38edo: Difference between revisions

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{{Infobox ET
| Prime factorization = 2 x 19
| Step size = 31.579¢
| Fifth = 22\19 = 694.737¢
| Major 2nd = 6\38 = 189¢
| Minor 2nd = 4\38 = 126¢
| Augmented 1sn = 2\38 = 63¢
}}
'''38edo''' divides the octave into 38 equal parts of 31.578947 [[cent]]s. Since 38 = 2*19, it can be thought of as two parallel [[19edo]]s. It [[tempers out]] the same [[5-limit]] commas as 19edo, namely [[81/80]], [[3125/3072]] and [[15625/15552]]. In the [[7-limit]], we can add [[50/49]], and tempering out 81/80 and 50/49 gives [[injera]] temperament, for which 38 is the [[optimal patent val]]. In the [[11-limit]], we can add 121/120 and 176/175.
'''38edo''' divides the octave into 38 equal parts of 31.578947 [[cent]]s. Since 38 = 2*19, it can be thought of as two parallel [[19edo]]s. It [[tempers out]] the same [[5-limit]] commas as 19edo, namely [[81/80]], [[3125/3072]] and [[15625/15552]]. In the [[7-limit]], we can add [[50/49]], and tempering out 81/80 and 50/49 gives [[injera]] temperament, for which 38 is the [[optimal patent val]]. In the [[11-limit]], we can add 121/120 and 176/175.
{{Primes in edo|38}}
{{Primes in edo|38}}
Line 8: Line 17:
! Step
! Step
! Cents
! Cents
! colspan="3" | [[Ups and Downs Notation]]*
|-
|-
| 0
| 0
| 0.0000
| 0.0000
| Perfect 1sn
| P1
| D
|-
|-
| 1
| 1
| 31.5789
| 31.5789
| Up 1sn
| ^1
| ^D
|-
|-
| 2
| 2
| 63.1579
| 63.1579
| Aug 1sn, dim 2nd
| A1, d2
| D#, Ebb
|-
|-
| 3
| 3
| 94.7368
| 94.7368
| Upaug 1sn, downminor 2nd
| ^A1, vm2
| ^D#, vEb
|-
|-
| 4
| 4
| 126.3157
| 126.3157
| Minor 2nd
| m2
| Eb
|-
|-
| 5
| 5
| 157.8947
| 157.8947
| Mid 2nd
| ~2
| vE
|-
|-
| 6
| 6
| 189.4737
| 189.4737
| Major 2nd
| M2
| E
|-
|-
| 7
| 7
| 221.0526
| 221.0526
| Upmajor 2nd
| ^M2
| ^E
|-
|-
| 8
| 8
| 252.6316
| 252.6316
| Aug 2nd, Dim 3rd
| A2, d3
| E#, Fb
|-
|-
| 9
| 9
| 284.2105
| 284.2105
| Downminor 3rd
| vm3
| vF
|-
|-
| 10
| 10
| 315.7895
| 315.7895
| Minor 3rd
| m3
| F
|-
|-
| 11
| 11
| 347.3684
| 347.3684
| Mid 3rd
| ~3
| ^F
|-
|-
| 12
| 12
| 378.9474
| 378.9474
| Major 3rd
| M3
| F#
|-
|-
| 13
| 13
| 410.5263
| 410.5263
| Upmajor 3rd, Downdim 4th
| ^M3, vd4
| ^F#, vGb
|-
|-
| 14
| 14
| 442.1053
| 442.1053
| Aug 3rd, dim 4th
| A3, d4
| Fx, Gb
|-
|-
| 15
| 15
| 473.6843
| 473.6843
| Down 4th
| v4
| vG
|-
|-
| 16
| 16
| 505.2632
| 505.2632
| Perfect 4th
| P4
| G
|-
|-
| 17
| 17
| 536.8421
| 536.8421
| Up 4th
| ^4
| ^G
|-
|-
| 18
| 18
| 568.4211
| 568.4211
| Aug 4th
| A4
| G#
|-
|-
| 19
| 19
| 600.0000
| 600.0000
| Upaug 4th, downdim 5th
| ^A4, vd5
| ^G#, vAb
|-
|-
| 20
| 20
| 631.5789
| 631.5789
| Dim 5th
| d5
| Ab
|-
|-
| 21
| 21
| 663.1579
| 663.1579
| Down 5th
| v5
| vA
|-
|-
| 22
| 22
| 694.7368
| 694.7368
| Perfect 5th
| P5
| A
|-
|-
| 23
| 23
| 726.3157
| 726.3157
| Up 5th
| ^5
| ^A
|-
|-
| 24
| 24
| 757.8947
| 757.8947
| Aug 5th, dim 6th
| A5, d6
| A#, Bbb
|-
|-
| 25
| 25
| 789.4737
| 789.4737
| Upaug 5th, downminor 6th
| ^A5, vm6
| ^A#, vBb
|-
|-
| 26
| 26
| 821.0526
| 821.0526
| Minor 6th
| m6
| Bb
|-
|-
| 27
| 27
| 852.6316
| 852.6316
| Mid 6th
| ~6
| Bv
|-
|-
| 28
| 28
| 884.2105
| 884.2105
| Major 6th
| M6
| B
|-
|-
| 29
| 29
| 915.7895
| 915.7895
| Upmajor 6th
| ^M6
| ^B
|-
|-
| 30
| 30
| 947.3684
| 947.3684
| Aug 6th, dim 7th
| A6, d7
| B#, Cb
|-
|-
| 31
| 31
| 978.9474
| 978.9474
| Downminor 7th
| vm7
| Cv
|-
|-
| 32
| 32
| 1010.5263
| 1010.5263
| Minor 7th
| m7
| C
|-
|-
| 33
| 33
| 1042.1053
| 1042.1053
| Mid 7th
| ~7
| C^
|-
|-
| 34
| 34
| 1073.6843
| 1073.6843
| Major 7th
| M7
| C#
|-
|-
| 35
| 35
| 1105.2632
| 1105.2632
| Upmajor 7th, Downdim 8ve
| ^M7, vd8
| ^C#, vDb
|-
|-
| 36
| 36
| 1136.8421
| 1136.8421
| Aug 7th, dim 8ve
| A7, d8
| Cx, Db
|-
|-
| 37
| 37
| 1168.4211
| 1168.4211
| Down 8ve
| v8
| vD
|-
|-
| 38
| 38
| 1200.0000
| 1200.0000
| Perfect 8ve
| P8
| D
|}
|}
 
<nowiki>*</nowiki> Ups and downs may be substituted with semi-sharps and semi-flats, respectively
[[Category:Theory]]
[[Category:Theory]]
[[Category:Equal divisions of the octave]]
[[Category:Equal divisions of the octave]]
[[Category:38edo| ]]  <!-- main article -->
[[Category:38edo| ]]  <!-- main article -->

Revision as of 17:27, 26 February 2021

← 37edo 38edo 39edo →
Prime factorization 2 x 19
Step size 31.5789 ¢ 
Fifth 22\38 (694.737 ¢) (→ 11\19)
Semitones (A1:m2) 2:4 (63.16 ¢ : 126.3 ¢)
Consistency limit 5
Distinct consistency limit 5

38edo divides the octave into 38 equal parts of 31.578947 cents. Since 38 = 2*19, it can be thought of as two parallel 19edos. It tempers out the same 5-limit commas as 19edo, namely 81/80, 3125/3072 and 15625/15552. In the 7-limit, we can add 50/49, and tempering out 81/80 and 50/49 gives injera temperament, for which 38 is the optimal patent val. In the 11-limit, we can add 121/120 and 176/175. Script error: No such module "primes_in_edo".

Intervals

Step Cents Ups and Downs Notation*
0 0.0000 Perfect 1sn P1 D
1 31.5789 Up 1sn ^1 ^D
2 63.1579 Aug 1sn, dim 2nd A1, d2 D#, Ebb
3 94.7368 Upaug 1sn, downminor 2nd ^A1, vm2 ^D#, vEb
4 126.3157 Minor 2nd m2 Eb
5 157.8947 Mid 2nd ~2 vE
6 189.4737 Major 2nd M2 E
7 221.0526 Upmajor 2nd ^M2 ^E
8 252.6316 Aug 2nd, Dim 3rd A2, d3 E#, Fb
9 284.2105 Downminor 3rd vm3 vF
10 315.7895 Minor 3rd m3 F
11 347.3684 Mid 3rd ~3 ^F
12 378.9474 Major 3rd M3 F#
13 410.5263 Upmajor 3rd, Downdim 4th ^M3, vd4 ^F#, vGb
14 442.1053 Aug 3rd, dim 4th A3, d4 Fx, Gb
15 473.6843 Down 4th v4 vG
16 505.2632 Perfect 4th P4 G
17 536.8421 Up 4th ^4 ^G
18 568.4211 Aug 4th A4 G#
19 600.0000 Upaug 4th, downdim 5th ^A4, vd5 ^G#, vAb
20 631.5789 Dim 5th d5 Ab
21 663.1579 Down 5th v5 vA
22 694.7368 Perfect 5th P5 A
23 726.3157 Up 5th ^5 ^A
24 757.8947 Aug 5th, dim 6th A5, d6 A#, Bbb
25 789.4737 Upaug 5th, downminor 6th ^A5, vm6 ^A#, vBb
26 821.0526 Minor 6th m6 Bb
27 852.6316 Mid 6th ~6 Bv
28 884.2105 Major 6th M6 B
29 915.7895 Upmajor 6th ^M6 ^B
30 947.3684 Aug 6th, dim 7th A6, d7 B#, Cb
31 978.9474 Downminor 7th vm7 Cv
32 1010.5263 Minor 7th m7 C
33 1042.1053 Mid 7th ~7 C^
34 1073.6843 Major 7th M7 C#
35 1105.2632 Upmajor 7th, Downdim 8ve ^M7, vd8 ^C#, vDb
36 1136.8421 Aug 7th, dim 8ve A7, d8 Cx, Db
37 1168.4211 Down 8ve v8 vD
38 1200.0000 Perfect 8ve P8 D

* Ups and downs may be substituted with semi-sharps and semi-flats, respectively

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