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'''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 ==
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|}
|}


== 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"
|-
|-
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| 1
| 1
| 4\77
| 4\77
| [[Unicorn]]/alicorn/camahueto/qilin
| [[Unicorn]] / alicorn / camahueto / qilin
|-
|-
| 1
| 1
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| 1
| 1
| 32\77
| 32\77
| [[Helmholtz]]/[[grackle]]
| [[Helmholtz]] / [[grackle]]
|-
|-
| 1
| 1
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== 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]]
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[[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]]
Retrieved from "https://en.xen.wiki/w/77edo"