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{{todo|merge articles|inline=1|text='''Do not merge until discussion is complete. See [[User talk:BudjarnLambeth/1ed28.7c]]'''. Merge into [[42edo]]? (in a section titled "[[stretched and compressed tuning]]s", most likely, focusing on the most important aspects)}}
{{Editable user page}}
{{Infobox ET|1ed31444/30927}}
{{Infobox ET|1ed31444/30927}}
'''Arithmetic pitch sequence of 720jot''' ('''APS720jot'''), also known as '''1 equal division of 720jot''' ('''1ed720jot''') or '''720-jot equal temperament''' ('''720jot-et'''), is an [[Equal-step tuning|equal-step]] scale with steps of 720 [[30103edo|jots]] (~28.7015 [[cents]]), or each 720th step of [[30103edo]].
'''Arithmetic pitch sequence of 720jot''' ('''APS720jot'''), also known as '''1 equal division of 720jot''' ('''1ed720jot''') or '''720-jot equal temperament''' ('''720jot-et'''), is an [[Equal-step tuning|equal-step]] scale with steps of 720 [[30103edo|jots]] (~28.7015 [[cents]]), or each 720th step of [[30103edo]].
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APS720jot divides [[2/1]] into ~41.8097 equal parts, making it close to [[42edo]], but with [[pseudo-octave|impure octaves]] and better approximations of harmonics [[3/1]] and [[5/1]].
APS720jot divides [[2/1]] into ~41.8097 equal parts, making it close to [[42edo]], but with [[pseudo-octave|impure octaves]] and better approximations of harmonics [[3/1]] and [[5/1]].


It is very similar to [[42ed257/128]], 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.
It is a kind of opposite twin to the scale [[1ed28.5c|APS715jot]], 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).


APS720jot is a kind of opposite twin to the scale [[191zpi]]. 191zpi makes [[42edo]] into a better 2.3.5.11.13 tuning, while APS720jot 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 [[42ed257/128]] and [[zpi|189zpi]].
 
See [[Table of stretched 42edo tunings]] for more.


===Harmonics===
===Harmonics===
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No other stretched or compressed version of 42edo has that property.
No other stretched or compressed version of 42edo has that property.
{{harmonics in cet|28.701432|intervals=prime|title=Harmonics in APS720jot}}
{{harmonics in cet|28.701432|intervals=prime|columns=16|title=Harmonics in APS720jot}}


42edo for comparison:
42edo for comparison:
{{harmonics in equal|42|intervals=prime|columns=10|title=Harmonics in 42edo}}
{{harmonics in equal|42|intervals=prime|columns=16|title=Harmonics in 42edo}}


== Notation ==
== Notation ==
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== Scales ==
== Scales ==
; [[MOS scale]]s
To generate these in [[Scale Workshop]], you can first generate them in 42edo, then apply a stretch of 1.00455012.
* Eugene/Tritikleismic[9]: '''3 8 3 3 8 3 3 8 3'''
 
 
; [[MOS scale]]s<ref>Seville/Sevond is not available because it is generated by 42edo’s sharp fifth, and APS720jot is designed to improve 42edo’s flat fifth instead.</ref>
* [[Augmented family#Eugene|Eugene]]/[[Tritikleismic]][9]: '''3 8 3 3 8 3 3 8 3'''
* Eugene/Tritikleismic[15]: '''3 3 2 3 3 3 3 2 3 3 3 3 2 3 3'''
* Eugene/Tritikleismic[15]: '''3 3 2 3 3 3 3 2 3 3 3 3 2 3 3'''
* Lemba[16]: '''3 2 3 2 3 3 2 3 3 2 3 2 3 3 2 3'''
* [[Lemba]][16]: '''3 2 3 2 3 3 2 3 3 2 3 2 3 3 2 3'''
* Qeema/Skateboard[15]: '''2 5 2 2 2 5 2 2 2 5 2 2 2 5 2'''
* [[Qeema]]/[[Skateboard]][15]: '''2 5 2 2 2 5 2 2 2 5 2 2 2 5 2'''
* Qeema/Skateboard[19]: '''2 2 3 2 2 2 2 3 2 2 2 3 2 2 2 2 3 2 2'''
* Qeema/Skateboard[19]: '''2 2 3 2 2 2 2 3 2 2 2 3 2 2 2 2 3 2 2'''
* ''Seville/Sevond is not available because it is generated by 42edo’s sharp fifth, and APS720jot is designed to improve 42edo’s flat fifth instead''
 
 
; Subsets of MOS scales
''(Names used are [[Template:Idiosyncratic|idiosyncratic]].)''
* Eugene/Tritikleismic[9]
** Groovy aeolian pentatonic: '''11 6 8 3 14'''
** [[Otonal]] mixolydian pentatonic: '''14 3 8 11 6'''
** Pseudo-[[equipentatonic]]: '''11 6 8 6 11'''
** Septimal melodic minor pentatonic: '''8 3 14 14 3'''
** Septimal Picardy pentatonic: '''8 6 11 3 14'''
** Undecimal lydian-aeolian pentatonic: '''8 14 3 11 6'''
** Yokai pentatonic: '''3 14 8 3 14'''
 


; [[Tonality diamond]]s
; [[Tonality diamond]]s
These take advantage of APS720jot’s improved approximations of the full 47-limit compared to 42edo, especially its improved 3/1 and 5/1.
These take advantage of APS720jot’s improved approximations of the full 47-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'''
 


; [[Combination product set]]s
; [[Combination product set]]s
These take advantage of APS720jot’s improved approximations of the full 47-limit compared to 42edo, especially its improved 3/1 and 5/1.
These take advantage of APS720jot’s improved approximations of the full 47-limit compared to 42edo, especially its improved 3/1 and 5/1.
* Pentany(4 of 1,3,5,7,11): '''6 13 8 10 5'''
* Pentany(4 of 1,3,5,7,11): '''6 13 8 10 5'''
* Pentany(4 of 5,17,23,29,37): '''5 10 14 8 5'''
* Pentany(4 of 5,17,23,29,37): '''10 14 8 5 5'''
* Pentany(4 of 5,17,23,29,41): '''2 9 10 14 7'''
* Pentany(4 of 5,17,23,29,41): '''10 14 7 2 9'''
* Pentany(4 of 5,17,23,31,41): '''2 9 6 18 7'''
* Pentany(4 of 5,17,23,31,41): '''7 3 8 6 18'''
* Pentany(4 of 5,17,23,31,43): '''4 10 6 18 4'''
* Hexany(2 of 1,3,5,7): '''10 3 10 8 3 8'''
* Hexany(2 of 1,3,5,7): '''10 3 10 8 3 8'''
* Hexany(2 of 5,17,23,29): '''8 10 4 10 5 5'''
* Hexany(2 of 5,17,23,31): '''10 6 3 5 10 8''' '
* Hexany(2 of 5,17,23,31): '''3 5 10 8 10 6'''
* Hexany(2 of 19,31,36,48): '''9 5 9 3 13 3'''
* Hexany(2 of 21,32,38,45): '''4 6 5 6 15 6'''


== Instruments ==
== Instruments ==
* [[Lumatone]]
* [[Lumatone]]
** APS720jot can use most types of [[Lumatone mapping for 42edo]]
** APS720jot can use most types of [[Lumatone mapping for 42edo]]
== Music ==
* ''[https://www.youtube.com/watch?v=vdjhC9i5KF4 Four Short Experiments in Octave Stretched 42edo (Dec 2024)]'' - [[Budjarn Lambeth]]
== See also ==
* [[Gallery of arithmetic pitch sequences]]
== Notes ==


[[Category:42-tone scales]]
[[Category:42-tone scales]]
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