38edo: Difference between revisions

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{{EDO intro|38}}
{{EDO intro|38}}


== Theory ==
==Theory==
Since 38 = 2*19, it can be thought of as two parallel [[19edo]]s. While the halving of the step size lowers [[consistency]] and leaves it only mediocre in terms of overall [[Relative_errors_of_small_EDOs|relative error]], the fact that the 3rd & 5th harmonics are flat by almost exactly the same amount, while the 11th is double that means there are quite a few near perfect composite ratios, such as the the [[6/5]] it shares with 19edo, plus [[11/9]], [[15/11]] & [[25/22]], (and their inversions) while a single step nears [[55/54]]. This gives several interesting possibilities for unusual near-just chords such as 15:18:22:25:30. 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.  
Since 38 = 2*19, it can be thought of as two parallel [[19edo]]s. While the halving of the step size lowers [[consistency]] and leaves it only mediocre in terms of overall [[Relative_errors_of_small_EDOs|relative error]], the fact that the 3rd & 5th harmonics are flat by almost exactly the same amount, while the 11th is double that means there are quite a few near perfect composite ratios, such as the the [[6/5]] it shares with 19edo, plus [[11/9]], [[15/11]] & [[25/22]], (and their inversions) while a single step nears [[55/54]]. This gives several interesting possibilities for unusual near-just chords such as 15:18:22:25:30. 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.  


Line 14: Line 14:


== Intervals ==
== Intervals ==
{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
|-
|-
! Step
! Step
! Cents
! Cents
! 19-odd-limit ratios,<br />treated as 38df
!19-odd-limit ratios,
treated as 38df
! colspan="3" | [[Ups and Downs Notation]]*
! colspan="3" | [[Ups and Downs Notation]]*
|-
|-
| 0
| 0
| 0.0000
| 0.0000
|
| Perfect 1sn
| Perfect 1sn
| P1
| P1
Line 29: Line 32:
| 1
| 1
| 31.5789
| 31.5789
|
| Up 1sn
| Up 1sn
| ^1
| ^1
Line 35: Line 39:
| 2
| 2
| 63.1579
| 63.1579
|
| Aug 1sn, dim 2nd
| Aug 1sn, dim 2nd
| A1, d2
| A1, d2
Line 41: Line 46:
| 3
| 3
| 94.7368
| 94.7368
| [[20/19]], [[19/18]], [[18/17]], [[17/16]]
|[[20/19]], [[19/18]], [[18/17]], [[17/16]]
| Upaug 1sn, downminor 2nd
| Upaug 1sn, downminor 2nd
| ^A1, vm2
| ^A1, vm2
Line 48: Line 53:
| 4
| 4
| 126.3157
| 126.3157
| [[16/15]], [[15/14]], [[14/13]], [[13/12]]
|[[16/15]], [[15/14]], [[14/13]], [[13/12]]
| Minor 2nd
| Minor 2nd
| m2
| m2
Line 55: Line 60:
| 5
| 5
| 157.8947
| 157.8947
| [[12/11]], [[11/10]]
|[[12/11]], [[11/10]]
| Mid 2nd
| Mid 2nd
| ~2
| ~2
Line 62: Line 67:
| 6
| 6
| 189.4737
| 189.4737
| [[10/9]], [[19/17]], [[9/8]]
|[[10/9]], [[19/17]], [[9/8]]
| Major 2nd
| Major 2nd
| M2
| M2
Line 69: Line 74:
| 7
| 7
| 221.0526
| 221.0526
| [[17/15]]
|[[17/15]]
| Upmajor 2nd
| Upmajor 2nd
| ^M2
| ^M2
Line 76: Line 81:
| 8
| 8
| 252.6316
| 252.6316
| [[8/7]], [[15/13]], [[22/19]], [[7/6]]
|[[8/7]], [[15/13]], [[22/19]], [[7/6]]
| Aug 2nd, Dim 3rd
| Aug 2nd, Dim 3rd
| A2, d3
| A2, d3
Line 83: Line 88:
| 9
| 9
| 284.2105
| 284.2105
| [[20/17]], [[13/11]], [[19/16]]
|[[20/17]], [[13/11]], [[19/16]]
| Downminor 3rd
| Downminor 3rd
| vm3
| vm3
Line 90: Line 95:
| 10
| 10
| 315.7895
| 315.7895
| [[6/5]]
|[[6/5]]
| Minor 3rd
| Minor 3rd
| m3
| m3
Line 97: Line 102:
| 11
| 11
| 347.3684
| 347.3684
| [[17/14]], [[11/9]]
|[[17/14]], [[11/9]]
| Mid 3rd
| Mid 3rd
| ~3
| ~3
Line 104: Line 109:
| 12
| 12
| 378.9474
| 378.9474
| [[16/13]], [[5/4]]
|[[16/13]], [[5/4]]
| Major 3rd
| Major 3rd
| M3
| M3
Line 111: Line 116:
| 13
| 13
| 410.5263
| 410.5263
| [[24/19]], [[19/15]], [[14/11]]
|[[24/19]], [[19/15]], [[14/11]]
| Upmajor 3rd, Downdim 4th
| Upmajor 3rd, Downdim 4th
| ^M3, vd4
| ^M3, vd4
Line 118: Line 123:
| 14
| 14
| 442.1053
| 442.1053
| [[9/7]], [[22/17]], [[13/10]]
|[[9/7]], [[22/17]], [[13/10]]
| Aug 3rd, dim 4th
| Aug 3rd, dim 4th
| A3, d4
| A3, d4
Line 125: Line 130:
| 15
| 15
| 473.6843
| 473.6843
| [[17/13]]
|[[17/13]]
| Down 4th
| Down 4th
| v4
| v4
Line 132: Line 137:
| 16
| 16
| 505.2632
| 505.2632
| [[4/3]]
|[[4/3]]
| Perfect 4th
| Perfect 4th
| P4
| P4
Line 139: Line 144:
| 17
| 17
| 536.8421
| 536.8421
| [[19/14]], [[15/11]], [[26/19]], [[11/8]]
|[[19/14]], [[15/11]], [[26/19]], [[11/8]]
| Up 4th
| Up 4th
| ^4
| ^4
Line 146: Line 151:
| 18
| 18
| 568.4211
| 568.4211
| [[18/13]], [[7/5]]
|[[18/13]], [[7/5]]
| Aug 4th
| Aug 4th
| A4
| A4
Line 153: Line 158:
| 19
| 19
| 600.0000
| 600.0000
| [[24/17]], [[17/12]]
|[[24/17]], [[17/12]]
| Upaug 4th, downdim 5th
| Upaug 4th, downdim 5th
| ^A4, vd5
| ^A4, vd5
Line 160: Line 165:
| 20
| 20
| 631.5789
| 631.5789
| [[10/7]], [[13/9]]
|[[10/7]], [[13/9]]
| Dim 5th
| Dim 5th
| d5
| d5
Line 167: Line 172:
| 21
| 21
| 663.1579
| 663.1579
| [[16/11]], [[19/13]], [[22/15]], [[28/19]]
|[[16/11]], [[19/13]], [[22/15]], [[28/19]]
| Down 5th
| Down 5th
| v5
| v5
Line 174: Line 179:
| 22
| 22
| 694.7368
| 694.7368
| [[3/2]]
|[[3/2]]
| Perfect 5th
| Perfect 5th
| P5
| P5
Line 181: Line 186:
| 23
| 23
| 726.3157
| 726.3157
| [[26/17]]
|[[26/17]]
| Up 5th
| Up 5th
| ^5
| ^5
Line 188: Line 193:
| 24
| 24
| 757.8947
| 757.8947
| [[20/13]], [[17/11]], [[14/9]]
|[[20/13]], [[17/11]], [[14/9]]
| Aug 5th, dim 6th
| Aug 5th, dim 6th
| A5, d6
| A5, d6
Line 195: Line 200:
| 25
| 25
| 789.4737
| 789.4737
| [[11/7]], [[30/19]], [[19/12]]
|[[11/7]], [[30/19]], [[19/12]]
| Upaug 5th, downminor 6th
| Upaug 5th, downminor 6th
| ^A5, vm6
| ^A5, vm6
Line 202: Line 207:
| 26
| 26
| 821.0526
| 821.0526
| [[8/5]], [[13/8]]
|[[8/5]], [[13/8]]
| Minor 6th
| Minor 6th
| m6
| m6
Line 209: Line 214:
| 27
| 27
| 852.6316
| 852.6316
| [[18/11]], [[28/17]]
|[[18/11]], [[28/17]]
| Mid 6th
| Mid 6th
| ~6
| ~6
Line 216: Line 221:
| 28
| 28
| 884.2105
| 884.2105
| [[5/3]]
|[[5/3]]
| Major 6th
| Major 6th
| M6
| M6
Line 223: Line 228:
| 29
| 29
| 915.7895
| 915.7895
| [[32/19]], [[22/13]], [[17/10]]
|[[32/19]], [[22/13]], [[17/10]]
| Upmajor 6th
| Upmajor 6th
| ^M6
| ^M6
Line 230: Line 235:
| 30
| 30
| 947.3684
| 947.3684
| [[12/7]], [[19/11]], [[26/15]], [[7/4]]
|[[12/7]], [[19/11]], [[26/15]], [[7/4]]
| Aug 6th, dim 7th
| Aug 6th, dim 7th
| A6, d7
| A6, d7
Line 237: Line 242:
| 31
| 31
| 978.9474
| 978.9474
| [[30/17]]
|[[30/17]]
| Downminor 7th
| Downminor 7th
| vm7
| vm7
Line 244: Line 249:
| 32
| 32
| 1010.5263
| 1010.5263
| [[16/9]], [[34/19]], [[9/5]]
|[[16/9]], [[34/19]], [[9/5]]
| Minor 7th
| Minor 7th
| m7
| m7
Line 251: Line 256:
| 33
| 33
| 1042.1053
| 1042.1053
| [[20/11]], [[11/6]]
|[[20/11]], [[11/6]]
| Mid 7th
| Mid 7th
| ~7
| ~7
Line 258: Line 263:
| 34
| 34
| 1073.6843
| 1073.6843
| [[24/13]], [[13/7]], [[28/15]], [[15/8]]
|[[24/13]], [[13/7]], [[28/15]], [[15/8]]
| Major 7th
| Major 7th
| M7
| M7
Line 265: Line 270:
| 35
| 35
| 1105.2632
| 1105.2632
| [[32/17]], [[17/9]], [[36/19]], [[19/10]]
|[[32/17]], [[17/9]], [[36/19]], [[19/10]]
| Upmajor 7th, Downdim 8ve
| Upmajor 7th, Downdim 8ve
| ^M7, vd8
| ^M7, vd8
Line 272: Line 277:
| 36
| 36
| 1136.8421
| 1136.8421
|
| Aug 7th, dim 8ve
| Aug 7th, dim 8ve
| A7, d8
| A7, d8
Line 278: Line 284:
| 37
| 37
| 1168.4211
| 1168.4211
|
| Down 8ve
| Down 8ve
| v8
| v8
Line 284: Line 291:
| 38
| 38
| 1200.0000
| 1200.0000
|
| Perfect 8ve
| Perfect 8ve
| P8
| P8
| D
| D
|}
|}
<nowiki />* Ups and downs may be substituted with semi-sharps and semi-flats, respectively.
<nowiki>*</nowiki> Ups and downs may be substituted with semi-sharps and semi-flats, respectively


== Instruments ==
== Instruments ==
* [[Lumatone mapping for 38edo]]
*[[Lumatone mapping for 38edo]]
* [[Skip fretting system 38 2 11]]
*[[Skip fretting system 38 2 11]]


== Music ==
== Music ==
* [https://www.youtube.com/watch?v=Cw1Cz1ojoSw Canon at the Semitone on The Mother's Malison Theme for Cor Anglais and Violin] by [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=Cw1Cz1ojoSw Canon at the Semitone on The Mother's Malison Theme for Cor Anglais and Violin] by [[Claudi Meneghin]]


[[Category:38edo| ]]  <!-- Main article -->
[[Category:38edo| ]]  <!-- main article -->
[[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->
[[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->
[[Category:Listen]]
[[Category:Listen]]
[[Category:Todo:add rank 2 temperaments table]]
[[Category:Todo:add rank 2 temperaments table]]

Revision as of 11:20, 27 October 2024

← 37edo 38edo 39edo →
Prime factorization 2 × 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

Template:EDO intro

Theory

Since 38 = 2*19, it can be thought of as two parallel 19edos. While the halving of the step size lowers consistency and leaves it only mediocre in terms of overall relative error, the fact that the 3rd & 5th harmonics are flat by almost exactly the same amount, while the 11th is double that means there are quite a few near perfect composite ratios, such as the the 6/5 it shares with 19edo, plus 11/9, 15/11 & 25/22, (and their inversions) while a single step nears 55/54. This gives several interesting possibilities for unusual near-just chords such as 15:18:22:25:30. 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.

In 38df, every prime interval from 3 to 19 is characterized by a flat intonation. Furthermore, the mapping of all 19-odd-limit intervals in 38df aligns with their closest approximations in 38edo, excepting for 7/4 and 13/8, along with their octave complements 8/7 and 16/13, which are by definition mapped to their secondary optimal steps within 38df. In other words, all 19-odd-limit intervals are consistent within the 38df val ⟨38 60 88 106 131 140 155 161].

The harmonic series from 1 to 20 is approximated within 38df by the sequence: 38 22 16 12 10 8 8 6 6 5 5 4 4 4 4 3 3 3 3

[Harmonic series 2-20 in 38df]


Approximation of prime harmonics in 38edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.0 -7.2 -7.4 +10.1 -14.5 +12.1 -10.2 -13.3 +3.3 +12.5 -8.2
Relative (%) +0.0 -22.9 -23.3 +32.1 -45.8 +38.3 -32.4 -42.1 +10.5 +39.7 -25.9
Steps
(reduced) 		38
(0) 		60
(22) 		88
(12) 		107
(31) 		131
(17) 		141
(27) 		155
(3) 		161
(9) 		172
(20) 		185
(33) 		188
(36)

Intervals

Step Cents 19-odd-limit ratios,

treated as 38df

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

Instruments

Music

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