28edo: Difference between revisions

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191 edits
→Table of intervals: Removed 7mus column
Line 16: Line 16:
|-
|-
| rowspan="2" style="text-align:center;" | Step #
| rowspan="2" style="text-align:center;" | Step #
| colspan="2" style="text-align:center;" | ET
| style="text-align:center;" | ET
| colspan="2" style="text-align:center;" | Just  
| colspan="2" style="text-align:center;" | Just  
| style="text-align:center;" | Difference
| style="text-align:center;" | Difference
Line 24: Line 24:
|-
|-
|Cents
|Cents
|7mus
|Interval
|Interval
|Cents
|Cents
Line 34: Line 33:
| style="text-align:center;" | 0
| style="text-align:center;" | 0
| style="text-align:center;" | 0¢
| style="text-align:center;" | 0¢
|0
| style="text-align:center;" |  
| style="text-align:center;" |  
| style="text-align:center;" |  
| style="text-align:center;" |  
Line 44: Line 42:
| style="text-align:center;" | 1
| style="text-align:center;" | 1
| style="text-align:center;" | 42.86
| style="text-align:center;" | 42.86
|54.86 (36.DC<sub>16</sub>)
| style="text-align:center;" |  
| style="text-align:center;" |  
| style="text-align:center;" |  
| style="text-align:center;" |  
Line 54: Line 51:
| style="text-align:center;" | 2
| style="text-align:center;" | 2
| style="text-align:center;" | 85.71
| style="text-align:center;" | 85.71
|109.71 (6D.B7<sub>16</sub>)
| style="text-align:center;" | 21:20
| style="text-align:center;" | 21:20
| style="text-align:center;" | 84.47
| style="text-align:center;" | 84.47
Line 64: Line 60:
| style="text-align:center;" | 3
| style="text-align:center;" | 3
| style="text-align:center;" | 128.57
| style="text-align:center;" | 128.57
|164.57 (A4.92<sub>16</sub>)
| style="text-align:center;" | 14:13
| style="text-align:center;" | 14:13
| style="text-align:center;" | 128.3
| style="text-align:center;" | 128.3
Line 74: Line 69:
| style="text-align:center;" | 4
| style="text-align:center;" | 4
| style="text-align:center;" | 171.43
| style="text-align:center;" | 171.43
|219.43 (DB.6E<sub>16</sub>)
| style="text-align:center;" | 11:10
| style="text-align:center;" | 11:10
| style="text-align:center;" | 165
| style="text-align:center;" | 165
Line 84: Line 78:
| style="text-align:center;" | 5
| style="text-align:center;" | 5
| style="text-align:center;" | 214.29
| style="text-align:center;" | 214.29
|274.27 (112.49<sub>16</sub>)
| style="text-align:center;" | 17:15
| style="text-align:center;" | 17:15
| style="text-align:center;" | 216.69
| style="text-align:center;" | 216.69
Line 94: Line 87:
| style="text-align:center;" | 6
| style="text-align:center;" | 6
| style="text-align:center;" | 257.14
| style="text-align:center;" | 257.14
|329.14 (149.25<sub>16</sub>)
| style="text-align:center;" | 7:6
| style="text-align:center;" | 7:6
| style="text-align:center;" | 266.87
| style="text-align:center;" | 266.87
Line 104: Line 96:
| style="text-align:center;" | 7
| style="text-align:center;" | 7
| style="text-align:center;" | 300
| style="text-align:center;" | 300
|384 (180<sub>16</sub>)
| style="text-align:center;" | 6:5
| style="text-align:center;" | 6:5
| style="text-align:center;" | 315.64
| style="text-align:center;" | 315.64
Line 114: Line 105:
| style="text-align:center;" | 8
| style="text-align:center;" | 8
| style="text-align:center;" | 342.86
| style="text-align:center;" | 342.86
|438.86 (1B6.DC<sub>16</sub>)
| style="text-align:center;" | 11:9
| style="text-align:center;" | 11:9
| style="text-align:center;" | 347.41
| style="text-align:center;" | 347.41
Line 124: Line 114:
| style="text-align:center;" | 9
| style="text-align:center;" | 9
| style="text-align:center;" | 385.71
| style="text-align:center;" | 385.71
|493.71 (1ED.B7<sub>16</sub>)
| style="text-align:center;" | 5:4
| style="text-align:center;" | 5:4
| style="text-align:center;" | 386.31
| style="text-align:center;" | 386.31
Line 134: Line 123:
| style="text-align:center;" | 10
| style="text-align:center;" | 10
| style="text-align:center;" | 428.57
| style="text-align:center;" | 428.57
|548.57 (224.92<sub>16</sub>)
| style="text-align:center;" | 9:7
| style="text-align:center;" | 9:7
| style="text-align:center;" | 435.08
| style="text-align:center;" | 435.08
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| style="text-align:center;" | 11
| style="text-align:center;" | 11
| style="text-align:center;" | 471.43
| style="text-align:center;" | 471.43
|603.43 (26B.6E<sub>16</sub>)
| style="text-align:center;" | 21:16
| style="text-align:center;" | 21:16
| style="text-align:center;" | 470.78
| style="text-align:center;" | 470.78
Line 154: Line 141:
| style="text-align:center;" | 12
| style="text-align:center;" | 12
| style="text-align:center;" | 514.29
| style="text-align:center;" | 514.29
|658.29 (292.49<sub>16</sub>)
| style="text-align:center;" | 4:3
| style="text-align:center;" | 4:3
| style="text-align:center;" | 498.045
| style="text-align:center;" | 498.045
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| style="text-align:center;" | 13
| style="text-align:center;" | 13
| style="text-align:center;" | 557.14
| style="text-align:center;" | 557.14
|713.14 (2C9.25<sub>16</sub>)
| style="text-align:center;" | 11:8
| style="text-align:center;" | 11:8
| style="text-align:center;" | 551.32
| style="text-align:center;" | 551.32
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| style="text-align:center;" | 14
| style="text-align:center;" | 14
| style="text-align:center;" | 600
| style="text-align:center;" | 600
|768 (300<sub>16</sub>)
| style="text-align:center;" | 7:5
| style="text-align:center;" | 7:5
| style="text-align:center;" | 582.51
| style="text-align:center;" | 582.51
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| style="text-align:center;" | 15
| style="text-align:center;" | 15
| style="text-align:center;" | 642.86
| style="text-align:center;" | 642.86
|822.86 (336.DC<sub>16</sub>)
| style="text-align:center;" | 16:11
| style="text-align:center;" | 16:11
| style="text-align:center;" | 648.68
| style="text-align:center;" | 648.68
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| style="text-align:center;" | 16
| style="text-align:center;" | 16
| style="text-align:center;" | 685.71
| style="text-align:center;" | 685.71
|877.71 (36D.B7<sub>16</sub>)
| style="text-align:center;" | 3:2
| style="text-align:center;" | 3:2
| style="text-align:center;" | 701.955
| style="text-align:center;" | 701.955
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| style="text-align:center;" | 17
| style="text-align:center;" | 17
| style="text-align:center;" | 728.57
| style="text-align:center;" | 728.57
|934.57 (3A4.92<sub>16</sub>)
| style="text-align:center;" | 32:21
| style="text-align:center;" | 32:21
| style="text-align:center;" | 729.22
| style="text-align:center;" | 729.22
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| style="text-align:center;" | 18
| style="text-align:center;" | 18
| style="text-align:center;" | 771.43
| style="text-align:center;" | 771.43
|987.43 (3DB.6E<sub>16</sub>)
| style="text-align:center;" | 14:9
| style="text-align:center;" | 14:9
| style="text-align:center;" | 764.92
| style="text-align:center;" | 764.92
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| style="text-align:center;" | 19
| style="text-align:center;" | 19
| style="text-align:center;" | 814.29
| style="text-align:center;" | 814.29
|1042.29 (412.49<sub>16</sub>)
| style="text-align:center;" | 8:5
| style="text-align:center;" | 8:5
| style="text-align:center;" | 813.68
| style="text-align:center;" | 813.68
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| style="text-align:center;" | 20
| style="text-align:center;" | 20
| style="text-align:center;" | 857.14
| style="text-align:center;" | 857.14
|1097.14 (449.24<sub>16</sub>)
| style="text-align:center;" | 18:11
| style="text-align:center;" | 18:11
| style="text-align:center;" | 852.59
| style="text-align:center;" | 852.59
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| style="text-align:center;" | 21
| style="text-align:center;" | 21
| style="text-align:center;" | 900
| style="text-align:center;" | 900
|1152 (480<sub>16</sub>)
| style="text-align:center;" | 5:3
| style="text-align:center;" | 5:3
| style="text-align:center;" | 884.36
| style="text-align:center;" | 884.36
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| style="text-align:center;" | 22
| style="text-align:center;" | 22
| style="text-align:center;" | 942.86
| style="text-align:center;" | 942.86
|1206.86 (4B6.DC<sub>16</sub>)
| style="text-align:center;" | 12:7
| style="text-align:center;" | 12:7
| style="text-align:center;" | 933.13
| style="text-align:center;" | 933.13
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| style="text-align:center;" | 23
| style="text-align:center;" | 23
| style="text-align:center;" | 985.71
| style="text-align:center;" | 985.71
|1261.71 (4DC.49<sub>16</sub>)
| style="text-align:center;" | 30:17
| style="text-align:center;" | 30:17
| style="text-align:center;" | 983.31
| style="text-align:center;" | 983.31
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| style="text-align:center;" | 24
| style="text-align:center;" | 24
| style="text-align:center;" | 1028.57
| style="text-align:center;" | 1028.57
|1316.57 (524.92<sub>16</sub>)
| style="text-align:center;" | 20:11
| style="text-align:center;" | 20:11
| style="text-align:center;" | 1035
| style="text-align:center;" | 1035
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| style="text-align:center;" | 25
| style="text-align:center;" | 25
| style="text-align:center;" | 1071.42
| style="text-align:center;" | 1071.42
|1371.43 (55A.6E<sub>16</sub>)
| style="text-align:center;" | 13:7
| style="text-align:center;" | 13:7
| style="text-align:center;" | 1071.70
| style="text-align:center;" | 1071.70
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| style="text-align:center;" | 26
| style="text-align:center;" | 26
| style="text-align:center;" | 1114.29
| style="text-align:center;" | 1114.29
|1426.29 (592.49<sub>16</sub>)
| style="text-align:center;" | 40:21
| style="text-align:center;" | 40:21
| style="text-align:center;" | 1115.53
| style="text-align:center;" | 1115.53
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| style="text-align:center;" | 27
| style="text-align:center;" | 27
| style="text-align:center;" | 1157.14
| style="text-align:center;" | 1157.14
|1471.14 (5C9.25<sub>16</sub>)
| style="text-align:center;" |  
| style="text-align:center;" |  
| style="text-align:center;" |  
| style="text-align:center;" |  
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| style="text-align:center;" | 28
| style="text-align:center;" | 28
| style="text-align:center;" | 1200
| style="text-align:center;" | 1200
|1536 (600<sub>16</sub>)
| style="text-align:center;" | 2:1
| style="text-align:center;" | 2:1
| style="text-align:center;" | 1200
| style="text-align:center;" | 1200

Revision as of 14:58, 12 December 2019

Basic properties

28edo, a multiple of both 7edo and 14edo (and of course 2edo and 4edo), has a step size of 42.857 cents. It shares three intervals with 12edo: the 300 cent minor third, the 600 cent tritone, and the 900 cent major sixth. Thus it tempers out the greater diesis 648:625. It does not however temper out the 128:125 lesser diesis, as its major third is less than 1 cent flat (and its inversion the minor sixth less than 1 cent sharp). It has the same perfect fourth and fifth as 7edo. It also has decent approximations of several septimal intervals, of which 9/7 and its inversion 14/9 are also found in 14edo.

Subgroups

28edo can approximate the 7-limit subgroup 2.27.5.21 quite well, and on this subgroup it has the same commas and tunings as 84edo. The temperament corresponding to orwell temperament now has a major third as generator, though as before 225/224, 1728/1715 and 6144/6125 are tempered out. The 225/224-tempered version of the augmented triad has a very low complexity, so many of them appear in the MOS scales for this temperament, which have sizes 7, 10, 13, 16, 19, 22, 25.

Another subgroup for which 28edo works quite well is 2.5.11.19.21.27.29.39.

Table of intervals

The following table compares it to potentially useful nearby just intervals.

Step # ET Just Difference

(ET minus Just)

Up/down Notation
Cents Interval Cents
0 0¢ unison 1 D
1 42.86 up-unison ^1 D^
2 85.71 21:20 84.47 1.24 double-up, double-down ^^1, vv2 D^^, Evv
3 128.57 14:13 128.3 0.27 down 2nd v2 Ev
4 171.43 11:10 165 6.43 2nd 2 E
5 214.29 17:15 216.69 -2.40 up 2nd ^2 E^
6 257.14 7:6 266.87 -9.73 double-up 2nd, double-down 3rd ^^2, vv3 E^^, Fvv
7 300 6:5 315.64 -15.64 down 3rd v3 Fv
8 342.86 11:9 347.41 -4.55 3rd 3 F
9 385.71 5:4 386.31 -0.60 up 3rd ^3 F^
10 428.57 9:7 435.08 -6.51 double-up 3rd, double-down 4th ^^3, vv4 F^^, Gvv
11 471.43 21:16 470.78 0.65 down 4th v4 Gv
12 514.29 4:3 498.045 16.245 4th 4 G
13 557.14 11:8 551.32 5.82 up 4th ^4 G^
14 600 7:5 582.51 17.49 double-up 4th, double-down 5th ^^4, vv5 G^^, vvA
15 642.86 16:11 648.68 -5.82 down 5th v5 Av
16 685.71 3:2 701.955 -16.245 5th 5 A
17 728.57 32:21 729.22 -0.65 up 5th ^5 A^
18 771.43 14:9 764.92 6.51 double-up 5th, double-down 6th ^^5, vv6 A^^, Bvv
19 814.29 8:5 813.68 0.61 down 6th v6 Bv
20 857.14 18:11 852.59 4.55 6th 6 B
21 900 5:3 884.36 15.64 up 6th ^6 B^
22 942.86 12:7 933.13 9.73 double-up 6th, double-down 7th ^^6, vv7 B^^, Cvv
23 985.71 30:17 983.31 2.40 down 7th v7 Cv
24 1028.57 20:11 1035 -6.43 7th 7 C
25 1071.42 13:7 1071.70 -0.27 up 7th ^7 C^
26 1114.29 40:21 1115.53 -1.24 double-up 7th, double-down 8ve ^^7, vv8 C^^, Dvv
27 1157.14 down 8ve v8 Dv
28 1200 2:1 1200 0 8ve 8 D

Chord Names

Ups and downs can be used to name 28edo chords. Because every interval is perfect, the quality can be omitted, and the words major, minor, augmented and diminished are never used.

0-8-16 = C E G = C = C or C perfect

0-7-16 = C Ev G = C(v3) = C down-three

0-9-16 = C E^ G = C(^3) = C up-three

0-8-15 = C E Gv = C(v5) = C down-five

0-9-17 = C E^ G^ = C(^3,^5) = C up-three up-five

0-8-16-24 = C E G B = C7 = C seven

0-8-16-23 = C E G Bv = C(v7) = C down-seven

0-7-16-24 = C Ev G B = C7(v3) = C seven down-three

0-7-16-23 = C Ev G Bv = C.v7 = C dot down seven

For a more complete list, see Ups and Downs Notation - Chord names in other EDOs.

Rank two temperaments

Periods

per octave

Generator Temperaments
1 1\28
1 3\28 Negri
1 5\28 Machine
1 9\28 Worschmidt
1 11\28
1 13\28 Thuja
2 1\28
2 3\28
2 5\28 Antikythera
4 1\28
4 2\28 Demolished
4 3\28
7 1\28 Whitewood
14 1\28

Commas

28 EDO tempers out the following commas. (Note: This assumes the val < 28 44 65 79 97 104 |.)

Comma Monzo Cents Name 1 Name 2
2187/2048 | -11 7 > 113.69 Apotome
648/625 | 3 4 -4 > 62.57 Major Diesis Diminished Comma
16875/16384 | -14 3 4 > 51.12 Negri Comma Double Augmentation Diesis
| 17 1 -8 > 11.45 Wuerschmidt Comma
36/35 | 2 2 -1 -1 > 48.77 Septimal Quarter Tone
50/49 | 1 0 2 -2 > 34.98 Tritonic Diesis Jubilisma
3125/3087 | 0 -2 5 -3 > 21.18 Gariboh
126/125 | 1 2 -3 1 > 13.79 Septimal Semicomma Starling Comma
65625/65536 | -16 1 5 1 > 2.35 Horwell
| 47 -7 -7 -7 > 0.34 Akjaysma 5\7 Octave Comma
176/175 | 4 0 -2 -1 1 > 9.86 Valinorsma
441/440 | -3 2 -1 2 -1 > 3.93 Werckisma
4000/3993 | 5 -1 3 0 -3 > 3.03 Wizardharry

Some scales

machine5

machine6

machine11

Compositions

28 tone Prelude by Kosmorksy

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