38edo: Difference between revisions

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== 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 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.

Since 38 factors as {{nowrap|2 × 19}}, 38edo can be thought of as two parallel chains of [[19edo]]. It provides a possible correction to the [[11/1|11th harmonic]] of 19edo, which works well with 19edo's flat approximations of the [[3/1|3rd]] and [[5/1|5th]] harmonics, making it a decent [[2.3.5.11 subgroup|2.3.5.11-subgroup]] system. Compared to 19edo, the halving of the step size lowers [[consistency]], and leaves it only mediocre in terms of overall [[relative interval error|relative error]]. However, the fact that the 3rd and 5th harmonics are flat by almost exactly the same amount, while the 11th is close to 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 [[octave complement]]s, 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 [[patent val]], 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 7-limit. In the [[11-limit]], we can add [[121/120]] and [[176/175]], and in the [[13-limit]] we can add [[66/65]] and [[144/143]]. 38edo patently supports [[mohajira]] up to the 13-limit. While the [[7/1|7th]] and [[13/1|13th]] harmonics themselves are improved compared to 19edo, many other intervals involving these harmonics become less accurate, so whether 38edo actually corrects them is debatable.

~~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 }}~~.

Instead, the [[val]] {{val| 38 60 88 '''106''' 131 '''140''' 155 161 }} (38df in [[wart notation]]) can be used, where primes [[7/1|7]] and [[13/1|13]] use their second-best approximations, and are mapped the same as in 19edo. The [[2.3.5.7.13 subgroup|2.3.5.7.13-subgroup]] mapping of 19edo is preserved in 38df, while harmonics [[11/1|11]], [[17/1|17]], and [[19/1|19]] are mapped between steps of 19edo. In 38df, every [[odd harmonic]] from 3 to 19 is characterized by a flat intonation. Furthermore, the [[mapping]]s of all [[19-odd-limit]] intervals in 38df align with their closest approximations in 38edo, except for [[7/4]], [[13/8]], and their octave complements [[8/7]] and [[16/13]], which are by definition mapped to their second-closest steps within 38df. The 38df mapping thus creates a natural full [[19-limit]] extension to the 2.3.5.7.13-subgroup mapping of 19edo. It tempers out [[49/48]], [[65/64]], [[81/80]], [[225/224]], etc. as in 19edo, as well as [[121/120]], [[289/288]], [[324/323]], [[361/360]], and many more.

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 }}

The harmonic series from 1 to 20 is approximated within 38df by the step 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]]]

Line 41:

| 1

| 31.6

|

| [[55/54]], [[45/44]], ''[[33/32]]''

|

| [[64/63]], ''[[36/35]]''

|

| [[56/55]]

| Up 1sn

| ^1

Line 50:

| 2

| 63.2

|

| [[25/24]], [[34/33]]

|

| [[22/21]]

|

| [[28/27]], [[26/25]], [[27/26]]

| Aug 1sn, dim 2nd

| A1, d2

Line 60:

| 94.7

| [[20/19]], [[19/18]], [[18/17]], [[17/16]]

| ''[[15/14]]''

| ''[[15/14]]'', [[21/20]]

|

| Upaug 1sn, downminor 2nd

Line 78:

| 157.9

| [[12/11]], [[11/10]]

| ''[[13/12]]''

| ''[[13/12]]'', [[35/32]]

|

| Mid 2nd

Line 88:

| [[10/9]], [[19/17]], [[9/8]]

|

| [[28/25]]

| Major 2nd

| M2

Line 95:

| 7

| 221.1

| [[17/15]]

| [[25/22]], [[17/15]]

| [[8/7]], ''[[15/13]]''

|

Line 158:

| 14

| 442.1

| [[22/17]]

| [[22/17]], [[32/25]]

| ''[[14/11]]'', ''[[17/13]]''

| [[9/7]], [[13/10]]

| [[9/7]], [[13/10]], ''[[21/16]]''

| Aug 3rd, dim 4th

| A3, d4

Line 167:

| 15

| 473.7

|

| [[25/19]]

| ''[[13/10]]''

| [[21/16]], ''[[13/10]]''

| [[17/13]]

| Down 4th

Line 185:

| 17

| 536.8

| [[15/11]], [[11/8]]

| [[15/11]], [[11/8]], [[34/25]]

| ''[[18/13]]''

| [[19/14]], [[26/19]]

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| 18

| 568.4

|

| [[25/18]]

| ''[[26/19]]''

| [[18/13]], [[7/5]]

Line 212:

| 20

| 631.6

|

| [[36/25]]

| ''[[19/13]]''

| [[10/7]], [[13/9]]

Line 221:

| 21

| 663.2

| [[16/11]], [[22/15]]

| [[22/15]], [[16/11]], [[25/17]]

| ''[[13/9]]''

| [[19/13]], [[28/19]]

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| 23

| 726.3

|

| [[38/25]]

| ''[[20/13]]''

| [[26/17]]

Line 248:

| 24

| 757.9

| [[17/11]]

| [[17/11]], [[25/16]]

| ''[[26/17]]'', ''[[11/7]]''

|

| [[14/9]], [[20/13]], ''[[32/21]]''

| Aug 5th, dim 6th

| A5, d6

Line 295:

| [[32/19]], [[17/10]]

|

| [[22/13]]

| Upmajor 6th

| ^M6

Line 311:

| 31

| 978.9

| [[30/17]]

| [[44/25]], [[30/17]]

| ''[[26/15]]'', [[7/4]]

|

Line 322:

| [[16/9]], [[34/19]], [[9/5]]

|

| [[25/14]]

| Minor 7th

| m7

Line 330:

| 1042.1

| [[20/11]], [[11/6]]

| ''[[24/13]]''

| ''[[24/13]]'', [[64/35]]

|

| Mid 7th

Line 356:

| 36

| 1136.8

| [[33/17]], [[48/25]]

|

| [[27/14]], [[25/13]], [[52/27]]

|

| Aug 7th, dim 8ve

| A7, d8

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| 37

| 1168.4

|

| ''[[64/33]]'', [[88/45]], [[108/55]]

|

| [[63/32]]

|

| [[55/28]]

| Down 8ve

| v8

@@ Line 3: / Line 3: @@
 == 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 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.
+Since 38 factors as {{nowrap|2 × 19}}, 38edo can be thought of as two parallel chains of [[19edo]]. It provides a possible correction to the [[11/1|11th harmonic]] of 19edo, which works well with 19edo's flat approximations of the [[3/1|3rd]] and [[5/1|5th]] harmonics, making it a decent [[2.3.5.11 subgroup|2.3.5.11-subgroup]] system. Compared to 19edo, the halving of the step size lowers [[consistency]], and leaves it only mediocre in terms of overall [[relative interval error|relative error]]. However, the fact that the 3rd and 5th harmonics are flat by almost exactly the same amount, while the 11th is close to 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 [[octave complement]]s, 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 [[patent val]], 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 7-limit. In the [[11-limit]], we can add [[121/120]] and [[176/175]], and in the [[13-limit]] we can add [[66/65]] and [[144/143]]. 38edo patently supports [[mohajira]] up to the 13-limit. While the [[7/1|7th]] and [[13/1|13th]] harmonics themselves are improved compared to 19edo, many other intervals involving these harmonics become less accurate, so whether 38edo actually corrects them is debatable.
-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 }}.
+Instead, the [[val]] {{val| 38 60 88 '''106''' 131 '''140''' 155 161 }} (38df in [[wart notation]]) can be used, where primes [[7/1|7]] and [[13/1|13]] use their second-best approximations, and are mapped the same as in 19edo. The [[2.3.5.7.13 subgroup|2.3.5.7.13-subgroup]] mapping of 19edo is preserved in 38df, while harmonics [[11/1|11]], [[17/1|17]], and [[19/1|19]] are mapped between steps of 19edo. In 38df, every [[odd harmonic]] from 3 to 19 is characterized by a flat intonation. Furthermore, the [[mapping]]s of all [[19-odd-limit]] intervals in 38df align with their closest approximations in 38edo, except for [[7/4]], [[13/8]], and their octave complements [[8/7]] and [[16/13]], which are by definition mapped to their second-closest steps within 38df. The 38df mapping thus creates a natural full [[19-limit]] extension to the 2.3.5.7.13-subgroup mapping of 19edo. It tempers out [[49/48]], [[65/64]], [[81/80]], [[225/224]], etc. as in 19edo, as well as [[121/120]], [[289/288]], [[324/323]], [[361/360]], and many more.
-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 }}
+The harmonic series from 1 to 20 is approximated within 38df by the step 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]]]
@@ Line 41: / Line 41: @@
 | 1
 | 31.6
-|
+| [[55/54]], [[45/44]], ''[[33/32]]''
-|
+| [[64/63]], ''[[36/35]]''
-|
+| [[56/55]]
 | Up 1sn
 | ^1
@@ Line 50: / Line 50: @@
 | 2
 | 63.2
-|
+| [[25/24]], [[34/33]]
-|
+| [[22/21]]
-|
+| [[28/27]], [[26/25]], [[27/26]]
 | Aug 1sn, dim 2nd
 | A1, d2
@@ Line 60: / Line 60: @@
 | 94.7
 | [[20/19]], [[19/18]], [[18/17]], [[17/16]]
-| ''[[15/14]]''
+| ''[[15/14]]'', [[21/20]]
 |
 | Upaug 1sn, downminor 2nd
@@ Line 78: / Line 78: @@
 | 157.9
 | [[12/11]], [[11/10]]
-| ''[[13/12]]''
+| ''[[13/12]]'', [[35/32]]
 |
 | Mid 2nd
@@ Line 88: / Line 88: @@
 | [[10/9]], [[19/17]], [[9/8]]
 |
-|
+| [[28/25]]
 | Major 2nd
 | M2
@@ Line 95: / Line 95: @@
 | 7
 | 221.1
-| [[17/15]]
+| [[25/22]], [[17/15]]
 | [[8/7]], ''[[15/13]]''
 |
@@ Line 158: / Line 158: @@
 | 14
 | 442.1
-| [[22/17]]
+| [[22/17]], [[32/25]]
 | ''[[14/11]]'', ''[[17/13]]''
-| [[9/7]], [[13/10]]
+| [[9/7]], [[13/10]], ''[[21/16]]''
 | Aug 3rd, dim 4th
 | A3, d4
@@ Line 167: / Line 167: @@
 | 15
 | 473.7
-|
+| [[25/19]]
-| ''[[13/10]]''
+| [[21/16]], ''[[13/10]]''
 | [[17/13]]
 | Down 4th
@@ Line 185: / Line 185: @@
 | 17
 | 536.8
-| [[15/11]], [[11/8]]
+| [[15/11]], [[11/8]], [[34/25]]
 | ''[[18/13]]''
 | [[19/14]], [[26/19]]
@@ Line 194: / Line 194: @@
 | 18
 | 568.4
-|
+| [[25/18]]
 | ''[[26/19]]''
 | [[18/13]], [[7/5]]
@@ Line 212: / Line 212: @@
 | 20
 | 631.6
-|
+| [[36/25]]
 | ''[[19/13]]''
 | [[10/7]], [[13/9]]
@@ Line 221: / Line 221: @@
 | 21
 | 663.2
-| [[16/11]], [[22/15]]
+| [[22/15]], [[16/11]], [[25/17]]
 | ''[[13/9]]''
 | [[19/13]], [[28/19]]
@@ Line 239: / Line 239: @@
 | 23
 | 726.3
-|
+| [[38/25]]
 | ''[[20/13]]''
 | [[26/17]]
@@ Line 248: / Line 248: @@
 | 24
 | 757.9
-| [[17/11]]
+| [[17/11]], [[25/16]]
 | ''[[26/17]]'', ''[[11/7]]''
-|
+| [[14/9]], [[20/13]], ''[[32/21]]''
 | Aug 5th, dim 6th
 | A5, d6
@@ Line 295: / Line 295: @@
 | [[32/19]], [[17/10]]
 |
-|
+| [[22/13]]
 | Upmajor 6th
 | ^M6
@@ Line 311: / Line 311: @@
 | 31
 | 978.9
-| [[30/17]]
+| [[44/25]], [[30/17]]
 | ''[[26/15]]'', [[7/4]]
 |
@@ Line 322: / Line 322: @@
 | [[16/9]], [[34/19]], [[9/5]]
 |
-|
+| [[25/14]]
 | Minor 7th
 | m7
@@ Line 330: / Line 330: @@
 | 1042.1
 | [[20/11]], [[11/6]]
-| ''[[24/13]]''
+| ''[[24/13]]'', [[64/35]]
 |
 | Mid 7th
@@ Line 356: / Line 356: @@
 | 36
 | 1136.8
+| [[33/17]], [[48/25]]
 |
-|
+| [[27/14]], [[25/13]], [[52/27]]
-|
 | Aug 7th, dim 8ve
 | A7, d8
@@ Line 365: / Line 365: @@
 | 37
 | 1168.4
-|
+| ''[[64/33]]'', [[88/45]], [[108/55]]
-|
+| [[63/32]]
-|
+| [[55/28]]
 | Down 8ve
 | v8