44edo: Difference between revisions

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did my best to add ratios
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→Intervals: sed -E "s#([0-9]+)/([0-9]+)#\[\[\1/\2\]\]#g" 44EDOintervals.txt > 44EDOintervals.new.txt # also adjust ToDo because consistency check and additional column for ratios of 7 are still needed; add subgroup specification; replaced some ratios, but needs further checking
Line 18: Line 18:


== Intervals ==
== Intervals ==
{{Todo|complete table|inline=1|text = Add a column of approximated JI ratios to the following table; recommend dual ratios for the 7th harmonic which has a lot of relative error.}}
{{Todo|complete table|inline=1|text = Consistency check in column of approximated JI ratios in the following table; recommend dual ratios for the 7th harmonic which has a lot of relative error; might need some additional ratios; 29th harmonic is not terrible, so either add ratios of 29 or remove ratios of 31.}}
In 44edo, sharps and flats alter pitch by 6 edosteps. This means intervals can be notated with half sharps and half flats equal to 3 EDOsteps, in addition to ups and downs. The table below uses only sharps, flats, and ups and downs. When translating music from 22edo to 44edo, single ups and downs simply become double ups and downs (vEb in 22edo would be vvEb in 44edo).  
In 44edo, sharps and flats alter pitch by 6 edosteps. This means intervals can be notated with half sharps and half flats equal to 3 EDOsteps, in addition to ups and downs. The table below uses only sharps, flats, and ups and downs. When translating music from 22edo to 44edo, single ups and downs simply become double ups and downs (vEb in 22edo would be vvEb in 44edo).  


Line 25: Line 25:
! Degrees
! Degrees
! Cents
! Cents
!Approximate JI ratios
!Approximate JI ratios*
! colspan="3" | [[Ups and downs notation]]
! colspan="3" | [[Ups and downs notation]]
|-
|-
| 0
| 0
| 0.000
| 0.000
|1/1
| [[1/1]]
| Perfect 1sn
| Perfect 1sn
| P1
| P1
Line 37: Line 37:
| 1
| 1
| 27.273
| 27.273
|64/63
| [[64/63]]
| Up 1sn
| Up 1sn
| ^1
| ^1
Line 44: Line 44:
| 2
| 2
| 54.545
| 54.545
|33/32
| [[33/32]], [[32/31]]
| Minor 2nd
| Minor 2nd
| m2
| m2
Line 51: Line 51:
| 3
| 3
| 81.818
| 81.818
|23/22
| [[23/22]]
| Upminor 2nd
| Upminor 2nd
| ^m2
| ^m2
Line 58: Line 58:
| 4
| 4
| 109.091
| 109.091
|16/15
| [[16/15]]
| Dupminor 2nd, Downmid 2nd
| Dupminor 2nd, Downmid 2nd
| ^^m2, v~2
| ^^m2, v~2
Line 65: Line 65:
| 5
| 5
| 136.364
| 136.364
|69/64
| [[13/12]]
| Mid 2nd
| Mid 2nd
| ~2
| ~2
Line 72: Line 72:
| 6
| 6
| 163.636
| 163.636
|11/10
| [[11/10]], [[32/29]]
| Dudmajor 2nd, Upmid 2nd
| Dudmajor 2nd, Upmid 2nd
| vvM2, ^~2
| vvM2, ^~2
Line 79: Line 79:
| 7
| 7
| 190.909
| 190.909
|28/25
| [[28/25]]
| Downmajor 2nd
| Downmajor 2nd
| vM2
| vM2
Line 86: Line 86:
| 8
| 8
| 218.182
| 218.182
|9/8
| [[9/8]]
| Major 2nd
| Major 2nd
| M2
| M2
Line 93: Line 93:
| 9
| 9
| 245.455
| 245.455
|8/7
| [[8/7]]
| Upmajor 2nd, Downminor 3rd
| Upmajor 2nd, Downminor 3rd
| ^M2, vm3
| ^M2, vm3
Line 100: Line 100:
| 10
| 10
| 272.727
| 272.727
|7/6, 32/27
| [[7/6]], [[32/27]]
| Minor 3rd
| Minor 3rd
| m3
| m3
Line 107: Line 107:
| 11
| 11
| 300.000
| 300.000
|19/16
| [[19/16]]
| Upminor 3rd
| Upminor 3rd
| ^m3
| ^m3
Line 114: Line 114:
| 12
| 12
| 327.273
| 327.273
|11/9, 76/63
| [[11/9]], [[29/24]]
| Dupminor 3rd, Downmid 3rd
| Dupminor 3rd, Downmid 3rd
| ^^m3, v~3
| ^^m3, v~3
Line 121: Line 121:
| 13
| 13
| 354.545
| 354.545
|16/13
| [[16/13]]
| Mid 3rd
| Mid 3rd
| ~3
| ~3
Line 128: Line 128:
| 14
| 14
| 381.818
| 381.818
|5/4
| [[5/4]]
| Dudmajor 3rd, Upmid 3rd
| Dudmajor 3rd, Upmid 3rd
| vvM3, ^~3
| vvM3, ^~3
Line 135: Line 135:
| 15
| 15
| 409.091
| 409.091
|80/63
| [[80/63]]
| Downmajor 3rd
| Downmajor 3rd
| vM3
| vM3
Line 142: Line 142:
| 16
| 16
| 436.364
| 436.364
|81/64, 9/7
| [[81/64]], [[9/7]]
| Major 3rd
| Major 3rd
| M3
| M3
Line 149: Line 149:
| 17
| 17
| 463.636
| 463.636
|21/16
| [[21/16]]
| Upmajor 3rd, Down 4th
| Upmajor 3rd, Down 4th
| ^M3, v4
| ^M3, v4
Line 156: Line 156:
| 18
| 18
| 490.909
| 490.909
|4/3
| [[4/3]]
| Perfect 4th
| Perfect 4th
| P4
| P4
Line 163: Line 163:
| 19
| 19
| 518.182
| 518.182
|4/3?
| [[19/14]]
| Up 4th
| Up 4th
| ^4
| ^4
Line 170: Line 170:
| 20
| 20
| 545.455
| 545.455
|11/8
| [[11/8]]
| Dup 4th, Downmid 4th, Dim 5th
| Dup 4th, Downmid 4th, Dim 5th
| ^^4, v~4, d5
| ^^4, v~4, d5
Line 177: Line 177:
| 21
| 21
| 572.727
| 572.727
|7/5
| [[7/5]]
| Mid 4th, Updim 5th
| Mid 4th, Updim 5th
| ~4, ^d5
| ~4, ^d5
Line 184: Line 184:
| 22
| 22
| 600.000
| 600.000
|99/70, 140/99
| [[99/70]], [[140/99]]
| Upmid 4th, Downmid 5th
| Upmid 4th, Downmid 5th
| ^~4, v~5
| ^~4, v~5
Line 191: Line 191:
| 23
| 23
| 627.273
| 627.273
|10/7
| [[10/7]]
| Downaug 4th, Mid 5th
| Downaug 4th, Mid 5th
| vA4, ~5
| vA4, ~5
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| 24
| 24
| 654.545
| 654.545
|16/11
| [[16/11]]
| Aug 4th, Upmid 5th, Dud 5th
| Aug 4th, Upmid 5th, Dud 5th
| A4, ^~5, vv5
| A4, ^~5, vv5
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| 25
| 25
| 681.818
| 681.818
|3/2?
| [[28/19]]
| Down 5th
| Down 5th
| v5
| v5
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| 26
| 26
| 709.091
| 709.091
|3/2
| [[3/2]]
| Perfect 5th
| Perfect 5th
| P5
| P5
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| 27
| 27
| 736.364
| 736.364
|32/21
| [[32/21]]
| Up 5th, Downminor 6th
| Up 5th, Downminor 6th
| ^5, vm6
| ^5, vm6
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| 28
| 28
| 763.636
| 763.636
|14/9
| [[14/9]]
| Minor 6th
| Minor 6th
| m6
| m6
Line 233: Line 233:
| 29
| 29
| 790.909
| 790.909
|128/81
| [[128/81]]
| Upminor 6th
| Upminor 6th
| ^m6
| ^m6
Line 240: Line 240:
| 30
| 30
| 818.182
| 818.182
|8/5
| [[8/5]]
| Dupminor 6th, Downmid 6th
| Dupminor 6th, Downmid 6th
| ^^m6, v~6
| ^^m6, v~6
Line 247: Line 247:
| 31
| 31
| 845.455
| 845.455
|13/8
| [[13/8]]
| Mid 6th
| Mid 6th
| ~6
| ~6
Line 254: Line 254:
| 32
| 32
| 872.727
| 872.727
|5/3
| [[5/3]], [[48/29]]
| Dudmajor 6th, Upmid 6th
| Dudmajor 6th, Upmid 6th
| vvM6, ^~6
| vvM6, ^~6
Line 261: Line 261:
| 33
| 33
| 900.000
| 900.000
|27/16
| [[27/16]]
| Downmajor 6th
| Downmajor 6th
| vM6
| vM6
Line 268: Line 268:
| 34
| 34
| 927.273
| 927.273
|12/7
| [[12/7]]
| Major 6th
| Major 6th
| M6
| M6
Line 275: Line 275:
| 35
| 35
| 954.545
| 954.545
|7/4?
| [[7/4]]?
| Upmajor 6th, Downminor 7th
| Upmajor 6th, Downminor 7th
| ^M6, vm7
| ^M6, vm7
Line 282: Line 282:
| 36
| 36
| 981.818
| 981.818
|7/4, 16/9
| [[7/4]], [[16/9]]
| Minor 7th
| Minor 7th
| m7
| m7
Line 289: Line 289:
| 37
| 37
| 1009.091
| 1009.091
|9/5
| [[9/5]]
| Upminor 7th
| Upminor 7th
| ^m7
| ^m7
Line 296: Line 296:
| 38
| 38
| 1036.364
| 1036.364
|20/11
| [[20/11]], [[29/16]]
| Dupminor 7th, Downmid 7th
| Dupminor 7th, Downmid 7th
| ^^m7, v~7
| ^^m7, v~7
Line 303: Line 303:
| 39
| 39
| 1063.636
| 1063.636
|37/20
| [[24/13]]
| Mid 7th
| Mid 7th
| ~7
| ~7
Line 310: Line 310:
| 40
| 40
| 1090.909
| 1090.909
|15/8
| [[15/8]]
| Dudmajor 7th, Upmid 7th
| Dudmajor 7th, Upmid 7th
| vvM7, ^~7
| vvM7, ^~7
Line 317: Line 317:
| 41
| 41
| 1118.182
| 1118.182
|40/21
| [[40/21]]
| Downmajor 7th
| Downmajor 7th
| vM7
| vM7
Line 324: Line 324:
| 42
| 42
| 1145.455
| 1145.455
|27/14, 31/16
| [[27/14]], [[31/16]]
| Major 7th
| Major 7th
| M7
| M7
Line 331: Line 331:
| 43
| 43
| 1172.727
| 1172.727
|63/32
| [[63/32]]
| Upmajor 7th, Down 8ve
| Upmajor 7th, Down 8ve
| ^M7, v8
| ^M7, v8
Line 338: Line 338:
| 44
| 44
| 1200.000
| 1200.000
|2/1
| [[2/1]]
| Perfect 8ve
| Perfect 8ve
| P8
| P8
| D
| D
|}
|}
<nowiki>* In 2.3.5.7.11.13.17.19.23.29.31 subgroup</nowiki>


== Notation ==
== Notation ==

Revision as of 16:43, 6 May 2026

← 43edo 44edo 45edo →
Prime factorization 22 × 11
Step size 27.2727 ¢ 
Fifth 26\44 (709.091 ¢) (→ 13\22)
Semitones (A1:m2) 6:2 (163.6 ¢ : 54.55 ¢)
Consistency limit 5
Distinct consistency limit 5

44 equal divisions of the octave (abbreviated 44edo or 44ed2), also called 44-tone equal temperament (44tet) or 44 equal temperament (44et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 44 equal parts of about 27.3 ¢ each. Each step represents a frequency ratio of 21/44, or the 44th root of 2.

Theory

44edo is a double of 22edo, to which it adds the ratios of 13, 19, and 23. While not the most accurate 2.3.5.7.11 tuning, 22edo is certainly a relatively compact one, and it's natural to extend it this way. The most practically useful of these additions is easily the 13th harmonic with its neutral intervals, but the 17th, 19th, and 23rd are not to be dismissed.

It is on the optimal ET sequence for 7-, 11- and 13-limit nautilus temperament, for 11-limit spell temperament, and for 13-limit cantrip temperament. In the 13-limit it supplies the optimal patent val for vigin temperament.

The 2*44 subgroup of 44edo is 2.9.5.21.11.13.17.19.23, on which 44 tempers out the same commas as the patent val for 88edo.

Harmonics

Approximation of odd harmonics in 44edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +7.1 -4.5 +13.0 -13.0 -5.9 +4.9 +2.6 +4.1 +2.5 -7.1 -1.0
Relative (%) +26.2 -16.5 +47.6 -47.7 -21.5 +18.1 +9.7 +15.2 +9.1 -26.2 -3.7
Steps
(reduced) 		70
(26) 		102
(14) 		124
(36) 		139
(7) 		152
(20) 		163
(31) 		172
(40) 		180
(4) 		187
(11) 		193
(17) 		199
(23)

Subsets and supersets

44edo has subsets 2, 4, 11, 22.

One step of 44edo is very close (only 0.0086 cents sharp) to 64/63 (the septimal comma). Ruthenium temperament realizes this proximity through a regular temperament perspective, and it is supported by a large number of edos which are a multiple of 44 - for example 1012edo, 1848edo, and 2684edo.

Intervals

Todo: complete table

Consistency check in column of approximated JI ratios in the following table; recommend dual ratios for the 7th harmonic which has a lot of relative error; might need some additional ratios; 29th harmonic is not terrible, so either add ratios of 29 or remove ratios of 31.

In 44edo, sharps and flats alter pitch by 6 edosteps. This means intervals can be notated with half sharps and half flats equal to 3 EDOsteps, in addition to ups and downs. The table below uses only sharps, flats, and ups and downs. When translating music from 22edo to 44edo, single ups and downs simply become double ups and downs (vEb in 22edo would be vvEb in 44edo).

Degrees Cents Approximate JI ratios* Ups and downs notation
0 0.000 1/1 Perfect 1sn P1 D
1 27.273 64/63 Up 1sn ^1 ^D
2 54.545 33/32, 32/31 Minor 2nd m2 Eb
3 81.818 23/22 Upminor 2nd ^m2 ^Eb
4 109.091 16/15 Dupminor 2nd, Downmid 2nd ^^m2, v~2 ^^Eb
5 136.364 13/12 Mid 2nd ~2 vvvE, ^^^Eb
6 163.636 11/10, 32/29 Dudmajor 2nd, Upmid 2nd vvM2, ^~2 vvE
7 190.909 28/25 Downmajor 2nd vM2 vE
8 218.182 9/8 Major 2nd M2 E
9 245.455 8/7 Upmajor 2nd, Downminor 3rd ^M2, vm3 ^E, vF
10 272.727 7/6, 32/27 Minor 3rd m3 F
11 300.000 19/16 Upminor 3rd ^m3 ^F
12 327.273 11/9, 29/24 Dupminor 3rd, Downmid 3rd ^^m3, v~3 ^^F
13 354.545 16/13 Mid 3rd ~3 ^^^F, vvvF#
14 381.818 5/4 Dudmajor 3rd, Upmid 3rd vvM3, ^~3 vvF#
15 409.091 80/63 Downmajor 3rd vM3 vF#
16 436.364 81/64, 9/7 Major 3rd M3 F#
17 463.636 21/16 Upmajor 3rd, Down 4th ^M3, v4 ^F#, vG
18 490.909 4/3 Perfect 4th P4 G
19 518.182 19/14 Up 4th ^4 ^G
20 545.455 11/8 Dup 4th, Downmid 4th, Dim 5th ^^4, v~4, d5 Ab, ^^G
21 572.727 7/5 Mid 4th, Updim 5th ~4, ^d5 ^^^G, vvvG#
22 600.000 99/70, 140/99 Upmid 4th, Downmid 5th ^~4, v~5 vvG#, ^^Ab
23 627.273 10/7 Downaug 4th, Mid 5th vA4, ~5 vvvA, ^^^Ab
24 654.545 16/11 Aug 4th, Upmid 5th, Dud 5th A4, ^~5, vv5 G#, vvA
25 681.818 28/19 Down 5th v5 vA
26 709.091 3/2 Perfect 5th P5 A
27 736.364 32/21 Up 5th, Downminor 6th ^5, vm6 ^A, vBb
28 763.636 14/9 Minor 6th m6 Bb
29 790.909 128/81 Upminor 6th ^m6 ^Bb
30 818.182 8/5 Dupminor 6th, Downmid 6th ^^m6, v~6 ^^Bb
31 845.455 13/8 Mid 6th ~6 ^^^Bb, vvvB
32 872.727 5/3, 48/29 Dudmajor 6th, Upmid 6th vvM6, ^~6 vvB
33 900.000 27/16 Downmajor 6th vM6 vB
34 927.273 12/7 Major 6th M6 B
35 954.545 7/4? Upmajor 6th, Downminor 7th ^M6, vm7 ^B, vC
36 981.818 7/4, 16/9 Minor 7th m7 C
37 1009.091 9/5 Upminor 7th ^m7 ^C
38 1036.364 20/11, 29/16 Dupminor 7th, Downmid 7th ^^m7, v~7 ^^C
39 1063.636 24/13 Mid 7th ~7 ^^^C, vvvC#
40 1090.909 15/8 Dudmajor 7th, Upmid 7th vvM7, ^~7 vvC#
41 1118.182 40/21 Downmajor 7th vM7 vC#
42 1145.455 27/14, 31/16 Major 7th M7 C#
43 1172.727 63/32 Upmajor 7th, Down 8ve ^M7, v8 ^C#, vD
44 1200.000 2/1 Perfect 8ve P8 D

* In 2.3.5.7.11.13.17.19.23.29.31 subgroup

Notation

Ups and downs notation

44edo can be notated with ups and downs, spoken as up, dup, trup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, trud, dupflat etc.

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12
Sharp symbol
         
             
Flat symbol          
             

Half-sharps and half-flats can be used to avoid triple arrows:

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12
Sharp symbol
 
    
    
    
Flat symbol  
    
    
    

Alternative ups and downs have sharps and flats with arrows borrowed from extended Helmholtz–Ellis notation:

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Sharp symbol  
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
Flat symbol  
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 

If double arrows are not desirable, arrows can be attached to quarter-tone accidentals:

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12 13
Sharp symbol
 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
Flat symbol  
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 

Sagittal notation

This notation uses the same sagittal sequence as edos 23b, 30, and 37, and is a superset of the notations for edos 22 and 11.

Evo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation513/51281/8027/26

Revo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation513/51281/8027/26

Evo-SZ flavor

Sagittal notationPeriodic table of EDOs with sagittal notation513/51281/8027/26

In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's primary comma (the comma it exactly represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it approximately represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this edo.

Regular temperament properties

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 3\44 81.82 22/21 Nautilus (44d)
1 7\44 190.91 9/8 Spell (44def) / cantrip (44de)
1 9\44 245.46 15/13 Immunity (44cff, 2.3.5.13)
1 13\44 354.55 11/9 Beatles / ringo (44e)
1 15\44 409.09 5/4 Hocus (44)
2 3\44 81.82 22/21 Harry (44ceff)
4 4\44 109.09 16/15 Bidia (44d, 7-limit)

* Octave-reduced form, reduced to the first half-octave

Scales

  • Evacuated planet[idiosyncratic term] (approximated from 66afdo): 5 13 8 12 6
  • Approximations of gamelan scales:
    • 5-tone pelog: 4 6 15 4 15
    • 7-tone pelog: 4 6 9 6 4 10 5
    • 5-tone slendro: 9 9 8 9 9

Instrument layouts

Music

Bryan Deister
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