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77edo: Difference between revisions

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Assessment of a full 19-limit interpretation
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With [[3/1|harmonic 3]] less than a cent flat, [[5/1|harmonic 5]] a bit over three cents sharp and [[7/1|7]]'s less flat than that, 77edo represents an excellent tuning choice for both [[valentine]], the 31 & 46 temperament, and [[starling]], the [[126/125]] [[planar temperament]], giving the [[optimal patent val]] for [[11-limit]] valentine and its [[13-limit]] extensions dwynwen and valentino, as well as 11-limit starling and [[oxpecker]] temperaments. It also gives the optimal patent val for [[grackle]] and various members of the [[unicorn family]], with a [[generator]] of 4\77 instead of the 5\77 (which gives valentine). These are 7-limit [[Unicorn family #Alicorn|alicorn]] and 11- and 13-limit [[Unicorn family #Camahueto|camahueto]].
With [[3/1|harmonic 3]] less than a cent flat, [[5/1|harmonic 5]] a bit over three cents sharp and [[7/1|7]]'s less flat than that, 77edo represents an excellent tuning choice for both [[valentine]], the 31 & 46 temperament, and [[starling]], the [[126/125]] [[planar temperament]], giving the [[optimal patent val]] for [[11-limit]] valentine and its [[13-limit]] extensions dwynwen and valentino, as well as 11-limit starling and [[oxpecker]] temperaments. It also gives the optimal patent val for [[grackle]] and various members of the [[unicorn family]], with a [[generator]] of 4\77 instead of the 5\77 (which gives valentine). These are 7-limit [[Unicorn family #Alicorn|alicorn]] and 11- and 13-limit [[Unicorn family #Camahueto|camahueto]].


77et tempers out [[32805/32768]] in the [[5-limit]], [[126/125]], [[1029/1024]] and [[6144/6125]] in the 7-limit, [[121/120]], [[176/175]], [[385/384]] and [[441/440]] in the 11-limit, and [[196/195]], [[351/350]], [[352/351]], [[676/675]] and [[729/728]] in the 13-limit.
77et tempers out [[32805/32768]] in the [[5-limit]], [[126/125]], [[1029/1024]] and [[6144/6125]] in the 7-limit, [[121/120]], [[176/175]], [[385/384]] and [[441/440]] in the 11-limit, and [[196/195]], [[351/350]], [[352/351]], [[676/675]] and [[729/728]] in the 13-limit.
 
The [[17/1|17]] and [[19/1|19]] are tuned fairly well, making it [[consistent]] to the no-11 [[21-odd-limit]]. The equal temperament tempers out [[256/255]] in the 17-limit; and [[171/170]], [[361/360]], [[513/512]], and [[1216/1215]] in the 19-limit.  


77edo is an excellent edo for [[Carlos Alpha]], since the difference between 5 steps of 77edo and 1 step of Carlos Alpha is only -0.042912 cents.
77edo is an excellent edo for [[Carlos Alpha]], since the difference between 5 steps of 77edo and 1 step of Carlos Alpha is only -0.042912 cents.
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! Degree
! Degree
! Cents
! Cents
! Approximate Ratios<br>in the 13-limit
! Approximate Ratios*
|-
|-
| 0
| 0
Line 45: Line 47:
| 6
| 6
| 93.506
| 93.506
| 135/128
| 18/17, 19/18, 20/19
|-
|-
| 7
| 7
| 109.091
| 109.091
| 16/15
| 16/15, 17/16
|-
|-
| 8
| 8
Line 65: Line 67:
| 11
| 11
| 171.429
| 171.429
| 72/65
| 21/19
|-
|-
| 12
| 12
Line 77: Line 79:
| 14
| 14
| 218.182
| 218.182
| 256/225
| 17/15
|-
|-
| 15
| 15
Line 85: Line 87:
| 16
| 16
| 249.351
| 249.351
| 15/13
| 15/13, 22/19
|-
|-
| 17
| 17
Line 93: Line 95:
| 18
| 18
| 280.519
| 280.519
| 33/28
| 20/17
|-
|-
| 19
| 19
| 296.104
| 296.104
| 13/11, 32/27
| 13/11, 19/16, 32/27
|-
|-
| 20
| 20
Line 109: Line 111:
| 22
| 22
| 342.857
| 342.857
| 11/9, 39/32
| 11/9, 17/14
|-
|-
| 23
| 23
| 358.442
| 358.442
| 16/13, 27/22
| 16/13, 21/17
|-
|-
| 24
| 24
Line 125: Line 127:
| 26
| 26
| 405.195
| 405.195
| 33/26, 81/64
| 19/15, 24/19, 33/26
|-
|-
| 27
| 27
Line 141: Line 143:
| 30
| 30
| 467.532
| 467.532
| 21/16
| 17/13, 21/16
|-
|-
| 31
| 31
Line 157: Line 159:
| 34
| 34
| 529.870
| 529.870
| 49/36
| 19/14
|-
|-
| 35
| 35
| 545.455
| 545.455
| 11/8, ''15/11''
| 11/8, ''15/11'', 26/19
|-
|-
| 36
| 36
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| 38
| 38
| 592.208
| 592.208
| 45/32
| 24/17, 38/27, 45/32
|-
|-
| 39
| 39
| 607.792
| 607.792
| 64/45
| 17/12, 27/19, 64/45
|-
|-
| 40
| 40
Line 189: Line 191:
| 42
| 42
| 654.545
| 654.545
| 16/11, ''22/15''
| 16/11, 19/13, ''22/15''
|-
|-
| 43
| 43
| 670.130
| 670.130
| 72/49
| 28/19
|-
|-
| 44
| 44
Line 209: Line 211:
| 47
| 47
| 732.468
| 732.468
| 32/21
| 26/17, 32/21
|-
|-
| 48
| 48
Line 225: Line 227:
| 51
| 51
| 794.805
| 794.805
| 52/33, 128/81
| 19/12, 30/19, 52/33
|-
|-
| 52
| 52
Line 237: Line 239:
| 54
| 54
| 841.558
| 841.558
| 13/8, 44/27
| 13/8, 34/21
|-
|-
| 55
| 55
| 857.143
| 857.143
| 18/11, 64/39
| 18/11, 28/17
|-
|-
| 56
| 56
Line 253: Line 255:
| 58
| 58
| 903.896
| 903.896
| 27/16, 22/13
| 22/13, 27/16, 32/19
|-
|-
| 59
| 59
| 919.481
| 919.481
| 56/33
| 17/10
|-
|-
| 60
| 60
Line 265: Line 267:
| 61
| 61
| 950.649
| 950.649
| 26/15
| 26/15, 19/11
|-
|-
| 62
| 62
Line 273: Line 275:
| 63
| 63
| 981.818
| 981.818
| 225/128
| 30/17
|-
|-
| 64
| 64
Line 285: Line 287:
| 66
| 66
| 1028.571
| 1028.571
| 65/36
| 38/21
|-
|-
| 67
| 67
Line 301: Line 303:
| 70
| 70
| 1090.909
| 1090.909
| 15/8
| 15/8, 32/17
|-
|-
| 71
| 71
| 1106.494
| 1106.494
| 256/135
| 17/9, 19/10, 36/19
|-
|-
| 72
| 72
Line 331: Line 333:
| 2/1
| 2/1
|}
|}
<nowiki>*</nowiki> as a 19-limit temperament


== Regular temperament properties ==
== Regular temperament properties ==
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