28edo

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Revision as of 20:43, 5 December 2020 by Yourmusic Productions (talk | contribs) (Another temperament I can properly link now it has it's own page.)
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Theory

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.

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.

Intervals

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

Step # ET Just Difference
(ET minus Just) 		Ups and Downs Notation
Cents Interval Cents
0 0¢ unison 1 D
1 42.86 41:40 42.74 0.12 up-unison ^1 ^D
2 85.71 21:20 84.47 1.24 double-up 1sn, double-down 2nd ^^1, vv2 ^^D, vvE
3 128.57 14:13 128.3 0.27 down 2nd v2 vE
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, vvF
7 300 6:5 315.64 -15.64 down 3rd v3 vF
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, vvG
11 471.43 21:16 470.78 0.65 down 4th v4 vG
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 vA
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, vvB
19 814.29 8:5 813.68 0.61 down 6th v6 vB
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, vvC
23 985.71 30:17 983.31 2.40 down 7th v7 vC
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, vvD
27 1157.14 80:41 1157.26 -0.12 down 8ve v8 vD
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 vE G = Cv = C down
  • 0-9-16 = C ^E G = C^ = C up
  • 0-8-15 = C E vG = C(v5) = C down-five
  • 0-9-17 = C ^E ^G = C^(^5) = C up up-five
  • 0-8-16-24 = C E G B = C7 = C seven
  • 0-8-16-23 = C E G vB = C,v7 = C add down-seven
  • 0-7-16-24 = C vE G B = Cv,7 = C down add seven
  • 0-7-16-23 = C vE G vB = Cv7 = C 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 Oneirotonic
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 |.)

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

Scales

28edo is particularly well suited to Whitewood in the same way that 15edo is for Blackwood, as it has one third that is heavily tempered, but in a familiar way shared with 12edo, while the other one is significantly closer to just. (Contrast with 20 & 21edo, where one third remains the same as in 12, and the other is pushed further away, making the overall sound considerably more xenharmonic) This makes the 5ths more out of tune, but in a useful way, as you can stack major and minor thirds indefinitely until they repeat 4 octaves and 14 notes up, producing one of the largest nonrepeating harmonious chords possible in an edo this low. This produces mode of symmetry scales with two different modes and 4 different keys, making it equally easy to establish any chord in the scale as the root and modulate between them.

  • Whitewood Major [14] 13131313131313
  • Whitewood Minor [14] 31313131313131
  • (Whitewood neutral is also theoretically possible, stacking neutral or subminor & supermajor thirds, but in practice that works out as 22222222222222, or 14edo, so it doesn't count as a 28edo scale)

If you're looking for something a little more xenharmonic sounding, but still relatively low in badness, Negri [9] & [10] function very similarly to Porcupine [7] & [8] in 15edo, further cementing the idea that 28 is to 15 what 29 is to 12 - a similar structure, but with tempering errors in the opposite direction and slightly greater complexity. Both have a decent selection of familiar major and minor chords clustered towards one end of the scale, while the way they modulate between one-another and the melodies they form is quite alien to people used to the diatonic scale. Most of the compositional notes on the Porcupine page can also be applied here.

  • Negri [9] 333343333
  • Negri [10] 3333333331

However, unlike 15, 28 is complex enough to do recognisable approximations of various diatonic scales and their modes, although they will sound noticeably out of tune and it's obviously not the best method of using the temperament.

  • Diatonic Major [7] 5434552
  • Diatonic Minor [7] 5254345
  • Harmonic Minor [7] 5254372
  • Harmonic Major [7] 5434372
  • Melodic Minor [7] 5254552
  • Melodic Major [7] 5434345

Interestingly, as it has a near perfect 21/16, 28edo can also generate Oneirotonic scales (see 13edo) by stacking it's 11th degree, and they actually sound better in this temperament.

Music

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