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m →Scales: Added subsets of tritikleismic9: will eventually change these to 1 2 1 2 format instead of 100.0 200.0 300.0 format but just getting them written down for now to start
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{{Infobox ET}}
{{Infobox ET|42ed257/128}}
'''42ed257/128''' is the [[Ed257/128|equal division of the interval 257/128]] into forty-two parts of 28.73 [[cents]] each, corresponding to ~41.77[[edo]].  
'''42ed257/128''' is the [[Ed257/128|equal division of the interval 257/128]] into forty-two parts of 28.73 [[cents]] each, corresponding to ~41.77[[edo]].  


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=== Harmonics ===
=== Harmonics ===
''See also: [[Table of stretched 42edo tunings]]''
42ed257/128 is a kind of opposite twin to the scale [[ed255/128#42ed255/128|42ed255/128]], as they improve 42edo’s [[JI]] approximation by about the same amount, but in opposite directions (those harmonics which are slightly sharp in one are slightly flat in the other).


42ed257/128 is a kind of opposite twin to the scale [[191zpi]]. 191zpi makes [[42edo]] into a better 2.3.5.11.13 tuning, while 42ed257/128 makes 42edo into a better 2.3.5.7.17 tuning. They both improve 42edo but in opposite directions.
42ed257/128’s step size is very close to that of [[1ed28.7c|APS720jot]] and [[zpi|189zpi]].


42ed257/128 is very similar to [[1ed28.7c|APS720jot]], which achieves many of the same aims. The difference is that 42ed257/128 has better harmonics 3, 5 and 7, while APS720jot has better harmonics 2, 11 and 13.
See [[Table of stretched 42edo tunings]] for more.


{{Harmonics in equal|42|257|128|intervals=prime}}
{{Harmonics in equal|42|257|128|intervals=prime}}
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== Scales ==
== Scales ==
To generate these in [[Scale Workshop]], you can first generate them in 42edo, then apply a stretch of 1.00455012.
; [[MOS scale]]s
; [[MOS scale]]s
* Eugene/Tritikleismic[9]: '''3 8 3 3 8 3 3 8 3'''
* Eugene/Tritikleismic[9]: '''3 8 3 3 8 3 3 8 3'''
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; Subsets of [[MOS scale]]s
; Subsets of MOS scales
''(Names used are [[Template:Idiosyncratic|idiosyncratic]]. Cent values rounded to 1 dec pl.)''
''(Names used are [[Template:Idiosyncratic|idiosyncratic]].)''
* Eugene/Tritikleismic[9]
* Eugene/Tritikleismic[9]
** Yokai pentatonic: 86.1, 487.9, 717.5, 803.6, 1205.5
** Groovy aeolian pentatonic: '''11 6 8 3 14'''
** Septimal Picardy pentatonic: 229.6, 401.8, 717.5, 803.6, 1205.5
** [[Otonal]] mixolydian pentatonic: '''14 3 8 11 6'''
** Groovy aeolian pentatonic: 315.7, 487.9, 717.5, 803.6, 1205.5
** Pseudo-[[equipentatonic]]: '''11 6 8 6 11'''
** Pseudo-[[equipentatoic]]: 315.7, 487.9, 717.5, 889.7, 1205.5
** Septimal melodic minor pentatonic: '''8 3 14 14 3'''
** Undecimal lydian-aeolian pentatonic: 229.6, 631.4, 1033.3, 1205.5
** Septimal Picardy pentatonic: '''8 6 11 3 14'''
** Septimal melodic minor pentatonic: 229.6, 315.7, 717.5, 1119.4, 1205.5
** Undecimal lydian-aeolian pentatonic: '''8 14 3 11 6'''
** Major pseudo-[[pelog]]: 401.8, 487.9, 717.5, 1119.4, 1205.5
** Yokai pentatonic: '''3 14 8 3 14'''




; [[Tonality diamond]]s
; [[Tonality diamond]]s
These take advantage of 42ed257/128’s improved approximations of the full 47-limit compared to 42edo, especially its improved 3/1 and 5/1.
These take advantage of 42ed257/128’s improved approximations of the full 7-limit compared to 42edo, especially its improved 3/1 and 5/1.
* [[Diamond7]]: '''8 1 2 2 4 3 2 2 4 3 2 2 8'''
* [[Diamond7]]: '''8 1 2 2 4 3 2 2 4 3 2 2 8'''
* [[Diamond9]]: '''6 1 1 1 2 2 2 2 3 2 2 3 1 3 1 2 1 1 6'''
* [[Diamond9]]: '''6 1 1 1 2 2 2 2 3 2 2 3 1 3 1 2 1 1 6'''
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