User:BudjarnLambeth/1ed28.7c: Difference between revisions

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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)}}

'''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~~ 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 in cet|28.701432|intervals=prime|title=Harmonics in APS720jot}}

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:

{{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 ==

APS720jot can use most notation systems designed for 42edo. See [[42edo#Notation]].

==Scala file==

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1205.460144

</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 ==

* [[Gallery of arithmetic pitch sequences]]

== Notes ==

[[Category:42-tone scales]]

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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}}
 '''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]].
@@ Line 7: / Line 9: @@
 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 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.
+ed257/128’s step size is very close to that of [[42ed257/128]] and [[zpi|189zpi]].
+See [[Table of stretched 42edo tunings]] for more.
 ===Harmonics===
-{{harmonics in cet|28.701432|intervals=prime|title=Harmonics in APS720jot}}
+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}}
 edo 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 ==
+APS720jot can use most notation systems designed for 42edo. See [[42edo#Notation]].
 ==Scala file==
@@ Line 71: / Line 81: @@
 .460144
 </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 ==
+* [[Gallery of arithmetic pitch sequences]]
+== Notes ==
 [[Category:42-tone scales]]