77edo: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
Contribution (talk | contribs)
2,880 edits
Cleanup
Line 1: Line 1:
'''77 equal temperament''', '''77-tET''' or '''77-EDO''' divides the octave into 77 steps of size 15.5844 [[cent|cents]] each.  
'''77 equal temperament''', '''77-tET''' or '''77-EDO''' divides the octave into 77 steps of size 15.5844 [[cent]]s each.  


== Theory ==
== Theory ==
With fifths less than a cent flat, major thirds a bit over three cents sharp and [[7/4|7/4s]] less flat than that, 77edo represents an excellent tuning choice for both [[valentine]], the 31&46 temperament, and [[Starling_family|starling]], the [[126/125]] [[Planar_Temperament|planar temperament]], giving the [[optimal patent val]] for [[11-limit]] valentine and its [[13-limit]] extensions dwynwen and valentino, as well as [[Starling_family #11-limit|11-limit starling]] and [[Starling_family #Oxpecker|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 [[Starling temperaments #camahueto|camahueto]].
With fifths less than a cent flat, major thirds a bit over three cents sharp and [[7/4]]'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.


77edo is an excellent EDO for [[Carlos Alpha]] scale, 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]] scale, since the difference between 5 steps of 77edo and 1 step of Carlos Alpha is only -0.042912 cents.
=== Prime harmonics ===
{{Primes in edo|77}}


== Intervals ==
== Intervals ==
Line 330: Line 332:
|}
|}


== Just approximation ==
== Regular temperament properties ==
=== Selected just intervals ===
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 77et.
{| class="wikitable center-all"
{| class="wikitable center-4 center-5 center-6"
! colspan="2" |
! rowspan="2" | Subgroup
! prime 2
! rowspan="2" | [[Comma list]]
! prime 3
! rowspan="2" | [[Mapping]]
! prime 5
! rowspan="2" | Optimal<br>8ve stretch (¢)
! prime 7
! colspan="2" | Tuning error
! prime 11
! prime 13
! prime 17
! prime 19
|-
|-
! rowspan="2" |Error
! [[TE error|Absolute]] (¢)
! absolute (¢)
! [[TE simple badness|Relative]] (%)
| 0.00
| -0.66
| +3.30
| -2.59
| -5.86
| +1.03
| +4.14
| -1.41
|-
|-
![[Relative error|relative]] (%)
| 2.3
| 0.0
| {{monzo| -122 77 }}
| -4.2
| [{{val| 77 122 }}]
| +21.2
| +0.207
| -16.6
| 0.207
| -37.6
| 1.33
| +6.6
| +26.5
| -9.0
|}
 
=== Temperament Measures ===
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 77et.  
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 11-limit
! 13-limit
! 17-limit
! 19-limit
|-
|-
! colspan="2" |Octave stretch (¢)
| 2.3.5
| +0.207
| 32805/32768, 1594323/1562500
| [{{val| 77 122 179 }}]
| -0.336
| -0.336
| 0.785
| 5.04
|-
| 2.3.5.7
| 126/125, 1029/1024, 10976/10935
| [{{val| 77 122 179 216 }}]
| -0.021
| -0.021
| 0.872
| 5.59
|-
| 2.3.5.7.11
| 121/120, 126/125, 176/175, 10976/10935
| [{{val| 77 122 179 216 266 }}]
| +0.322
| +0.322
| 1.039
| 6.66
|-
| 2.3.5.7.11.13
| 121/120, 126/125, 176/175, 196/195, 676/675
| [{{val| 77 122 179 216 266 285 }}]
| +0.222
| +0.222
| +0.045
| +0.081
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.207
| 0.785
| 0.872
| 1.039
| 0.974
| 0.974
| 1.000
| 0.940
|-
! [[TE simple badness|relative]] (%)
| 1.33
| 5.04
| 5.59
| 6.66
| 6.25
| 6.25
| 6.42
| 6.03
|}
|}


== Linear temperaments ==
=== Rank-2 temperaments ===
 
{| class="wikitable center-all left-3"
{| class="wikitable center-all left-3"
|-
|-
Line 428: Line 401:
| 1
| 1
| 4\77
| 4\77
| [[Unicorn]]/alicorn/camahueto/qilin
| [[Unicorn]] / alicorn / camahueto / qilin
|-
|-
| 1
| 1
Line 516: Line 489:
| 1
| 1
| 32\77
| 32\77
| [[Helmholtz]]/[[grackle]]
| [[Helmholtz]] / [[grackle]]
|-
|-
| 1
| 1
Line 568: Line 541:


== Music ==
== Music ==
* ''[http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/star_1-GrimaldiA+Bmod.mp3 Star 1-GrimaldiA+Bmod]'' by [[Joel Grant Taylor]]
* ''[http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/star_1-GrimaldiA+Bmod.mp3 Star 1-GrimaldiA+Bmod]'' by [[Joel Grant Taylor]]
* ''[http://micro.soonlabel.com/star/20120830-77et-star.mp3 77et Star]'' by [[Chris Vaisvil]]
* ''[http://micro.soonlabel.com/star/20120830-77et-star.mp3 77et Star]'' by [[Chris Vaisvil]]
Line 574: Line 546:


[[Category:Equal divisions of the octave]]
[[Category:Equal divisions of the octave]]
[[Category:listen]]
[[Category:Listen]]
[[Category:star]]
[[Category:Star]]
[[Category:starling]]
[[Category:Starling]]
[[Category:valentine]]
[[Category:Valentine]]

Revision as of 18:38, 26 December 2021

77 equal temperament, 77-tET or 77-EDO divides the octave into 77 steps of size 15.5844 cents each.

Theory

With fifths less than a cent flat, major thirds a bit over three cents sharp and 7/4'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 alicorn and 11- and 13-limit 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.

77edo is an excellent EDO for Carlos Alpha scale, since the difference between 5 steps of 77edo and 1 step of Carlos Alpha is only -0.042912 cents.

Prime harmonics

Script error: No such module "primes_in_edo".

Intervals

Degree Cents Approximate Ratios
in the 13-limit
0 0.000 1/1
1 15.584 81/80, 99/98
2 31.169 64/63, 49/48
3 46.753 33/32, 36/35
4 62.338 28/27, 26/25
5 77.922 21/20, 25/24
6 93.506 135/128
7 109.091 16/15
8 124.675 15/14
9 140.260 13/12
10 155.844 12/11, 11/10
11 171.429 72/65
12 187.013 10/9
13 202.597 9/8
14 218.182 256/225
15 233.766 8/7
16 249.351 15/13
17 264.935 7/6
18 280.519 33/28
19 296.104 32/27, 13/11
20 311.688 6/5
21 327.273 98/81
22 342.857 11/9, 39/32
23 358.442 16/13
24 374.026 56/45, 26/21
25 389.610 5/4
26 405.195 33/26, 81/64
27 420.779 14/11, 32/25
28 436.364 9/7
29 451.948 13/10
30 467.532 21/16
31 483.117 120/91
32 498.701 4/3
33 514.286 27/20
34 529.870 49/36
35 545.455 11/8, 15/11
36 561.039 18/13
37 576.623 7/5
38 592.208 45/32
39 607.792 64/45
40 623.377 10/7
41 638.961 13/9
42 654.545 16/11, 22/15
43 670.130 72/49
44 685.714 40/27
45 701.299 3/2
46 716.883 91/60
47 732.468 32/21
48 748.052 20/13
49 763.636 14/9
50 779.221 11/7, 25/16
51 794.805 52/33, 128/81
52 810.390 8/5
53 825.974 45/28, 21/13
54 841.558 13/8
55 857.143 18/11, 64/39
56 872.727 81/49
57 888.312 5/3
58 903.896 27/16, 22/13
59 919.481 56/33
60 935.065 12/7
61 950.649 26/15
62 966.234 7/4
63 981.818 225/128
64 997.403 16/9
65 1012.987 9/5
66 1028.571 65/36
67 1044.156 11/6, 20/11
68 1059.740 24/13
69 1075.325 28/15
70 1090.909 15/8
71 1106.494 256/135
72 1122.078 40/21, 48/25
73 1137.662 27/14, 25/13
74 1153.247 64/33, 35/18
75 1168.831 63/32, 96/49
76 1184.416 160/81, 196/99
77 1200.000 2/1

Regular temperament properties

The following table shows TE temperament measures (RMS normalized by the rank) of 77et.

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-122 77⟩ [⟨77 122]] +0.207 0.207 1.33
2.3.5 32805/32768, 1594323/1562500 [⟨77 122 179]] -0.336 0.785 5.04
2.3.5.7 126/125, 1029/1024, 10976/10935 [⟨77 122 179 216]] -0.021 0.872 5.59
2.3.5.7.11 121/120, 126/125, 176/175, 10976/10935 [⟨77 122 179 216 266]] +0.322 1.039 6.66
2.3.5.7.11.13 121/120, 126/125, 176/175, 196/195, 676/675 [⟨77 122 179 216 266 285]] +0.222 0.974 6.25

Rank-2 temperaments

Periods
per Octave 		Generator Temperaments
1 1\77
1 2\77
1 3\77
1 4\77 Unicorn / alicorn / camahueto / qilin
1 5\77 Valentine
1 6\77
1 8\77
1 9\77 Tsaharuk
1 10\77
1 12\77
1 13\77
1 15\77 Guiron
1 16\77 Hemischis
1 17\77
1 18\77
1 19\77
1 20\77 Oolong
1 23\77 Restles
1 24\77
1 25\77
1 26\77
1 27\77
1 29\77
1 30\77
1 31\77 Hemiseven
1 32\77 Helmholtz / grackle
1 34\77 Mabila
1 36\77
1 37\77
1 38\77
7 1\77 Absurdity
7 2\77
7 3\77
7 4\77
7 5\77
11 1\77 Hendecatonic
11 2\77
11 3\77

Music

Retrieved from "https://en.xen.wiki/index.php?title=77edo&oldid=82656"