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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|38}}
{{ED intro}}


== Theory ==
== Theory ==
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]]; 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.  
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 [[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].  
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.  


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}}
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]]]
[[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}}


Line 18: Line 21:
! 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]]*
([[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
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
Line 57: Line 61:
|-
|-
| 5
| 5
| 157.8947
| 157.9
| [[12/11]], [[11/10]]
| [[12/11]], [[11/10]]
| Mid 2nd
| Mid 2nd
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
Line 71: Line 75:
|-
|-
| 7
| 7
| 221.0526
| 221.1
| [[17/15]]
| [[17/15]]
| Upmajor 2nd
| Upmajor 2nd
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
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
Line 92: Line 96:
|-
|-
| 10
| 10
| 315.7895
| 315.8
| [[6/5]]
| [[6/5]]
| Minor 3rd
| Minor 3rd
Line 99: Line 103:
|-
|-
| 11
| 11
| 347.3684
| 347.4
| [[17/14]], [[11/9]]
| [[17/14]], [[11/9]]
| Mid 3rd
| Mid 3rd
Line 106: Line 110:
|-
|-
| 12
| 12
| 378.9474
| 378.9
| [[16/13]], [[5/4]]
| [[16/13]], [[5/4]]
| Major 3rd
| Major 3rd
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
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
Line 127: Line 131:
|-
|-
| 15
| 15
| 473.6843
| 473.7
| [[17/13]]
| [[17/13]]
| Down 4th
| Down 4th
Line 134: Line 138:
|-
|-
| 16
| 16
| 505.2632
| 505.3
| [[4/3]]
| [[4/3]]
| Perfect 4th
| Perfect 4th
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|-
|-
| 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
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|-
|-
| 18
| 18
| 568.4211
| 568.4
| [[18/13]], [[7/5]]
| [[18/13]], [[7/5]]
| Aug 4th
| Aug 4th
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|-
|-
| 19
| 19
| 600.0000
| 600.0
| [[24/17]], [[17/12]]
| [[24/17]], [[17/12]]
| Upaug 4th, downdim 5th
| Upaug 4th, downdim 5th
Line 162: Line 166:
|-
|-
| 20
| 20
| 631.5789
| 631.6
| [[10/7]], [[13/9]]
| [[10/7]], [[13/9]]
| Dim 5th
| Dim 5th
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
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|-
|-
| 22
| 22
| 694.7368
| 694.7
| [[3/2]]
| [[3/2]]
| Perfect 5th
| Perfect 5th
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|-
|-
| 23
| 23
| 726.3157
| 726.3
| [[26/17]]
| [[26/17]]
| Up 5th
| Up 5th
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|-
|-
| 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 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
Line 204: Line 208:
|-
|-
| 26
| 26
| 821.0526
| 821.1
| [[8/5]], [[13/8]]
| [[8/5]], [[13/8]]
| Minor 6th
| Minor 6th
Line 211: Line 215:
|-
|-
| 27
| 27
| 852.6316
| 852.6
| [[18/11]], [[28/17]]
| [[18/11]], [[28/17]]
| Mid 6th
| Mid 6th
Line 218: Line 222:
|-
|-
| 28
| 28
| 884.2105
| 884.2
| [[5/3]]
| [[5/3]]
| Major 6th
| Major 6th
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
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
Line 239: Line 243:
|-
|-
| 31
| 31
| 978.9474
| 978.9
| [[30/17]]
| [[30/17]]
| Downminor 7th
| Downminor 7th
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
Line 253: Line 257:
|-
|-
| 33
| 33
| 1042.1053
| 1042.1
| [[20/11]], [[11/6]]
| [[20/11]], [[11/6]]
| Mid 7th
| Mid 7th
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
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
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 />* 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 ==
Line 301: Line 353:


== 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 -->
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