38edo: Difference between revisions
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== 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]] | 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 every [[ | 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 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]]] | ||