User:BudjarnLambeth/1ed28.7c: Difference between revisions
Created page with "{{Infobox ET|1ed31444/30927}} '''Arithmetic pitch sequence of 720jot''' ('''APS720jot'''), also known as '''1 equal division of 720jot''' ('''1ed720jot''') or '''720-jot equal..." |
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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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Although it is not exactly 28.7 cents, it is close enough that it can reasonably be referred to as '''1 equal division of 28.7 cents''' ('''1ed28.7c'''). This title is used for ease of navigation and comparison with other [[arithmetic pitch sequence]]s on the wiki. | Although it is not exactly 28.7 cents, it is close enough that it can reasonably be referred to as '''1 equal division of 28.7 cents''' ('''1ed28.7c'''). This title is used for ease of navigation and comparison with other [[arithmetic pitch sequence]]s on the wiki. | ||
==Theory== | |||
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]]. | ||
==Harmonics== | 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). | ||
{{harmonics in cet|28.701432|intervals=prime|title=Harmonics in APS720jot}} | |||
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=== | |||
APS720jot has less than 40% relative error on every single prime harmonic from 2 up to 47, making it a suitable full [[47-limit]] tuning. | |||
No other stretched or compressed version of 42edo has that property. | |||
{{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= | {{harmonics in equal|42|intervals=prime|columns=16|title=Harmonics in 42edo}} | ||
== Notation == | |||
APS720jot can use most notation systems designed for 42edo. See [[42edo#Notation]]. | |||
==Scala file== | ==Scala file== | ||
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1205.460144 | 1205.460144 | ||
</pre> | </pre> | ||
== Scales == | |||
To generate these in [[Scale Workshop]], you can first generate them in 42edo, then apply a stretch of 1.00455012. | |||
; [[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''' | |||
* [[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[19]: '''2 2 3 2 2 2 2 3 2 2 2 3 2 2 2 2 3 2 2''' | |||
; 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 | |||
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''' | |||
* [[Diamond9]]: '''6 1 1 1 2 2 2 2 3 2 2 3 1 3 1 2 1 1 6''' | |||
; [[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. | |||
* Pentany(4 of 1,3,5,7,11): '''6 13 8 10 5''' | |||
* Pentany(4 of 5,17,23,29,37): '''10 14 8 5 5''' | |||
* Pentany(4 of 5,17,23,29,41): '''10 14 7 2 9''' | |||
* Pentany(4 of 5,17,23,31,41): '''7 3 8 6 18''' | |||
* Hexany(2 of 1,3,5,7): '''10 3 10 8 3 8''' | |||
* Hexany(2 of 5,17,23,31): '''10 6 3 5 10 8''' ' | |||
== Instruments == | |||
* [[Lumatone]] | |||
** 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 == | == See also == | ||
* [[ | * [[Gallery of arithmetic pitch sequences]] | ||
== Notes == | |||
[[Category:42-tone scales]] | [[Category:42-tone scales]] | ||