84edo: Difference between revisions
Jump to navigation
Jump to search
→Interval table: +some basic ratios |
Expand on theory |
||
| Line 5: | Line 5: | ||
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]]. 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''' }} supporting [[newspeak]]. 84edo orwell offers mosses of size 9, 13, 22, and 31, of which the 31-note scale is the [[maximal evenness]] scale. | 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]]. 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''' }} supporting [[newspeak]]. 84edo orwell offers mosses of size 9, 13, 22, and 31, of which the 31-note scale is the [[maximal evenness]] scale. | ||
In the [[13-limit]] it is the [[optimal patent val]] for the rank-5 temperament tempering out [[144/143]]. | It has fairly good approximation to higher [[prime harmonic]]s such as [[13/1|13]], [[19/1|19]], [[23/1|23]], [[29/1|29]], and [[31/1|31]]. In fact, it is [[consistent]] to the no-11 no-17 [[25-odd-limit]]. In the [[13-limit]] it is the [[optimal patent val]] for the rank-5 temperament tempering out [[144/143]]. | ||
=== Prime harmonics === | === Prime harmonics === | ||
| Line 14: | Line 14: | ||
84edo is a [[largely composite]] number. Since 84 factors as {{factorization|84}}, 84edo has subset edos {{EDOs| 2, 3, 4, 6, 7, 12, 14, 21, 28, 42 }}. Being a small multiple of 12, 84et tempers out the [[Pythagorean comma]], thus supporting the 1/12-octave temperament [[compton]]. Being a small multiple of 28, it tempers out the [[oquatonic|oquatonic comma]], which maps 5/4 to 9\28. | 84edo is a [[largely composite]] number. Since 84 factors as {{factorization|84}}, 84edo has subset edos {{EDOs| 2, 3, 4, 6, 7, 12, 14, 21, 28, 42 }}. Being a small multiple of 12, 84et tempers out the [[Pythagorean comma]], thus supporting the 1/12-octave temperament [[compton]]. Being a small multiple of 28, it tempers out the [[oquatonic|oquatonic comma]], which maps 5/4 to 9\28. | ||
== | == 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 a "at" (the symbol "@" unlike in the 4L 5s page cannot be used because of technical details). | 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 a "at" (the symbol "@" unlike in the 4L 5s page cannot be used because of technical details). | ||
{| class="wikitable" | {| class="wikitable" | ||
Revision as of 08:14, 24 June 2024
| ← 83edo | 84edo | 85edo → |
Theory
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. 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] supporting newspeak. 84edo orwell offers mosses of size 9, 13, 22, and 31, of which the 31-note scale is the maximal evenness scale.
It has fairly good approximation to higher prime harmonics such as 13, 19, 23, 29, and 31. In fact, it is consistent to the no-11 no-17 25-odd-limit. In the 13-limit it is the optimal patent val for the rank-5 temperament tempering out 144/143.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -1.96 | -0.60 | +2.60 | +5.82 | +2.33 | -4.96 | +2.49 | +0.30 | -1.01 | -2.18 | +5.80 |
| Relative (%) | +0.0 | -13.7 | -4.2 | +18.2 | +40.8 | +16.3 | -34.7 | +17.4 | +2.1 | -7.0 | -15.2 | +40.6 | |
| Steps (reduced) |
84 (0) |
133 (49) |
195 (27) |
236 (68) |
291 (39) |
311 (59) |
343 (7) |
357 (21) |
380 (44) |
408 (72) |
416 (80) |
438 (18) | |
| Harmonic | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | 83 | 89 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.49 | +2.77 | +5.92 | -2.08 | -2.03 | -2.60 | +6.41 | +6.02 | +0.78 | +6.89 | +7.10 | +0.55 |
| Relative (%) | -3.4 | +19.4 | +41.5 | -14.5 | -14.2 | -18.2 | +44.9 | +42.1 | +5.5 | +48.2 | +49.7 | +3.8 | |
| Steps (reduced) |
450 (30) |
456 (36) |
467 (47) |
481 (61) |
494 (74) |
498 (78) |
510 (6) |
517 (13) |
520 (16) |
530 (26) |
536 (32) |
544 (40) | |
Subsets and supersets
84edo is a largely composite number. Since 84 factors as 22 × 3 × 7, 84edo has subset edos 2, 3, 4, 6, 7, 12, 14, 21, 28, 42. Being a small multiple of 12, 84et tempers out the Pythagorean comma, thus supporting the 1/12-octave temperament compton. Being a small multiple of 28, it tempers out the oquatonic comma, which maps 5/4 to 9\28.
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 a "at" (the symbol "@" unlike in the 4L 5s page cannot be used because of technical details).
| # | Cents | Approximate Ratios* | Ups and Downs Notation | 4L 5s Notation | ||||
|---|---|---|---|---|---|---|---|---|
| 0 | 0.000 | 1/1 | Perfect 1sn | P1 | D | Perfect 1sn | P1 | J |
| 1 | 14.286 | 81/80, 126/125 | Up 1sn | ^1 | ^D | Up 1sn | ^1 | J^ |
| 2 | 28.571 | 50/49, 64/63 | Dup 1sn | ^^1 | ^^D | Downaug 1sn | vA1 | Jv& |
| 3 | 42.857 | 36/35, 40/39, 49/48 | Trup 1sn | ^^^1 | ^^^D | Aug 1sn | A1 | J& |
| 4 | 57.143 | 27/26 | Trudminor 2nd | vvvm2 | vvvEb | Upaug 1sn, Downdim 2nd | ^A1, vd2 | J^&, Kvaa |
| 5 | 71.429 | 25/24, 26/25, 28/27 | Dudminor 2nd | vvm2 | vvEb | Dim 2nd | d2 | Kaa |
| 6 | 85.714 | 21/20 | Downminor 2nd | vm2 | vEb | Updim 2nd | ^d2 | K^aa |
| 7 | 100.000 | Minor 2nd | m2 | Eb | Downminor 2nd | vm2 | Kva | |
| 8 | 114.286 | 15/14, 16/15 | Upminor 2nd | ^m2 | ^Eb | Minor 2nd | m2 | Ka |
| 9 | 128.571 | 14/13 | Dupminor 2nd | ^^m2 | ^^Eb | Upminor 2nd | ^m2 | K^a |
| 10 | 142.857 | 13/12 | Trupminor 2nd | ^^^m2 | ^^^Eb | Downmajor 2nd | vM2 | Kv |
| 11 | 157.143 | Trudmajor 2nd | vvvM2 | vvvE | Major 2nd | M2 | K | |
| 12 | 171.429 | Dudmajor 2nd | vvM2 | vvE | Upmajor 2nd | ^M2 | K^ | |
| 13 | 185.714 | 10/9 | Downmajor 2nd | vM2 | vE | Downaug 2nd | vA2 | Kv& |
| 14 | 200.000 | 9/8 | Major 2nd | M2 | E | Aug 2nd | A2 | K& |
| 15 | 214.286 | Upmajor 2nd | ^M2 | ^E | Upaug 2nd, Downdim 3rd | ^A2, vd3 | K^&, Lva | |
| 16 | 228.571 | 8/7 | Dupmajor 2nd | ^^M2 | ^^E | Dim 3rd | d3 | La |
| 17 | 242.857 | 15/13 | Trupmajor 2nd | ^^^M2 | ^^^E | Updim 3rd | ^d3 | L^a |
| 18 | 257.143 | Trudminor 3rd | vvvm3 | vvvF | Down 3rd | v3 | Lv | |
| 19 | 271.429 | 7/6 | Dudminor 3rd | vvm2 | vvF | Perfect 3rd | P3 | L |
| 20 | 285.714 | Downminor 3rd | vm3 | vF | Up 3rd | ^3 | L^ | |
| 21 | 300.000 | 32/27 | Minor 3rd | m3 | F | Downaug 3rd | vA3 | Lv& |
| 22 | 314.286 | 6/5 | Upminor 3rd | ^m3 | ^F | Aug 3rd | A3 | L& |
| 23 | 328.571 | Dupminor 3rd | ^^m3 | ^^F | Upaug 3rd, Downdim 4th | ^A3, vd4 | L^&, Mvaa | |
| 24 | 342.857 | 39/32 | Trupminor 3rd | ^^^m3 | ^^^F | Dim 4th | d4 | Maa |
| 25 | 357.143 | 16/13 | Trudmajor 3rd | vvvM3 | vvvF# | Updim 4th | ^d4 | M^aa |
| 26 | 371.429 | 26/21 | Dudmajor 3rd | vvM3 | vvF# | Downminor 4th | vm4 | Mva |
| 27 | 385.714 | 5/4 | Downmajor 3rd | vM3 | vF# | Minor 4th | m4 | Ma |
| 28 | 400.000 | Major 3rd | M3 | F# | Upminor 4th | ^m4 | M^a | |
| 29 | 414.286 | Upmajor 3rd | ^M3 | ^F# | Downmajor 4th | vM4 | Mv | |
| 30 | 428.571 | 9/7 | Dupmajor 3rd | ^^M3 | ^^F# | Major 4th | M4 | M |
| 31 | 442.857 | Trupmajor 3rd | ^^^M3 | ^^^F# | Upmajor 4th | ^M4 | M^ | |
| 32 | 457.143 | 13/10 | Trud 4th | vvv4 | vvvG | Downaug 4th | vA4 | Mv& |
| 33 | 471.429 | 21/16 | Dud 4th | vv4 | vvG | Aug 4th | A4 | M& |
| 34 | 485.714 | Down 4th | v4 | vG | Downminor 5th | vm5 | Nva | |
| 35 | 500.000 | 4/3 | Perfect 4th | P4 | G | Minor 5th | m5 | Na |
| 36 | 514.286 | 27/20 | Up 4th | ^4 | ^G | Upminor 5th | ^m5 | N^a |
| 37 | 528.571 | Dup 4th | ^^4 | ^^G | Downmajor 5th | vM5 | Nv | |
| 38 | 542.857 | Trup 4th | ^^^4 | ^^^G | Major 5th | M5 | N | |
| 39 | 557.143 | 18/13 | Trudaug 4th | vvvA4 | vvvG# | Upmajor 5th | ^M5 | N^ |
| 40 | 571.429 | Dudaug 4th | vvA4 | vvG# | Downaug 5th | vA5 | Nv& | |
| 41 | 585.714 | 7/5 | Downaug 4th | vA4 | vG# | Aug 5th | A5 | N& |
| 42 | 600.000 | Aug 4th, Dim 5th | A4, d5 | G#, Ab | Upaug 5th, Downdim 6th | ^A5, vd6 | N^&, Ovaa | |
| 43 | 614.286 | 10/7 | Updim 5th | ^d5 | ^Ab | Dim 6th | d6 | Oaa |
| 44 | 628.571 | Dupdim 5th | ^^d5 | ^^Ab | Updim 6th | ^d6 | O^aa | |
| 45 | 642.857 | 13/9 | Trupdim 5th | ^^^d5 | ^^^Ab | Downminor 6th | vm6 | Ova |
| 46 | 657.143 | Trud 5th | vvv5 | vvvA | Minor 6th | m6 | Oa | |
| 47 | 671.429 | Dud 5th | vv5 | vvA | Upminor 6th | ^m6 | O^a | |
| 48 | 685.714 | 40/27 | Down 5th | v5 | vA | Downmajor 6th | vM6 | Ov |
| 49 | 700.000 | 3/2 | Perfect 5th | P5 | A | Major 6th | M6 | O |
| 50 | 714.286 | Up 5th | ^5 | ^A | Upmajor 6th | ^M6 | O^ | |
| 51 | 728.571 | 32/21 | Dup 5th | ^^5 | ^^A | Dim 7th | d7 | Paa |
| 52 | 742.857 | 20/13 | Trup 5th | ^^^5 | ^^^A | Aug 6th | A6 | O& |
| 53 | 757.143 | Trudminor 6th | vvvm6 | vvvBb | Downminor 7th | vm7 | Pva | |
| 54 | 771.429 | 14/9 | Dudminor 6th | vvm6 | vvBb | Minor 7th | m7 | Pa |
| 55 | 785.714 | Downminor 6th | vm6 | vBb | Upminor 7th | ^m7 | P^a | |
| 56 | 800.000 | Minor 6th | m6 | Bb | Downmajor 7th | vM7 | Pv | |
| 57 | 814.286 | 8/5 | Upminor 6th | ^m6 | ^Bb | Major 7th | M7 | P |
| 58 | 828.571 | 21/13 | Dupminor 6th | ^^m6 | ^^Bb | Upmajor 7th | ^M7 | P^ |
| 59 | 842.857 | 13/8 | Trupminor 6th | ^^^m6 | ^^^Bb | Downaug 7th | vA7 | Pv& |
| 60 | 857.143 | 64/39 | Trudmajor 6th | vvvM6 | vvvB | Aug 7th | A7 | P& |
| 61 | 871.429 | Dudmajor 6th | vvM6 | vvB | Upaug 7th, Downdim 8th | ^A7, vd8 | P^&, Qvaa | |
| 62 | 885.714 | 5/3 | Downmajor 6th | vM6 | vB | Dim 8th | d8 | Qaa |
| 63 | 900.000 | 27/16 | Major 6th | M6 | B | Updim 8th | ^d8 | Q^aa |
| 64 | 914.286 | Upmajor 6th | ^M6 | ^B | Down 8th | v8 | Qva | |
| 65 | 928.571 | 12/7 | Dupmajor 6th | ^^M6 | ^^B | Perfect 8th | P8 | Qa |
| 66 | 942.857 | Trupmajor 6th | ^^^M6 | ^^^B | Up 8th | ^8 | Q^a | |
| 67 | 957.143 | 26/15 | Trudminor 7th | vvvm7 | vvvC | Downaug 8th | vA8 | Qv |
| 68 | 971.429 | 7/4 | Dudminor 7th | vvm7 | vvC | Aug 8th | A8 | Q |
| 69 | 985.714 | Downminor 7th | vm7 | vC | Upaug 8th, Downdim 9th | ^A8, vd9 | Q^, Rvaa | |
| 70 | 1000.000 | 16/9 | Minor 7th | m7 | C | Dim 9th | d9 | Raa |
| 71 | 1014.286 | 9/5 | Upminor 7th | ^m7 | ^C | Updim 9th | ^d9 | R^aa |
| 72 | 1028.571 | Dupminor 7th | ^^m7 | ^^C | Downminor 9th | vm9 | Rva | |
| 73 | 1042.857 | Trupminor 7th | ^^^m7 | ^^^C | Minor 9th | m9 | Ra | |
| 74 | 1057.143 | 24/13 | Trudmajor 7th | vvvM7 | vvvC# | Upminor 9th | ^m9 | R^a |
| 75 | 1071.429 | 13/7 | Dudmajor 7th | vvM7 | vvC# | Downmajor 9th | vM9 | Rv |
| 76 | 1085.714 | 15/8, 28/15 | Downmajor 7th | vM7 | vC# | Major 9th | M9 | R |
| 77 | 1100.000 | Major 7th | M7 | C# | Upmajor 9th | ^M9 | R^ | |
| 78 | 1114.286 | 40/21 | Upmajor 7th | ^M7 | ^C# | Downaug 9th | vA9 | Rv& |
| 79 | 1128.571 | 25/13, 27/14, 48/25 | Dupmajor 7th | ^^M7 | ^^C# | Aug 9th | A9 | R& |
| 80 | 1142.857 | 52/27 | Trupmajor 7th | ^^^M7 | ^^^C# | Upaug 9th, Downdim 10th | ^A9, vd10 | R^&, Jva |
| 81 | 1157.143 | 35/18, 39/20, 96/49 | Trud 8ve | vvv8 | vvvD | Dim 10th | d10 | Ja |
| 82 | 1171.429 | 49/25, 63/32 | Dud 8ve | vv8 | vvD | Updim 10th | ^d10 | J^a |
| 83 | 1185.714 | 125/63, 160/81 | Down 8ve | v8 | vD | Down 10th | v10 | Jv |
| 84 | 1200.000 | 2/1 | Perfect 8ve | P8 | D | Perfect 10th | P10 | J |
* as a 2.3.5.7.13-subgroup temperament
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 | 3.72 |
| 2.3.5.7 | 225/224, 1728/1715, 78732/78125 | [⟨84 133 195 236]] | +0.141 | 0.769 | 5.39 |
| 2.3.5.7.11 | 225/224, 441/440, 1344/1331, 1728/1715 | [⟨84 133 195 236 291]] (84) | -0.225 | 1.003 | 7.02 |
| 2.3.5.7.11 | 99/98, 121/120, 176/175, 78732/78125 | [⟨84 133 195 236 290]] (84e) | +0.601 | 1.151 | 8.05 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 19\84 | 271.43 | 7/6 | Orwell (84e) |
| Newspeak (84) | ||||
| 1 | 25\84 | 357.14 | 768/625 | Dodifo |
| 1 | 27\84 | 385.71 | 5/4 | Mutt |
| 1 | 31\84 | 442.86 | 125/81 | Sensei |
| 1 | 41\84 | 585.71 | 7/5 | Merman |
| 2 | 5\84 | 71.43 | 25/24 | Narayana |
| 2 | 11\84 | 157.14 | 35/32 | Bison |
| 2 | 13\84 | 185.71 | 10/9 | Secant |
| 3 | 11\84 | 157.14 | 35/32 | Nessafof |
| 7 | 5\84 | 500.00 (14.29) |
4/3 (81/80) |
Absurdity |
| 12 | 27\84 (1\84) |
385.71 (14.29) |
5/4 (126/125) |
Compton |
| 21 | 41\84 (1\84) |
585.71 (14.29) |
91875/65536 (126/125) |
Akjayland |
| 28 | 49\84 (1\84) |
500.00 (14.29) |
4/3 (105/104) |
Oquatonic |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct
Scales
MOS
Brightest mode is listed.
Other
Music
- Ten for chamber ensemble (1991) Ives Ensemble recording (YouTube) [dead link]
- Two4 for violin and piano or shō (1991) Harr & Miyata recording (YouTube)
- Two5 for tenor trombone and piano (1991) Fulkerson & Denyer recording (YouTube)
- Two6 for violin and piano (1992) Haar & Snijders recording (YouTube)
- Requiem in Gb 1/7 Orwell (2023)
- Undiminished (2023)