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chain

[Joseph Monzo]

A chain is a type of linear tuning which is theoretically infinite on both ends. Good examples, which are all "chain-of-5th" temperaments, are pythagorean, meantone, and schismic (such as those of Helmholtz and Groven).

Below are shown some examples of 31-note "pythagorean" chain-of-5th tunings for various EDOs. Each example uses the patent-val mapping of prime-factor 3. Column 1 shows the generator number, column 2 shows the pythagorean-based notation, and column 3 shows in x\EDO form how many degrees of the EDO map to that note.


 41edo "pythagorean" chain-of-5ths

 gen note  x\41
 17  Ax   39
 16  Dx   15
 15  Gx   32
 14  Cx    8
 13  Fx   25
 12  B#    1
 11  E#   18
 10  A#   35
  9  D#   11
  8  G#   28
  7  C#    4
  6  F#   21
  5  B    38
  4  E    14
  3  A    31
  2  D     7
  1  G    24
  0  C     0
 -1  F    17
 -2  Bb   34
 -3  Eb   10
 -4  Ab   27
 -5  Db    3
 -6  Gb   20
 -7  Cb   37
 -8  Fb   13
 -9  Bbb  30
-10  Ebb   6
-11  Abb  23
-12  Dbb  40
-13  Gbb  16


87edo "pythagorean" chain-of-5ths

 gen note  x\87
  17  Ax   84
  16  Dx   33
  15  Gx   69
  14  Cx   18
  13  Fx   54
  12  B#    3
  11  E#   39
  10  A#   75
   9  D#   24
   8  G#   60
   7  C#    9
   6  F#   45
   5  B    81
   4  E    30
   3  A    66
   2  D    15
   1  G    51
   0  C     0
  -1  F    36
  -2  Bb   72
  -3  Eb   21
  -4  Ab   57
  -5  Db    6
  -6  Gb   42
  -7  Cb   78
  -8  Fb   27
  -9  Bbb  63
 -10  Ebb  12
 -11  Abb  48
 -12  Dbb  84
 -13  Gbb  33


 311edo "pythagorean" chain-of-5ths

 gen note  x\311
  17  Ax   295
  16  Dx   113
  15  Gx   242
  14  Cx    60
  13  Fx   189
  12  B#     7
  11  E#   136
  10  A#   265
   9  D#    83
   8  G#   212
   7  C#    30
   6  F#   159
   5  B    288
   4  E    106
   3  A    235
   2  D     53
   1  G    182
   0  C      0
  -1  F    129
  -2  Bb   258
  -3  Eb    76
  -4  Ab   205
  -5  Db    23
  -6  Gb   152
  -7  Cb   281
  -8  Fb    99
  -9  Bbb  228
 -10  Ebb   46
 -11  Abb  175
 -12  Dbb  304
 -13  Gbb  122


 540edo "pythagorean" chain-of-5ths

 gen note  x\540
  17  Ax   512
  16  Dx   196
  15  Gx   420
  14  Cx   104
  13  Fx   328
  12  B#    12
  11  E#   236
  10  A#   460
   9  D#   144
   8  G#   368
   7  C#    52
   6  F#   276
   5  B    500
   4  E    184
   3  A    408
   2  D     92
   1  G    316
   0  C      0
  -1  F    224
  -2  Bb   448
  -3  Eb   132
  -4  Ab   356
  -5  Db   40
  -6  Gb   264
  -7  Cb   488
  -8  Fb   172
  -9  Bbb  396
 -10  Ebb   80
 -11  Abb  304
 -12  Dbb  528
 -13  Gbb  212

. . . . . . . . .

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Encyclopedia of Microtonal Music Theory

chain