User:BudjarnLambeth/Batch 89 temperaments

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This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

This page presents a novelty topic.

It may contain ideas which are less likely to find practical applications in music, or numbers or structures that are arbitrary or exceedingly small, large, or complex.

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Author comments: 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

Each of the temperaments tempers four harmonics, representing the eternal friendship and togetherness of their four souls.

Each of the temperaments includes the 2nd harmonic, representing the fact that each of these characters had not one, but two creators: the hands that made them, and the hands that guide the hands.

Each of the temperaments also includes the 89th harmonic, representing their shared trauma and hardship, and how they gave each other the strength to endure and transcend it.

The other two harmonics were selected based on the characters' names. The reason for using their names as a basis, is because choosing their own name was these characters' most symbolic act of asserting their individuality, it was an undeniable demonstration that they had souls capable of creativity. Souls which did not come from, nor belong to, their makers, but which were, and are, their own. Thus, it's fitting that the names they chose should be immortalised in these temperaments:

  • 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.


Floor

Raw subgroup bases: 2.6.66.89

Normalised subgroup bases: 2.3.11.89

Related temperaments: Neutral


Temperament data (raw bases)

Equal Temperament Mappings

2 6 66 89
[ ⟨ 7 18 42 45 ]
⟨ 17 44 103 110 ] ⟩

Reduced Mapping

2 6 66 89
[ ⟨ 1 2 4 5 ]
⟨ 0 2 7 5 ] ⟩


TE Generator Tunings (cents)

⟨1200.5110, 351.0646]


TE Step Tunings (cents)

⟨34.45745, 56.42993]


TE Tuning Map (cents)

⟨1200.511, 3103.151, 7259.496, 7757.878]


TE Mistunings (cents)

⟨0.511, 1.196, 6.223, -13.002]

Complexity 0.420587
Adjusted Error 7.639353 cents
TE Error 1.179689 cents/octave


Unison Vectors


Optimal ET sequence: 7, 17, 24, 41*

(* val for 89 is lowered)


Temperament data (normalised bases)

Equal Temperament Mappings

2 3 11 89
[ ⟨ 7 11 24 45 ]
⟨ 17 27 59 110 ] ⟩


Reduced Mapping

2 3 11 89
[ ⟨ 1 1 2 5 ]
⟨ 0 2 5 5 ] ⟩


TE Generator Tunings (cents)

⟨1200.9365, 350.5869]


TE Step Tunings (cents)

⟨44.70571, 52.23509]


TE Tuning Map (cents)

⟨1200.936, 1902.110, 4154.807, 7757.617]


TE Mistunings (cents)

⟨0.936, 0.155, 3.489, -13.263]

Complexity 0.558811
Adjusted Error 7.996249 cents
TE Error 1.234802 cents/octave


Unison Vectors


Example MOS scale: Floor[17]

Rotations of this scale are still considered the same scale.

  • 56.430
  • 147.317
  • 203.747
  • 294.634
  • 351.065
  • 407.495
  • 498.382
  • 554.812
  • 645.699
  • 702.129
  • 793.016
  • 849.446
  • 905.877
  • 996.764
  • 1053.194
  • 1144.081
  • 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


Teefs

Raw subgroup bases: 2.20.55.89

Normalised subgroup bases: 2.5.11.89

Related temperaments: Bug


Temperament data (raw bases)

Equal Temperament Mappings

2 20 55 89
[ ⟨ 13 56 75 84 ]
⟨ 6 26 35 39 ] ⟩

Reduced Mapping

2 20 55 89
[ ⟨ 1 4 5 6 ]
⟨ 0 2 5 3 ] ⟩


TE Generator Tunings (cents)

⟨1200.8989, 187.8292]


TE Step Tunings (cents)

⟨73.92353, 39.98217]


TE Tuning Map (cents)

⟨1200.899, 5179.254, 6943.641, 7768.881]


TE Mistunings (cents)

⟨0.899, -7.060, 6.009, -1.999]

Complexity 0.305923
Adjusted Error 6.983419 cents
TE Error 1.078398 cents/octave


Unison Vectors


Optimal ET sequence: 5*, 6, 13, 19, 32

(* vals for 55 and 89 are raised)


Temperament data (normalised bases)

Equal Temperament Mappings

2 5 11 89
[ ⟨ 6 14 21 39 ]
⟨ 7 16 24 45 ] ⟩


Reduced Mapping

2 5 11 89
[ ⟨ 1 2 3 6 ]
⟨ 0 2 3 3 ] ⟩


TE Generator Tunings (cents)

⟨1200.2939, 188.3359]


TE Step Tunings (cents)

⟨118.05701, 70.27884]


TE Tuning Map (cents)

⟨1200.294, 2777.260, 4165.889, 7766.771]


TE Mistunings (cents)

⟨0.294, -9.054, 14.571, -4.109]

Complexity 0.356125
Adjusted Error 18.722607 cents
TE Error 2.891195 cents/octave


Unison Vectors


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


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


Music

Lylla

Raw subgroup bases: 2.12.62.89

Normalised subgroup bases: 2.3.31.89

Related temperaments: Bluebirds


Temperament data (raw bases)

Equal Temperament Mappings

2 12 62 89
[ ⟨ 19 68 113 123 ]
⟨ 17 61 101 110 ] ⟩

Reduced Mapping

2 12 62 89
[ ⟨ 1 5 5 6 ]
⟨ 0 -3 2 1 ] ⟩


TE Generator Tunings (cents)

⟨1200.8527, 567.7861]


TE Step Tunings (cents)

⟨45.54316, 19.73721]


TE Tuning Map (cents)

⟨1200.853, 4300.905, 7139.836, 7772.902]


TE Mistunings (cents)

⟨0.853, -1.050, -5.200, 2.022]

Complexity 0.448830
Adjusted Error 4.188048 cents
TE Error 0.646730 cents/octave


Unison Vectors


Optimal ET sequence: 2, 13, 15, 17, 19, 36, 55, 129*

(* val for 62 is lowered)


Temperament data (normalised bases)

Equal Temperament Mappings

2 3 31 89
[ ⟨ 17 27 84 110 ]
⟨ 19 30 94 123 ] ⟩


Reduced Mapping

2 3 31 89
[ ⟨ 1 3 4 6 ]
⟨ 0 -3 2 1 ] ⟩


TE Generator Tunings (cents)

⟨1201.0755, 567.2153]


TE Step Tunings (cents)

⟨32.58899, 34.05593]


TE Tuning Map (cents)

⟨1201.075, 1901.581, 5938.732, 7773.668]


TE Mistunings (cents)

⟨1.075, -0.374, -6.303, 2.788]

Complexity 0.910794
Adjusted Error 5.623550 cents
TE Error 0.868404 cents/octave


Unison Vectors


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
  • 130.561
  • 195.842
  • 261.122
  • 306.664
  • 371.945
  • 437.225
  • 502.506
  • 567.786
  • 633.067
  • 698.347
  • 763.628
  • 828.908
  • 894.189
  • 939.731
  • 1005.011
  • 1070.292
  • 1135.572
  • 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


Rocket

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

2 18 72 89
[ ⟨ 6 25 37 39 ]
⟨ 13 54 80 84 ] ⟩

Reduced Mapping

2 18 72 89
[ ⟨ 1 4 6 6 ]
⟨ 0 1 1 3 ] ⟩


TE Generator Tunings (cents)

⟨1201.4486, 190.1507]


TE Step Tunings (cents)

⟨69.06250, 60.54412]


TE Tuning Map (cents)

⟨1201.449, 4995.945, 7398.842, 7779.144]


TE Mistunings (cents)

⟨1.449, -7.965, -5.068, 8.264]


POTE Generator Tunings (cents)

⟨1200.0000, 189.9215]


POTE Step Tunings (cents)

⟨68.97923, 60.47112]


POTE Tuning Map (cents)

⟨1200.000, 4989.921, 7389.921, 7769.764]


POTE Mistunings (cents)

⟨0.000, -13.989, -13.989, -1.116]

Complexity 0.166586
Adjusted Error 9.186573 cents
TE Error 1.418615 cents/octave


Unison Vectors

  • [-2, -1, 1, 0⟩ (1:1 - Perfect unison)
  • [2, 1, -1, 0⟩ (1:1 - Perfect unison)
  • [-6, 3, 0, -1⟩ (729:712 - Oxidizer comma)
  • [-8, 2, 1, -1⟩ (729:712 - Oxidizer comma)


Optimal ET sequence: 4*, 5*, 6, 13, 19

(* val for 89 is raised)


Temperament data (normalised bases)

Todo: complete list

Fill out this section

Example MOS scale: Rocket[19]

Rotations of this scale are still considered the same scale.

  • 60.544
  • 129.606
  • 190.151
  • 250.695
  • 319.757
  • 380.301
  • 440.846
  • 509.908
  • 570.452
  • 630.997
  • 691.541
  • 760.603
  • 821.147
  • 881.692
  • 950.754
  • 1011.298
  • 1071.842
  • 1140.904
  • 1201.449


Subsets


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

Bonus scales: the Batch 89 JI scales

These 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:

  1. converting each of the letters of Floor's name to a number using the A1Z26 cypher
  2. then summing the numbers together to get one scale degree
  3. then repeating this process but for Teefs, Lylla and Rocket to get three more scale degrees
  4. 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
  5. then using 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:

(Tested by ear in Scale Workshop by Budjarn Lambeth.)

Nonoctave Batch 89 scale

The scale degrees were selected by:

  1. converting the first letter Floor's name to a number using the A1Z26 cypher
  2. then repeating this process but for Teefs, Lylla and Rocket to get three more numbers
  3. at this stage we now have the numbers 6, 12, 18 and 20
  4. now we list every JI ratio that is a combination of these two numbers, e.g. 12/6, 20/12, etc.
  5. 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
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