38edo: Difference between revisions

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 }}.  

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 }}

! Step

! Cents

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

! colspan="3" | [[Ups and downs notation]]*

([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and vvd2)

|

|

|

| [[16/15]], [[15/14]], [[14/13]], [[13/12]]

| [[8/7]], [[15/13]], [[22/19]], [[7/6]]

| [[20/17]], [[13/11]], [[19/16]]

| [[17/14]], [[11/9]]

| [[16/13]], [[5/4]]

| [[24/19]], [[19/15]], [[14/11]]

| [[9/7]], [[22/17]], [[13/10]]

| [[19/14]], [[15/11]], [[26/19]], [[11/8]]

| [[16/11]], [[19/13]], [[22/15]], [[28/19]]

| [[20/13]], [[17/11]], [[14/9]]

| [[11/7]], [[30/19]], [[19/12]]

| [[8/5]], [[13/8]]

| [[18/11]], [[28/17]]

| [[32/19]], [[22/13]], [[17/10]]

| [[12/7]], [[19/11]], [[26/15]], [[7/4]]

| [[24/13]], [[13/7]], [[28/15]], [[15/8]]

|

|

{{sharpness-sharp2a}}

| [[Delorean]] / [[Subkla]] || 9\38 || 1

| [[Migration]] / [[Mohajira]] / [[Nethertone]] / [[Ptolemy]] / [[Subklei]] || 11\38 || 1

| [[Maquila]] / [[Wilsec]] || 17\38 || 1

| [[Bimeantone]] / [[Injera]] || 3\38 || 2

| [[Bison]] / [[Hemikleismic]] || 5\38 || 2

| [[Astrology]] / [[Divination]] / [[Horoscope]] || 7\38 || 2

38edo's approximation of [[JI]] can be improved by slightly [[octave stretch|stretching the octave]].

What follows is a comparison of stretched-octave 38edo tunings.

; 38edo

* Step size: 31.579{{c}}, octave size: 1200.00{{c}}  

{{Harmonics in equal|38|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 38edo}}

{{Harmonics in equal|38|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 38edo (continued)}}

; [[WE|38et, 13-limit WE tuning]]  

* Step size: 31.599{{c}}, octave size: 1200.77{{c}}

Stretching the octave of 38edo by around 1{{c}} results in improved primes 3, 5, 11, 17 and 19, but worse primes 2, 7 and 13. This approximates all harmonics up to 16 within 14.9{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.

; [[ed5|88ed5]]  

* Step size: 31.663{{c}}, octave size: 1203.18{{c}}

Stretching the octave of 38edo by around 3{{c}} results in improved primes 3, 5, 11, 13, 17 and 19 but worse primes 2 and 7. This approximates all harmonics up to 16 within 12.7{{c}}. The tuning 88ed5 does this.

{{Harmonics in equal|88|5|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 88ed5}}

{{Harmonics in equal|88|5|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 88ed5 (continued)}}

; [[zpi|166zpi]]  

* Step size: 31.671{{c}}, octave size: 1203.48{{c}}

Stretching the octave of 38edo by around 3.5{{c}} results in improved primes 3, 5, 11, 13, 17 and 19, but worse primes 2 and 7. This approximates all harmonics up to 16 within 14.0{{c}}. This results in the same [[mapping]] as [[wart|38df]], which is the mapping used by most of 38edo's lowest-[[badness]] temperaments. The tuning 166zpi does this.

; [[60edt]]

* Step size: 31.699{{c}}, octave size: 1204.57{{c}}

Stretching the octave of 38edo by around 4.5{{c}} results in improved primes 3, 5, 7, 11, 13, 17 and 19, but a much worse prime 2. This approximates all harmonics up to 16 within 13.7{{c}}. This results in the same [[mapping]] as [[wart|38df]], which is the mapping used by most of 38edo's lowest-[[badness]] temperaments. The tuning 60edt does this.

{{Harmonics in equal|60|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 60edt}}