38edo: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Contribution (talk | contribs)
2,880 edits
No edit summary
Sintel (talk | contribs)
autopatrolled, emailconfirmed
1,548 edits
→Approximation to JI: -zeta peak index
 
(34 intermediate revisions by 9 users not shown)
Line 1: Line 1:
{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|38}}
{{ED intro}}


==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 interval error|relative error]], the fact that the 3rd and 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.  


In [[Wart notation|38df]], all primes are flat from 3 to 19.  
It [[tempering out|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, the mapping of all [[19-odd-limit]] intervals is identical to their direct best approximation in 38edo, excepting, of course, for 7/4 and 13/8, as well as their octave complements 8/7 and 16/13, which are mapped to their second-best steps within 38df.  
Using the [[Warts|38df]] mapping, 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]] {{val| 38 60 88 106 131 140 155 161 }}.  
 
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]]]
 
=== Prime harmonics ===
{{Harmonics in equal|38}}
{{Harmonics in equal|38}}


== Intervals ==
== Intervals ==
{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
|-
|-
! Step
! Step
! Cents
! Cents
!19-limit ratios,
! 19-odd-limit ratios,<br>in 38df val
treated as 38df
! colspan="3" | [[Ups and downs notation]]*
! colspan="3" | [[Ups and Downs Notation]]*
([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and vvd2)
|-
|-
| 0
| 0
| 0.0000
| 0.0
|
|
| Perfect 1sn
| Perfect 1sn
Line 29: Line 33:
|-
|-
| 1
| 1
| 31.5789
| 31.6
|
|
| Up 1sn
| Up 1sn
Line 36: Line 40:
|-
|-
| 2
| 2
| 63.1579
| 63.2
|
|
| Aug 1sn, dim 2nd
| Aug 1sn, dim 2nd
Line 43: Line 47:
|-
|-
| 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
| ^A1, vm2
| ^A1, vm2
Line 50: Line 54:
|-
|-
| 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
| m2
| m2
Line 57: Line 61:
|-
|-
| 5
| 5
| 157.8947
| 157.9
|[[12/11]], [[11/10]]
| [[12/11]], [[11/10]]
| Mid 2nd
| Mid 2nd
| ~2
| ~2
Line 64: Line 68:
|-
|-
| 6
| 6
| 189.4737
| 189.5
|[[10/9]], [[19/17]], [[9/8]]
| [[10/9]], [[19/17]], [[9/8]]
| Major 2nd
| Major 2nd
| M2
| M2
Line 71: Line 75:
|-
|-
| 7
| 7
| 221.0526
| 221.1
|[[17/15]]
| [[17/15]]
| Upmajor 2nd
| Upmajor 2nd
| ^M2
| ^M2
Line 78: Line 82:
|-
|-
| 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
| A2, d3
| A2, d3
Line 85: Line 89:
|-
|-
| 9
| 9
| 284.2105
| 284.2
|[[20/17]], [[13/11]], [[19/16]]
| [[20/17]], [[13/11]], [[19/16]]
| Downminor 3rd
| Downminor 3rd
| vm3
| vm3
Line 92: Line 96:
|-
|-
| 10
| 10
| 315.7895
| 315.8
|[[6/5]]
| [[6/5]]
| Minor 3rd
| Minor 3rd
| m3
| m3
Line 99: Line 103:
|-
|-
| 11
| 11
| 347.3684
| 347.4
|[[17/14]], [[11/9]]
| [[17/14]], [[11/9]]
| Mid 3rd
| Mid 3rd
| ~3
| ~3
Line 106: Line 110:
|-
|-
| 12
| 12
| 378.9474
| 378.9
|[[16/13]], [[5/4]]
| [[16/13]], [[5/4]]
| Major 3rd
| Major 3rd
| M3
| M3
Line 113: Line 117:
|-
|-
| 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
| ^M3, vd4
| ^M3, vd4
Line 120: Line 124:
|-
|-
| 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
| A3, d4
| A3, d4
Line 127: Line 131:
|-
|-
| 15
| 15
| 473.6843
| 473.7
|[[17/13]]
| [[17/13]]
| Down 4th
| Down 4th
| v4
| v4
Line 134: Line 138:
|-
|-
| 16
| 16
| 505.2632
| 505.3
|[[4/3]]
| [[4/3]]
| Perfect 4th
| Perfect 4th
| P4
| P4
Line 141: Line 145:
|-
|-
| 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
| ^4
| ^4
Line 148: Line 152:
|-
|-
| 18
| 18
| 568.4211
| 568.4
|[[18/13]], [[7/5]]
| [[18/13]], [[7/5]]
| Aug 4th
| Aug 4th
| A4
| A4
Line 155: Line 159:
|-
|-
| 19
| 19
| 600.0000
| 600.0
|[[24/17]], [[17/12]]
| [[24/17]], [[17/12]]
| Upaug 4th, downdim 5th
| Upaug 4th, downdim 5th
| ^A4, vd5
| ^A4, vd5
Line 162: Line 166:
|-
|-
| 20
| 20
| 631.5789
| 631.6
|[[10/7]], [[13/9]]
| [[10/7]], [[13/9]]
| Dim 5th
| Dim 5th
| d5
| d5
Line 169: Line 173:
|-
|-
| 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
| v5
| v5
Line 176: Line 180:
|-
|-
| 22
| 22
| 694.7368
| 694.7
|[[3/2]]
| [[3/2]]
| Perfect 5th
| Perfect 5th
| P5
| P5
Line 183: Line 187:
|-
|-
| 23
| 23
| 726.3157
| 726.3
|[[26/17]]
| [[26/17]]
| Up 5th
| Up 5th
| ^5
| ^5
Line 190: Line 194:
|-
|-
| 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
| A5, d6
| A5, d6
Line 197: Line 201:
|-
|-
| 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
| ^A5, vm6
| ^A5, vm6
Line 204: Line 208:
|-
|-
| 26
| 26
| 821.0526
| 821.1
|[[8/5]], [[13/8]]
| [[8/5]], [[13/8]]
| Minor 6th
| Minor 6th
| m6
| m6
Line 211: Line 215:
|-
|-
| 27
| 27
| 852.6316
| 852.6
|[[18/11]], [[28/17]]
| [[18/11]], [[28/17]]
| Mid 6th
| Mid 6th
| ~6
| ~6
Line 218: Line 222:
|-
|-
| 28
| 28
| 884.2105
| 884.2
|[[5/3]]
| [[5/3]]
| Major 6th
| Major 6th
| M6
| M6
Line 225: Line 229:
|-
|-
| 29
| 29
| 915.7895
| 915.8
|[[32/19]], [[22/13]], [[17/10]]
| [[32/19]], [[22/13]], [[17/10]]
| Upmajor 6th
| Upmajor 6th
| ^M6
| ^M6
Line 232: Line 236:
|-
|-
| 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
| A6, d7
| A6, d7
Line 239: Line 243:
|-
|-
| 31
| 31
| 978.9474
| 978.9
|[[30/17]]
| [[30/17]]
| Downminor 7th
| Downminor 7th
| vm7
| vm7
Line 246: Line 250:
|-
|-
| 32
| 32
| 1010.5263
| 1010.5
|[[16/9]], [[34/19]], [[9/5]]
| [[16/9]], [[34/19]], [[9/5]]
| Minor 7th
| Minor 7th
| m7
| m7
Line 253: Line 257:
|-
|-
| 33
| 33
| 1042.1053
| 1042.1
|[[20/11]], [[11/6]]
| [[20/11]], [[11/6]]
| Mid 7th
| Mid 7th
| ~7
| ~7
Line 260: Line 264:
|-
|-
| 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
| M7
| M7
Line 267: Line 271:
|-
|-
| 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
| ^M7, vd8
| ^M7, vd8
Line 274: Line 278:
|-
|-
| 36
| 36
| 1136.8421
| 1136.8
|
|
| Aug 7th, dim 8ve
| Aug 7th, dim 8ve
Line 281: Line 285:
|-
|-
| 37
| 37
| 1168.4211
| 1168.4
|
|
| Down 8ve
| Down 8ve
Line 288: Line 292:
|-
|-
| 38
| 38
| 1200.0000
| 1200.0
|
|
| Perfect 8ve
| Perfect 8ve
Line 294: Line 298:
| 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
 
== Notation ==
=== Ups and downs notation ===
Spoken as up, sharp, upsharp, etc. Note that up can be respelled as downsharp.
{{sharpness-sharp2a}}
 
=== Quarter-tone notation ===
Since a sharp raises by two steps, [[24edo#Notation|quarter-tone accidentals]] can also be used:
{{sharpness-sharp2}}
 
=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[17edo#Sagittal notation|17]], [[24edo#Sagittal notation|24]], and [[31edo#Sagittal notation|31]], is a subset of the notation for [[76edo#Sagittal notation|76-EDO]], and is a superset of the notation for [[19edo#Sagittal notation|19-EDO]].
 
==== Evo flavor ====
<imagemap>
File:38-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 679 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[33/32]]
default [[File:38-EDO_Evo_Sagittal.svg]]
</imagemap>
 
==== Revo flavor ====
<imagemap>
File:38-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 703 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[33/32]]
default [[File:38-EDO_Revo_Sagittal.svg]]
</imagemap>
 
==== Evo-SZ flavor ====
<imagemap>
File:38-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 631 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[33/32]]
default [[File:38-EDO_Evo-SZ_Sagittal.svg]]
</imagemap>
 
Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein-Zimmerman notation.
 
== Approximation to JI ==
=== Interval mappings ===
{{Q-odd-limit intervals}}


== 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]]
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/rewy-32BfRs ''Spirit of the Night - Secret of Mana (microtonal cover in 38edo)''] (2025)
 
; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=Cw1Cz1ojoSw Canon at the Semitone on The Mother's Malison Theme for Cor Anglais and Violin] (2022)


[[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]]

Latest revision as of 00:05, 16 August 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

38 equal divisions of the octave (abbreviated 38edo or 38ed2), also called 38-tone equal temperament (38tet) or 38 equal temperament (38et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 38 equal parts of about 31.6 ¢ each. Each step represents a frequency ratio of 21/38, or the 38th root of 2.

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 and 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.

Using the 38df mapping, 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*

(EUs: vvA1 and vvd2)

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

Ups and downs notation

Spoken as up, sharp, upsharp, etc. Note that up can be respelled as downsharp.

Step offset −4 −3 −2 −1 0 +1 +2 +3 +4
Symbol  
 		    
 		 
 		 
 		 
 		 
 		    

Quarter-tone notation

Since a sharp raises by two steps, quarter-tone accidentals can also be used:

Step offset −4 −3 −2 −1 0 +1 +2 +3 +4
Symbol  
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 

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.

Approximation to JI

Interval mappings

The following tables show how 15-odd-limit intervals are represented in 38edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 38edo (direct approximation, even if inconsistent)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
11/9, 18/11 0.040 0.1
15/11, 22/15 0.109 0.3
5/3, 6/5 0.148 0.5
13/7, 14/13 1.982 6.3
15/13, 26/15 4.891 15.5
13/11, 22/13 4.999 15.8
13/9, 18/13 5.039 16.0
15/14, 28/15 6.873 21.8
11/7, 14/11 6.982 22.1
9/7, 14/9 7.021 22.2
9/5, 10/9 7.070 22.4
11/10, 20/11 7.109 22.5
3/2, 4/3 7.218 22.9
11/6, 12/11 7.258 23.0
5/4, 8/5 7.366 23.3
7/4, 8/7 10.121 32.1
13/8, 16/13 12.104 38.3
13/10, 20/13 12.109 38.3
13/12, 24/13 12.257 38.8
7/5, 10/7 14.091 44.6
7/6, 12/7 14.239 45.1
9/8, 16/9 14.436 45.7
11/8, 16/11 14.476 45.8
15/8, 16/15 14.585 46.2
15-odd-limit intervals in 38edo (patent val mapping)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
11/9, 18/11 0.040 0.1
15/11, 22/15 0.109 0.3
5/3, 6/5 0.148 0.5
13/7, 14/13 1.982 6.3
9/5, 10/9 7.070 22.4
11/10, 20/11 7.109 22.5
3/2, 4/3 7.218 22.9
11/6, 12/11 7.258 23.0
5/4, 8/5 7.366 23.3
7/4, 8/7 10.121 32.1
13/8, 16/13 12.104 38.3
9/8, 16/9 14.436 45.7
11/8, 16/11 14.476 45.8
15/8, 16/15 14.585 46.2
7/6, 12/7 17.340 54.9
7/5, 10/7 17.488 55.4
13/12, 24/13 19.322 61.2
13/10, 20/13 19.470 61.7
9/7, 14/9 24.558 77.8
11/7, 14/11 24.597 77.9
15/14, 28/15 24.706 78.2
13/9, 18/13 26.540 84.0
13/11, 22/13 26.580 84.2
15/13, 26/15 26.688 84.5

Instruments

Music

Bryan Deister
Claudi Meneghin
Retrieved from "https://en.xen.wiki/index.php?title=38edo&oldid=207052"