38edo - Revision history

Overthink: /* Theory */ expand 38df paragraph

2026-06-04T20:40:58Z

Theory: expand 38df paragraph

← Older revision		Revision as of 20:40, 4 June 2026
Line 7:		Line 7:
	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 [[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.

	Instead, the [[val]] {{val\| 38 60 88 '''106''' 131 '''140''' 155 161 }} (38df in [[wart notation]]) can be used, where the [[2.3.5.7.13 subgroup\|2.3.5.7.13-subgroup]] mapping of 19edo is preserved, while harmonics [[11/1\|11]], [[17/1\|17]], and [[19/1\|19]] are ~~corrected~~. 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.		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 step 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 }}

Overthink: /* Theory */ fix typo

2026-06-04T20:30:20Z

Theory: fix typo

← Older revision		Revision as of 20:30, 4 June 2026
Line 7:		Line 7:
	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 [[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.

	Instead, the [[val]] {{val\| 38 60 88 '''106''' 131 '''140''' 155 161 }} (38df in [[wart notation]]) can be used, where the [[2.3.5.7.13 subgroup\|2.3.5.13-subgroup]] mapping of 19edo is preserved, while harmonics [[11/1\|11]], [[17/1\|17]], and [[19/1\|19]] are corrected. 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.		Instead, the [[val]] {{val\| 38 60 88 '''106''' 131 '''140''' 155 161 }} (38df in [[wart notation]]) can be used, where the [[2.3.5.7.13 subgroup\|2.3.5.7.13-subgroup]] mapping of 19edo is preserved, while harmonics [[11/1\|11]], [[17/1\|17]], and [[19/1\|19]] are corrected. 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.

	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 }}		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 }}

Overthink: /* Intervals */ place this first

2026-06-04T20:18:38Z

Intervals: place this first

← Older revision		Revision as of 20:18, 4 June 2026
Line 52:		Line 52:
	\| [[25/24]], [[34/33]]		\| [[25/24]], [[34/33]]
	\| [[22/21]]		\| [[22/21]]
	\| [[26/25]], [[27/26]], [[28/27]]		\| [[28/27]], [[26/25]], [[27/26]]
	\| Aug 1sn, dim 2nd		\| Aug 1sn, dim 2nd
	\| A1, d2		\| A1, d2

Overthink: /* Intervals */ add more ratios

2026-06-04T20:18:03Z

Intervals: add more ratios

← Older revision		Revision as of 20:18, 4 June 2026
Line 41:		Line 41:
	\| 1		\| 1
	\| 31.6		\| 31.6
	\|		\| [[55/54]], [[45/44]], ''[[33/32]]''
	\|		\| [[64/63]], ''[[36/35]]''
	\|		\| [[56/55]]
	\| Up 1sn		\| Up 1sn
	\| ^1		\| ^1
Line 50:		Line 50:
	\| 2		\| 2
	\| 63.2		\| 63.2
	\|		\| [[25/24]], [[34/33]]
	\|		\| [[22/21]]
	\|		\| [[26/25]], [[27/26]], [[28/27]]
	\| Aug 1sn, dim 2nd		\| Aug 1sn, dim 2nd
	\| A1, d2		\| A1, d2
Line 60:		Line 60:
	\| 94.7		\| 94.7
	\| [[20/19]], [[19/18]], [[18/17]], [[17/16]]		\| [[20/19]], [[19/18]], [[18/17]], [[17/16]]
	\| ''[[15/14]]''		\| ''[[15/14]]'', [[21/20]]
	\|		\|
	\| Upaug 1sn, downminor 2nd		\| Upaug 1sn, downminor 2nd
Line 78:		Line 78:
	\| 157.9		\| 157.9
	\| [[12/11]], [[11/10]]		\| [[12/11]], [[11/10]]
	\| ''[[13/12]]''		\| ''[[13/12]]'', [[35/32]]
	\|		\|
	\| Mid 2nd		\| Mid 2nd
Line 88:		Line 88:
	\| [[10/9]], [[19/17]], [[9/8]]		\| [[10/9]], [[19/17]], [[9/8]]
	\|		\|
	\|		\| [[28/25]]
	\| Major 2nd		\| Major 2nd
	\| M2		\| M2
Line 95:		Line 95:
	\| 7		\| 7
	\| 221.1		\| 221.1
	\| [[17/15]]		\| [[25/22]], [[17/15]]
	\| [[8/7]], ''[[15/13]]''		\| [[8/7]], ''[[15/13]]''
	\|		\|
Line 158:		Line 158:
	\| 14		\| 14
	\| 442.1		\| 442.1
	\| [[22/17]]		\| [[22/17]], [[32/25]]
	\| ''[[14/11]]'', ''[[17/13]]''		\| ''[[14/11]]'', ''[[17/13]]''
	\| [[9/7]], [[13/10]]		\| [[9/7]], [[13/10]], ''[[21/16]]''
	\| Aug 3rd, dim 4th		\| Aug 3rd, dim 4th
	\| A3, d4		\| A3, d4
Line 167:		Line 167:
	\| 15		\| 15
	\| 473.7		\| 473.7
	\|		\| [[25/19]]
	\| ''[[13/10]]''		\| [[21/16]], ''[[13/10]]''
	\| [[17/13]]		\| [[17/13]]
	\| Down 4th		\| Down 4th
Line 185:		Line 185:
	\| 17		\| 17
	\| 536.8		\| 536.8
	\| [[15/11]], [[11/8]]		\| [[15/11]], [[11/8]], [[34/25]]
	\| ''[[18/13]]''		\| ''[[18/13]]''
	\| [[19/14]], [[26/19]]		\| [[19/14]], [[26/19]]
Line 194:		Line 194:
	\| 18		\| 18
	\| 568.4		\| 568.4
	\|		\| [[25/18]]
	\| ''[[26/19]]''		\| ''[[26/19]]''
	\| [[18/13]], [[7/5]]		\| [[18/13]], [[7/5]]
Line 212:		Line 212:
	\| 20		\| 20
	\| 631.6		\| 631.6
	\|		\| [[36/25]]
	\| ''[[19/13]]''		\| ''[[19/13]]''
	\| [[10/7]], [[13/9]]		\| [[10/7]], [[13/9]]
Line 221:		Line 221:
	\| 21		\| 21
	\| 663.2		\| 663.2
	\| [[16/11]], [[22/15]]		\| [[22/15]], [[16/11]], [[25/17]]
	\| ''[[13/9]]''		\| ''[[13/9]]''
	\| [[19/13]], [[28/19]]		\| [[19/13]], [[28/19]]
Line 239:		Line 239:
	\| 23		\| 23
	\| 726.3		\| 726.3
	\|		\| [[38/25]]
	\| ''[[20/13]]''		\| ''[[20/13]]''
	\| [[26/17]]		\| [[26/17]]
Line 248:		Line 248:
	\| 24		\| 24
	\| 757.9		\| 757.9
	\| [[17/11]]		\| [[17/11]], [[25/16]]
	\| ''[[26/17]]'', ''[[11/7]]''		\| ''[[26/17]]'', ''[[11/7]]''
	\|		\| [[14/9]], [[20/13]], ''[[32/21]]''
	\| Aug 5th, dim 6th		\| Aug 5th, dim 6th
	\| A5, d6		\| A5, d6
Line 295:		Line 295:
	\| [[32/19]], [[17/10]]		\| [[32/19]], [[17/10]]
	\|		\|
	\|		\| [[22/13]]
	\| Upmajor 6th		\| Upmajor 6th
	\| ^M6		\| ^M6
Line 311:		Line 311:
	\| 31		\| 31
	\| 978.9		\| 978.9
	\| [[30/17]]		\| [[44/25]], [[30/17]]
	\| ''[[26/15]]'', [[7/4]]		\| ''[[26/15]]'', [[7/4]]
	\|		\|
Line 322:		Line 322:
	\| [[16/9]], [[34/19]], [[9/5]]		\| [[16/9]], [[34/19]], [[9/5]]
	\|		\|
	\|		\| [[25/14]]
	\| Minor 7th		\| Minor 7th
	\| m7		\| m7
Line 330:		Line 330:
	\| 1042.1		\| 1042.1
	\| [[20/11]], [[11/6]]		\| [[20/11]], [[11/6]]
	\| ''[[24/13]]''		\| ''[[24/13]]'', [[64/35]]
	\|		\|
	\| Mid 7th		\| Mid 7th
Line 356:		Line 356:
	\| 36		\| 36
	\| 1136.8		\| 1136.8
			\| [[33/17]], [[48/25]]
	\|		\|
	\|		\| [[27/14]], [[25/13]], [[52/27]]
	\|
	\| Aug 7th, dim 8ve		\| Aug 7th, dim 8ve
	\| A7, d8		\| A7, d8
Line 365:		Line 365:
	\| 37		\| 37
	\| 1168.4		\| 1168.4
	\|		\| ''[[64/33]]'', [[88/45]], [[108/55]]
	\|		\| [[63/32]]
	\|		\| [[55/28]]
	\| Down 8ve		\| Down 8ve
	\| v8		\| v8

Overthink: /* Theory */Grammar changes

2026-05-30T17:55:25Z

Theory: Grammar changes

← Older revision		Revision as of 17:55, 30 May 2026
Line 3:		Line 3:

	== Theory ==		== Theory ==
	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]] ~~and~~ [[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.

	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 [[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.

	Instead, the [[val]] {{val\| 38 60 88 '''106''' 131 '''140''' 155 161 }} (38df in [[wart notation]]) can be used, where the [[2.3.5.7.13 subgroup\|2.3.5.13-subgroup]] mapping of 19edo is preserved, while harmonics [[11/1\|11]], [[17/1\|17]], and [[19/1\|19]] are corrected. In 38df, every [[odd harmonic]] 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, except for 7/4 ~~and~~ 13/8, ~~along with~~ 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.		Instead, the [[val]] {{val\| 38 60 88 '''106''' 131 '''140''' 155 161 }} (38df in [[wart notation]]) can be used, where the [[2.3.5.7.13 subgroup\|2.3.5.13-subgroup]] mapping of 19edo is preserved, while harmonics [[11/1\|11]], [[17/1\|17]], and [[19/1\|19]] are corrected. 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.

	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 }}		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 }}

Overthink: /* Theory */Mention that 38df preserves 19edo's 2.3.5.7.13 earlier

2026-05-30T17:50:53Z

Theory: Mention that 38df preserves 19edo's 2.3.5.7.13 earlier

← Older revision		Revision as of 17:50, 30 May 2026
Line 7:		Line 7:
	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 [[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.

	Instead, the [[val]] {{val\| 38 60 88 '''106''' 131 '''140''' 155 161 }} (38df in [[wart notation]]) can be used, where every [[~~prime~~ harmonic]] 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, except for 7/4 and 13/8, along with their octave complements 8/7 and 16/13, which are by definition mapped to their second-closest steps within 38df. The 38df mapping ~~this~~ creates a natural full [[19-limit]] extension to the ~~[[2.3.5.7.13 subgroup\|~~2.3.5.7.13-subgroup]] mapping of 19edo.		Instead, the [[val]] {{val\| 38 60 88 '''106''' 131 '''140''' 155 161 }} (38df in [[wart notation]]) can be used, where the [[2.3.5.7.13 subgroup\|2.3.5.13-subgroup]] mapping of 19edo is preserved, while harmonics [[11/1\|11]], [[17/1\|17]], and [[19/1\|19]] are corrected. In 38df, every [[odd harmonic]] 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, except for 7/4 and 13/8, along with 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.

	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 }}		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 }}

Overthink: /* Theory */Improve writing in 38df paragraph

2026-05-30T17:46:01Z

Theory: Improve writing in 38df paragraph

← Older revision		Revision as of 17:46, 30 May 2026
Line 7:		Line 7:
	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 [[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.

	Instead, the [[val]] {{val\| 38 60 88 '''106''' 131 '''140''' 155 161 }} (38df in [[wart notation]]) can be used, where every [[prime harmonic]] 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. ~~Thus~~ 38df creates a natural full [[19-limit]] extension to the [[2.3.5.7.13 subgroup\|2.3.5.7.13-subgroup]] mapping of 19edo.		Instead, the [[val]] {{val\| 38 60 88 '''106''' 131 '''140''' 155 161 }} (38df in [[wart notation]]) can be used, where every [[prime harmonic]] 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, except for 7/4 and 13/8, along with their octave complements 8/7 and 16/13, which are by definition mapped to their second-closest steps within 38df. The 38df mapping this creates a natural full [[19-limit]] extension to the [[2.3.5.7.13 subgroup\|2.3.5.7.13-subgroup]] mapping of 19edo.

	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]]]		[[File:Harmonic_series_38df.mp3]] [[:File:Harmonic_series_38df.mp3\|[Harmonic series 2-20 in 38df]]]

Overthink: /* Theory */Rewrite

2026-05-30T17:41:31Z

Theory: Rewrite

← Older revision		Revision as of 17:41, 30 May 2026
Line 3:		Line 3:

	== 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 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]] and [[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 [[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 every [[prime harmonic]] 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. Thus 38df creates a natural full [[19-limit]] extension to the [[2.3.5.7.13 subgroup\|2.3.5.7.13-subgroup]] mapping of 19edo.

	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 sequence: {{nowrap\| 38 22 16 12 10 8 8 6 6 5 5 4 4 4 4 3 3 3 3 }}

Overthink: /* Intervals */ split ratios by subgroup

2026-05-22T16:01:27Z

Intervals: split ratios by subgroup

Show changes

FloraC: /* Intervals */ formatting

2026-05-16T07:06:18Z

Intervals: formatting

Show changes

← Older revision		Revision as of 20:40, 4 June 2026
Line 7:		Line 7:
	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 [[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.

	Instead, the [[val]] {{val\| 38 60 88 '''106''' 131 '''140''' 155 161 }} (38df in [[wart notation]]) can be used, where the [[2.3.5.7.13 subgroup\|2.3.5.7.13-subgroup]] mapping of 19edo is preserved, while harmonics [[11/1\|11]], [[17/1\|17]], and [[19/1\|19]] are ~~corrected~~. 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.		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 step 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 }}

← Older revision		Revision as of 20:30, 4 June 2026
Line 7:		Line 7:
	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 [[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.

	Instead, the [[val]] {{val\| 38 60 88 '''106''' 131 '''140''' 155 161 }} (38df in [[wart notation]]) can be used, where the [[2.3.5.7.13 subgroup\|2.3.5.13-subgroup]] mapping of 19edo is preserved, while harmonics [[11/1\|11]], [[17/1\|17]], and [[19/1\|19]] are corrected. 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.		Instead, the [[val]] {{val\| 38 60 88 '''106''' 131 '''140''' 155 161 }} (38df in [[wart notation]]) can be used, where the [[2.3.5.7.13 subgroup\|2.3.5.7.13-subgroup]] mapping of 19edo is preserved, while harmonics [[11/1\|11]], [[17/1\|17]], and [[19/1\|19]] are corrected. 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.

	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 }}		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 }}

← Older revision		Revision as of 17:55, 30 May 2026
Line 3:		Line 3:

	== Theory ==		== Theory ==
	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]] ~~and~~ [[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.

	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 [[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.

	Instead, the [[val]] {{val\| 38 60 88 '''106''' 131 '''140''' 155 161 }} (38df in [[wart notation]]) can be used, where the [[2.3.5.7.13 subgroup\|2.3.5.13-subgroup]] mapping of 19edo is preserved, while harmonics [[11/1\|11]], [[17/1\|17]], and [[19/1\|19]] are corrected. In 38df, every [[odd harmonic]] 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, except for 7/4 ~~and~~ 13/8, ~~along with~~ 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.		Instead, the [[val]] {{val\| 38 60 88 '''106''' 131 '''140''' 155 161 }} (38df in [[wart notation]]) can be used, where the [[2.3.5.7.13 subgroup\|2.3.5.13-subgroup]] mapping of 19edo is preserved, while harmonics [[11/1\|11]], [[17/1\|17]], and [[19/1\|19]] are corrected. 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.

	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 }}		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 }}

← Older revision		Revision as of 17:50, 30 May 2026
Line 7:		Line 7:
	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 [[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.

	Instead, the [[val]] {{val\| 38 60 88 '''106''' 131 '''140''' 155 161 }} (38df in [[wart notation]]) can be used, where every [[~~prime~~ harmonic]] 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, except for 7/4 and 13/8, along with their octave complements 8/7 and 16/13, which are by definition mapped to their second-closest steps within 38df. The 38df mapping ~~this~~ creates a natural full [[19-limit]] extension to the ~~[[2.3.5.7.13 subgroup\|~~2.3.5.7.13-subgroup]] mapping of 19edo.		Instead, the [[val]] {{val\| 38 60 88 '''106''' 131 '''140''' 155 161 }} (38df in [[wart notation]]) can be used, where the [[2.3.5.7.13 subgroup\|2.3.5.13-subgroup]] mapping of 19edo is preserved, while harmonics [[11/1\|11]], [[17/1\|17]], and [[19/1\|19]] are corrected. In 38df, every [[odd harmonic]] 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, except for 7/4 and 13/8, along with 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.

	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 }}		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 }}

← Older revision		Revision as of 17:46, 30 May 2026
Line 7:		Line 7:
	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 [[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.

	Instead, the [[val]] {{val\| 38 60 88 '''106''' 131 '''140''' 155 161 }} (38df in [[wart notation]]) can be used, where every [[prime harmonic]] 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. ~~Thus~~ 38df creates a natural full [[19-limit]] extension to the [[2.3.5.7.13 subgroup\|2.3.5.7.13-subgroup]] mapping of 19edo.		Instead, the [[val]] {{val\| 38 60 88 '''106''' 131 '''140''' 155 161 }} (38df in [[wart notation]]) can be used, where every [[prime harmonic]] 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, except for 7/4 and 13/8, along with their octave complements 8/7 and 16/13, which are by definition mapped to their second-closest steps within 38df. The 38df mapping this creates a natural full [[19-limit]] extension to the [[2.3.5.7.13 subgroup\|2.3.5.7.13-subgroup]] mapping of 19edo.

	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]]]		[[File:Harmonic_series_38df.mp3]] [[:File:Harmonic_series_38df.mp3\|[Harmonic series 2-20 in 38df]]]