44edo: Difference between revisions

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removed ratios of 7 (for now); checked consistency
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MisterShafXen (talk | contribs)
1,074 edits
added some ratios of 7, can't finish rn bc I'm running low on battery
Tags: Visual edit Mobile edit Mobile web edit Advanced mobile edit
Line 18: Line 18:


== Intervals ==
== Intervals ==
{{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.}}
{{Todo|complete table|inline=1|text = Recommend dual ratios for the 7th harmonic which has a lot of relative error; might need some additional ratios.}}


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 27: Line 27:
! Cents
! Cents
! Approximate ratios*
! Approximate ratios*
!Ratios of 7 (patent val)
!Ratios of 7 (44d val)
! colspan="3" | [[Ups and downs notation]]
! colspan="3" | [[Ups and downs notation]]
|-
|-
Line 32: Line 34:
| 0.0
| 0.0
| [[1/1]]
| [[1/1]]
|
|
| Perfect 1sn
| Perfect 1sn
| P1
| P1
Line 39: Line 43:
| 27.3
| 27.3
| [[65/64]]
| [[65/64]]
|
|[[64/63]]
| Up 1sn
| Up 1sn
| ^1
| ^1
Line 46: Line 52:
| 54.5
| 54.5
| [[32/31]], [[33/32]]
| [[32/31]], [[33/32]]
|51/49
|
| Minor 2nd
| Minor 2nd
| m2
| m2
Line 53: Line 61:
| 81.8
| 81.8
| [[23/22]]
| [[23/22]]
|
|[[22/21]]
| Upminor 2nd
| Upminor 2nd
| ^m2
| ^m2
Line 60: Line 70:
| 109.1
| 109.1
| [[16/15]]
| [[16/15]]
|
|147/136
| Dupminor 2nd, Downmid 2nd
| Dupminor 2nd, Downmid 2nd
| ^^m2, v~2
| ^^m2, v~2
Line 67: Line 79:
| 136.4
| 136.4
| [[13/12]]
| [[13/12]]
|[[14/13]]
|
| Mid 2nd
| Mid 2nd
| ~2
| ~2
Line 74: Line 88:
| 163.6
| 163.6
| [[11/10]], [[32/29]]
| [[11/10]], [[32/29]]
|31/28
|551/504
| Dudmajor 2nd, Upmid 2nd
| Dudmajor 2nd, Upmid 2nd
| vvM2, ^~2
| vvM2, ^~2
Line 81: Line 97:
| 190.9
| 190.9
| [[19/17]]
| [[19/17]]
|143/126
|352/315
| Downmajor 2nd
| Downmajor 2nd
| vM2
| vM2
Line 88: Line 106:
| 218.2
| 218.2
| [[9/8]]
| [[9/8]]
|[[8/7]]
|143/126
| Major 2nd
| Major 2nd
| M2
| M2
Line 95: Line 115:
| 245.5
| 245.5
| 52/45
| 52/45
|65/56
|[[8/7]]
| Upmajor 2nd, Downminor 3rd
| Upmajor 2nd, Downminor 3rd
| ^M2, vm3
| ^M2, vm3
Line 102: Line 124:
| 272.7
| 272.7
| [[32/27]]
| [[32/27]]
|[[7/6]]
|221/189
| Minor 3rd
| Minor 3rd
| m3
| m3
Line 109: Line 133:
| 300.0
| 300.0
| [[19/16]]
| [[19/16]]
|
|896/729
| Upminor 3rd
| Upminor 3rd
| ^m3
| ^m3
Line 116: Line 142:
| 327.3
| 327.3
| [[11/9]], [[29/24]]
| [[11/9]], [[29/24]]
|896/729
|76/63
| Dupminor 3rd, Downmid 3rd
| Dupminor 3rd, Downmid 3rd
| ^^m3, v~3
| ^^m3, v~3
Line 123: Line 151:
| 354.5
| 354.5
| [[16/13]]
| [[16/13]]
|26/21
|
| Mid 3rd
| Mid 3rd
| ~3
| ~3
Line 130: Line 160:
| 381.8
| 381.8
| [[5/4]]
| [[5/4]]
|
|26/21
| Dudmajor 3rd, Upmid 3rd
| Dudmajor 3rd, Upmid 3rd
| vvM3, ^~3
| vvM3, ^~3
Line 137: Line 169:
| 409.1
| 409.1
| 171/136
| 171/136
|
|[[80/63]]
| Downmajor 3rd
| Downmajor 3rd
| vM3
| vM3
Line 144: Line 178:
| 436.4
| 436.4
| [[81/64]]
| [[81/64]]
|[[9/7]]
|
| Major 3rd
| Major 3rd
| M3
| M3
Line 151: Line 187:
| 463.6
| 463.6
| [[13/10]]
| [[13/10]]
|
|[[9/7]]
| Upmajor 3rd, Down 4th
| Upmajor 3rd, Down 4th
| ^M3, v4
| ^M3, v4
Line 158: Line 196:
| 490.9
| 490.9
| [[4/3]]
| [[4/3]]
|62/49
|
| Perfect 4th
| Perfect 4th
| P4
| P4
Line 165: Line 205:
| 518.2
| 518.2
| 171/128
| 171/128
|189/143
|256/189
| Up 4th
| Up 4th
| ^4
| ^4
Line 172: Line 214:
| 545.5
| 545.5
| [[11/8]]
| [[11/8]]
|29/21
|
| Dup 4th, Downmid 4th, Dim 5th
| Dup 4th, Downmid 4th, Dim 5th
| ^^4, v~4, d5
| ^^4, v~4, d5
Line 179: Line 223:
| 572.7
| 572.7
| 46/33
| 46/33
|
|29/21
| Mid 4th, Updim 5th
| Mid 4th, Updim 5th
| ~4, ^d5
| ~4, ^d5
Line 186: Line 232:
| 600.0
| 600.0
| 44/31, 31/22
| 44/31, 31/22
|[[7/5]], [[10/7]]
|
| Upmid 4th, Downmid 5th
| Upmid 4th, Downmid 5th
| ^~4, v~5
| ^~4, v~5
Line 193: Line 241:
| 627.3
| 627.3
| 33/23
| 33/23
|
|[[10/7]]
| Downaug 4th, Mid 5th
| Downaug 4th, Mid 5th
| vA4, ~5
| vA4, ~5
Line 200: Line 250:
| 654.5
| 654.5
| [[16/11]]
| [[16/11]]
|
|
| Aug 4th, Upmid 5th, Dud 5th
| Aug 4th, Upmid 5th, Dud 5th
| A4, ^~5, vv5
| A4, ^~5, vv5
Line 207: Line 259:
| 681.8
| 681.8
| 256/171
| 256/171
|
|
| Down 5th
| Down 5th
| v5
| v5
Line 214: Line 268:
| 709.1
| 709.1
| [[3/2]]
| [[3/2]]
|
|
| Perfect 5th
| Perfect 5th
| P5
| P5
Line 221: Line 277:
| 736.4
| 736.4
| 195/128
| 195/128
|
|
| Up 5th, Downminor 6th
| Up 5th, Downminor 6th
| ^5, vm6
| ^5, vm6
Line 228: Line 286:
| 763.6
| 763.6
| 99/64, [[128/81]]
| 99/64, [[128/81]]
|
|
| Minor 6th
| Minor 6th
| m6
| m6
Line 235: Line 295:
| 790.9
| 790.9
| 69/44
| 69/44
|
|
| Upminor 6th
| Upminor 6th
| ^m6
| ^m6
Line 242: Line 304:
| 818.2
| 818.2
| [[8/5]]
| [[8/5]]
|
|
| Dupminor 6th, Downmid 6th
| Dupminor 6th, Downmid 6th
| ^^m6, v~6
| ^^m6, v~6
Line 249: Line 313:
| 845.5
| 845.5
| [[13/8]]
| [[13/8]]
|
|
| Mid 6th
| Mid 6th
| ~6
| ~6
Line 256: Line 322:
| 872.7
| 872.7
| [[5/3]], [[48/29]]
| [[5/3]], [[48/29]]
|
|
| Dudmajor 6th, Upmid 6th
| Dudmajor 6th, Upmid 6th
| vvM6, ^~6
| vvM6, ^~6
Line 263: Line 331:
| 900.0
| 900.0
| [[32/19]]
| [[32/19]]
|
|
| Downmajor 6th
| Downmajor 6th
| vM6
| vM6
Line 270: Line 340:
| 927.3
| 927.3
| [[27/16]]
| [[27/16]]
|
|
| Major 6th
| Major 6th
| M6
| M6
Line 277: Line 349:
| 954.5
| 954.5
| 45/26
| 45/26
|
|
| Upmajor 6th, Downminor 7th
| Upmajor 6th, Downminor 7th
| ^M6, vm7
| ^M6, vm7
Line 284: Line 358:
| 981.8
| 981.8
| [[16/9]]
| [[16/9]]
|
|
| Minor 7th
| Minor 7th
| m7
| m7
Line 291: Line 367:
| 1009.1
| 1009.1
| [[9/5]]
| [[9/5]]
|
|
| Upminor 7th
| Upminor 7th
| ^m7
| ^m7
Line 298: Line 376:
| 1036.4
| 1036.4
| [[20/11]], [[29/16]]
| [[20/11]], [[29/16]]
|
|
| Dupminor 7th, Downmid 7th
| Dupminor 7th, Downmid 7th
| ^^m7, v~7
| ^^m7, v~7
Line 305: Line 385:
| 1063.6
| 1063.6
| [[24/13]]
| [[24/13]]
|
|
| Mid 7th
| Mid 7th
| ~7
| ~7
Line 312: Line 394:
| 1090.9
| 1090.9
| [[15/8]]
| [[15/8]]
|
|
| Dudmajor 7th, Upmid 7th
| Dudmajor 7th, Upmid 7th
| vvM7, ^~7
| vvM7, ^~7
Line 319: Line 403:
| 1118.2
| 1118.2
| 44/23
| 44/23
|
|
| Downmajor 7th
| Downmajor 7th
| vM7
| vM7
Line 326: Line 412:
| 1145.5
| 1145.5
| [[31/16]]
| [[31/16]]
|
|
| Major 7th
| Major 7th
| M7
| M7
Line 333: Line 421:
| 1172.7
| 1172.7
| 128/65
| 128/65
|
|
| Upmajor 7th, Down 8ve
| Upmajor 7th, Down 8ve
| ^M7, v8
| ^M7, v8
Line 340: Line 430:
| 1200.0
| 1200.0
| [[2/1]]
| [[2/1]]
|
|
| Perfect 8ve
| Perfect 8ve
| P8
| P8

Revision as of 18:11, 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

Recommend dual ratios for the 7th harmonic which has a lot of relative error; might need some additional ratios.

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).

# Cents Approximate ratios* Ratios of 7 (patent val) Ratios of 7 (44d val) Ups and downs notation
0 0.0 1/1 Perfect 1sn P1 D
1 27.3 65/64 64/63 Up 1sn ^1 ^D
2 54.5 32/31, 33/32 51/49 Minor 2nd m2 Eb
3 81.8 23/22 22/21 Upminor 2nd ^m2 ^Eb
4 109.1 16/15 147/136 Dupminor 2nd, Downmid 2nd ^^m2, v~2 ^^Eb
5 136.4 13/12 14/13 Mid 2nd ~2 vvvE, ^^^Eb
6 163.6 11/10, 32/29 31/28 551/504 Dudmajor 2nd, Upmid 2nd vvM2, ^~2 vvE
7 190.9 19/17 143/126 352/315 Downmajor 2nd vM2 vE
8 218.2 9/8 8/7 143/126 Major 2nd M2 E
9 245.5 52/45 65/56 8/7 Upmajor 2nd, Downminor 3rd ^M2, vm3 ^E, vF
10 272.7 32/27 7/6 221/189 Minor 3rd m3 F
11 300.0 19/16 896/729 Upminor 3rd ^m3 ^F
12 327.3 11/9, 29/24 896/729 76/63 Dupminor 3rd, Downmid 3rd ^^m3, v~3 ^^F
13 354.5 16/13 26/21 Mid 3rd ~3 ^^^F, vvvF#
14 381.8 5/4 26/21 Dudmajor 3rd, Upmid 3rd vvM3, ^~3 vvF#
15 409.1 171/136 80/63 Downmajor 3rd vM3 vF#
16 436.4 81/64 9/7 Major 3rd M3 F#
17 463.6 13/10 9/7 Upmajor 3rd, Down 4th ^M3, v4 ^F#, vG
18 490.9 4/3 62/49 Perfect 4th P4 G
19 518.2 171/128 189/143 256/189 Up 4th ^4 ^G
20 545.5 11/8 29/21 Dup 4th, Downmid 4th, Dim 5th ^^4, v~4, d5 Ab, ^^G
21 572.7 46/33 29/21 Mid 4th, Updim 5th ~4, ^d5 ^^^G, vvvG#
22 600.0 44/31, 31/22 7/5, 10/7 Upmid 4th, Downmid 5th ^~4, v~5 vvG#, ^^Ab
23 627.3 33/23 10/7 Downaug 4th, Mid 5th vA4, ~5 vvvA, ^^^Ab
24 654.5 16/11 Aug 4th, Upmid 5th, Dud 5th A4, ^~5, vv5 G#, vvA
25 681.8 256/171 Down 5th v5 vA
26 709.1 3/2 Perfect 5th P5 A
27 736.4 195/128 Up 5th, Downminor 6th ^5, vm6 ^A, vBb
28 763.6 99/64, 128/81 Minor 6th m6 Bb
29 790.9 69/44 Upminor 6th ^m6 ^Bb
30 818.2 8/5 Dupminor 6th, Downmid 6th ^^m6, v~6 ^^Bb
31 845.5 13/8 Mid 6th ~6 ^^^Bb, vvvB
32 872.7 5/3, 48/29 Dudmajor 6th, Upmid 6th vvM6, ^~6 vvB
33 900.0 32/19 Downmajor 6th vM6 vB
34 927.3 27/16 Major 6th M6 B
35 954.5 45/26 Upmajor 6th, Downminor 7th ^M6, vm7 ^B, vC
36 981.8 16/9 Minor 7th m7 C
37 1009.1 9/5 Upminor 7th ^m7 ^C
38 1036.4 20/11, 29/16 Dupminor 7th, Downmid 7th ^^m7, v~7 ^^C
39 1063.6 24/13 Mid 7th ~7 ^^^C, vvvC#
40 1090.9 15/8 Dudmajor 7th, Upmid 7th vvM7, ^~7 vvC#
41 1118.2 44/23 Downmajor 7th vM7 vC#
42 1145.5 31/16 Major 7th M7 C#
43 1172.7 128/65 Upmajor 7th, Down 8ve ^M7, v8 ^C#, vD
44 1200.0 2/1 Perfect 8ve P8 D

* As a 2.3.5.11.13.17.19.23.29.31-subgroup temperament

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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