051
|{{val|107 170 248 300 370}}
| {{mapping| 107 339 248 300 370 }}
| +0.2622
| +0.973
| 1.7513
| 0.774
| 15.62
| 6.90
|-
|-
|2.3.5.7.11.13
| 2.9.5.7.11.13
|100/99, 196/195, 275/273, 1232/1215, 1029/1024
| 225/224, 325/324, 441/440, 847/845, 24500/24167
|{{val|107 170 248 300 370 396}}
| {{mapping| 107 339 248 300 370 396 }}
| +0.1917
| +0.783
| 1.6065
| 0.823
| 14.32
| 7.33
|-
|-
|2.3.5.7.11.13.17
| 2.9.5.7.11.13.17
|100/99, 196/195, 136/135, 275/273, 1232/1215, 1547/1536
| 170/169, 225/224, 325/324, 441/440, 847/845, 2000/1989
|{{val|107 170 248 300 370 396}}
| {{mapping| 107 339 248 300 370 396 437 }}
| +0.3048
| +0.812
| 1.5129
| 0.765
| 13.49
| 6.82
|}
|}
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Prime EDO]]
Retrieved from "https://en.xen.wiki/w/107edo"