74edo: Difference between revisions

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{{EDO intro|74}}
{{EDO intro|74}}


It is most notable as a [[meantone]] tuning, tempering out [[81/80]] in the [[5-limit]]; 81/80 and [[126/125]] (and hence [[225/224]]) in the [[7-limit]]; [[99/98]], 176/175 and 441/440 in the [[11-limit]]; and [[144/143]] and 847/845 in the [[13-limit]]. Discarding 847/845 from that gives [[Meantone_family|13-limit meantone]], aka 13-limit [[huygens]], for which 74edo gives the [[optimal patent val]]; and discarding 144/143 gives a 13-limit 62&74 temperament with half-octave period and two parallel tracks of meantone.
== Theory ==
74edo is most notable as a [[meantone]] tuning, [[tempering out]] [[81/80]] in the [[5-limit]]; [[126/125]] and [[225/224]] in the [[7-limit]]; [[99/98]], [[176/175]] and [[441/440]] in the [[11-limit]]; and [[144/143]] and [[847/845]] in the [[13-limit]]. Discarding 847/845 from that gives 13-limit meantone, a.k.a. 13-limit [[huygens]], for which 74edo gives the [[optimal patent val]]; and discarding 144/143 gives a 13-limit 62 & 74 temperament with half-octave period and two parallel tracks of meantone.


74edo tunes [[harmonic]] [[11/1|11]] only 1/30 of a cent sharp, and [[13/1|13]] 2.7 cents sharp, making it a distinctly interesting choice for higher-limit meantone.
=== Odd harmonics ===
{{Harmonics in equal|74}}
{{Harmonics in equal|74}}
74 tunes 11 only 1/30 of a cent sharp, and 13 2.7 cents sharp, making it a distinctly interesting choice for higher-limit meantone.


== Intervals ==
== Intervals ==
Line 12: Line 14:


== Music ==
== Music ==
* [http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-74-edo.mp3 Twinkle canon – 74 edo] by  [http://soonlabel.com/xenharmonic/archives/573 Claudi Meneghin] {{dead link}}
; [[Claudi Meneghin]]
* ''Twinkle canon'' (2012) – [https://web.archive.org/web/20171009205013/http://soonlabel.com/xenharmonic/archives/573 detail] | [https://web.archive.org/web/20201127015514/http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-74-edo.mp3 play]


[[Category:74edo| ]] <!-- main article -->
[[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->
[[Category:Meantone]]
[[Category:Meantone]]
[[Category:Listen]]
[[Category:Listen]]
[[Category:Historical]]
[[Category:Historical]]

Revision as of 10:06, 27 June 2024

← 73edo 74edo 75edo →
Prime factorization 2 × 37
Step size 16.2162 ¢ 
Fifth 43\74 (697.297 ¢)
Semitones (A1:m2) 5:7 (81.08 ¢ : 113.5 ¢)
Consistency limit 5
Distinct consistency limit 5

Template:EDO intro

Theory

74edo is most notable as a meantone tuning, tempering out 81/80 in the 5-limit; 126/125 and 225/224 in the 7-limit; 99/98, 176/175 and 441/440 in the 11-limit; and 144/143 and 847/845 in the 13-limit. Discarding 847/845 from that gives 13-limit meantone, a.k.a. 13-limit huygens, for which 74edo gives the optimal patent val; and discarding 144/143 gives a 13-limit 62 & 74 temperament with half-octave period and two parallel tracks of meantone.

74edo tunes harmonic 11 only 1/30 of a cent sharp, and 13 2.7 cents sharp, making it a distinctly interesting choice for higher-limit meantone.

Odd harmonics

Approximation of odd harmonics in 74edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -4.66 +2.88 +4.15 +6.90 +0.03 +2.72 -1.78 -7.66 -5.62 -0.51 +4.16
Relative (%) -28.7 +17.7 +25.6 +42.6 +0.2 +16.7 -11.0 -47.2 -34.7 -3.1 +25.6
Steps
(reduced) 		117
(43) 		172
(24) 		208
(60) 		235
(13) 		256
(34) 		274
(52) 		289
(67) 		302
(6) 		314
(18) 		325
(29) 		335
(39)

Intervals

Steps Cents Approximate ratios Ups and downs notation
0 0 1/1 D
1 16.2 ^D, vE♭♭
2 32.4 ^^D, E♭♭
3 48.6 34/33, 37/36, 38/37 vvD♯, ^E♭♭
4 64.9 26/25 vD♯, ^^E♭♭
5 81.1 21/20, 22/21 D♯, vvE♭
6 97.3 18/17 ^D♯, vE♭
7 113.5 16/15 ^^D♯, E♭
8 129.7 14/13 vvD𝄪, ^E♭
9 145.9 25/23, 37/34 vD𝄪, ^^E♭
10 162.2 11/10 D𝄪, vvE
11 178.4 31/28 ^D𝄪, vE
12 194.6 19/17, 28/25, 37/33 E
13 210.8 26/23, 35/31 ^E, vF♭
14 227 33/29 ^^E, F♭
15 243.2 23/20, 38/33 vvE♯, ^F♭
16 259.5 vE♯, ^^F♭
17 275.7 34/29 E♯, vvF
18 291.9 13/11 ^E♯, vF
19 308.1 F
20 324.3 29/24 ^F, vG♭♭
21 340.5 28/23 ^^F, G♭♭
22 356.8 16/13 vvF♯, ^G♭♭
23 373 26/21, 31/25, 36/29 vF♯, ^^G♭♭
24 389.2 5/4 F♯, vvG♭
25 405.4 24/19 ^F♯, vG♭
26 421.6 37/29 ^^F♯, G♭
27 437.8 vvF𝄪, ^G♭
28 454.1 13/10 vF𝄪, ^^G♭
29 470.3 21/16, 38/29 F𝄪, vvG
30 486.5 ^F𝄪, vG
31 502.7 G
32 518.9 31/23 ^G, vA♭♭
33 535.1 15/11 ^^G, A♭♭
34 551.4 11/8 vvG♯, ^A♭♭
35 567.6 vG♯, ^^A♭♭
36 583.8 7/5 G♯, vvA♭
37 600 17/12, 24/17 ^G♯, vA♭
38 616.2 10/7 ^^G♯, A♭
39 632.4 vvG𝄪, ^A♭
40 648.6 16/11 vG𝄪, ^^A♭
41 664.9 22/15 G𝄪, vvA
42 681.1 ^G𝄪, vA
43 697.3 A
44 713.5 ^A, vB♭♭
45 729.7 29/19, 32/21, 35/23 ^^A, B♭♭
46 745.9 20/13, 37/24 vvA♯, ^B♭♭
47 762.2 31/20 vA♯, ^^B♭♭
48 778.4 A♯, vvB♭
49 794.6 19/12 ^A♯, vB♭
50 810.8 8/5 ^^A♯, B♭
51 827 21/13, 29/18 vvA𝄪, ^B♭
52 843.2 13/8 vA𝄪, ^^B♭
53 859.5 23/14 A𝄪, vvB
54 875.7 ^A𝄪, vB
55 891.9 B
56 908.1 22/13 ^B, vC♭
57 924.3 29/17 ^^B, C♭
58 940.5 vvB♯, ^C♭
59 956.8 33/19 vB♯, ^^C♭
60 973 B♯, vvC
61 989.2 23/13 ^B♯, vC
62 1005.4 25/14, 34/19 C
63 1021.6 ^C, vD♭♭
64 1037.8 20/11 ^^C, D♭♭
65 1054.1 vvC♯, ^D♭♭
66 1070.3 13/7 vC♯, ^^D♭♭
67 1086.5 15/8 C♯, vvD♭
68 1102.7 17/9 ^C♯, vD♭
69 1118.9 21/11 ^^C♯, D♭
70 1135.1 25/13 vvC𝄪, ^D♭
71 1151.4 33/17, 37/19 vC𝄪, ^^D♭
72 1167.6 C𝄪, vvD
73 1183.8 ^C𝄪, vD
74 1200 2/1 D

Music

Claudi Meneghin
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