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'''105edo''' is the [[equal division of the octave]] into 105 equal parts of 11.429 [[cent]]s each. It is most notable as a tuning of meantone and in particular higher limit extensions of meantone, as it is the highest edo that strictly fulfills both criteria of meantone - ie, all intervals can be reached by stacking it's best fifth, and stacking four of them equals it's best major third. It [[tempers out]] [[81/80]] in the [[5-limit]]; 81/80, [[126/125]] and hence 225/224 in the [[7-limit]]; 99/98, 176/175 and 441/440 in the [[11-limit]]; and if we want to push that far, 144/143 in the [[13-limit]]. This is the sharper fifth mapping (aka "huygens") of 11-limit meantone.
{{Infobox ET}}
{{Primes in edo|105}}
{{ED intro}}  


105edo gives the [[optimal patent val]] for 11-limit meantone (ie huygens rather than meanpop) and provides a good tuning in the 13-limit, though [[74edo]] is in that case the optimal patent val. 105 is highly composite, being the product 3*5*7 (i. e. (14+1)*14/2) of the three smallest odd primes, with other divisors being 15, 21 and 35. As the common multiple of these three primes and the triangular number closest to 100, 105 is a perfect substitute for it when a "cent" is desired to include them all or be a triangular number.
== Theory ==
105edo is most notable as a tuning of [[meantone]] and in particular higher-limit extensions of meantone, such as [[grosstone]] and [[huygens]]. It [[tempering out|tempers out]] [[81/80]] in the [[5-limit]]; 81/80, [[126/125]] and hence 225/224 in the [[7-limit]]; 99/98, 176/175 and 441/440 in the [[11-limit]]; and if we want to push that far, 144/143 in the [[13-limit]]. This is the sharper fifth mapping of 11-limit meantone (a.k.a. huygens rather than meanpop), for which it gives the [[optimal patent val]], and provides a good tuning for the 13-limit extension, though [[74edo]] is in that case the optimal patent val. 105edo's meantone fifth is nearly identical to the [[CTE tuning|CTE generator]] for meantone.


== 105edo close-up ==
=== Odd harmonics ===
{{Harmonics in equal|105}}


<pre>C . . Dbb B## . . C# . . Db . . . C## . . D</pre>
=== Subsets and supersets ===
105 is the product of 3 × 5 × 7, the three smallest odd primes, with other divisors being 15, 21 and 35.


As such, the val [105 165 245 294], which is contorted in 2.n for each prime n in the subgroup, may be used to extend the concept of 21edo's 5-limit harmony to the 7-limit, producing an independent dimension for each prime.


[[Category:Theory]]
== Intervals ==
{{Main|Table of 105edo intervals}}
=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|105}}
 
== Instruments ==
=== Lumatone ===
The [[lumatone]] can be used to play 105edo. For key mappings, see: [[Lumatone mapping for 105edo]].
 
[[Category:105edo| ]] <!-- main article -->
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Huygens]]
[[Category:Meantone]]
[[Category:Meantone]]
[[Category:Equal divisions of the octave]]
[[Category:105edo| ]]
Since 105edo has a step of 11.429 cents, it also allows one to use its MOS scales as circulating temperaments.
{| class="wikitable"
|+Circulating temperaments in 105edo
!Tones
!Pattern
!L:s
|-
|5
|[[5edo]]
|equal
|-
|6
|[[3L 3s]]
|18:17
|-
|7
|[[7edo]]
|equal
|-
|8
|[[1L 7s]]
|14:13
|-
|9
|[[6L 3s]]
|12:11
|-
|10
|[[5L 5s]]
|11:10
|-
|11
|[[6L 5s]]
|10:9
|-
|12
|[[9L 3s]]
|9:8
|-
|13
|[[1L 12s]]
| rowspan="2" |8:7
|-
|14
|[[7L 7s]]
|-
|15
|[[15edo]]
|equal
|-
|16
|[[9L 7s]]
| rowspan="2" |7:6
|-
|17
|[[3L 14s]]
|-
|18
|15L 3s
| rowspan="3" |6:5
|-
|19
|[[10L 9s]]
|-
|20
|5L 15s
|-
|21
|[[21edo]]
|equal
|-
|22
|[[17L 5s]]
| rowspan="5" |5:4
|-
|23
|13L 10s
|-
|24
|9L 15s
|-
|25
|5L 20s
|-
|26
|1L 25s
|-
|27
|24L 3s
| rowspan="8" |4:3
|-
|28
|21L 7s
|-
|29
|18L 11s
|-
|30
|15L 15s
|-
|31
|12L 19s
|-
|32
|9L 23s
|-
|33
|6L 27s
|-
|34
|3L 31s
|-
|35
|[[35edo]]
|equal
|-
|36
|33L 3s
| rowspan="17" |3:2
|-
|37
|31L 6s
|-
|38
|29L 9s
|-
|39
|27L 12s
|-
|40
|25L 15s
|-
|41
|23L 18s
|-
|42
|21L 21s
|-
|43
|19L 24s
|-
|44
|17L 27s
|-
|45
|15L 30s
|-
|46
|13L 33s
|-
|47
|11L 36s
|-
|48
|9L 39s
|-
|49
|7L 42s
|-
|50
|5L 45s
|-
|51
|3L 48s
|-
|52
|1L 51s
|-
|53
|52L 1s
| rowspan="32" |2:1
|-
|54
|51L 3s
|-
|55
|50L 5s
|-
|56
|49L 7s
|-
|57
|48L 9s
|-
|58
|47L 11s
|-
|59
|46L 13s
|-
|60
|45L 15s
|-
|61
|44L 17s
|-
|62
|43L 19s
|-
|63
|42L 21s
|-
|64
|41L 23s
|-
|65
|40L 25s
|-
|66
|39L 27s
|-
|67
|38L 29s
|-
|68
|37L 31s
|-
|69
|36L 33s
|-
|70
|35L 35s
|-
|71
|34L 37s
|-
|72
|33L 39s
|-
|73
|32L 41s
|-
|74
|31L 43s
|-
|75
|30L 45s
|-
|76
|29L 47s
|-
|77
|28L 49s
|-
|78
|27L 51s
|-
|79
|26L 53s
|-
|80
|25L 55s
|-
|81
|24L 57s
|-
|82
|23L 59s
|-
|83
|22L 61s
|-
|84
|21L 63s
|}<!-- main article -->
[[Category:Huygens]]

Latest revision as of 19:44, 30 April 2025

← 104edo 105edo 106edo →
Prime factorization 3 × 5 × 7
Step size 11.4286 ¢ 
Fifth 61\105 (697.143 ¢)
Semitones (A1:m2) 7:10 (80 ¢ : 114.3 ¢)
Dual sharp fifth 62\105 (708.571 ¢)
Dual flat fifth 61\105 (697.143 ¢)
Dual major 2nd 18\105 (205.714 ¢) (→ 6\35)
Consistency limit 3
Distinct consistency limit 3

105 equal divisions of the octave (abbreviated 105edo or 105ed2), also called 105-tone equal temperament (105tet) or 105 equal temperament (105et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 105 equal parts of about 11.4 ¢ each. Each step represents a frequency ratio of 21/105, or the 105th root of 2.

Theory

105edo is most notable as a tuning of meantone and in particular higher-limit extensions of meantone, such as grosstone and huygens. It tempers out 81/80 in the 5-limit; 81/80, 126/125 and hence 225/224 in the 7-limit; 99/98, 176/175 and 441/440 in the 11-limit; and if we want to push that far, 144/143 in the 13-limit. This is the sharper fifth mapping of 11-limit meantone (a.k.a. huygens rather than meanpop), for which it gives the optimal patent val, and provides a good tuning for the 13-limit extension, though 74edo is in that case the optimal patent val. 105edo's meantone fifth is nearly identical to the CTE generator for meantone.

Odd harmonics

Approximation of odd harmonics in 105edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -4.81 +2.26 +2.60 +1.80 -2.75 +5.19 -2.55 -2.10 -0.37 -2.21 +0.30
Relative (%) -42.1 +19.8 +22.8 +15.8 -24.0 +45.4 -22.4 -18.4 -3.2 -19.3 +2.6
Steps
(reduced) 		166
(61) 		244
(34) 		295
(85) 		333
(18) 		363
(48) 		389
(74) 		410
(95) 		429
(9) 		446
(26) 		461
(41) 		475
(55)

Subsets and supersets

105 is the product of 3 × 5 × 7, the three smallest odd primes, with other divisors being 15, 21 and 35.

As such, the val [105 165 245 294], which is contorted in 2.n for each prime n in the subgroup, may be used to extend the concept of 21edo's 5-limit harmony to the 7-limit, producing an independent dimension for each prime.

Intervals

Main article: Table of 105edo intervals

15-odd-limit interval mappings

The following tables show how 15-odd-limit intervals are represented in 105edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 105edo (direct approximation, even if inconsistent)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
15/11, 22/15 0.192 1.7
7/5, 10/7 0.345 3.0
9/5, 10/9 0.453 4.0
9/7, 14/9 0.798 7.0
13/12, 24/13 1.430 12.5
9/8, 16/9 1.804 15.8
11/6, 12/11 2.066 18.1
5/4, 8/5 2.258 19.8
15/8, 16/15 2.554 22.4
13/7, 14/13 2.584 22.6
7/4, 8/7 2.603 22.8
11/8, 16/11 2.747 24.0
13/10, 20/13 2.929 25.6
13/9, 18/13 3.382 29.6
13/11, 22/13 3.495 30.6
15/13, 26/15 3.688 32.3
7/6, 12/7 4.014 35.1
5/3, 6/5 4.359 38.1
11/9, 18/11 4.551 39.8
3/2, 4/3 4.812 42.1
11/10, 20/11 5.004 43.8
15/14, 28/15 5.157 45.1
13/8, 16/13 5.187 45.4
11/7, 14/11 5.349 46.8
15-odd-limit intervals in 105edo (patent val mapping)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
15/11, 22/15 0.192 1.7
7/5, 10/7 0.345 3.0
11/6, 12/11 2.066 18.1
5/4, 8/5 2.258 19.8
15/8, 16/15 2.554 22.4
13/7, 14/13 2.584 22.6
7/4, 8/7 2.603 22.8
11/8, 16/11 2.747 24.0
13/10, 20/13 2.929 25.6
3/2, 4/3 4.812 42.1
11/10, 20/11 5.004 43.8
15/14, 28/15 5.157 45.1
13/8, 16/13 5.187 45.4
11/7, 14/11 5.349 46.8
11/9, 18/11 6.878 60.2
5/3, 6/5 7.070 61.9
7/6, 12/7 7.415 64.9
15/13, 26/15 7.741 67.7
13/11, 22/13 7.933 69.4
9/8, 16/9 9.624 84.2
13/12, 24/13 9.999 87.5
9/5, 10/9 11.882 104.0
9/7, 14/9 12.227 107.0
13/9, 18/13 14.811 129.6

Instruments

Lumatone

The lumatone can be used to play 105edo. For key mappings, see: Lumatone mapping for 105edo.

Retrieved from "https://en.xen.wiki/index.php?title=105edo&oldid=194827"