38edo: Difference between revisions

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m →Theory: cleanup
m →Intervals: reduce to one decimal place for cent values, per discussion on Discord
Line 19: Line 19:
! Step
! Step
! Cents
! Cents
! 19-odd-limit ratios,<br />treated as 38df
! 19-odd-limit ratios,<br>in 38df val
! colspan="3" | [[Ups and Downs Notation]]*
! colspan="3" | [[Ups and downs notation]]*
|-
|-
| 0
| 0
| 0.0000
| 0.0
|
|
| Perfect 1sn
| Perfect 1sn
Line 30: Line 30:
|-
|-
| 1
| 1
| 31.5789
| 31.6
|
|
| Up 1sn
| Up 1sn
Line 37: Line 37:
|-
|-
| 2
| 2
| 63.1579
| 63.2
|
|
| Aug 1sn, dim 2nd
| Aug 1sn, dim 2nd
Line 44: Line 44:
|-
|-
| 3
| 3
| 94.7368
| 94.7
| [[20/19]], [[19/18]], [[18/17]], [[17/16]]
| [[20/19]], [[19/18]], [[18/17]], [[17/16]]
| Upaug 1sn, downminor 2nd
| Upaug 1sn, downminor 2nd
Line 51: Line 51:
|-
|-
| 4
| 4
| 126.3157
| 126.3
| [[16/15]], [[15/14]], [[14/13]], [[13/12]]
| [[16/15]], [[15/14]], [[14/13]], [[13/12]]
| Minor 2nd
| Minor 2nd
Line 58: Line 58:
|-
|-
| 5
| 5
| 157.8947
| 157.9
| [[12/11]], [[11/10]]
| [[12/11]], [[11/10]]
| Mid 2nd
| Mid 2nd
Line 65: Line 65:
|-
|-
| 6
| 6
| 189.4737
| 189.5
| [[10/9]], [[19/17]], [[9/8]]
| [[10/9]], [[19/17]], [[9/8]]
| Major 2nd
| Major 2nd
Line 72: Line 72:
|-
|-
| 7
| 7
| 221.0526
| 221.1
| [[17/15]]
| [[17/15]]
| Upmajor 2nd
| Upmajor 2nd
Line 79: Line 79:
|-
|-
| 8
| 8
| 252.6316
| 252.6
| [[8/7]], [[15/13]], [[22/19]], [[7/6]]
| [[8/7]], [[15/13]], [[22/19]], [[7/6]]
| Aug 2nd, Dim 3rd
| Aug 2nd, Dim 3rd
Line 86: Line 86:
|-
|-
| 9
| 9
| 284.2105
| 284.2
| [[20/17]], [[13/11]], [[19/16]]
| [[20/17]], [[13/11]], [[19/16]]
| Downminor 3rd
| Downminor 3rd
Line 93: Line 93:
|-
|-
| 10
| 10
| 315.7895
| 315.8
| [[6/5]]
| [[6/5]]
| Minor 3rd
| Minor 3rd
Line 100: Line 100:
|-
|-
| 11
| 11
| 347.3684
| 347.4
| [[17/14]], [[11/9]]
| [[17/14]], [[11/9]]
| Mid 3rd
| Mid 3rd
Line 107: Line 107:
|-
|-
| 12
| 12
| 378.9474
| 378.9
| [[16/13]], [[5/4]]
| [[16/13]], [[5/4]]
| Major 3rd
| Major 3rd
Line 114: Line 114:
|-
|-
| 13
| 13
| 410.5263
| 410.5
| [[24/19]], [[19/15]], [[14/11]]
| [[24/19]], [[19/15]], [[14/11]]
| Upmajor 3rd, Downdim 4th
| Upmajor 3rd, Downdim 4th
Line 121: Line 121:
|-
|-
| 14
| 14
| 442.1053
| 442.1
| [[9/7]], [[22/17]], [[13/10]]
| [[9/7]], [[22/17]], [[13/10]]
| Aug 3rd, dim 4th
| Aug 3rd, dim 4th
Line 128: Line 128:
|-
|-
| 15
| 15
| 473.6843
| 473.7
| [[17/13]]
| [[17/13]]
| Down 4th
| Down 4th
Line 135: Line 135:
|-
|-
| 16
| 16
| 505.2632
| 505.3
| [[4/3]]
| [[4/3]]
| Perfect 4th
| Perfect 4th
Line 142: Line 142:
|-
|-
| 17
| 17
| 536.8421
| 536.8
| [[19/14]], [[15/11]], [[26/19]], [[11/8]]
| [[19/14]], [[15/11]], [[26/19]], [[11/8]]
| Up 4th
| Up 4th
Line 149: Line 149:
|-
|-
| 18
| 18
| 568.4211
| 568.4
| [[18/13]], [[7/5]]
| [[18/13]], [[7/5]]
| Aug 4th
| Aug 4th
Line 156: Line 156:
|-
|-
| 19
| 19
| 600.0000
| 600.0
| [[24/17]], [[17/12]]
| [[24/17]], [[17/12]]
| Upaug 4th, downdim 5th
| Upaug 4th, downdim 5th
Line 163: Line 163:
|-
|-
| 20
| 20
| 631.5789
| 631.6
| [[10/7]], [[13/9]]
| [[10/7]], [[13/9]]
| Dim 5th
| Dim 5th
Line 170: Line 170:
|-
|-
| 21
| 21
| 663.1579
| 663.2
| [[16/11]], [[19/13]], [[22/15]], [[28/19]]
| [[16/11]], [[19/13]], [[22/15]], [[28/19]]
| Down 5th
| Down 5th
Line 177: Line 177:
|-
|-
| 22
| 22
| 694.7368
| 694.7
| [[3/2]]
| [[3/2]]
| Perfect 5th
| Perfect 5th
Line 184: Line 184:
|-
|-
| 23
| 23
| 726.3157
| 726.3
| [[26/17]]
| [[26/17]]
| Up 5th
| Up 5th
Line 191: Line 191:
|-
|-
| 24
| 24
| 757.8947
| 757.9
| [[20/13]], [[17/11]], [[14/9]]
| [[20/13]], [[17/11]], [[14/9]]
| Aug 5th, dim 6th
| Aug 5th, dim 6th
Line 198: Line 198:
|-
|-
| 25
| 25
| 789.4737
| 789.5
| [[11/7]], [[30/19]], [[19/12]]
| [[11/7]], [[30/19]], [[19/12]]
| Upaug 5th, downminor 6th
| Upaug 5th, downminor 6th
Line 205: Line 205:
|-
|-
| 26
| 26
| 821.0526
| 821.1
| [[8/5]], [[13/8]]
| [[8/5]], [[13/8]]
| Minor 6th
| Minor 6th
Line 212: Line 212:
|-
|-
| 27
| 27
| 852.6316
| 852.6
| [[18/11]], [[28/17]]
| [[18/11]], [[28/17]]
| Mid 6th
| Mid 6th
Line 219: Line 219:
|-
|-
| 28
| 28
| 884.2105
| 884.2
| [[5/3]]
| [[5/3]]
| Major 6th
| Major 6th
Line 226: Line 226:
|-
|-
| 29
| 29
| 915.7895
| 915.8
| [[32/19]], [[22/13]], [[17/10]]
| [[32/19]], [[22/13]], [[17/10]]
| Upmajor 6th
| Upmajor 6th
Line 233: Line 233:
|-
|-
| 30
| 30
| 947.3684
| 947.4
| [[12/7]], [[19/11]], [[26/15]], [[7/4]]
| [[12/7]], [[19/11]], [[26/15]], [[7/4]]
| Aug 6th, dim 7th
| Aug 6th, dim 7th
Line 240: Line 240:
|-
|-
| 31
| 31
| 978.9474
| 978.9
| [[30/17]]
| [[30/17]]
| Downminor 7th
| Downminor 7th
Line 247: Line 247:
|-
|-
| 32
| 32
| 1010.5263
| 1010.5
| [[16/9]], [[34/19]], [[9/5]]
| [[16/9]], [[34/19]], [[9/5]]
| Minor 7th
| Minor 7th
Line 254: Line 254:
|-
|-
| 33
| 33
| 1042.1053
| 1042.1
| [[20/11]], [[11/6]]
| [[20/11]], [[11/6]]
| Mid 7th
| Mid 7th
Line 261: Line 261:
|-
|-
| 34
| 34
| 1073.6843
| 1073.7
| [[24/13]], [[13/7]], [[28/15]], [[15/8]]
| [[24/13]], [[13/7]], [[28/15]], [[15/8]]
| Major 7th
| Major 7th
Line 268: Line 268:
|-
|-
| 35
| 35
| 1105.2632
| 1105.3
| [[32/17]], [[17/9]], [[36/19]], [[19/10]]
| [[32/17]], [[17/9]], [[36/19]], [[19/10]]
| Upmajor 7th, Downdim 8ve
| Upmajor 7th, Downdim 8ve
Line 275: Line 275:
|-
|-
| 36
| 36
| 1136.8421
| 1136.8
|
|
| Aug 7th, dim 8ve
| Aug 7th, dim 8ve
Line 282: Line 282:
|-
|-
| 37
| 37
| 1168.4211
| 1168.4
|
|
| Down 8ve
| Down 8ve
Line 289: Line 289:
|-
|-
| 38
| 38
| 1200.0000
| 1200.0
|
|
| Perfect 8ve
| Perfect 8ve
Line 295: Line 295:
| D
| D
|}
|}
<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


== Notation ==
== Notation ==

Revision as of 14:50, 17 January 2025

← 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; the approximation to 11/9 in particular should be noted for forming a 10-strong consistent circle. 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]

Prime harmonics

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,
in 38df val 		Ups and downs notation*
0 0.0 Perfect 1sn P1 D
1 31.6 Up 1sn ^1 ^D
2 63.2 Aug 1sn, dim 2nd A1, d2 D#
3 94.7 20/19, 19/18, 18/17, 17/16 Upaug 1sn, downminor 2nd ^A1, vm2 ^D#, vEb
4 126.3 16/15, 15/14, 14/13, 13/12 Minor 2nd m2 Eb
5 157.9 12/11, 11/10 Mid 2nd ~2 vE
6 189.5 10/9, 19/17, 9/8 Major 2nd M2 E
7 221.1 17/15 Upmajor 2nd ^M2 ^E
8 252.6 8/7, 15/13, 22/19, 7/6 Aug 2nd, Dim 3rd A2, d3 E#, Fb
9 284.2 20/17, 13/11, 19/16 Downminor 3rd vm3 vF
10 315.8 6/5 Minor 3rd m3 F
11 347.4 17/14, 11/9 Mid 3rd ~3 ^F
12 378.9 16/13, 5/4 Major 3rd M3 F#
13 410.5 24/19, 19/15, 14/11 Upmajor 3rd, Downdim 4th ^M3, vd4 ^F#, vGb
14 442.1 9/7, 22/17, 13/10 Aug 3rd, dim 4th A3, d4 Gb
15 473.7 17/13 Down 4th v4 vG
16 505.3 4/3 Perfect 4th P4 G
17 536.8 19/14, 15/11, 26/19, 11/8 Up 4th ^4 ^G
18 568.4 18/13, 7/5 Aug 4th A4 G#
19 600.0 24/17, 17/12 Upaug 4th, downdim 5th ^A4, vd5 ^G#, vAb
20 631.6 10/7, 13/9 Dim 5th d5 Ab
21 663.2 16/11, 19/13, 22/15, 28/19 Down 5th v5 vA
22 694.7 3/2 Perfect 5th P5 A
23 726.3 26/17 Up 5th ^5 ^A
24 757.9 20/13, 17/11, 14/9 Aug 5th, dim 6th A5, d6 A#
25 789.5 11/7, 30/19, 19/12 Upaug 5th, downminor 6th ^A5, vm6 ^A#, vBb
26 821.1 8/5, 13/8 Minor 6th m6 Bb
27 852.6 18/11, 28/17 Mid 6th ~6 vB
28 884.2 5/3 Major 6th M6 B
29 915.8 32/19, 22/13, 17/10 Upmajor 6th ^M6 ^B
30 947.4 12/7, 19/11, 26/15, 7/4 Aug 6th, dim 7th A6, d7 B#, Cb
31 978.9 30/17 Downminor 7th vm7 vC
32 1010.5 16/9, 34/19, 9/5 Minor 7th m7 C
33 1042.1 20/11, 11/6 Mid 7th ~7 ^C
34 1073.7 24/13, 13/7, 28/15, 15/8 Major 7th M7 C#
35 1105.3 32/17, 17/9, 36/19, 19/10 Upmajor 7th, Downdim 8ve ^M7, vd8 ^C#, vDb
36 1136.8 Aug 7th, dim 8ve A7, d8 Db
37 1168.4 Down 8ve v8 vD
38 1200.0 Perfect 8ve P8 D

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

Notation

Sagittal notation

This notation uses the same sagittal sequence as EDOs 17, 24, and 31, is a subset of the notation for 76-EDO, and is a superset of the notation for 19-EDO.

Evo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation33/32

Revo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation33/32

Evo-SZ flavor

Sagittal notationPeriodic table of EDOs with sagittal notation33/32

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein-Zimmerman notation.

Instruments

Music

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