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User:BudjarnLambeth/Batch 89 temperaments: Difference between revisions

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Added normalised subgroup bases as per recommendation on the talk page; will further edit the rest of the page to use the normalised bases in the near future
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Line 1: Line 1:
{{Technical data page}}<br><br>
{{novelty}}
{{Editable user page|Feel free to add examples of music made with these temperaments, and feel free to add any new scales, approaches or other concepts you develop based on these ideas.}}
These temperaments were documented in honour of the members of Batch 89, as depicted in the 2023 film ''Guardians of the Galaxy Vol. 3''. The temperaments were first documented in 2023.
== Methodology ==
== Methodology ==
These temperaments were documented in honour of the members of Batch 89, as depicted in ''Guardians of the Galaxy Vol. 3'' (2023).
Each of the temperaments tempers four harmonics, representing the eternal friendship and togetherness of their four souls.
Each of the temperaments tempers four harmonics, representing the eternal friendship and togetherness of their four souls.


Line 12: Line 17:
* One harmonic was chosen by converting the first letter of the character's name to a number using the A1Z26 cypher, and using that number.
* One harmonic was chosen by converting the first letter of the character's name to a number using the A1Z26 cypher, and using that number.
* The other was chosen by converting each of the letters of the character's name to a number using the A1Z26 cypher, and then summing the numbers together.  
* The other was chosen by converting each of the letters of the character's name to a number using the A1Z26 cypher, and then summing the numbers together.  


== Floor ==
== Floor ==
Raw subgroup bases: 2.6.66.89
Raw subgroup bases: '''2.6.66.89'''


Normalised subgroup bases: 2.3.11.89
Normalised subgroup bases: '''2.3.11.89'''


Related temperaments: [[Neutral]]


=== Temperament data (raw bases) ===
Equal Temperament Mappings
Equal Temperament Mappings
{| class="wikitable"
{| class="wikitable"
Line 42: Line 51:
|] ⟩
|] ⟩
|}
|}


Reduced Mapping
Reduced Mapping
Line 86: Line 94:
TE Mistunings (cents)
TE Mistunings (cents)


⟨0.511, 1.196, 6.223, -13.002]
⟨0.511, 1.196, 6.223, -13.002]  
 
{| class="wikitable"
|Complexity
|0.420587
|-
|Adjusted Error
|7.639353 cents
|-
|TE Error
|1.179689 cents/octave
|}
 
 
Unison Vectors
* [-3, 1, -1, 1⟩ (89:88 - [[89/88|Tailwind comma]])
* [0, -5, 0, 2⟩ (7921:7776 - [[7921/7776|Attic comma]])
* [3, -6, 1, 1⟩ (979:972 - [[979/972|Basement comma]])
* [-6, 7, -2, 0⟩ (243:242 - [[243/242|Rastma]])
 
 
{{Optimal ET sequence|legend=1| 7, 17, 24, 41* }}
 
(* val for 89 is lowered)
 


=== Temperament data (normalised bases) ===


Equal Temperament Mappings
Equal Temperament Mappings
Line 93: Line 126:
!
!
!2
!2
!6
!3
!66
!11
!89
!89
!
!
Line 100: Line 133:
|[ ⟨
|[ ⟨
|7
|7
|18
|11
|42
|24
|45
|45
|]
|]
Line 107: Line 140:
|⟨
|⟨
|17
|17
|44
|27
|103
|59
|110
|110
|] ⟩
|] ⟩
Line 118: Line 151:
!
!
!2
!2
!6
!3
!66
!11
!89
!89
!
!
|-
|-
|[ ⟨
|[ ⟨
|1
|1
|1
|2
|2
|4
|5
|5
|]
|]
Line 133: Line 166:
|0
|0
|2
|2
|7
|5
|5
|5
|] ⟩
|] ⟩
Line 139: Line 172:




POTE Generator Tunings (cents)
TE Generator Tunings (cents)


⟨1200.0000, 350.9151]
⟨1200.9365, 350.5869]




POTE Step Tunings (cents)
TE Step Tunings (cents)


⟨34.44279, 56.40591]
⟨44.70571, 52.23509]




POTE Tuning Map (cents)
TE Tuning Map (cents)


⟨1200.000, 3101.830, 7256.406, 7754.576]
⟨1200.936, 1902.110, 4154.807, 7757.617]




POTE Mistunings (cents)
TE Mistunings (cents)


⟨0.000, -0.125, 3.133, -16.304]
⟨0.936, 0.155, 3.489, -13.263]


{| class="wikitable"
{| class="wikitable"
|Complexity
|Complexity
|0.420587
|0.558811
|-
|-
|Adjusted Error
|Adjusted Error
|7.639353 cents
|7.996249 cents
|-
|-
|TE Error
|TE Error
|1.179689 cents/octave
|1.234802 cents/octave
|}
|}


Line 172: Line 205:
Unison Vectors
Unison Vectors


* [-3, 1, -1, 1⟩ (89:88)
* [-3, 0, -1, 1⟩ (89:88) - ([[89/88|Tailwind comma]])
* [0, -5, 0, 2⟩ (7921:7776)
* [-1, 5, -2, 0⟩ (243:242) - ([[243/242|Rastma]])
* [3, -6, 1, 1⟩ (979:972)
* [-2, -5, 1, 1⟩ (979:972) - ([[979/972|Basement comma]])
* [-6, 7, -2, 0⟩ (243:242)
* [-5, -5, 0, 2⟩ (7921:7776) - ([[7921/7776|Attic comma]])


{{Optimal ET sequence|legend=1| 7, 17, 24, 41* }}
(* val for 89 is lowered)


=== Example MOS scale: Floor[17] ===
=== Example MOS scale: Floor[17] ===
Rotations of this scale are still considered the same scale.


* 56.430
* 56.430
Line 200: Line 231:
* 1144.081
* 1144.081
* 1200.511
* 1200.511
'''Subsets'''
Cosmic (approximated from [[32afdo]])
* 498.382
* 702.129
* 793.016
* 996.764
* 1200.511
Major (approximated from [[12edo]])
(rotate to get its modes e.g. minor)
* 203.747
* 407.494
* 498.381
* 702.129
* 905.876
* 1109.624
* 1200.511
Minor Hexatonic
* 203.747
* 294.634
* 498.382
* 702.129
* 996.764
* 1200.511
=== Music ===
* ''[https://www.youtube.com/watch?v=YHbBJOHdD_E Improvisation in Floor(17) in Scale Workshop]'' by [[Budjarn Lambeth]] (Jun 2023)


== Teefs ==
== Teefs ==
Raw subgroup bases: 2.20.55.89
Raw subgroup bases: '''2.20.55.89'''
 
Normalised subgroup bases: '''2.5.11.89'''


Normalised subgroup bases: 2.5.11.89
Related temperaments: [[Bug]]


=== Temperament data (raw bases) ===
Equal Temperament Mappings
Equal Temperament Mappings
{| class="wikitable"
{| class="wikitable"
Line 229: Line 302:
|] ⟩
|] ⟩
|}
|}


Reduced Mapping
Reduced Mapping
Line 275: Line 347:
⟨0.899, -7.060, 6.009, -1.999]
⟨0.899, -7.060, 6.009, -1.999]


{| class="wikitable"
|Complexity
|0.305923
|-
|Adjusted Error
|6.983419 cents
|-
|TE Error
|1.078398 cents/octave
|}
Unison Vectors
* [-5, 1, -1, 1⟩ (89:88 - [[89/88|Tailwind comma]])
* [0, 3, 0, -2⟩ (8000:7921 - [[8000/7921|Incisor comma]])
* [-5, 4, -1, -1⟩ (1000:979 - [[1000/979|Canine comma]])
* [-5, -2, -1, 3⟩ (704969:704000 - [[704969/704000|Molar comma]])
{{Optimal ET sequence|legend=1| 5*, 6, 13, 19, 32 }}


POTE Generator Tunings (cents)
(* vals for 55 and 89 are raised)
 
 
=== Temperament data (normalised bases) ===
 
Equal Temperament Mappings
{| class="wikitable"
!
!2
!5
!11
!89
!
|-
|[ ⟨
|6
|14
|21
|39
|]
|-
|⟨
|7
|16
|24
|45
|] ⟩
|}
 
 
Reduced Mapping
{| class="wikitable"
!
!2
!5
!11
!89
!
|-
|[ ⟨
|1
|2
|3
|6
|]
|-
|⟨
|0
|2
|3
|3
|] ⟩
|}
 
 
TE Generator Tunings (cents)


⟨1200.0000, 187.6886]
⟨1200.2939, 188.3359]




POTE Step Tunings (cents)
TE Step Tunings (cents)


⟨73.86819, 39.95225]
⟨118.05701, 70.27884]




POTE Tuning Map (cents)
TE Tuning Map (cents)


⟨1200.000, 5175.377, 6938.443, 7763.066]
⟨1200.294, 2777.260, 4165.889, 7766.771]




POTE Mistunings (cents)
TE Mistunings (cents)


⟨-0.000, -10.936, 0.812, -7.814]
⟨0.294, -9.054, 14.571, -4.109]


{| class="wikitable"
{| class="wikitable"
|Complexity
|Complexity
|0.305923
|0.356125
|-
|-
|Adjusted Error
|Adjusted Error
|6.983419 cents
|18.722607 cents
|-
|-
|TE Error
|TE Error
|1.078398 cents/octave
|2.891195 cents/octave
|}
|}


Line 309: Line 456:
Unison Vectors
Unison Vectors


* [-5, 1, -1, 1⟩ (89:88)
* [-3, 0, -1, 1⟩ (89:88) - ([[89/88|Tailwind comma]])
* [0, 3, 0, -2⟩ (8000:7921)
* [0, 3, -2, 0⟩ (125:121) - ([[125/121|Jug comma]])
* [-5, 4, -1, -1⟩ (1000:979)
* [3, 3, -1, -1⟩ (1000:979) - ([[1000/979|Canine comma]])
* [-5, -2, -1, 3⟩ (704969:704000)
* [6, 3, 0, -2⟩ (8000:7921) - ([[8000/7921|Incisor comma]])
 
 
=== Example inflected MOS scale: UFO scale ===
An [[inflected MOS]] of the Teefs[19] [[MOS scale]]. Also a subset of the Teefs[51] MOS scale, and it can be approximated closely by [[51edo]]. Rotations of this scale are still considered the same scale.
 
* 47.780
* 92.776
* 188.336
* 210.834
* 258.614
* 306.394
* 399.170
* 494.730
* 542.510
* 657.784
* 705.564
* 801.124
* 893.900
* 941.680
* 989.460
* 1011.958
* 1107.518
* 1152.514
* 1200.294
 
 
'''Subsets'''


{{Optimal ET sequence|legend=1| 5*, 6, 13, 19, 32 }}


(* vals for 55 and 89 are raised)
Cosmic (approximated from [[32afdo]])
* 494.730
* 705.564
* 801.124
* 1011.958
* 1200.294
 
 
Major (approximated from [[12edo]])
 
(rotate to get its modes e.g. minor)
* 188.336
* 399.170
* 494.730
* 705.564
* 893.900
* 1107.518
* 1200.294
 
 
Minor Hexatonic (approximated from [[12edo]])
* 188.336
* 306.394
* 494.730
* 705.564
* 1011.958
* 1200.294
 


=== Example MOS scale: Teefs[19] ===
=== Music ===


* 73.924
* 113.906
* 187.829
* 261.753
* 301.735
* 375.658
* 449.582
* 489.564
* 563.488
* 637.411
* 711.335
* 751.317
* 825.241
* 899.164
* 939.146
* 1013.070
* 1086.993
* 1126.975
* 1200.899


== Lylla ==
== Lylla ==
Raw subgroup bases: 2.12.62.89
Raw subgroup bases: '''2.12.62.89'''
 
Normalised subgroup bases: '''2.3.31.89'''


Normalised subgroup bases: 2.3.31.89
Related temperaments: [[Bluebirds]]
 
 
=== Temperament data (raw bases) ===


Equal Temperament Mappings
Equal Temperament Mappings
Line 368: Line 554:
|] ⟩
|] ⟩
|}
|}


Reduced Mapping
Reduced Mapping
Line 414: Line 599:
⟨0.853, -1.050, -5.200, 2.022]
⟨0.853, -1.050, -5.200, 2.022]


{| class="wikitable"
|Complexity
|0.448830
|-
|Adjusted Error
|4.188048 cents
|-
|TE Error
|0.646730 cents/octave
|}
Unison Vectors
* [7, 0, 1, -2⟩ (7936:7921 - [[7936/7921|Lily comma]])
* [-9, 1, 2, -1⟩ (2883:2848 - [[2883/2848|Lilac comma]])
* [-16, 1, 1, 1⟩ (8277:8192 - [[8277/8192|Lilly pilly comma]])
* [-2, 1, 3, -3⟩ (714984:704969 - [[714984/704969|Lightyear comma]])
{{Optimal ET sequence|legend=1| 2, 13, 15, 17, 19, 36, 55, 129* }}
(* val for 62 is lowered)
=== Temperament data (normalised bases) ===


Equal Temperament Mappings
Equal Temperament Mappings
Line 419: Line 629:
!
!
!2
!2
!12
!3
!62
!31
!89
!89
!
!
|-
|-
|[ ⟨
|[ ⟨
|19
|17
|68
|27
|113
|84
|123
|110
|]
|]
|-
|-
|⟨
|⟨
|17
|19
|61
|30
|101
|94
|110
|123
|] ⟩
|] ⟩
|}
|}
Line 444: Line 654:
!
!
!2
!2
!12
!3
!62
!31
!89
!89
!
!
Line 451: Line 661:
|[ ⟨
|[ ⟨
|1
|1
|5
|3
|5
|4
|6
|6
|]
|]
Line 465: Line 675:




POTE Generator Tunings (cents)
TE Generator Tunings (cents)


⟨1200.0000, 567.3830]
⟨1201.0755, 567.2153]




POTE Step Tunings (cents)
TE Step Tunings (cents)


⟨45.51082, 19.72320]
⟨32.58899, 34.05593]




POTE Tuning Map (cents)
TE Tuning Map (cents)


⟨1200.000, 4297.851, 7134.766, 7767.383]
⟨1201.075, 1901.581, 5938.732, 7773.668]




POTE Mistunings (cents)
TE Mistunings (cents)


⟨0.000, -4.104, -10.270, -3.497]
⟨1.075, -0.374, -6.303, 2.788]


{| class="wikitable"
{| class="wikitable"
|Complexity
|Complexity
|0.448830
|0.910794
|-
|-
|Adjusted Error
|Adjusted Error
|4.188048 cents
|5.623550 cents
|-
|-
|TE Error
|TE Error
|0.646730 cents/octave
|0.868404 cents/octave
|}
|}


Line 498: Line 708:
Unison Vectors
Unison Vectors


* [7, 0, 1, -2⟩ (7936:7921)
* [-5, 1, 2, -1⟩ (2883:2848) - ([[2883/2848|Lilac comma]])
* [-9, 1, 2, -1⟩ (2883:2848)
* [8, 0, 1, -2⟩ (7936:7921) - ([[7936/7921|Lily comma]])
* [-16, 1, 1, 1⟩ (8277:8192)
* [-13, 1, 1, 1⟩ (8277:8192) - ([[8277/8192|Lilly pilly comma]])
* [-2, 1, 3, -3⟩ (714984:704969)
* [-18, 2, 3, 0⟩ (268119:262144) - ([[268119/262144|Lutra comma]])


{{Optimal ET sequence|legend=1| 2, 13, 15, 17, 19, 36, 55, 129* }}
(* val for 62 is lowered)


=== Example MOS scale: Lylla[19] ===
=== Example MOS scale: Lylla[19] ===
Rotations of this scale are still considered the same scale. Note the 829 cent interval, a solid approximation of [[acoustic phi]].


* 65.281
* 65.281
Line 528: Line 736:
* 1135.572
* 1135.572
* 1200.853
* 1200.853
'''Subsets'''
Cosmic (approximated from [[32afdo]])
* 502.506
* 698.347
* 828.908
* 1005.011
* 1200.853
Minor Hexatonic (approximated from [[12edo]])
* 195.842
* 306.664
* 502.506
* 698.347
* 1005.011
* 1200.853
=== Music ===
* ''[https://www.youtube.com/watch?v=kcXcSIWOWaI Improvisation in Lylla(19) in Scale Workshop]'' by [[Budjarn Lambeth]] (Jun 2023)


== Rocket ==
== Rocket ==
Raw subgroup bases: 2.18.72.89


Normalised subgroup bases: 2.9+.9-.89 (contains a dual harmonic 9)
Raw subgroup bases: '''2.18.72.89'''
 
Normalised subgroup bases: '''2.9+.9-.89''' (contains a dual harmonic 9)
 
Related temperaments: [[Lu]], [[Deutone]]
 
 
=== Temperament data (raw bases) ===


Equal Temperament Mappings
Equal Temperament Mappings
Line 557: Line 796:
|] ⟩
|] ⟩
|}
|}


Reduced Mapping
Reduced Mapping
Line 621: Line 859:
POTE Mistunings (cents)
POTE Mistunings (cents)


⟨0.000, -13.989, -13.989, -1.116]
⟨0.000, -13.989, -13.989, -1.116]  


{| class="wikitable"
{| class="wikitable"
Line 637: Line 875:
Unison Vectors
Unison Vectors


* [-2, -1, 1, 0⟩ (1:1)
* [-2, -1, 1, 0⟩ (1:1 - [[1/1|Perfect unison]])
* [2, 1, -1, 0⟩ (1:1)
* [2, 1, -1, 0⟩ (1:1 - Perfect unison)
* [-6, 3, 0, -1⟩ (729:712)
* [-6, 3, 0, -1⟩ (729:712 - [[729/712|Oxidizer comma]])
* [-8, 2, 1, -1⟩ (729:712)
* [-8, 2, 1, -1⟩ (729:712 - Oxidizer comma)
 


{{Optimal ET sequence|legend=1| 4*, 5*, 6, 13, 19 }}
{{Optimal ET sequence|legend=1| 4*, 5*, 6, 13, 19 }}


(* val for 89 is raised)
(* val for 89 is raised)
=== Temperament data (normalised bases) ===
{{todo|complete list|inline=yes|comment=Fill out this section}}


=== Example MOS scale: Rocket[19] ===
=== Example MOS scale: Rocket[19] ===
Rotations of this scale are still considered the same scale.


* 60.544
* 60.544
Line 668: Line 912:
* 1201.449
* 1201.449


[[Category:Lists of temperaments]]
 
[[Category:17-tone scales]]
'''Subsets'''
[[Category:19-tone scales]]
 
[[Category:MOS scales]]
 
[[Category:89-limit]]
Cosmic (approximated from [[32afdo]])
* 509.908
* 691.541
* 821.147
* 1011.298
* 1201.449
 
 
Minor Hexatonic (approximated from [[12edo]])
* 190.151
* 319.757
* 509.908
* 691.541
* 1011.298
* 1201.449
 
 
=== Music ===
* ''[https://www.youtube.com/watch?v=_NepcrTf9ng The Wound]'' - [[Budjarn Lambeth]] (Nov 2024)
 
== Bonus scales: the Batch 89 JI scales ==
These [[just intonation|just intonation scales]] were documented in honour of the members of Batch 89, as depicted in the 2023 film ''Guardians of the Galaxy Vol. 3''. The scales were first documented in 2026.
 
=== Otonal Batch 89 scale ===
The scale degrees were selected by:
# ''converting each of the letters of Floor's name to a number using the A1Z26 cypher''
# ''then summing the numbers together to get one scale degree''
# ''then repeating this process but for Teefs, Lylla and Rocket to get three more scale degrees''
# ''then adding the octave, so that identical copies of the scale are infinitely reflected up and down the pitch spectrum, reflecting that these characters will be together forever''
# ''then using {{w|negative harmony}} to change the scale from utonal to otonal, to reflect how the characters found wonder even in a dark situation and used it to make the galaxy a brighter place''
 
This results in the following scale:
* 110/89
* 124/89
* 132/89
* 144/89
* 178/89
 
Which is a subset of [[89afdo]] (''see [[afdo]]'').
 
==== Edo approximations ====
 
Edos with nice-sounding approximations of this scale include:
* [[14edo]]: 4 3 1 2 4
* [[17edo]]: 5 3 2 2 5
* [[20edo]]: 6 4 1 3 6
* [[21edo]]: 6 4 2 3 6
* [[23edo]]: 7 4 2 3 7 ''(closest to JI until 65edo)''
* [[24edo]]: 7 4 3 3 7
* [[26edo]]: 8 4 1 3 8
* [[27edo]]: 8 5 2 4 8
* [[30edo]]: 9 5 3 4 9
* [[31edo]]: 9 6 3 4 9
* [[34edo]]: 10 6 3 5 10
* [[37edo]]: 11 7 3 5 11
* [[40edo]]: 12 7 4 5 12
* [[44edo]]: 13 8 4 6 13
* [[46edo]]: 14 8 4 6 14
* [[53edo]]: 16 9 5 7 16
* [[60edo]]: 18 9 5 8 18
* [[65edo]]: 20 11 6 8 20
 
(Tested by ear in [[Scale Workshop]] by [[Budjarn Lambeth]].)
 
=== Nonoctave Batch 89 scale ===
The scale degrees were selected by:
# ''converting the first letter Floor's name to a number using the A1Z26 cypher''
# ''then repeating this process but for Teefs, Lylla and Rocket to get three more numbers''
# ''at this stage we now have the numbers 6, 12, 18 and 20''
# ''now we list every JI ratio that is a combination of these two numbers, e.g. 12/6, 20/12, etc.''
# ''then we sort these in order from the smallest interval to the largest, using the largest as the scale [[period]]''
 
This results in the following scale:
* 20/18
* 18/12
* 20/12
* 12/6
* 18/6
* 20/6
Retrieved from "https://en.xen.wiki/w/User:BudjarnLambeth/Batch_89_temperaments"