78edo: Difference between revisions

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Much like [[100edo|100bddd]], the 78ddd val can be used to construct an alternative to 22edo for pajara. The large and small step sizes in this case have ratio 4:3. The width of the tempered perfect fifth is 707.7{{cent}}. The major third is 384.6{{cent}}; less than two cents flat of just. The harmonic seventh is 984.6{{cent}}, or about 15.8{{cent}} sharp; hence this tuning prioritizes the 3- and 5-limits over the 7-limit, while still ensuring that no basic 7-limit intervals other than the tritones are more than 16{{cent}} off. The 22-note 2MOS generated in this way could be used to build straight-fretted guitars that would be [https://en.wikipedia.org/wiki/Augmented-fourths_tuning tuned in tritones]. The appeal of this scale is that it is less xenharmonic than [[22edo]] is, for listeners accustomed to 12edo. In particular, the 163.6{{cent}} "flat minor whole tone" of 22edo is now 169.2{{cent}}, making it more clearly a ''whole'' tone (albeit noticeably flat), rather than a neutral second.
Much like [[100edo|100bddd]], the 78ddd val can be used to construct an alternative to 22edo for pajara. The large and small step sizes in this case have ratio 4:3. The width of the tempered perfect fifth is 707.7{{cent}}. The major third is 384.6{{cent}}; less than two cents flat of just. The harmonic seventh is 984.6{{cent}}, or about 15.8{{cent}} sharp; hence this tuning prioritizes the 3- and 5-limits over the 7-limit, while still ensuring that no basic 7-limit intervals other than the tritones are more than 16{{cent}} off. The 22-note 2MOS generated in this way could be used to build straight-fretted guitars that would be [https://en.wikipedia.org/wiki/Augmented-fourths_tuning tuned in tritones]. The appeal of this scale is that it is less xenharmonic than [[22edo]] is, for listeners accustomed to 12edo. In particular, the 163.6{{cent}} "flat minor whole tone" of 22edo is now 169.2{{cent}}, making it more clearly a ''whole'' tone (albeit noticeably flat), rather than a neutral second.
== Intervals ==
{|class="wikitable"
|-
!#
!Cents
!Diatonic interval category
|-
|0
|0.0
|perfect unison
|-
|1
|15.4
|superunison
|-
|2
|30.8
|superunison
|-
|3
|46.2
|subminor second
|-
|4
|61.5
|subminor second
|-
|5
|76.9
|subminor second
|-
|6
|92.3
|minor second
|-
|7
|107.7
|minor second
|-
|8
|123.1
|supraminor second
|-
|9
|138.5
|supraminor second
|-
|10
|153.8
|neutral second
|-
|11
|169.2
|submajor second
|-
|12
|184.6
|major second
|-
|13
|200.0
|major second
|-
|14
|215.4
|major second
|-
|15
|230.8
|supermajor second
|-
|16
|246.2
|ultramajor second
|-
|17
|261.5
|subminor third
|-
|18
|276.9
|subminor third
|-
|19
|292.3
|minor third
|-
|20
|307.7
|minor third
|-
|21
|323.1
|supraminor third
|-
|22
|338.5
|supraminor third
|-
|23
|353.8
|neutral third
|-
|24
|369.2
|submajor third
|-
|25
|384.6
|major third
|-
|26
|400.0
|major third
|-
|27
|415.4
|major third
|-
|28
|430.8
|supermajor third
|-
|29
|446.2
|ultramajor third
|-
|30
|461.5
|subfourth
|-
|31
|476.9
|subfourth
|-
|32
|492.3
|perfect fourth
|-
|33
|507.7
|perfect fourth
|-
|34
|523.1
|superfourth
|-
|35
|538.5
|superfourth
|-
|36
|553.8
|superfourth
|-
|37
|569.2
|low tritone
|-
|38
|584.6
|low tritone
|-
|39
|600.0
|high tritone
|-
|40
|615.4
|high tritone
|-
|41
|630.8
|high tritone
|-
|42
|646.2
|subfifth
|-
|43
|661.5
|subfifth
|-
|44
|676.9
|subfifth
|-
|45
|692.3
|perfect fifth
|-
|46
|707.7
|perfect fifth
|-
|47
|723.1
|superfifth
|-
|48
|738.5
|superfifth
|-
|49
|753.8
|ultrafifth
|-
|50
|769.2
|subminor sixth
|-
|51
|784.6
|minor sixth
|-
|52
|800.0
|minor sixth
|-
|53
|815.4
|minor sixth
|-
|54
|830.8
|supraminor sixth
|-
|55
|846.2
|neutral sixth
|-
|56
|861.5
|submajor sixth
|-
|57
|876.9
|submajor sixth
|-
|58
|892.3
|major sixth
|-
|59
|907.7
|major sixth
|-
|60
|923.1
|supermajor sixth
|-
|61
|938.5
|supermajor sixth
|-
|62
|953.8
|ultramajor sixth
|-
|63
|969.2
|subminor seventh
|-
|64
|984.6
|minor seventh
|-
|65
|1000.0
|minor seventh
|-
|66
|1015.4
|minor seventh
|-
|67
|1030.8
|supraminor seventh
|-
|68
|1046.2
|neutral seventh
|-
|69
|1061.5
|submajor seventh
|-
|70
|1076.9
|submajor seventh
|-
|71
|1092.3
|major seventh
|-
|72
|1107.7
|major seventh
|-
|73
|1123.1
|supermajor seventh
|-
|74
|1138.5
|supermajor seventh
|-
|75
|1153.8
|ultramajor seventh
|-
|76
|1169.2
|suboctave
|-
|77
|1184.6
|suboctave
|-
|78
|1200.0
|perfect octave
|}


[[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->
[[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->
[[Category:Keen]]
[[Category:Keen]]

Revision as of 21:58, 9 March 2023

← 77edo 78edo 79edo →
Prime factorization 2 × 3 × 13
Step size 15.3846 ¢ 
Fifth 46\78 (707.692 ¢) (→ 23\39)
Semitones (A1:m2) 10:4 (153.8 ¢ : 61.54 ¢)
Dual sharp fifth 46\78 (707.692 ¢) (→ 23\39)
Dual flat fifth 45\78 (692.308 ¢) (→ 15\26)
Dual major 2nd 13\78 (200 ¢) (→ 1\6)
Consistency limit 7
Distinct consistency limit 7

Template:EDO intro

Theory

Approximation of odd harmonics in 78edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +5.74 -1.70 +0.40 -3.91 +2.53 +5.63 +4.04 +2.74 -5.21 +6.14 +2.49
Relative (%) +37.3 -11.0 +2.6 -25.4 +16.4 +36.6 +26.3 +17.8 -33.8 +39.9 +16.2
Steps
(reduced) 		124
(46) 		181
(25) 		219
(63) 		247
(13) 		270
(36) 		289
(55) 		305
(71) 		319
(7) 		331
(19) 		343
(31) 		353
(41)

This tuning tempers out 2048/2025 in the 5-limit; 875/864 and 2401/2400 in the 7-limit; and 100/99, 385/384 and 1375/1372 in the 11-limit. It provides the optimal patent val for 11-limit keen temperament.

Much like 100bddd, the 78ddd val can be used to construct an alternative to 22edo for pajara. The large and small step sizes in this case have ratio 4:3. The width of the tempered perfect fifth is 707.7 ¢. The major third is 384.6 ¢; less than two cents flat of just. The harmonic seventh is 984.6 ¢, or about 15.8 ¢ sharp; hence this tuning prioritizes the 3- and 5-limits over the 7-limit, while still ensuring that no basic 7-limit intervals other than the tritones are more than 16 ¢ off. The 22-note 2MOS generated in this way could be used to build straight-fretted guitars that would be tuned in tritones. The appeal of this scale is that it is less xenharmonic than 22edo is, for listeners accustomed to 12edo. In particular, the 163.6 ¢ "flat minor whole tone" of 22edo is now 169.2 ¢, making it more clearly a whole tone (albeit noticeably flat), rather than a neutral second.

Intervals

# Cents Diatonic interval category
0 0.0 perfect unison
1 15.4 superunison
2 30.8 superunison
3 46.2 subminor second
4 61.5 subminor second
5 76.9 subminor second
6 92.3 minor second
7 107.7 minor second
8 123.1 supraminor second
9 138.5 supraminor second
10 153.8 neutral second
11 169.2 submajor second
12 184.6 major second
13 200.0 major second
14 215.4 major second
15 230.8 supermajor second
16 246.2 ultramajor second
17 261.5 subminor third
18 276.9 subminor third
19 292.3 minor third
20 307.7 minor third
21 323.1 supraminor third
22 338.5 supraminor third
23 353.8 neutral third
24 369.2 submajor third
25 384.6 major third
26 400.0 major third
27 415.4 major third
28 430.8 supermajor third
29 446.2 ultramajor third
30 461.5 subfourth
31 476.9 subfourth
32 492.3 perfect fourth
33 507.7 perfect fourth
34 523.1 superfourth
35 538.5 superfourth
36 553.8 superfourth
37 569.2 low tritone
38 584.6 low tritone
39 600.0 high tritone
40 615.4 high tritone
41 630.8 high tritone
42 646.2 subfifth
43 661.5 subfifth
44 676.9 subfifth
45 692.3 perfect fifth
46 707.7 perfect fifth
47 723.1 superfifth
48 738.5 superfifth
49 753.8 ultrafifth
50 769.2 subminor sixth
51 784.6 minor sixth
52 800.0 minor sixth
53 815.4 minor sixth
54 830.8 supraminor sixth
55 846.2 neutral sixth
56 861.5 submajor sixth
57 876.9 submajor sixth
58 892.3 major sixth
59 907.7 major sixth
60 923.1 supermajor sixth
61 938.5 supermajor sixth
62 953.8 ultramajor sixth
63 969.2 subminor seventh
64 984.6 minor seventh
65 1000.0 minor seventh
66 1015.4 minor seventh
67 1030.8 supraminor seventh
68 1046.2 neutral seventh
69 1061.5 submajor seventh
70 1076.9 submajor seventh
71 1092.3 major seventh
72 1107.7 major seventh
73 1123.1 supermajor seventh
74 1138.5 supermajor seventh
75 1153.8 ultramajor seventh
76 1169.2 suboctave
77 1184.6 suboctave
78 1200.0 perfect octave
Retrieved from "https://en.xen.wiki/index.php?title=78edo&oldid=105181"