84edo: Difference between revisions

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Eliora (talk | contribs)
autopatrolled, emailconfirmed
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enrichment with information, emphasizing relation to orwell, i'll add relative error later because it isn't simply absolute calibrated to edo size, it's by complexity
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Its [[patent val]] {{val| 84 133 195 236 291}} makes it an excellent orwell tuning and also a good one for compton, and the 84e val, {{val| 84 133 195 236 290 }}, is almost identical to the 11-limit POTE tuning for orwell. In the [[13-limit]] it is the [[optimal patent val]] for the rank five temperament tempering out [[144/143]].
Its [[patent val]] {{val| 84 133 195 236 291}} makes it an excellent orwell tuning and also a good one for compton, and the 84e val, {{val| 84 133 195 236 290 }}, is almost identical to the 11-limit POTE tuning for orwell. In the [[13-limit]] it is the [[optimal patent val]] for the rank five temperament tempering out [[144/143]].


84edo is where the '''[[orwell]]''' temperament takes its name from, since the generator of 7/6 is equal to 19 steps of the EDO, referencing the [[Wikipedia:Nineteen Eighty-Four|book 1984]]. The maximum evenness orwell in this temperament is a 31 note scale. Orwell in 84edo comes in two varieties - 31e & 84, being the proper orwell, and 31 & 84, being [[Newspeak|newspeak.]]
84edo is where the '''[[orwell]]''' temperament takes its name from, since the generator of 7/6 is equal to 19 steps of the EDO, referencing the [[Wikipedia:Nineteen Eighty-Four|book 1984]]. The maximum evenness orwell in this temperament is a 31 note scale, and other MOS are of size 9, 13, and 22. Orwell in 84edo comes in two varieties - 31e & 84, being the proper orwell, and 31 & 84, being [[Newspeak|newspeak.]]


{{Primes in edo|84}}
{{Primes in edo|84}}


== Table of intervals ==
== Table of intervals ==
For this table, the notation of Orwell[9] from the [[4L 5s]] page is taken. Notes are denoted as LsLsLsLss = JKLMNOPQRJ, and raising and lowering by a chroma (L − s), 3 steps in this instance, is denoted by & "amp" and @ "at".
{| class="wikitable"
{| class="wikitable"
|+Table of 84edo intervals
|+Table of 84edo intervals
!Step
!Step
!Size (Cents)
!Size (Cents)
!Orwell[31] Name
!Orwell note
(if tonic is J)
!Orwellian Name
!Associated ratio
!Associated ratio
|-
|-
|0
|0
|0.000
|0.000
|J
|unison, prime
|unison, prime
|1/1
|1/1 exact
|-
|-
|3
|3
|42.857
|42.857
|J&
|
|
|-
|11
|157.143
|K
|second
|second
|
|
|-
|-
|6
|19
|85.714
|271.429
|L
|third
|third
|[[7/6]]
|-
|22
|314.286
|L&
|major third
|
|
|-
|-
|9
|30
|128.571
|428.571
|M
|fourth
|
|
|-
|38
|542.857
|N
|fifth
|[[11/8]] in the 84b val
|-
|41
|585.714
|N&
|
|
|
|-
|49
|700.000
|O
|sixth
|[[3/2]]
|-
|57
|814.286
|P
|seventh
|[[5/3]]
|-
|-
|11
|60
|157.142
|857.143
|P&
|
|
|[[105/64]]
|-
|68
|971.429
|Q
|eighth
|
|
|-
|-
|14
|76
|200.000
|1085.714
|
|R
|ninth
|
|
|-
|-
|17
|79
|242.857
|1128.571
|R&
|
|
|
|
|-
|-
|19
|84
|271.428
|1200.000
|eighth
|J (tenth above)
|7/6
|perfect tenth
|[[2/1]] exact
|}
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" |Subgroup
! rowspan="2" |[[Comma list]]
! rowspan="2" |[[Mapping]]
! rowspan="2" |Optimal
8ve stretch (¢)
! colspan="2" |Tuning error
|-
![[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
|-
|2.3.5
|78732/78125, 531441/524288
|{{val|84 133 195}}
|0.498
|0.531
|
|-
|-
|38
|2.3.5.7
|225/224, 1728/1715, 321489/320000
|{{val|84 133 195 236}}
|0.141
|0.769
|
|
|minor fifteenth
|11/8 (84e val)
|-
|-
|39
|2.3.5.7.11
|225/224, 441/440, 1944/1925, 8019/8000
|{{val|84 133 195 236 291}}
| -0.225
|1.003
|
|
|major fifteenth
|11/8
|-
|-
|57
|2.3.5.7.11
|99/98, 121/120, 1728/1715, 321489/320000
|{{val|84 133 195 236 290}} (84e)
|0.601
|1.151
|
|
|twenty-second
|5/3
|}
|}
== Tempered commas ==
[[5-limit]] commas: 78732/78125, 531441/524288, 2109375/2097152
[[7-limit]] commas: 225/224, 1728/1715, 2430/2401, 6144/6125
[[11-limit]] commas: 441/440, 1344/1331, 1375/1372
84e: 99/98, 121/120, 176/175, 385/384, 540/539, 5632/5625
[[13-limit]] commas: 144/143, 351/350, 364/363, 625/625
84e: 275/273, 640/637, 351/350, 352/351, 625/624, 1001/1000
== Scales ==
== Scales ==
* Orwell[9] - [[4L 5s]]
* Orwell[9] - [[4L 5s]]

Revision as of 18:21, 13 September 2022

← 83edo 84edo 85edo →
Prime factorization 22 × 3 × 7
Step size 14.2857 ¢ 
Fifth 49\84 (700 ¢) (→ 7\12)
Semitones (A1:m2) 7:7 (100 ¢ : 100 ¢)
Consistency limit 7
Distinct consistency limit 7

84edo divides the octave into 84 equal parts of size 14.286 cents each.

Theory

Its patent val ⟨84 133 195 236 291] makes it an excellent orwell tuning and also a good one for compton, and the 84e val, ⟨84 133 195 236 290], is almost identical to the 11-limit POTE tuning for orwell. In the 13-limit it is the optimal patent val for the rank five temperament tempering out 144/143.

84edo is where the orwell temperament takes its name from, since the generator of 7/6 is equal to 19 steps of the EDO, referencing the book 1984. The maximum evenness orwell in this temperament is a 31 note scale, and other MOS are of size 9, 13, and 22. Orwell in 84edo comes in two varieties - 31e & 84, being the proper orwell, and 31 & 84, being newspeak.

Script error: No such module "primes_in_edo".

Table of intervals

For this table, the notation of Orwell[9] from the 4L 5s page is taken. Notes are denoted as LsLsLsLss = JKLMNOPQRJ, and raising and lowering by a chroma (L − s), 3 steps in this instance, is denoted by & "amp" and @ "at".

Table of 84edo intervals
Step Size (Cents) Orwell note

(if tonic is J)

Orwellian Name Associated ratio
0 0.000 J unison, prime 1/1 exact
3 42.857 J&
11 157.143 K second
19 271.429 L third 7/6
22 314.286 L& major third
30 428.571 M fourth
38 542.857 N fifth 11/8 in the 84b val
41 585.714 N&
49 700.000 O sixth 3/2
57 814.286 P seventh 5/3
60 857.143 P& 105/64
68 971.429 Q eighth
76 1085.714 R ninth
79 1128.571 R&
84 1200.000 J (tenth above) perfect tenth 2/1 exact

Regular temperament properties

Subgroup Comma list Mapping Optimal

8ve stretch (¢)

Tuning error
Absolute (¢) Relative (%)
2.3.5 78732/78125, 531441/524288 ⟨84 133 195] 0.498 0.531
2.3.5.7 225/224, 1728/1715, 321489/320000 ⟨84 133 195 236] 0.141 0.769
2.3.5.7.11 225/224, 441/440, 1944/1925, 8019/8000 ⟨84 133 195 236 291] -0.225 1.003
2.3.5.7.11 99/98, 121/120, 1728/1715, 321489/320000 ⟨84 133 195 236 290] (84e) 0.601 1.151

Scales

Music

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