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 {{nowrap|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.  


In [[Warts|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 [[Consistency|consistent]] within the 38df [[val]] ⟨38 60 88 106 131 140 155 161].  
In [[Warts|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 [[Consistency|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
The harmonic series from 1 to 20 is approximated within 38df by the sequence: {{nowrap|38 22 16 12 10 8 8 6 6 5 5 4 4 4 4 3 3 3 3}}


[[File:Harmonic_series_38df.mp3]] [[:File:Harmonic_series_38df.mp3|[Harmonic series 2-20 in 38df]]]
[[File:Harmonic_series_38df.mp3]] [[:File:Harmonic_series_38df.mp3|[Harmonic series 2-20 in 38df]]]
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,
! 19-odd-limit ratios,<br />treated as 38df
treated as 38df
! colspan="3" | [[Ups and Downs Notation]]*
! colspan="3" | [[Ups and Downs Notation]]*
|-
|-
Line 46: Line 44:
| 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 53: Line 51:
| 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 60: Line 58:
| 5
| 5
| 157.8947
| 157.8947
|[[12/11]], [[11/10]]
| [[12/11]], [[11/10]]
| Mid 2nd
| Mid 2nd
| ~2
| ~2
Line 67: Line 65:
| 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 74: Line 72:
| 7
| 7
| 221.0526
| 221.0526
|[[17/15]]
| [[17/15]]
| Upmajor 2nd
| Upmajor 2nd
| ^M2
| ^M2
Line 81: Line 79:
| 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 88: Line 86:
| 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 95: Line 93:
| 10
| 10
| 315.7895
| 315.7895
|[[6/5]]
| [[6/5]]
| Minor 3rd
| Minor 3rd
| m3
| m3
Line 102: Line 100:
| 11
| 11
| 347.3684
| 347.3684
|[[17/14]], [[11/9]]
| [[17/14]], [[11/9]]
| Mid 3rd
| Mid 3rd
| ~3
| ~3
Line 109: Line 107:
| 12
| 12
| 378.9474
| 378.9474
|[[16/13]], [[5/4]]
| [[16/13]], [[5/4]]
| Major 3rd
| Major 3rd
| M3
| M3
Line 116: Line 114:
| 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 123: Line 121:
| 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 130: Line 128:
| 15
| 15
| 473.6843
| 473.6843
|[[17/13]]
| [[17/13]]
| Down 4th
| Down 4th
| v4
| v4
Line 137: Line 135:
| 16
| 16
| 505.2632
| 505.2632
|[[4/3]]
| [[4/3]]
| Perfect 4th
| Perfect 4th
| P4
| P4
Line 144: Line 142:
| 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 151: Line 149:
| 18
| 18
| 568.4211
| 568.4211
|[[18/13]], [[7/5]]
| [[18/13]], [[7/5]]
| Aug 4th
| Aug 4th
| A4
| A4
Line 158: Line 156:
| 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 165: Line 163:
| 20
| 20
| 631.5789
| 631.5789
|[[10/7]], [[13/9]]
| [[10/7]], [[13/9]]
| Dim 5th
| Dim 5th
| d5
| d5
Line 172: Line 170:
| 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 179: Line 177:
| 22
| 22
| 694.7368
| 694.7368
|[[3/2]]
| [[3/2]]
| Perfect 5th
| Perfect 5th
| P5
| P5
Line 186: Line 184:
| 23
| 23
| 726.3157
| 726.3157
|[[26/17]]
| [[26/17]]
| Up 5th
| Up 5th
| ^5
| ^5
Line 193: Line 191:
| 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 200: Line 198:
| 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 207: Line 205:
| 26
| 26
| 821.0526
| 821.0526
|[[8/5]], [[13/8]]
| [[8/5]], [[13/8]]
| Minor 6th
| Minor 6th
| m6
| m6
Line 214: Line 212:
| 27
| 27
| 852.6316
| 852.6316
|[[18/11]], [[28/17]]
| [[18/11]], [[28/17]]
| Mid 6th
| Mid 6th
| ~6
| ~6
Line 221: Line 219:
| 28
| 28
| 884.2105
| 884.2105
|[[5/3]]
| [[5/3]]
| Major 6th
| Major 6th
| M6
| M6
Line 228: Line 226:
| 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 235: Line 233:
| 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 242: Line 240:
| 31
| 31
| 978.9474
| 978.9474
|[[30/17]]
| [[30/17]]
| Downminor 7th
| Downminor 7th
| vm7
| vm7
Line 249: Line 247:
| 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 256: Line 254:
| 33
| 33
| 1042.1053
| 1042.1053
|[[20/11]], [[11/6]]
| [[20/11]], [[11/6]]
| Mid 7th
| Mid 7th
| ~7
| ~7
Line 263: Line 261:
| 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 270: Line 268:
| 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 296: Line 294:
| D
| D
|}
|}
<nowiki>*</nowiki> Ups and downs may be substituted with semi-sharps and semi-flats, respectively
<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 15:33, 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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