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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]]]
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== 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]]
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