3159811edo: Difference between revisions

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Line 27: Line 27:
! 38
! 38
! 39
! 39
! 40
|-
|-
! JI Ratios
! JI Ratios
Line 37: Line 38:
| 19/16
| 19/16
| 39/32
| 39/32
| 5/4
|-
|-
! … in cents
! … in cents
Line 47: Line 49:
| 297.513
| 297.513
| 342.483
| 342.483
| 386.314
|-
|-
! Degrees in 3159811edo
! Degrees in 3159811edo
Line 57: Line 60:
| 783404
| 783404
| 901817
| 901817
| 1017232
|}
|}


Line 62: Line 66:
|-
|-
! Overtones
! Overtones
! 40
! 41
! 41
! 42
! 42
Line 70: Line 73:
! 46
! 46
! 47
! 47
! 48
|-
|-
! JI Ratios
! JI Ratios
| 5/4
| 41/32
| 41/32
| 21/16
| 21/16
Line 80: Line 83:
| 23/16
| 23/16
| 47/32
| 47/32
| 3/2
|-
|-
! … in cents
! … in cents
| 386.314
| 429.062
| 429.062
| 470.781
| 470.781
Line 90: Line 93:
| 628.274
| 628.274
| 665.507
| 665.507
| 701.955
|-
|-
! Degrees in 3159811edo
! Degrees in 3159811edo
| 1017232
| 1129797
| 1129797
| 1239649
| 1239649
Line 100: Line 103:
| 1654357
| 1654357
| 1752396
| 1752396
| 1848371
|}
|}


Line 105: Line 109:
|-
|-
! Overtones
! Overtones
! 48
! 49
! 49
! 50
! 50
Line 113: Line 116:
! 54
! 54
! 55
! 55
! 56
|-
|-
! JI Ratios
! JI Ratios
| 3/2
| 49/32
| 49/32
| 25/16
| 25/16
Line 123: Line 126:
| 27/16
| 27/16
| 55/32
| 55/32
| 7/4
|-
|-
! … in cents
! … in cents
| 701.955
| 737.652
| 737.652
| 772.627
| 772.627
Line 133: Line 136:
| 905.865
| 905.865
| 937.632
| 937.632
| 968.826
|-
|-
! Degrees in 3159811edo
! Degrees in 3159811edo
| 1848371
| 1942367
| 1942367
| 2034464
| 2034464
Line 143: Line 146:
| 2385302
| 2385302
| 2468949
| 2468949
| 2551089
|}
|}


Line 148: Line 152:
|-
|-
! Overtones
! Overtones
! 56
! 57
! 57
! 58
! 58
Line 159: Line 162:
|-
|-
! JI Ratios
! JI Ratios
| 7/4
| 57/32
| 57/32
| 29/16
| 29/16
Line 170: Line 172:
|-
|-
! … in cents
! … in cents
| 968.826
| 999.468
| 999.468
| 1029.577
| 1029.577
Line 181: Line 182:
|-
|-
! Degrees in 3159811edo
! Degrees in 3159811edo
| 2551089
| 2631775
| 2631775
| 2711058
| 2711058

Revision as of 13:28, 7 May 2025

Template:Niche is deprecated. Please use Template:Mathematical interest instead.
← 3159810edo 3159811edo 3159812edo →
Prime factorization 29 × 108959
Step size 0.00037977 ¢ 
Fifth 1848371\3159811 (701.955 ¢)
Semitones (A1:m2) 299353:237578 (113.7 ¢ : 90.22 ¢)
Consistency limit 65
Distinct consistency limit 65

3159811 equal divisions of the octave (abbreviated 3159811edo or 3159811ed2), also called 3159811-tone equal temperament (3159811tet) or 3159811 equal temperament (3159811et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 3159811 equal parts of about 0.00038 ¢ each. Each step represents a frequency ratio of 21/3159811, or the 3159811th root of 2.

3159811edo is consistent in the 65-odd-limit with a lower relative error than any previous equal temperaments in the 61-limit. It is the smallest EDO which is purely consistent[idiosyncratic term] in the 63-odd-limit (i.e. does not exceed 25% relative error on the first 63 harmonics of the harmonic series).

Theory

Prime harmonics

Approximation of odd harmonics in 3159811edo (3–15)
Harmonic 3 5 7 9 11 13 15
Error Absolute (¢) +0.00002113 +0.00002452 +0.00001382 +0.00004226 -0.00003126 -0.00004793 +0.00004565
Relative (%) +5.6 +6.5 +3.6 +11.1 -8.2 -12.6 +12.0
Steps
(reduced) 		5008182
(1848371) 		7336854
(1017232) 		8870711
(2551089) 		10016364
(536931) 		10931150
(1451717) 		11692690
(2213257) 		12345036
(2865603)
Approximation of odd harmonics in 3159811edo (17–31)
Harmonic 17 19 21 23 25 27 29 31
Error Absolute (¢) -0.00001818 -0.00003197 +0.00003494 +0.00006535 +0.00004904 +0.00006338 +0.00008120 +0.00000087
Relative (%) -4.8 -8.4 +9.2 +17.2 +12.9 +16.7 +21.4 +0.2
Steps
(reduced) 		12915610
(276366) 		13422648
(783404) 		13878893
(1239649) 		14293601
(1654357) 		14673708
(2034464) 		15024546
(2385302) 		15350302
(2711058) 		15654324
(3015080)
Approximation of odd harmonics in 3159811edo (33–47)
Harmonic 33 35 37 39 41 43 45 47
Error Absolute (¢) -0.00001013 +0.00003834 -0.00001850 -0.00002681 +0.00009218 -0.00002310 +0.00006678 +0.00001747
Relative (%) -2.7 +10.1 -4.9 -7.1 +24.3 -6.1 +17.6 +4.6
Steps
(reduced) 		15939332
(140277) 		16207565
(408510) 		16460888
(661833) 		16700872
(901817) 		16928852
(1129797) 		17145971
(1346916) 		17353218
(1554163) 		17551451
(1752396)
Approximation of odd harmonics in 3159811edo (49–63)
Harmonic 49 51 53 55 57 59 61 63
Error Absolute (¢) +0.00002763 +0.00000295 -0.00002258 -0.00000674 -0.00001084 -0.00008220 -0.00002937 +0.00005607
Relative (%) +7.3 +0.8 -5.9 -1.8 -2.9 -21.6 -7.7 +14.8
Steps
(reduced) 		17741422
(1942367) 		17923792
(2124737) 		18099146
(2300091) 		18268004
(2468949) 		18430830
(2631775) 		18588040
(2788985) 		18740009
(2940954) 		18887075
(3088020)

Scales

Harmonic scales

3159811edo accurately approximates the mode 32 of harmonic series. All interval pairs are distinguished.

Overtones 32 33 34 35 36 37 38 39 40
JI Ratios 1/1 33/32 17/16 35/32 9/8 37/32 19/16 39/32 5/4
… in cents 0 53.273 104.955 155.14 203.91 251.344 297.513 342.483 386.314
Degrees in 3159811edo 0 140277 276366 408510 536931 661833 783404 901817 1017232
Overtones 41 42 43 44 45 46 47 48
JI Ratios 41/32 21/16 43/32 11/8 45/32 23/16 47/32 3/2
… in cents 429.062 470.781 511.518 551.318 590.224 628.274 665.507 701.955
Degrees in 3159811edo 1129797 1239649 1346916 1451717 1554163 1654357 1752396 1848371
Overtones 49 50 51 52 53 54 55 56
JI Ratios 49/32 25/16 51/32 13/8 53/32 27/16 55/32 7/4
… in cents 737.652 772.627 806.91 840.528 873.505 905.865 937.632 968.826
Degrees in 3159811edo 1942367 2034464 2124737 2213257 2300091 2385302 2468949 2551089
Overtones 57 58 59 60 61 62 63 64
JI Ratios 57/32 29/16 59/32 15/8 61/32 31/16 63/32 2/1
… in cents 999.468 1029.577 1059.172 1088.269 1116.885 1145.036 1172.736 1200
Degrees in 3159811edo 2631775 2711058 2788985 2865603 2940954 3015080 3088020 3159811
  • The scale in adjacent steps is 140277, 136089, 132144, 128421, 124902, 121571, 118413, 115415, 112565, 109852, 107267, 104801, 102446, 100194, 98039, 95975, 93996, 92097, 90273, 88520, 86834, 85211, 83647, 82140, 80686, 79283, 77927, 76618, 75351, 74126, 72940, 71791.
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