84edo: Difference between revisions

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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
Eliora (talk | contribs)
autopatrolled, emailconfirmed
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→Regular temperament properties: stylistic, explaining relation to orwell, adding rank 2 temps and stuff
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|Semitones = 7:7 (100¢:100¢)
|Semitones = 7:7 (100¢:100¢)
|Consistency = 7}}
|Consistency = 7}}
'''84edo''' divides the [[octave]] into 84 equal parts of size 14.286 [[cent]]s each.
{{EDO intro|84}}


== Theory ==
== Theory ==
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]].
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, 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.]]
=== Orwell ===
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]].  


{{Primes in edo|84}}
From a regular temperament perspective, orwell in 84edo comes in two varieties - the 84e val {{val| 84 133 195 236 290 }}, supporting the original orwell, and its [[patent val]] {{val| 84 133 195 236 291}} representing [[newspeak]]. 84edo orwell offers MOS of size 9, 13, 22, and 31, of which the 31 note scale is the maximum evenness scale.
 
=== Other ===
84edo is a significantly composite number, with divisors 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84. Being a small multiple of 12, it tempres out the [[Pythagorean comma]], thus supporting period-12 temperament [[compton]]. Being a small multiple of 28, it tempers out the [[oquatonic comma]], which maps 5/4 to 9\28.{{Primes in edo|84}}


== Table of intervals ==
== Table of intervals ==
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|1.151
|1.151
|
|
|}
=== Rank-2 temperaments by generator ===
{| class="wikitable center-all left-5"
!Periods
per octave
!Generator
!Cents
!Associated
ratio
!Temperaments
|-
| rowspan="2" |1
| rowspan="2" |[[Wikipedia:Nineteen Eighty-Four|19\84]]
| rowspan="2" |271.428
| rowspan="2" | 7/6
|[[Orwell]] (84e val)
|-
|[[Newspeak]] (84p val)
|-
|1
|27\84
|385.714
|5/4
|[[Mutt]]
|-
|12
|27\84
(6\84)
|385.714
(85.714)
|5/4
(20480/19683)
|[[Compton]]
|-
|28
|49\84
(1\84)
|500.000
(14.286)
|4/3
(105/104)
|[[Oquatonic]]
|}
|}
== Scales ==
== Scales ==

Revision as of 18:36, 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

Template:EDO intro

Theory

In the 13-limit it is the optimal patent val for the rank five temperament tempering out 144/143.

Orwell

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.

From a regular temperament perspective, orwell in 84edo comes in two varieties - the 84e val ⟨84 133 195 236 290], supporting the original orwell, and its patent val ⟨84 133 195 236 291] representing newspeak. 84edo orwell offers MOS of size 9, 13, 22, and 31, of which the 31 note scale is the maximum evenness scale.

Other

84edo is a significantly composite number, with divisors 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84. Being a small multiple of 12, it tempres out the Pythagorean comma, thus supporting period-12 temperament compton. Being a small multiple of 28, it tempers out the oquatonic comma, which maps 5/4 to 9\28.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

Rank-2 temperaments by generator

Periods

per octave

Generator Cents Associated

ratio

Temperaments
1 19\84 271.428 7/6 Orwell (84e val)
Newspeak (84p val)
1 27\84 385.714 5/4 Mutt
12 27\84

(6\84)

385.714

(85.714)

5/4

(20480/19683)

Compton
28 49\84

(1\84)

500.000

(14.286)

4/3

(105/104)

Oquatonic

Scales

Music

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