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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 interval error|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 [[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.  
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.  
 
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.  


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 }}.  
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 }}.  
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! 19-odd-limit ratios,<br>in 38df val
! 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
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<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


== Zeta properties ==
== Notation ==
=== Zeta peak index ===
=== Ups and downs notation ===
{| class="wikitable"
Spoken as up, sharp, upsharp, etc. Note that up can be respelled as downsharp.
|-
{{sharpness-sharp2a}}
! colspan="3" | Tuning
! colspan="3" | Strength
! colspan="2" | Closest EDO
! colspan="2" | Integer limit
|-
! ZPI
! Steps per octave
! Step size (cents)
! Height
! Integral
! Gap
! EDO
! Octave (cents)
! Consistent
! Distinct
|-
| [[166zpi]]
| 37.8901105027757
| 31.6705331305933
| 5.808723
| 0.986660
| 15.046792
| 38edo
| 1203.48025896255
| 6
| 6
|}


== Notation ==
=== Quarter-tone notation ===
=== Quarter-tone notation ===
Since a sharp raises by two steps, [[24edo#Notation|quarter-tone accidentals]] can also be used:
Since a sharp raises by two steps, [[24edo#Notation|quarter-tone accidentals]] can also be used:
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Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein-Zimmerman notation.
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 ==
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== 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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