Ed255/128

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An equal division of reduced harmonic 255 (ed255/128) is an equal-step tuning in which the octave-reduced 255th harmonic (255/128) is justly tuned and is divided in a given number of equal steps. 255/128 is very close to the octave, 2/1, but it is slightly flatter. This makes it suitable as an alternative to edos whose consonances are too sharp, such as 6edo.

Ed255/128s really only make sense for that purpose with 65 or fewer tones per pseudo-octave. With more tones than that, the relative error on 2/1 becomes unacceptably high and it makes more sense to switch to a different tuning like a zpi or ed511/256.

Ed255/128s are the complementary opposite of ed257/128s.

6ed255/128

Harmonics

Approximation of odd harmonics in 6ed255/128
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +86.8 -2.1 +12.0 -25.4 +25.0 -65.4 +84.6 +66.8 +73.1 +98.7 -58.8
Relative (%) +43.6 -1.1 +6.0 -12.8 +12.6 -32.9 +42.6 +33.6 +36.8 +49.6 -29.5
Steps
(reduced) 		10
(4) 		14
(2) 		17
(5) 		19
(1) 		21
(3) 		22
(4) 		24
(0) 		25
(1) 		26
(2) 		27
(3) 		27
(3)


6edo for comparison:

Approximation of odd harmonics in 6edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +98.0 +13.7 +31.2 -3.9 +48.7 -40.5 -88.3 +95.0 -97.5 -70.8 -28.3
Relative (%) +49.0 +6.8 +15.6 -2.0 +24.3 -20.3 -44.1 +47.5 -48.8 -35.4 -14.1
Steps
(reduced) 		10
(4) 		14
(2) 		17
(5) 		19
(1) 		21
(3) 		22
(4) 		23
(5) 		25
(1) 		25
(1) 		26
(2) 		27
(3)

Intervals

  • 198.871
  • 397.741
  • 596.612
  • 795.483
  • 994.353
  • 1193.224


11ed255/128

Harmonics

Approximation of odd harmonics in 11ed255/128
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +50.6 +34.0 -6.1 -7.3 -29.3 +6.9 -23.8 -23.6 +0.8 +44.5 -4.5
Relative (%) +46.6 +31.4 -5.6 -6.7 -27.0 +6.4 -22.0 -21.7 +0.7 +41.0 -4.2
Steps
(reduced) 		18
(7) 		26
(4) 		31
(9) 		35
(2) 		38
(5) 		41
(8) 		43
(10) 		45
(1) 		47
(3) 		49
(5) 		50
(6)


11edo for comparison:

Approximation of odd harmonics in 11edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -47.4 +50.0 +13.0 +14.3 -5.9 +32.2 +2.6 +4.1 +29.8 -34.4 +26.3
Relative (%) -43.5 +45.9 +11.9 +13.1 -5.4 +29.5 +2.4 +3.8 +27.3 -31.5 +24.1
Steps
(reduced) 		17
(6) 		26
(4) 		31
(9) 		35
(2) 		38
(5) 		41
(8) 		43
(10) 		45
(1) 		47
(3) 		48
(4) 		50
(6)

Intervals

  • 108.475
  • 216.95
  • 325.425
  • 433.9
  • 542.375
  • 650.85
  • 759.324
  • 867.799
  • 976.274
  • 1084.749
  • 1193.224


15ed255/128

15ed255/128 is very close to 47zpi. The 5- to 10-tone scales in 47zpi are also useable in 15ed255/128.

Harmonics

Approximation of prime harmonics in 15ed255/128
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -6.8 +7.2 -2.1 -27.8 -14.8 +14.2 +27.0 -6.4 -19.0 -22.6 +21.1
Relative (%) -8.5 +9.1 -2.7 -34.9 -18.6 +17.8 +34.0 -8.1 -23.9 -28.4 +26.5
Steps
(reduced) 		15
(0) 		24
(9) 		35
(5) 		42
(12) 		52
(7) 		56
(11) 		62
(2) 		64
(4) 		68
(8) 		73
(13) 		75
(0)


15edo for comparison:

Approximation of prime harmonics in 15edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.0 +18.0 +13.7 -8.8 +8.7 +39.5 -25.0 +22.5 +11.7 +10.4 -25.0
Relative (%) +0.0 +22.6 +17.1 -11.0 +10.9 +49.3 -31.2 +28.1 +14.7 +13.0 -31.3
Steps
(reduced) 		15
(0) 		24
(9) 		35
(5) 		42
(12) 		52
(7) 		56
(11) 		61
(1) 		64
(4) 		68
(8) 		73
(13) 		74
(14)

Intervals

  • 79.548
  • 159.097
  • 238.645
  • 318.193
  • 397.741
  • 477.29
  • 556.838
  • 636.386
  • 715.934
  • 795.483
  • 875.031
  • 954.579
  • 1034.128
  • 1113.676
  • 1193.224


17ed255/128

Harmonics

Approximation of prime harmonics in 17ed255/128
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -6.8 -6.8 +21.3 +0.3 -10.1 -18.6 +8.3 +26.3 -23.7 -3.8 +21.1
Relative (%) -9.7 -9.7 +30.3 +0.4 -14.4 -26.5 +11.9 +37.5 -33.7 -5.5 +30.0
Steps
(reduced) 		17
(0) 		27
(10) 		40
(6) 		48
(14) 		59
(8) 		63
(12) 		70
(2) 		73
(5) 		77
(9) 		83
(15) 		85
(0)


17edo for comparison:

Approximation of prime harmonics in 17edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.0 +3.9 -33.4 +19.4 +13.4 +6.5 -34.4 -15.2 +7.0 +29.2 -15.6
Relative (%) +0.0 +5.6 -47.3 +27.5 +19.0 +9.3 -48.7 -21.5 +9.9 +41.4 -22.1
Steps
(reduced) 		17
(0) 		27
(10) 		39
(5) 		48
(14) 		59
(8) 		63
(12) 		69
(1) 		72
(4) 		77
(9) 		83
(15) 		84
(16)

Intervals

  • 70.19
  • 140.379
  • 210.569
  • 280.759
  • 350.948
  • 421.138
  • 491.328
  • 561.517
  • 631.707
  • 701.897
  • 772.086
  • 842.276
  • 912.466
  • 982.655
  • 1052.845
  • 1123.034
  • 1193.224


18ed255/128

Harmonics

Approximation of odd harmonics in 18ed255/128
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +20.5 -2.1 +12.0 -25.4 +25.0 +0.9 +18.3 +0.5 +6.8 +32.4 +7.5
Relative (%) +30.9 -3.2 +18.1 -38.3 +37.7 +1.4 +27.7 +0.8 +10.3 +48.9 +11.4
Steps
(reduced) 		29
(11) 		42
(6) 		51
(15) 		57
(3) 		63
(9) 		67
(13) 		71
(17) 		74
(2) 		77
(5) 		80
(8) 		82
(10)


18edo for comparison:

Approximation of odd harmonics in 18edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +31.4 +13.7 +31.2 -3.9 -18.0 +26.1 -21.6 +28.4 -30.8 -4.1 -28.3
Relative (%) +47.1 +20.5 +46.8 -5.9 -27.0 +39.2 -32.4 +42.6 -46.3 -6.2 -42.4
Steps
(reduced) 		29
(11) 		42
(6) 		51
(15) 		57
(3) 		62
(8) 		67
(13) 		70
(16) 		74
(2) 		76
(4) 		79
(7) 		81
(9)

Intervals

  • 66.29
  • 132.58
  • 198.871
  • 265.161
  • 331.451
  • 397.741
  • 464.032
  • 530.322
  • 596.612
  • 662.902
  • 729.193
  • 795.483
  • 861.773
  • 928.063
  • 994.353
  • 1060.644
  • 1126.934
  • 1193.224


27ed255/128

Harmonics

Approximation of prime harmonics in 27ed255/128
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -6.8 -1.6 -2.1 -10.1 +2.9 -21.2 +0.5 -15.3 +7.5 +4.0 +21.1
Relative (%) -15.3 -3.7 -4.8 -22.9 +6.5 -47.9 +1.2 -34.5 +17.0 +9.0 +47.7
Steps
(reduced) 		27
(0) 		43
(16) 		63
(9) 		76
(22) 		94
(13) 		100
(19) 		111
(3) 		115
(7) 		123
(15) 		132
(24) 		135
(0)


27edo for comparison:

Approximation of prime harmonics in 27edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.0 +9.2 +13.7 +9.0 -18.0 +3.9 -16.1 +13.6 -6.1 -7.4 +10.5
Relative (%) +0.0 +20.6 +30.8 +20.1 -40.5 +8.8 -36.1 +30.6 -13.6 -16.5 +23.7
Steps
(reduced) 		27
(0) 		43
(16) 		63
(9) 		76
(22) 		93
(12) 		100
(19) 		110
(2) 		115
(7) 		122
(14) 		131
(23) 		134
(26)

Intervals

  • 44.193
  • 88.387
  • 132.58
  • 176.774
  • 220.967
  • 265.161
  • 309.354
  • 353.548
  • 397.741
  • 441.935
  • 486.128
  • 530.322
  • 574.515
  • 618.709
  • 662.902
  • 707.096
  • 751.289
  • 795.483
  • 839.676
  • 883.87
  • 928.063
  • 972.257
  • 1016.45
  • 1060.644
  • 1104.837
  • 1149.031
  • 1193.224


39ed255/128

Harmonics

Approximation of prime harmonics in 39ed255/128
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -6.8 -5.0 -2.1 -3.3 +9.7 -4.2 -9.7 +11.9 -12.9 +14.2 -9.5
Relative (%) -22.1 -16.5 -6.9 -10.9 +31.6 -13.7 -31.6 +39.0 -42.1 +46.3 -31.1
Steps
(reduced) 		39
(0) 		62
(23) 		91
(13) 		110
(32) 		136
(19) 		145
(28) 		160
(4) 		167
(11) 		177
(21) 		191
(35) 		194
(38)


39edo for comparison:

Approximation of prime harmonics in 39edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.0 +5.7 +13.7 -15.0 +2.5 -9.8 -12.6 +10.2 -12.9 -14.2 -6.6
Relative (%) +0.0 +18.6 +44.5 -48.7 +8.2 -31.7 -41.1 +33.1 -41.9 -46.1 -21.4
Steps
(reduced) 		39
(0) 		62
(23) 		91
(13) 		109
(31) 		135
(18) 		144
(27) 		159
(3) 		166
(10) 		176
(20) 		189
(33) 		193
(37)


42ed255/128

42ed255/128 is a kind of opposite twin to the scale 42ed257/128, as they improve 42edo’s JI approximation by about the same amount, but in opposite directions (those harmonics which are slightly sharp in one are slightly flat in the other).

42ed255/128’s step size is very close to that of APS715jot and 191zpi.

See Table of stretched 42edo tunings for more.

Harmonics

Approximation of prime harmonics in 42ed255/128
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -6.8 +1.5 -2.1 +12.0 -3.4 -8.6 +10.0 -12.1 -1.9 -5.5 -7.3
Relative (%) -23.9 +5.4 -7.5 +42.2 -12.1 -30.1 +35.2 -42.6 -6.8 -19.4 -25.8
Steps
(reduced) 		42
(0) 		67
(25) 		98
(14) 		119
(35) 		146
(20) 		156
(30) 		173
(5) 		179
(11) 		191
(23) 		205
(37) 		209
(41)


42edo for comparison:

Approximation of prime harmonics in 42edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.0 +12.3 +13.7 +2.6 -8.5 -12.0 +9.3 -11.8 +0.3 -1.0 -2.2
Relative (%) +0.0 +43.2 +47.9 +9.1 -29.6 -41.8 +32.7 -41.3 +1.0 -3.5 -7.6
Steps
(reduced) 		42
(0) 		67
(25) 		98
(14) 		118
(34) 		145
(19) 		155
(29) 		172
(4) 		178
(10) 		190
(22) 		204
(36) 		208
(40)

Scales


MOS scales
  • Eugene/Tritikleismic[9]: 3 8 3 3 8 3 3 8 3
  • Eugene/Tritikleismic[15]: 3 3 2 3 3 3 3 2 3 3 3 3 2 3 3
  • Lemba[16]: 3 2 3 2 3 3 2 3 3 2 3 2 3 3 2 3
  • Qeema/Skateboard[15]: 2 5 2 2 2 5 2 2 2 5 2 2 2 5 2
  • Qeema/Skateboard[19]: 2 2 3 2 2 2 2 3 2 2 2 3 2 2 2 2 3 2 2
  • Seville/Sevond[14] 1st mode: 1 5 1 5 1 5 1 5 1 5 1 5 1 5
  • Seville/Sevond[14] 2nd mode: 5 1 5 1 5 1 5 1 5 1 5 1 5 1
  • Seville/Sevond[21]: 1 4 1 1 4 1 1 4 1 1 4 1 1 4 1 1 4 1 1 4


Subsets of MOS scales

(Names used are idiosyncratic.)

  • Eugene/Tritikleismic[9]
    • Groovy aeolian pentatonic: 11 6 8 3 14
    • Otonal mixolydian pentatonic: 14 3 8 11 6
    • Pseudo-equipentatonic: 11 6 8 6 11
    • Septimal melodic minor pentatonic: 8 3 14 14 3
    • Septimal Picardy pentatonic: 8 6 11 3 14
    • Undecimal lydian-aeolian pentatonic: 8 14 3 11 6
    • Yokai pentatonic: 3 14 8 3 14


49ed255/128

Harmonics

Approximation of prime harmonics in 49ed255/128
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -6.8 -2.5 -10.2 -8.3 -11.6 -8.6 -10.3 -8.0 +2.1 -9.6 -3.3
Relative (%) -27.8 -10.4 -42.1 -34.2 -47.5 -35.1 -42.3 -33.0 +8.7 -39.3 -13.4
Steps
(reduced) 		49
(0) 		78
(29) 		114
(16) 		138
(40) 		170
(23) 		182
(35) 		201
(5) 		209
(13) 		223
(27) 		239
(43) 		244
(48)


49edo for comparison:

Approximation of prime harmonics in 49edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.0 +8.2 +5.5 +10.8 +11.9 -7.9 -7.0 -3.6 +8.5 -1.0 +6.0
Relative (%) +0.0 +33.7 +22.6 +44.0 +48.8 -32.2 -28.6 -14.8 +34.5 -4.1 +24.4
Steps
(reduced) 		49
(0) 		78
(29) 		114
(16) 		138
(40) 		170
(23) 		181
(34) 		200
(4) 		208
(12) 		222
(26) 		238
(42) 		243
(47)


54ed255/128

Harmonics

Approximation of prime harmonics in 54ed255/128
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -6.78 -1.64 -2.12 -10.12 +2.87 +0.92 +0.52 +6.83 +7.52 +3.96 -1.01
Relative (%) -30.7 -7.4 -9.6 -45.8 +13.0 +4.2 +2.4 +30.9 +34.1 +17.9 -4.6
Steps
(reduced) 		54
(0) 		86
(32) 		126
(18) 		152
(44) 		188
(26) 		201
(39) 		222
(6) 		231
(15) 		246
(30) 		264
(48) 		269
(53)


54edo for comparison:

Approximation of prime harmonics in 54edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +9.16 -8.54 +8.95 +4.24 +3.92 +6.16 -8.62 -6.05 -7.35 +10.52
Relative (%) +0.0 +41.2 -38.4 +40.3 +19.1 +17.6 +27.7 -38.8 -27.2 -33.1 +47.3
Steps
(reduced) 		54
(0) 		86
(32) 		125
(17) 		152
(44) 		187
(25) 		200
(38) 		221
(5) 		229
(13) 		244
(28) 		262
(46) 		268
(52)


Related concepts

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