80edo: Difference between revisions

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== Theory ==

80edo is the first edo that represents the [[19-odd-limit]] [[tonality diamond]] [[consistent]]ly, though it barely manages to do so. Despite this, a large number of intervals in higher odd limits ~~in the [[29-limit|29-prime-limit]]~~ are consistent, and its [[patent val]] generally does well at approximating (29-prime-limited) [[harmonic series]] segments, such as modes 16 through 30 but especially modes 8 through 15. It achieves this much consistency ~~because~~ all primes in the 29-limit except 13 ~~are~~ sharp of just~~, with~~ inconsistencies usually ~~arising~~ through not cancelling the over-sharpness of compound harmonics 21, 27, 35, 45, ~~and~~ 49 (and their octave-equivalents), which may be seen as an interesting limitation. This means it can be used as a general-purpose approximate 29~~-prime~~-limit system with a relatively manageable number of tones, with some care taken around inconsistency. It can also model larger primes if one is willing to accept their sharpness and for this purpose it does well for its size at the no-31's [[41-limit]], or even the [[43-limit]] if you are fine with [[43/32]] being slightly flat causing more inconsistencies. ~~In fact, except for [[26/25]], it is consistent in the no-21's no-27's no-31's no-35's [[41-odd-limit]]!~~ If one wants higher precision as one goes to higher primes to try to convey the subtle harmonic qualities of those primes, 80et arguably fails in general, although many specific cases may be convincing. A promising alternative is using 80et as a model in which to fit higher-limit JI by way of approximating as much of the harmonic series as possible, for which it can model the 125-odd-limit quite well (corresponding to the 113-prime-limit), leading to an excellent [[Ringer scale]] described in the [[#Ringer 80|Ringer 80 section of this article]].

80edo is the first edo that represents the [[19-odd-limit]] [[tonality diamond]] [[consistent]]ly, though it barely manages to do so. Despite this, a large number of intervals in higher odd limits are consistent, and its [[patent val]] generally does well at approximating the [[29-limit|29-prime-limited]] [[harmonic series]] segments, such as modes 16 through 30 but especially modes 8 through 15. It achieves this much consistency with all primes in the 29-limit except 13 being sharp of just; the inconsistencies usually arise through not cancelling the over-sharpness of compound harmonics [[21/1|21]], [[27/1|27]], [[35/1|35]], [[45/1|45]], [[49/1|49]], and their octave-equivalents, which may be seen as an interesting limitation. This means it can be used as a general-purpose approximate 29-limit system with a relatively manageable number of tones, with some care taken around inconsistency. In fact, it is almost consistent to the no-21 no-27 [[29-odd-limit]], with the exception of [[25/13]] and its octave complement, meaning it makes a surprisingly reasonable [[25-odd-limit]] system, with only [[26/21]], [[21/17]], [[21/16]] and their [[octave complement]]s as extra inconsistencies, which a theorist might find various justifications for. Possible additions to this include [[33/1|33]], [[37/1|37]], [[39/1|39]], and [[41/1|41]]. Thus, it can also model larger primes if one is willing to accept their sharpness, and for this purpose, it does well for its size at the no-31's [[41-limit]], or even the [[43-limit]] if you are fine with [[43/32]] being slightly flat causing more inconsistencies.

If one wants higher precision as one goes to higher primes to try to convey the subtle harmonic qualities of those primes, 80et arguably fails in general, although many specific cases may be convincing. A promising alternative is using 80et as a model in which to fit higher-limit JI by way of approximating as much of the harmonic series as possible, for which it can model the 125-odd-limit quite well (corresponding to the 113-prime-limit), leading to an excellent [[Ringer scale]] described in the [[#Ringer 80|Ringer 80 section of this article]].

=== As a tuning of other temperaments ===

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=== Potential for a general-purpose system ===

Though a strange tuning in lower prime limits, 80edo offers a very unique composite structure that can aid with familiarization/conceptualization by way of its subset edos of 2, 4, 5, 8, 10, 16 and 20. 80edo supports a plethora of multiperiod temperaments with accurate JI interval interpretations based on these edos; to see a fairly comprehensive list of these temperaments and of their most accurate JI interpretations with respect to integer multiples of their period, see [[#Consistent circles]]. These represent a large number of practically completely unexplored and novel high-limit temperaments with varying musical potential.

<nowiki />* The strangeness of its tuning can largely be explained by the addition of vals [[80edo]] = [[53edo]] + [[27edo]], where [[27edo]] exaggerates the idiosyncratic mapping of the 2.3.5.7.13 subgroup, as while 53edo tempers {[[625/624|S25]], [[676/675|S26]], [[729/728|S27]]} (supporting [[catakleismic]]) and {[[1728/1715|S6/S7]], [[5120/5103|S8/S9]]} (supporting [[buzzard]]), 27edo tempers {[[1728/1715|S6/S7]], [[64/63|S8]], [[325/324|S25*S26]], [[351/350|S26*S27]]} [[Square superparticular|implying]] {[[4375/4374|S25/S27]], [[169/168|S13]]} but maps S25~S27 positively and S26 negatively, which 80et thus inherits though with less damage. This is not insignificant, because this plays a special role (as we'll see in the next section on subsets).

=== Prime harmonics ===

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Since 80 factors into primes as 24 × 5, 80edo has subset edos {{EDOs| 2, 4, 5, 8, 10, 16, 20, and 40 }}.

As a composite edo, the main subsets it lacks are subsets of [[3edo|3]] and [[9edo|9]], but 9\80 = 135{{cent}} offers a good approximation to 1\9 = 133.33…{{c}}, and ~~one could argue that~~ 1\3 = 400{{cent}} ~~is the most difficult small edo interval to interpret (assuming interpreting~~ it ~~as [[5/4]] is not convincing or pleasing enough) in that its interpretations tend to be a large variety of high-complexity intervals, though if one wants a similar sound there is~~ 27\80 = 405{{cent}} as ~[[24/19]]~[[19~~/15~~]] ~~(though 24/19 is more accurate)~~, thus serving a similar function to the [[nestoria]] major third. As a result, 80edo is in some sense uniquely tasked with approximating small edos because it will often share subsets that can help make the approximation feel more regular and consistent by interpreting it as a near-equal multiperiod mos. This has the benefit of offering a relatively unexplored strategy of "tempered [[detempering]]", a sort of middle path between complete detempering to JI (which lacks the simplifications and unique comma pumping and structural opportunities of tempering) and not detempering the small edo at all (which can lead to challenging interpretation of harmony if one's goal is approximation to JI).

As a composite edo, the main subsets it lacks are subsets of [[3edo|3]] and [[9edo|9]], but 9\80 = 135{{cent}} offers a good approximation to 1\9 = 133.33…{{c}}, and instead of 1\3 = 400{{cent}}, it has 27\80 = 405{{cent}} as [[19/15]]~[[24/19]], thus serving a similar function to the [[nestoria]] major third. As a result, 80edo is in some sense uniquely tasked with approximating small edos because it will often share subsets that can help make the approximation feel more regular and consistent by interpreting it as a near-equal multiperiod mos. This has the benefit of offering a relatively unexplored strategy of "tempered [[detempering]]", a sort of middle path between complete detempering to JI (which lacks the simplifications and unique comma pumping and structural opportunities of tempering) and not detempering the small edo at all (which can lead to challenging interpretation of harmony if one's goal is approximation to JI).

Even if one finds this reasoning about not having subsets of 3 and 9 unconvincing, there is the fact that the idiosyncracies in the tuning profile of 80edo is intimately related to those of 27edo, so that it shares a deep logic with it through the 13-limit {{nowrap| 27e & 53 }} temperament [[quartonic]]. Even the sharp 7 is explained by 27edo being a sharp [[superpyth]] system. More mysterious is that the approximation of 1\9 at 9\80 = 135{{cent}}, when taken as a generator, is related to the shared [[41-limit]] structure between 80edo and the ultimate general purpose system, [[311edo]], through the {{nowrap|80 & 231}} temperament [[superlimmal]], where it represents [[27/25]]~[[40/37]], implying a slightly sharp tuning for 27/25, which is characteristic.

80edo is notable in not only it is consistent in the 19-odd-limit, but a large number of its supersets are also consistent in at least 19-odd-limit, if not larger. These are {{EDOs| 320, 400, 1600, 1920, 2000, 2320, 3920, 4320 }}. Temperament mergers of these produce various [[80th-octave temperaments]].

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| 24

| 360

| [[16/13]]

| [[16/13]], ''[[26/21]]''

|-

| 25

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| …

|}

<nowiki>*</nowiki> {{sg|no-31's [[37-limit]]}} Inconsistent interpretations in ''italic''.

<nowiki>*</nowiki> {{sg|80edo|limit=no-31's [[37-limit]]}} Inconsistent interpretations in ''italic''.

== Notation ==

=== Ups and downs ===

80edo can be notated using [[Kite's ups and downs notation]]. Note that quudsharp (quadruple-down sharp) is equivalent to quip (quintuple-up) and that quupflat (quadruple-up flat) is equivalent to quid (quintuple-down):

=== Sagittal ===

Notating 80edo in Sagittal (with diatonic whole tone equal to 14 edosteps, diatonic semitone equal to 5 edosteps):

{| class="wikitable" style="text-align: center;"

|-

! Degree

! −9

| '''−9'''

! −8

| −8

! −7

| −7

! −6

| −6

! −5

| −5

! −4

| −4

! −3

| −3

! −2

| −2

! −1

| −1

! 0

| '''0'''

! +1

| +1

! +2

| +2

! +3

| +3

! +4

| +4

! +5

| +5

! +6

| +6

! +7

| +7

! +8

| +8

! +9

| '''+9'''

|-

! Evo

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| [[Srutal archagall]]

| [[Bidia]]

| [[~~Pentorwell~~]]

| [[Pentaorwell]]

| 80 & 104

| [[Linus]] retraction

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| 44.9%

| Normal

| [[Supermajor]]

| [[Supermajor (temperament)|Supermajor]]

| [[Echidna]], [[semisupermajor]]

| ?

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! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal 8ve stretch (¢)

! rowspan="2" | Optimal 8ve stretch (¢)

! colspan="2" | Tuning error

|-

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|-

| 2.3

| {{~~monzo~~| 127 -80 }}

| {{Monzo| 127 -80 }}

| {{~~mapping~~| 80 127 }}

| {{Mapping| 80 127 }}

| −0.961

| 0.960

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| 2.3.5

| 2048/2025, 390625000/387420489

| {{~~mapping~~| 80 127 186}}

| {{Mapping| 80 127 186 }}

| −1.169

| 0.837

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| 2.3.5.7

| 1728/1715, 2048/2025, 3136/3125

| {{~~mapping~~| 80 127 186 225 }}

| {{Mapping| 80 127 186 225 }}

| −1.426

| 0.851

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| 2.3.5.7.11

| 176/175, 540/539, 896/891, 1331/1323

| {{~~mapping~~| 80 127 186 225 277 }}

| {{Mapping| 80 127 186 225 277 }}

| −1.353

| 0.775

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| 2.3.5.7.11.13

| 169/168, 176/175, 325/324, 364/363, 540/539

| {{~~mapping~~| 80 127 186 225 277 296 }}

| {{Mapping| 80 127 186 225 277 296 }}

| −1.105

| 0.901

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| 2.3.5.7.11.13.17

| 136/135, 169/168, 176/175, 221/220, 364/363, 540/539

| {{~~mapping~~| 80 127 186 225 277 296 327 }}

| {{Mapping| 80 127 186 225 277 296 327 }}

| −0.949

| 0.917

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| 2.3.5.7.11.13.17.19

| 136/135, 169/168, 176/175, 190/189, 221/220, 364/363, 400/399

| {{~~mapping~~| 80 127 186 225 277 296 327 340 }}

| {{Mapping| 80 127 186 225 277 296 327 340 }}

| −0.903

| 0.867

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=== Rank-2 temperaments ===

80et [[support]]s a profusion of 19-limit (and lower) rank-2 temperaments which have mostly not been explored~~. We might mention:~~

80et [[support]]s a profusion of 19-limit (and lower) rank-2 temperaments which have mostly not been explored.

* {{nowrap|31 & 80}}

* {{nowrap|72 & 80}}

* {{nowrap|34 & 80}}

* {{nowrap|46 & 80}}

* {{nowrap|29 & 80}}

* {{nowrap|12 & 80}}

* {{nowrap|22 & 80}}

* {{nowrap|58 & 80}}

* {{nowrap|41 & 80}}

~~In each case, the numbers joined by an ampersand represent 19-limit [[patent val]]s (meaning obtained by rounding to the nearest integer)~~.

{| class="wikitable center-all left-5"

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| 36/35~40/39

| [[Quartonic]]

|-

| 1

| 7\80

| 105

| 17/16

| [[Lucite]]

|-

| 1

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| 435

| 9/7

| [[Supermajor]]

| [[Supermajor (temperament)|Supermajor]]

|-

| 1

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| 17/13

| [[Semisept]]

|-

| 1

| 33\80

| 495

| 4/3

| [[Leapmonth]]

|-

| 1

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| 225 (15)

| 8/7 (64/63)

| [[~~Pentorwell~~]]

| [[Pentaorwell]]

|-

| 5

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| [[Degrees]]

|}

<nowiki/>* [[Normal ~~lists~~|Octave-reduced form]], reduced to the first half-octave, and [[~~Normal lists~~|minimal form]] in parentheses if distinct

<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

== Detemperaments ==

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Mode 63 of the harmonic series (corresponding to 125-odd-limit) with added odds from mode 63 × 2 = 126 in square brackets:

: 63:64:[129]:65:[131]:66:[133]:67:68:[137]:69:[139]:70:71:[143]:72:[145]:73:74:[149]

<pre>

: 75:76:[153]:77:78:[157]:79:80:[161]:81:82:83:[167]:84:85:86:[173]:87:88:89

63:64:[129]:65:[131]:66:[133]:67:68:[137]:69:[139]:70:71:[143]:72:[145]:73:74:[149]

: [179]:90:91:92:[185]:93:94:95:96:97:[195]:98:99:100:101:102:103:104:[209]:105

75:76:[153]:77:78:[157]:79:80:[161]:81:82:83:[167]:84:85:86:[173]:87:88:89

: 106:107:108:109:110:111:112:113:114:115:116:117:118:119:120:121:122:123:124:125(:126)

[179]:90:91:92:[185]:93:94:95:96:97:[195]:98:99:100:101:102:103:104:[209]:105

106:107:108:109:110:111:112:113:114:115:116:117:118:119:120:121:122:123:124:125(:126)

</pre>

The above is split into 20 harmonics per line a.k.a. ~300¢ worth of harmonic content per line.

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In lowest terms as a /105 scale corresponding to a [[primodal]] /53 scale, among other possible interpretations:

: 105:106:107:108:109:110:111:112:113:114:115:116:117:118:119:120:121:122:123:124:125:126:128:129:130:131:132:133:134:136:137:138:139:140:142:143:144:145:146:148:149:150:152:153:154:156:157:158:160:161:162:164:166:167:168:170:172:173:174:176:178:179:180:182:184:185:186:188:190:192:194:195:196:198:200:202:204:206:208:209:210

<pre>

105:106:107:108:109:110:111:112:113:114:115:116:117:118:119:120:121:122:123:124:125:126:128:129:130:131:132:133:134:136:137:138:139:140:142:143:144:145:146:148:149:150:152:153:154:156:157:158:160:161:162:164:166:167:168:170:172:173:174:176:178:179:180:182:184:185:186:188:190:192:194:195:196:198:200:202:204:206:208:209:210

</pre>

This form is useful for copy-pasting into tools that accept colon-separated harmonic series chord enumerations as scales.

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As reduced, rooted intervals (16 intervals per line):

: 129/128, 65/64, 131/128, 33/32, 133/128, 67/64, 17/16, 137/128, 69/64, 139/128, 35/32, 71/64, 143/128, 9/8, 145/128, 73/64,

<pre>

: 37/32, 75/64, 19/16, 153/128, 77/64, 39/32, 157/128, 79/64, 5/4, 161/128, 81/64, 41/32, 83/64, 167/128, 21/16, 85/64,

129/128, 65/64, 131/128, 33/32, 133/128, 67/64, 17/16, 137/128, 69/64, 139/128, 35/32, 71/64, 143/128, 9/8, 145/128, 73/64,

: 43/32, 173/128, 87/64, 11/8, 89/64, 179/128, 45/32, 91/64, 23/16, 185/128, 93/64, 47/32, 95/64, 3/2, 97/64, 195/128,

37/32, 75/64, 19/16, 153/128, 77/64, 39/32, 157/128, 79/64, 5/4, 161/128, 81/64, 41/32, 83/64, 167/128, 21/16, 85/64,

: 49/32, 99/64, 25/16, 101/64, 51/32, 103/64, 13/8, 209/128, 105/64, 53/32, 107/64, 27/16, 109/64, 55/32, 111/64, 7/4,

43/32, 173/128, 87/64, 11/8, 89/64, 179/128, 45/32, 91/64, 23/16, 185/128, 93/64, 47/32, 95/64, 3/2, 97/64, 195/128,

: 113/64, 57/32, 115/64, 29/16, 117/64, 59/32, 119/64, 15/8, 121/64, 61/32, 123/64, 31/16, 125/64, 63/32, 2/1

49/32, 99/64, 25/16, 101/64, 51/32, 103/64, 13/8, 209/128, 105/64, 53/32, 107/64, 27/16, 109/64, 55/32, 111/64, 7/4,

113/64, 57/32, 115/64, 29/16, 117/64, 59/32, 119/64, 15/8, 121/64, 61/32, 123/64, 31/16, 125/64, 63/32, 2/1

</pre>

== Scales ==

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* [[Echidna]][22]: 4 3 4 4 3 3 3 4 4 3 4 4 3 4 4 3 4 3 4 4 3 4

* Echidna[36]: 3 1 3 1 3 3 1 3 3 1 3 1 3 3 1 3 1 3 3 1 3 1 3 3 1 3 3 1 3 1 3 3 1 3 1 3

* [[Leapfrog]][12]: 5 9 5 9 5 5 9 5 9 5 9 5

* Leapfrog[17]: 5 4 5 5 5 4 5 5 4 5 5 4 5 5 5 4 5

* Leapfrog[29]: 4 1 4 1 4 4 1 4 1 4 1 4 4 1 4 1 4 4 1 4 1 4 1 4 4 1 4 1 4

* Leapfrog[46] : 1 3 1 1 3 1 3 1 1 3 1 1 3 1 3 1 1 3 1 1 3 1 1 3 1 3 1 1 3 1 1 3 1 3 1 1 3 1 1 3 1 3 1 1 3 1

* [[Octopus]][40]: 1 1 6 1 1 1 1 6 1 1 1 1 6 1 1 1 1 6 1 1 1 1 6 1 1 1 1 6 1 1 1 1 6 1 1 1 1 6 1 1

* [[Parakleismic]][23]: 4 4 1 4 4 4 4 4 1 4 4 4 4 4 1 4 4 4 4 4 1 4 4

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* Trisedodge[20]: 5 1 5 5 5 1 5 5 5 1 5 5 5 1 5 5 5 1 5 5

* Trisedodge[35]: 1 4 1 4 1 4 1 1 4 1 4 1 4 1 1 4 1 4 1 4 1 1 4 1 4 1 4 1 1 4 1 4 1 4 1

; [[Polymicrotonal]] scales

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* 12-tone 8&10edo scale: 10 6 4 4 8 8 8 8 4 4 6 10

* 12-tone 8&20edo scale: 4 6 10 4 8 8 8 8 4 10 6 4

; [[Combination product set]]s

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* 5 9 8 4 7 4 10 6 6 11 10

''Some of its interesting subsets:''

* 5 21 13 14 12 11 10 (''closely resembles [[14edo#scales|fennec scale]]{{idio}} from [[14edo]]'')

* 5 21 7 14 12 11 10 (''closely resembles [[14edo#scales|fennec scale]]{{idio}} from [[14edo]]'')

* 14 12 11 14 12 11 10 (''loosely resembles porcupine[7] or [[7edo]]'')

* 14 12 11 10 12 11 10 (''loosely resembles porcupine[7] or [[7edo]]'')

* 22 11 4 10 23 10 (''loosely resembles minor blues scale'')

* 22 11 14 12 11 10 (''loosely resembles [[porcupine]][6] or [[6afdo]]'')

* 26 7 14 6 27 (''sounds regal but brooding'')

* 26 7 14 12 21 (''sounds sparkly and delicate'')

* 26 11 10 ~~37 10~~ (''closely resembles [[6afdo#scales|geode]]{{idio}} subset of [[6afdo]]'')

* 22 11 14 23 10 (''closely resembles [[6afdo#scales|geode]]{{idio}} subset of [[6afdo]]'')

11-tone CPS ''(10-of-1,3,5,9,11,15,19,25,27,29,33)''

* 10 11 6 6 10 4 7 4 8 9 5

12-tone CPS ''(1-of-3,5,9,11,15,19,25,27,29,33,37,41)''

* 3 3 6 3 8 10 5 9 8 4 11 10

12-tone CPS ''(11-of-3,5,9,11,15,19,25,27,29,33,37,41)''

* 10 11 4 8 9 5 10 8 3 6 3 3

15-tone CPS ''(1-of-1,3,5,9,11,15,19,25,27,29,33,37,41,55,57)''

* 5 9 2 4 2 4 7 4 10 3 3 6 3 8 10

15-tone CPS ''(14-of-1,3,5,9,11,15,19,25,27,29,33,37,41,55,57)''

* 10 8 3 6 3 3 10 4 7 4 2 4 2 9 5

13-tone degen. [[eikosany]] ''(1,3,5,9,15,25)''

* 5 2 5 9 5 7 14 7 5 9 5 2 5

14-tone degen. eikosany ''(3,5,9,15,25,27)''

* 9 3 2 7 5 7 2 3 9 7 5 9 5 7

16-tone degen. eikosany ''(1,3,5,9,11,15)''

* 4 7 3 2 5 2 3 7 4 7 3 11 1 11 3 7

18-tone degen. eikosany ''(3,5,9,11,15,19)''

* 4 2 4 6 5 6 3 3 3 6 5 6 4 2 4 6 5 6

; Other scales

* [[Equipentatonic]] (exactly [[5edo]]): 16 16 16 16 16

* [[Equiheptatonic]] (approximate): 11 12 11 12 11 12 11

* [[Maeve Gutierrez|Gutierrez Moonglade scale]]

== Music ==

[[Bryan Deister]]

=== Modern renditions ===

; {{w|Frédéric Chopin}}

* Prelude Op. 28, No. 4 in E minor « Suffocation » (1839), arranged for harpsichord, tuned into 80-edo – rendered by [[Claudi Meneghin]] (2025)

** [https://www.youtube.com/watch?v=ng1UyvhHcrQ Quasi-Pythagorean version]

** [https://www.youtube.com/shorts/NBptgeIfReo Diaschismic version]

=== 21st century ===

; [[Bryan Deister]]

* [https://www.youtube.com/shorts/H6DlCHKii-o ''microtonal improvisation in 80edo''] (2025)

; [[Francium]]

* [https://www.youtube.com/watch?v=-MRhrpzRSC8 ''Itself''] (2024) – semisept in 80edo

* [https://www.youtube.com/watch?v=-MRhrpzRSC8 ''Itself''] (2024) – in semisept, 80edo tuning

* [https://www.youtube.com/watch?v=QuRHzoIozwo ''the circular one''] (2024)

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* [https://luphoria.com/xenpaper/#%23_licensed_under_CC-BY-4.0%2C_by_User%3AGodtone_(en.xen.wiki)%0A(osc%3Asawtooth12)%7B80edo%7D%7Br253Hz%7D%0A(bpm%3A96)(env%3A3085)%0A%5B0%2C47%5D-%5B0%2C14%2C47%5D-%5B0%2C14%2C26%2C47%5D-%0A%5B0%2C33%2C47%5D-%5B0%2C47%5D-%5B0%5D-%0A%5B0%2C47%5D-%5B0%2C14%2C47%5D-%5B0%2C14%2C26%2C47%5D-%0A%5B0%2C33%2C47%5D-%5B0%2C47%5D-%5B0%2C14%5D-%0A%5B%6061%2C14%2C28%5D-%5B14%2C28%2C61%5D-%5B1%2C40%2C61%5D-%0A%5B%6075%2C14%2C47%2C61%5D-%5B14%2C61%2C74%5D%5B14%5D--%0A%5B%6061%2C14%2C28%5D-%5B14%2C28%2C61%5D-%5B14%2C40%2C61%5D-%0A%5B%6075%2C14%2C47%2C61%5D--%5B%6066%2C14%2C33%2C80%5D--%0A%0A(bpm%3A128)(env%3A1282)%0A%5B0%5D-%5B0%2C6%5D._%5B0%2C18%5D-%5B0%2C6%2C47%5D.%0A%5B0%2C6%2C40%5D-%5B7%2C40%2C66%5D.%0A%5B0%2C7%2C26%2C59%5D-%5B7%2C26%2C59%2C80%5D.%0A%5B0%2C21%2C47%2C80%5D-%5B7%5D._%5B0%2C21%5D-%5B7%2C21%2C47%5D.%0A%5B0%2C7%2C40%5D-%5B0%2C7%2C40%2C66%5D._%5B0%2C7%2C26%2C61%5D-%5B7%2C26%2C61%2C80%5D.%0A%7Br%6061%7D%5B26%2C47%2C80%2C95%2C108%5D-%5B0%2C7%5D.%0A%5B0%2C20%5D-%5B%6047%2C0%2C20%2C47%5D._%5B%6040%2C%6066%2C7%2C40%5D-%5B7%2C40%2C66%5D._%5B%6060%2C6%2C27%2C60%5D-%5B27%2C60%2C80%5D.%0A%5B0%2C26%2C47%2C80%5D-%5B0%2C7%5D.%0A%5B0%2C20%5D-%5B%6047%2C0%2C20%2C47%5D._%5B%6040%2C%6066%2C7%2C40%5D-%5B7%2C40%2C66%5D._%5B%6060%2C6%2C27%2C60%5D-%5B27%2C60%2C80%5D.%0A%5B7%2C27%2C60%2C88%5D--%5B7%2C27%5D%5B27%2C60%2C88%5D---..%0A%7Br21%7D%5B0%2C26%2C47%5D-%7Br%6059%7D%5B0%2C26%2C47%5D-%7Br21%7D%5B0%2C26%2C47%5D-%0A%7Br%6059%7D%5B7%2C27%2C60%2C88%5D--%5B7%2C27%2C60%5D%5B27%2C60%2C88%5D---..%0A%7Br21%7D%5B0%2C26%2C47%5D-%7Br%6059%7D%5B0%2C26%2C47%2C80%5D-%7Br21%7D%5B0%2C26%2C47%5D-%0A%5B7%2C27%2C60%2C88%5D--%5B7%2C27%5D%5B27%2C60%2C88%5D---..%0A%7Br21%7D%5B0%2C26%2C47%5D-%7Br%6059%7D%5B0%2C26%2C47%5D-%7Br21%7D%5B0%2C26%2C47%5D-%0A%7Br%6059%7D%5B0%2C27%2C47%2C80%5D-%5B%6047%2C0%2C28%2C47%5D-%7Br%6040%7D%5B0%2C26%2C47%2C65%5D-%5B47%2C65%2C80%2C106%5D-%0A%7Br20%7D%5B0%2C27%2C47%2C80%5D-%5B%6047%2C0%2C28%2C47%5D-%7Br%6040%7D%5B0%2C26%2C47%2C65%5D-%5B47%2C65%2C80%2C106%5D-%0A%7Br20%7D%5B0%2C27%2C47%2C80%5D-%5B%6047%2C0%2C28%2C47%5D-%7Br%6040%7D%5B0%2C26%2C47%2C65%5D-%5B47%2C65%2C80%2C106%5D-%0A%7Br20%7D%5B0%2C27%2C47%2C80%5D-%5B%6047%2C0%2C28%2C47%5D-%7Br%6040%7D%5B0%2C26%2C47%2C65%5D-%5B47%2C65%2C80%2C106%5D-%0A%7Br20%7D%5B0%2C26%2C47%2C80%5D- unnamed xenpaper sketch] licensed under [https://creativecommons.org/licenses/by/4.0/ CC-BY-4.0]

* [https://luphoria.com/xenpaper/#%23_PLEASE_play_this_80_EDO_xenpaper_piece_out_loud%0A%23_PREFERABLY_on_mediocre_laptop_speakers%2C%0A%23_as_it_sounds_BETTER_acoustically!%0A%23_licensed_under_CC-BY-4.0%2C_by_User%3AGodtone_(en.xen.wiki)%0A(osc%3Asawtooth24)(bpm%3A161)%0A%7B80edo%7D_%23_inspiration%3A%0A%23_%7B44_%3A_54_%3A_56_%3A___58_%3A_60_%3A__69__%3A__74__%3A_82_%3A_85%7D%0A%23_%7B0%5C1_24%5C80_28%5C80_32%5C80_36%5C80_52%5C80_60%5C80_72%5C80_76%5C80%7D%0A%5B0_24_32_60%5D---%0A%5B0_23_36_52%5D---%0A%5B%6078_24_45_60%5D---%0A%5B%6075_24_46_61%5D---%0A%5B%6072_11_24_46_70%5D-------%0A%5B%6046_%6072_11_24_46%5D-------%0A%5B%6040_%6072_11_26_60%5D----%0A%5B%6040_%6072_11_26_52%5D--%0A%5B%6060_%6072_11_24_60%5D---%0A%5B%6055_%6072_11_24_62%5D---%0A%5B%6050_%6072_11_24_70%5D-------%0A%5B%6024_%6050_%6072_11_24_46%5D-------%0A%7Br20%7D%0A%5B%6024_%6050_%6072_11_24_46%5D-------%0A%6024_%6050_%6072_11_24_46%0A%5B%6024_%6055_%6072_11_24%5D-%0A%5B%6011_%6045_%6072_11_24_45%5D-------%0A%6011_%6045_%6072_11_24_45%0A%5B%6024_%6050_%6072_11_24_46%5D---------%0A%6024_%6050_%6072_11_24_46%0A%5B%6024_%6055_%6072_11_24%5D-%0A%5B%6011_%6045_%6072_11_24_45%5D---------%0A%6011_%6045_%6072_11_24_45%0A%5B%6024_%6050_%6072_11_24_46%5D-------%0A%5B%6024_%6055_%6072_11_24%5D---%0A%5B%6055_%6072_11_24_62%5D---%0A%5B%6040_%6074_25_44_59%5D%0A%5B%6040_%6072_25_44_59%5D-----%0A%5B%6042_%6072_25_42%5D%0A%5B%6060_%6072_25_36%5D------%0A%5B%6042_%6072_25_42%5D%0A%7Br%6060%7D%0A%5B%6055_%6072_11_24_62%5D%0A%5B%6050_%6072_11_24_70%5D----%0A%5B%6024_%6050_%6072_11_24_46%5D-%0A%5B%6072_11_24_46_70%5D-------%0A%5B%6050_%6072_11_24_70%5D-------%0A%5B%6072_11_24_46_70%5D-------%0A%5B%6040_%6072_11_26_52%5D-------%0A%5B%6060_%6072_11_24_60%5D----%0A%5B%6055_%6072_11_24_62%5D--%0A%5B%6050_%6072_11_24_62%5D%0A%5B%6050_%6072_11_24_70%5D----%0A%5B%6024_%6050_%6072_11_24_46%5D-%0A%5B%6072_11_24_46_70%5D------- unnamed piece] licensed under [https://creativecommons.org/licenses/by/4.0/ CC-BY-4.0]

~~[[User:Tristanbay|'''Tristan Bay''']]~~

* ''Subtract Hominem'' (2025) [https://tristanbay.bandcamp.com/track/subtract-hominem Bandcamp] | [https://youtu.be/JhGvrJ86jLU YouTube]

; [[Budjarn Lambeth]]

* [https://youtu.be/6N_8QM2UK5I ''Improvisation in compressed 80edo'' (435zpi)] (2025)

; [[Claudi Meneghin]]

* [https://www.youtube.com/watch?v=TgD7cN8a5D8 ''Lytel Twyelyghte Musicke (Little Twilight Music), for Brass, Winds, Strings, and Timpani, in 80-equal division of the octave, as the linear temperament generated by its fifth''] (2025)

; [[Tristan Bay]]

* ''Subtract Hominem'' (2025) – [https://tristanbay.bandcamp.com/track/subtract-hominem Bandcamp] | [https://youtu.be/JhGvrJ86jLU YouTube]

; [[Xotla]]

* "Mollusc Merchant" from ''Jazzbeetle'' (2023) [https://xotla.bandcamp.com/track/mollusc-merchant-80edo Bandcamp] | [https://www.youtube.com/watch?v=5cb0WHAwVuM YouTube]

* "Mollusc Merchant" from ''Jazzbeetle'' (2023) [https://xotla.bandcamp.com/track/mollusc-merchant-80edo Bandcamp] | [https://www.youtube.com/watch?v=5cb0WHAwVuM Original YouTube video] [https://www.youtube.com/watch?v=LnWJzffO7dY YouTube video without AI visuals] (2025)

==Instruments==

@@ Line 3: / Line 3: @@
 == Theory ==
-edo is the first edo that represents the [[19-odd-limit]] [[tonality diamond]] [[consistent]]ly, though it barely manages to do so. Despite this, a large number of intervals in higher odd limits in the [[29-limit|29-prime-limit]] are consistent, and its [[patent val]] generally does well at approximating (29-prime-limited) [[harmonic series]] segments, such as modes 16 through 30 but especially modes 8 through 15. It achieves this much consistency because all primes in the 29-limit except 13 are sharp of just, with inconsistencies usually arising through not cancelling the over-sharpness of compound harmonics 21, 27, 35, 45, and 49 (and their octave-equivalents), which may be seen as an interesting limitation. This means it can be used as a general-purpose approximate 29-prime-limit system with a relatively manageable number of tones, with some care taken around inconsistency. It can also model larger primes if one is willing to accept their sharpness and for this purpose it does well for its size at the no-31's [[41-limit]], or even the [[43-limit]] if you are fine with [[43/32]] being slightly flat causing more inconsistencies. In fact, except for [[26/25]], it is consistent in the no-21's no-27's no-31's no-35's [[41-odd-limit]]! If one wants higher precision as one goes to higher primes to try to convey the subtle harmonic qualities of those primes, 80et arguably fails in general, although many specific cases may be convincing. A promising alternative is using 80et as a model in which to fit higher-limit JI by way of approximating as much of the harmonic series as possible, for which it can model the 125-odd-limit quite well (corresponding to the 113-prime-limit), leading to an excellent [[Ringer scale]] described in the [[#Ringer 80|Ringer 80 section of this article]].
+edo is the first edo that represents the [[19-odd-limit]] [[tonality diamond]] [[consistent]]ly, though it barely manages to do so. Despite this, a large number of intervals in higher odd limits are consistent, and its [[patent val]] generally does well at approximating the [[29-limit|29-prime-limited]] [[harmonic series]] segments, such as modes 16 through 30 but especially modes 8 through 15. It achieves this much consistency with all primes in the 29-limit except 13 being sharp of just; the inconsistencies usually arise through not cancelling the over-sharpness of compound harmonics [[21/1|21]], [[27/1|27]], [[35/1|35]], [[45/1|45]], [[49/1|49]], and their octave-equivalents, which may be seen as an interesting limitation. This means it can be used as a general-purpose approximate 29-limit system with a relatively manageable number of tones, with some care taken around inconsistency. In fact, it is almost consistent to the no-21 no-27 [[29-odd-limit]], with the exception of [[25/13]] and its octave complement, meaning it makes a surprisingly reasonable [[25-odd-limit]] system, with only [[26/21]], [[21/17]], [[21/16]] and their [[octave complement]]s as extra inconsistencies, which a theorist might find various justifications for. Possible additions to this include [[33/1|33]], [[37/1|37]], [[39/1|39]], and [[41/1|41]]. Thus, it can also model larger primes if one is willing to accept their sharpness, and for this purpose, it does well for its size at the no-31's [[41-limit]], or even the [[43-limit]] if you are fine with [[43/32]] being slightly flat causing more inconsistencies.
+If one wants higher precision as one goes to higher primes to try to convey the subtle harmonic qualities of those primes, 80et arguably fails in general, although many specific cases may be convincing. A promising alternative is using 80et as a model in which to fit higher-limit JI by way of approximating as much of the harmonic series as possible, for which it can model the 125-odd-limit quite well (corresponding to the 113-prime-limit), leading to an excellent [[Ringer scale]] described in the [[#Ringer 80|Ringer 80 section of this article]].
 === As a tuning of other temperaments ===
@@ Line 12: / Line 14: @@
 === Potential for a general-purpose system ===
 Though a strange tuning in lower prime limits, 80edo offers a very unique composite structure that can aid with familiarization/conceptualization by way of its subset edos of 2, 4, 5, 8, 10, 16 and 20. 80edo supports a plethora of multiperiod temperaments with accurate JI interval interpretations based on these edos; to see a fairly comprehensive list of these temperaments and of their most accurate JI interpretations with respect to integer multiples of their period, see [[#Consistent circles]]. These represent a large number of practically completely unexplored and novel high-limit temperaments with varying musical potential.
-<nowiki />* The strangeness of its tuning can largely be explained by the addition of vals [[80edo]] = [[53edo]] + [[27edo]], where [[27edo]] exaggerates the idiosyncratic mapping of the 2.3.5.7.13 subgroup, as while 53edo tempers {[[625/624|S25]], [[676/675|S26]], [[729/728|S27]]} (supporting [[catakleismic]]) and {[[1728/1715|S6/S7]], [[5120/5103|S8/S9]]} (supporting [[buzzard]]), 27edo tempers {[[1728/1715|S6/S7]], [[64/63|S8]], [[325/324|S25*S26]], [[351/350|S26*S27]]} [[Square superparticular|implying]] {[[4375/4374|S25/S27]], [[169/168|S13]]} but maps S25~S27 positively and S26 negatively, which 80et thus inherits though with less damage. This is not insignificant, because this plays a special role (as we'll see in the next section on subsets).
 === Prime harmonics ===
@@ Line 21: / Line 21: @@
 Since 80 factors into primes as 2<sup>4</sup> × 5, 80edo has subset edos {{EDOs| 2, 4, 5, 8, 10, 16, 20, and 40 }}.
-As a composite edo, the main subsets it lacks are subsets of [[3edo|3]] and [[9edo|9]], but 9\80 = 135{{cent}} offers a good approximation to 1\9 = 133.33…{{c}}, and one could argue that 1\3 = 400{{cent}} is the most difficult small edo interval to interpret (assuming interpreting it as [[5/4]] is not convincing or pleasing enough) in that its interpretations tend to be a large variety of high-complexity intervals, though if one wants a similar sound there is 27\80 = 405{{cent}} as ~[[24/19]]~[[19/15]] (though 24/19 is more accurate), thus serving a similar function to the [[nestoria]] major third. As a result, 80edo is in some sense uniquely tasked with approximating small edos because it will often share subsets that can help make the approximation feel more regular and consistent by interpreting it as a near-equal multiperiod mos. This has the benefit of offering a relatively unexplored strategy of "tempered [[detempering]]", a sort of middle path between complete detempering to JI (which lacks the simplifications and unique comma pumping and structural opportunities of tempering) and not detempering the small edo at all (which can lead to challenging interpretation of harmony if one's goal is approximation to JI).
+As a composite edo, the main subsets it lacks are subsets of [[3edo|3]] and [[9edo|9]], but 9\80 = 135{{cent}} offers a good approximation to 1\9 = 133.33…{{c}}, and instead of 1\3 = 400{{cent}}, it has 27\80 = 405{{cent}} as [[19/15]]~[[24/19]], thus serving a similar function to the [[nestoria]] major third. As a result, 80edo is in some sense uniquely tasked with approximating small edos because it will often share subsets that can help make the approximation feel more regular and consistent by interpreting it as a near-equal multiperiod mos. This has the benefit of offering a relatively unexplored strategy of "tempered [[detempering]]", a sort of middle path between complete detempering to JI (which lacks the simplifications and unique comma pumping and structural opportunities of tempering) and not detempering the small edo at all (which can lead to challenging interpretation of harmony if one's goal is approximation to JI).
-Even if one finds this reasoning about not having subsets of 3 and 9 unconvincing, there is the fact that the idiosyncracies in the tuning profile of 80edo is intimately related to those of 27edo, so that it shares a deep logic with it through the 13-limit {{nowrap| 27e & 53 }} temperament [[quartonic]]. Even the sharp 7 is explained by 27edo being a sharp [[superpyth]] system. More mysterious is that the approximation of 1\9 at 9\80 = 135{{cent}}, when taken as a generator, is related to the shared [[41-limit]] structure between 80edo and the ultimate general purpose system, [[311edo]], through the {{nowrap|80 &amp; 231}} temperament [[superlimmal]], where it represents [[27/25]]~[[40/37]], implying a slightly sharp tuning for 27/25, which is characteristic.
 edo is notable in not only it is consistent in the 19-odd-limit, but a large number of its supersets are also consistent in at least 19-odd-limit, if not larger. These are {{EDOs| 320, 400, 1600, 1920, 2000, 2320, 3920, 4320 }}. Temperament mergers of these produce various [[80th-octave temperaments]].
@@ Line 132: / Line 130: @@
 | 24
 | 360
-| [[16/13]]
+| [[16/13]], ''[[26/21]]''
 |-
 | 25
@@ Line 202: / Line 200: @@
 | …
 |}
-<nowiki>*</nowiki> {{sg|no-31's [[37-limit]]}} Inconsistent interpretations in ''italic''.
+<nowiki>*</nowiki> {{sg|80edo|limit=no-31's [[37-limit]]}} Inconsistent interpretations in ''italic''.
 == Notation ==
+=== Ups and downs ===
+edo can be notated using [[Kite's ups and downs notation]]. Note that quudsharp (quadruple-down sharp) is equivalent to quip (quintuple-up) and that quupflat (quadruple-up flat) is equivalent to quid (quintuple-down):
+{{Ups and downs sharpness}}
+=== Sagittal ===
 Notating 80edo in Sagittal (with diatonic whole tone equal to 14 edosteps, diatonic semitone equal to 5 edosteps):
 {| class="wikitable" style="text-align: center;"
 |-
 ! Degree
-! −9
+| '''−9'''
-! −8
+| −8
-! −7
+| −7
-! −6
+| −6
-! −5
+| −5
-! −4
+| −4
-! −3
+| −3
-! −2
+| −2
-! −1
+| −1
-! 0
+| '''0'''
-! +1
+| +1
-! +2
+| +2
-! +3
+| +3
-! +4
+| +4
-! +5
+| +5
-! +6
+| +6
-! +7
+| +7
-! +8
+| +8
-! +9
+| '''+9'''
 |-
 ! Evo
@@ Line 311: / Line 314: @@
 | [[Srutal archagall]]
 | [[Bidia]]
-| [[Pentorwell]]
+| [[Pentaorwell]]
 | 80 & 104
 | [[Linus]] retraction
@@ Line 332: / Line 335: @@
 | 44.9%
 | Normal
-| [[Supermajor]]
+| [[Supermajor (temperament)|Supermajor]]
 | [[Echidna]], [[semisupermajor]]
 | ?
@@ Line 556: / Line 559: @@
 ! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br />8ve stretch (¢)
+! rowspan="2" | Optimal<br>8ve stretch (¢)
 ! colspan="2" | Tuning error
 |-
@@ Line 563: / Line 566: @@
 |-
 | 2.3
-| {{monzo| 127 -80 }}
+| {{Monzo| 127 -80 }}
-| {{mapping| 80 127 }}
+| {{Mapping| 80 127 }}
 | −0.961
 | 0.960
@@ Line 571: / Line 574: @@
 | 2.3.5
 | 2048/2025, 390625000/387420489
-| {{mapping| 80 127 186}}
+| {{Mapping| 80 127 186 }}
 | −1.169
 | 0.837
@@ Line 578: / Line 581: @@
 | 2.3.5.7
 | 1728/1715, 2048/2025, 3136/3125
-| {{mapping| 80 127 186 225 }}
+| {{Mapping| 80 127 186 225 }}
 | −1.426
 | 0.851
@@ Line 585: / Line 588: @@
 | 2.3.5.7.11
 | 176/175, 540/539, 896/891, 1331/1323
-| {{mapping| 80 127 186 225 277 }}
+| {{Mapping| 80 127 186 225 277 }}
 | −1.353
 | 0.775
@@ Line 592: / Line 595: @@
 | 2.3.5.7.11.13
 | 169/168, 176/175, 325/324, 364/363, 540/539
-| {{mapping| 80 127 186 225 277 296 }}
+| {{Mapping| 80 127 186 225 277 296 }}
 | −1.105
 | 0.901
@@ Line 599: / Line 602: @@
 | 2.3.5.7.11.13.17
 | 136/135, 169/168, 176/175, 221/220, 364/363, 540/539
-| {{mapping| 80 127 186 225 277 296 327 }}
+| {{Mapping| 80 127 186 225 277 296 327 }}
 | −0.949
 | 0.917
@@ Line 606: / Line 609: @@
 | 2.3.5.7.11.13.17.19
 | 136/135, 169/168, 176/175, 190/189, 221/220, 364/363, 400/399
-| {{mapping| 80 127 186 225 277 296 327 340 }}
+| {{Mapping| 80 127 186 225 277 296 327 340 }}
 | −0.903
 | 0.867
@@ Line 613: / Line 616: @@
 === Rank-2 temperaments ===
-et [[support]]s a profusion of 19-limit (and lower) rank-2 temperaments which have mostly not been explored. We might mention:
+et [[support]]s a profusion of 19-limit (and lower) rank-2 temperaments which have mostly not been explored.
-* {{nowrap|31 & 80}}
-* {{nowrap|72 & 80}}
-* {{nowrap|34 & 80}}
-* {{nowrap|46 & 80}}
-* {{nowrap|29 & 80}}
-* {{nowrap|12 & 80}}
-* {{nowrap|22 & 80}}
-* {{nowrap|58 & 80}}
-* {{nowrap|41 & 80}}
-In each case, the numbers joined by an ampersand represent 19-limit [[patent val]]s (meaning obtained by rounding to the nearest integer).
 {| class="wikitable center-all left-5"
@@ Line 647: / Line 638: @@
 | 36/35~40/39
 | [[Quartonic]]
+|-
+| 1
+| 7\80
+| 105
+| 17/16
+| [[Lucite]]
 |-
 | 1
@@ Line 664: / Line 661: @@
 | 435
 | 9/7
-| [[Supermajor]]
+| [[Supermajor (temperament)|Supermajor]]
 |-
 | 1
@@ Line 671: / Line 668: @@
 | 17/13
 | [[Semisept]]
+|-
+| 1
+| 33\80
+| 495
+| 4/3
+| [[Leapmonth]]
 |-
 | 1
@@ Line 706: / Line 709: @@
 | 225<br>(15)
 | 8/7<br>(64/63)
-| [[Pentorwell]]
+| [[Pentaorwell]]
 |-
 | 5
@@ Line 732: / Line 735: @@
 | [[Degrees]]
 |}
-<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
+<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
 == Detemperaments ==
@@ Line 742: / Line 745: @@
 Mode 63 of the harmonic series (corresponding to 125-odd-limit) with added odds from mode 63 × 2 = 126 in square brackets:
-: 63:64:[129]:65:[131]:66:[133]:67:68:[137]:69:[139]:70:71:[143]:72:[145]:73:74:[149]
+<pre>
-: 75:76:[153]:77:78:[157]:79:80:[161]:81:82:83:[167]:84:85:86:[173]:87:88:89
+:64:[129]:65:[131]:66:[133]:67:68:[137]:69:[139]:70:71:[143]:72:[145]:73:74:[149]
-: [179]:90:91:92:[185]:93:94:95:96:97:[195]:98:99:100:101:102:103:104:[209]:105
+:76:[153]:77:78:[157]:79:80:[161]:81:82:83:[167]:84:85:86:[173]:87:88:89
-: 106:107:108:109:110:111:112:113:114:115:116:117:118:119:120:121:122:123:124:125(:126)
+[179]:90:91:92:[185]:93:94:95:96:97:[195]:98:99:100:101:102:103:104:[209]:105
+:107:108:109:110:111:112:113:114:115:116:117:118:119:120:121:122:123:124:125(:126)
+</pre>
 The above is split into 20 harmonics per line a.k.a. ~300¢ worth of harmonic content per line.
@@ Line 751: / Line 756: @@
 In lowest terms as a /105 scale corresponding to a [[primodal]] /53 scale, among other possible interpretations:
-: 105:106:107:108:109:110:111:112:113:114:115:116:117:118:119:120:121:122:123:124:125:126:128:129:130:131:132:133:134:136:137:138:139:140:142:143:144:145:146:148:149:150:152:153:154:156:157:158:160:161:162:164:166:167:168:170:172:173:174:176:178:179:180:182:184:185:186:188:190:192:194:195:196:198:200:202:204:206:208:209:210
+<pre>
+:106:107:108:109:110:111:112:113:114:115:116:117:118:119:120:121:122:123:124:125:126:128:129:130:131:132:133:134:136:137:138:139:140:142:143:144:145:146:148:149:150:152:153:154:156:157:158:160:161:162:164:166:167:168:170:172:173:174:176:178:179:180:182:184:185:186:188:190:192:194:195:196:198:200:202:204:206:208:209:210
+</pre>
 This form is useful for copy-pasting into tools that accept colon-separated harmonic series chord enumerations as scales.
@@ Line 757: / Line 764: @@
 As reduced, rooted intervals (16 intervals per line):
-: 129/128, 65/64, 131/128, 33/32, 133/128, 67/64, 17/16, 137/128, 69/64, 139/128, 35/32, 71/64, 143/128, 9/8, 145/128, 73/64,
+<pre>
-: 37/32, 75/64, 19/16, 153/128, 77/64, 39/32, 157/128, 79/64, 5/4, 161/128, 81/64, 41/32, 83/64, 167/128, 21/16, 85/64,
+/128, 65/64, 131/128, 33/32, 133/128, 67/64, 17/16, 137/128, 69/64, 139/128, 35/32, 71/64, 143/128, 9/8, 145/128, 73/64,
-: 43/32, 173/128, 87/64, 11/8, 89/64, 179/128, 45/32, 91/64, 23/16, 185/128, 93/64, 47/32, 95/64, 3/2, 97/64, 195/128,
+/32, 75/64, 19/16, 153/128, 77/64, 39/32, 157/128, 79/64, 5/4, 161/128, 81/64, 41/32, 83/64, 167/128, 21/16, 85/64,
-: 49/32, 99/64, 25/16, 101/64, 51/32, 103/64, 13/8, 209/128, 105/64, 53/32, 107/64, 27/16, 109/64, 55/32, 111/64, 7/4,
+/32, 173/128, 87/64, 11/8, 89/64, 179/128, 45/32, 91/64, 23/16, 185/128, 93/64, 47/32, 95/64, 3/2, 97/64, 195/128,
-: 113/64, 57/32, 115/64, 29/16, 117/64, 59/32, 119/64, 15/8, 121/64, 61/32, 123/64, 31/16, 125/64, 63/32, 2/1
+/32, 99/64, 25/16, 101/64, 51/32, 103/64, 13/8, 209/128, 105/64, 53/32, 107/64, 27/16, 109/64, 55/32, 111/64, 7/4,
+/64, 57/32, 115/64, 29/16, 117/64, 59/32, 119/64, 15/8, 121/64, 61/32, 123/64, 31/16, 125/64, 63/32, 2/1
+</pre>
 == Scales ==
@@ Line 771: / Line 780: @@
 * [[Echidna]][22]: 4 3 4 4 3 3 3 4 4 3 4 4 3 4 4 3 4 3 4 4 3 4
 * Echidna[36]: 3 1 3 1 3 3 1 3 3 1 3 1 3 3 1 3 1 3 3 1 3 1 3 3 1 3 3 1 3 1 3 3 1 3 1 3
+* [[Leapfrog]][12]: 5 9 5 9 5 5 9 5 9 5 9 5
+* Leapfrog[17]: 5 4 5 5 5 4 5 5 4 5 5 4 5 5 5 4 5
+* Leapfrog[29]: 4 1 4 1 4 4 1 4 1 4 1 4 4 1 4 1 4 4 1 4 1 4 1 4 4 1 4 1 4
+* Leapfrog[46] : 1 3 1 1 3 1 3 1 1 3 1 1 3 1 3 1 1 3 1 1 3 1 1 3 1 3 1 1 3 1 1 3 1 3 1 1 3 1 1 3 1 3 1 1 3 1
 * [[Octopus]][40]: 1 1 6 1 1 1 1 6 1 1 1 1 6 1 1 1 1 6 1 1 1 1 6 1 1 1 1 6 1 1 1 1 6 1 1 1 1 6 1 1
 * [[Parakleismic]][23]: 4 4 1 4 4 4 4 4 1 4 4 4 4 4 1 4 4 4 4 4 1 4 4
@@ Line 781: / Line 794: @@
 * Trisedodge[20]: 5 1 5 5 5 1 5 5 5 1 5 5 5 1 5 5 5 1 5 5
 * Trisedodge[35]: 1 4 1 4 1 4 1 1 4 1 4 1 4 1 1 4 1 4 1 4 1 1 4 1 4 1 4 1 1 4 1 4 1 4 1
 ; [[Polymicrotonal]] scales
@@ Line 791: / Line 803: @@
 * 12-tone 8&10edo scale: 10 6 4 4 8 8 8 8 4 4 6 10
 * 12-tone 8&20edo scale: 4 6 10 4 8 8 8 8 4 10 6 4
 ; [[Combination product set]]s
@@ Line 798: / Line 809: @@
 * 5 9 8 4 7 4 10 6 6 11 10
 ''Some of its interesting subsets:''
-* 5 21 13 14 12 11 10 (''closely resembles [[14edo#scales|fennec scale]]{{idio}} from [[14edo]]'')
+* 5 21 7 14 12 11 10 (''closely resembles [[14edo#scales|fennec scale]]{{idio}} from [[14edo]]'')
-* 14 12 11 14 12 11 10 (''loosely resembles porcupine[7] or [[7edo]]'')
+* 14 12 11 10 12 11 10 (''loosely resembles porcupine[7] or [[7edo]]'')
 * 22 11 4 10 23 10 (''loosely resembles minor blues scale'')
 * 22 11 14 12 11 10 (''loosely resembles [[porcupine]][6] or [[6afdo]]'')
 * 26 7 14 6 27 (''sounds regal but brooding'')
 * 26 7 14 12 21 (''sounds sparkly and delicate'')
-* 26 11 10 37 10 (''closely resembles [[6afdo#scales|geode]]{{idio}} subset of [[6afdo]]'')
+* 22 11 14 23 10  (''closely resembles [[6afdo#scales|geode]]{{idio}} subset of [[6afdo]]'')
 -tone CPS ''(10-of-1,3,5,9,11,15,19,25,27,29,33)''
 * 10 11 6 6 10 4 7 4 8 9 5
 -tone CPS ''(1-of-3,5,9,11,15,19,25,27,29,33,37,41)''
 * 3 3 6 3 8 10 5 9 8 4 11 10
 -tone CPS ''(11-of-3,5,9,11,15,19,25,27,29,33,37,41)''
 * 10 11 4 8 9 5 10 8 3 6 3 3
 -tone CPS ''(1-of-1,3,5,9,11,15,19,25,27,29,33,37,41,55,57)''
 * 5 9 2 4 2 4 7 4 10 3 3 6 3 8 10
 -tone CPS ''(14-of-1,3,5,9,11,15,19,25,27,29,33,37,41,55,57)''
 * 10 8 3 6 3 3 10 4 7 4 2 4 2 9 5
 -tone degen. [[eikosany]] ''(1,3,5,9,15,25)''
 * 5 2 5 9 5 7 14 7 5 9 5 2 5
 -tone degen. eikosany ''(3,5,9,15,25,27)''
 * 9 3 2 7 5 7 2 3 9 7 5 9 5 7
 -tone degen. eikosany ''(1,3,5,9,11,15)''
 * 4 7 3 2 5 2 3 7 4 7 3 11 1 11 3 7
 -tone degen. eikosany ''(3,5,9,11,15,19)''
 * 4 2 4 6 5 6 3 3 3 6 5 6 4 2 4 6 5 6
 ; Other scales
 * [[Equipentatonic]] (exactly [[5edo]]): 16 16 16 16 16
 * [[Equiheptatonic]] (approximate): 11 12 11 12 11 12 11
+* [[Maeve Gutierrez|Gutierrez Moonglade scale]]
 == Music ==
-[[Bryan Deister]]
+=== Modern renditions ===
+; {{w|Frédéric Chopin}}
+* Prelude Op. 28, No. 4 in E minor « Suffocation » (1839), arranged for harpsichord, tuned into 80-edo – rendered by [[Claudi Meneghin]] (2025)
+** [https://www.youtube.com/watch?v=ng1UyvhHcrQ Quasi-Pythagorean version]
+** [https://www.youtube.com/shorts/NBptgeIfReo Diaschismic version]
+=== 21st century ===
+; [[Bryan Deister]]
 * [https://www.youtube.com/shorts/H6DlCHKii-o ''microtonal improvisation in 80edo''] (2025)
 ; [[Francium]]
-* [https://www.youtube.com/watch?v=-MRhrpzRSC8 ''Itself''] (2024) – semisept in 80edo
+* [https://www.youtube.com/watch?v=-MRhrpzRSC8 ''Itself''] (2024) – in semisept, 80edo tuning
 * [https://www.youtube.com/watch?v=QuRHzoIozwo ''the circular one''] (2024)
@@ Line 858: / Line 867: @@
 * [https://luphoria.com/xenpaper/#%23_licensed_under_CC-BY-4.0%2C_by_User%3AGodtone_(en.xen.wiki)%0A(osc%3Asawtooth12)%7B80edo%7D%7Br253Hz%7D%0A(bpm%3A96)(env%3A3085)%0A%5B0%2C47%5D-%5B0%2C14%2C47%5D-%5B0%2C14%2C26%2C47%5D-%0A%5B0%2C33%2C47%5D-%5B0%2C47%5D-%5B0%5D-%0A%5B0%2C47%5D-%5B0%2C14%2C47%5D-%5B0%2C14%2C26%2C47%5D-%0A%5B0%2C33%2C47%5D-%5B0%2C47%5D-%5B0%2C14%5D-%0A%5B%6061%2C14%2C28%5D-%5B14%2C28%2C61%5D-%5B1%2C40%2C61%5D-%0A%5B%6075%2C14%2C47%2C61%5D-%5B14%2C61%2C74%5D%5B14%5D--%0A%5B%6061%2C14%2C28%5D-%5B14%2C28%2C61%5D-%5B14%2C40%2C61%5D-%0A%5B%6075%2C14%2C47%2C61%5D--%5B%6066%2C14%2C33%2C80%5D--%0A%0A(bpm%3A128)(env%3A1282)%0A%5B0%5D-%5B0%2C6%5D._%5B0%2C18%5D-%5B0%2C6%2C47%5D.%0A%5B0%2C6%2C40%5D-%5B7%2C40%2C66%5D.%0A%5B0%2C7%2C26%2C59%5D-%5B7%2C26%2C59%2C80%5D.%0A%5B0%2C21%2C47%2C80%5D-%5B7%5D._%5B0%2C21%5D-%5B7%2C21%2C47%5D.%0A%5B0%2C7%2C40%5D-%5B0%2C7%2C40%2C66%5D._%5B0%2C7%2C26%2C61%5D-%5B7%2C26%2C61%2C80%5D.%0A%7Br%6061%7D%5B26%2C47%2C80%2C95%2C108%5D-%5B0%2C7%5D.%0A%5B0%2C20%5D-%5B%6047%2C0%2C20%2C47%5D._%5B%6040%2C%6066%2C7%2C40%5D-%5B7%2C40%2C66%5D._%5B%6060%2C6%2C27%2C60%5D-%5B27%2C60%2C80%5D.%0A%5B0%2C26%2C47%2C80%5D-%5B0%2C7%5D.%0A%5B0%2C20%5D-%5B%6047%2C0%2C20%2C47%5D._%5B%6040%2C%6066%2C7%2C40%5D-%5B7%2C40%2C66%5D._%5B%6060%2C6%2C27%2C60%5D-%5B27%2C60%2C80%5D.%0A%5B7%2C27%2C60%2C88%5D--%5B7%2C27%5D%5B27%2C60%2C88%5D---..%0A%7Br21%7D%5B0%2C26%2C47%5D-%7Br%6059%7D%5B0%2C26%2C47%5D-%7Br21%7D%5B0%2C26%2C47%5D-%0A%7Br%6059%7D%5B7%2C27%2C60%2C88%5D--%5B7%2C27%2C60%5D%5B27%2C60%2C88%5D---..%0A%7Br21%7D%5B0%2C26%2C47%5D-%7Br%6059%7D%5B0%2C26%2C47%2C80%5D-%7Br21%7D%5B0%2C26%2C47%5D-%0A%5B7%2C27%2C60%2C88%5D--%5B7%2C27%5D%5B27%2C60%2C88%5D---..%0A%7Br21%7D%5B0%2C26%2C47%5D-%7Br%6059%7D%5B0%2C26%2C47%5D-%7Br21%7D%5B0%2C26%2C47%5D-%0A%7Br%6059%7D%5B0%2C27%2C47%2C80%5D-%5B%6047%2C0%2C28%2C47%5D-%7Br%6040%7D%5B0%2C26%2C47%2C65%5D-%5B47%2C65%2C80%2C106%5D-%0A%7Br20%7D%5B0%2C27%2C47%2C80%5D-%5B%6047%2C0%2C28%2C47%5D-%7Br%6040%7D%5B0%2C26%2C47%2C65%5D-%5B47%2C65%2C80%2C106%5D-%0A%7Br20%7D%5B0%2C27%2C47%2C80%5D-%5B%6047%2C0%2C28%2C47%5D-%7Br%6040%7D%5B0%2C26%2C47%2C65%5D-%5B47%2C65%2C80%2C106%5D-%0A%7Br20%7D%5B0%2C27%2C47%2C80%5D-%5B%6047%2C0%2C28%2C47%5D-%7Br%6040%7D%5B0%2C26%2C47%2C65%5D-%5B47%2C65%2C80%2C106%5D-%0A%7Br20%7D%5B0%2C26%2C47%2C80%5D- unnamed xenpaper sketch] licensed under [https://creativecommons.org/licenses/by/4.0/ CC-BY-4.0]
 * [https://luphoria.com/xenpaper/#%23_PLEASE_play_this_80_EDO_xenpaper_piece_out_loud%0A%23_PREFERABLY_on_mediocre_laptop_speakers%2C%0A%23_as_it_sounds_BETTER_acoustically!%0A%23_licensed_under_CC-BY-4.0%2C_by_User%3AGodtone_(en.xen.wiki)%0A(osc%3Asawtooth24)(bpm%3A161)%0A%7B80edo%7D_%23_inspiration%3A%0A%23_%7B44_%3A_54_%3A_56_%3A___58_%3A_60_%3A__69__%3A__74__%3A_82_%3A_85%7D%0A%23_%7B0%5C1_24%5C80_28%5C80_32%5C80_36%5C80_52%5C80_60%5C80_72%5C80_76%5C80%7D%0A%5B0_24_32_60%5D---%0A%5B0_23_36_52%5D---%0A%5B%6078_24_45_60%5D---%0A%5B%6075_24_46_61%5D---%0A%5B%6072_11_24_46_70%5D-------%0A%5B%6046_%6072_11_24_46%5D-------%0A%5B%6040_%6072_11_26_60%5D----%0A%5B%6040_%6072_11_26_52%5D--%0A%5B%6060_%6072_11_24_60%5D---%0A%5B%6055_%6072_11_24_62%5D---%0A%5B%6050_%6072_11_24_70%5D-------%0A%5B%6024_%6050_%6072_11_24_46%5D-------%0A%7Br20%7D%0A%5B%6024_%6050_%6072_11_24_46%5D-------%0A%6024_%6050_%6072_11_24_46%0A%5B%6024_%6055_%6072_11_24%5D-%0A%5B%6011_%6045_%6072_11_24_45%5D-------%0A%6011_%6045_%6072_11_24_45%0A%5B%6024_%6050_%6072_11_24_46%5D---------%0A%6024_%6050_%6072_11_24_46%0A%5B%6024_%6055_%6072_11_24%5D-%0A%5B%6011_%6045_%6072_11_24_45%5D---------%0A%6011_%6045_%6072_11_24_45%0A%5B%6024_%6050_%6072_11_24_46%5D-------%0A%5B%6024_%6055_%6072_11_24%5D---%0A%5B%6055_%6072_11_24_62%5D---%0A%5B%6040_%6074_25_44_59%5D%0A%5B%6040_%6072_25_44_59%5D-----%0A%5B%6042_%6072_25_42%5D%0A%5B%6060_%6072_25_36%5D------%0A%5B%6042_%6072_25_42%5D%0A%7Br%6060%7D%0A%5B%6055_%6072_11_24_62%5D%0A%5B%6050_%6072_11_24_70%5D----%0A%5B%6024_%6050_%6072_11_24_46%5D-%0A%5B%6072_11_24_46_70%5D-------%0A%5B%6050_%6072_11_24_70%5D-------%0A%5B%6072_11_24_46_70%5D-------%0A%5B%6040_%6072_11_26_52%5D-------%0A%5B%6060_%6072_11_24_60%5D----%0A%5B%6055_%6072_11_24_62%5D--%0A%5B%6050_%6072_11_24_62%5D%0A%5B%6050_%6072_11_24_70%5D----%0A%5B%6024_%6050_%6072_11_24_46%5D-%0A%5B%6072_11_24_46_70%5D------- unnamed piece] licensed under [https://creativecommons.org/licenses/by/4.0/ CC-BY-4.0]
-[[User:Tristanbay|'''Tristan Bay''']]
-* ''Subtract Hominem'' (2025) [https://tristanbay.bandcamp.com/track/subtract-hominem Bandcamp] | [https://youtu.be/JhGvrJ86jLU YouTube]
+; [[Budjarn Lambeth]]
+* [https://youtu.be/6N_8QM2UK5I ''Improvisation in compressed 80edo'' (435zpi)] (2025)
+; [[Claudi Meneghin]]
+* [https://www.youtube.com/watch?v=TgD7cN8a5D8 ''Lytel Twyelyghte Musicke (Little Twilight Music), for Brass, Winds, Strings, and Timpani, in 80-equal division of the octave, as the linear temperament generated by its fifth''] (2025)
+; [[Tristan Bay]]
+* ''Subtract Hominem'' (2025) – [https://tristanbay.bandcamp.com/track/subtract-hominem Bandcamp] | [https://youtu.be/JhGvrJ86jLU YouTube]
 ; [[Xotla]]
-* "Mollusc Merchant" from ''Jazzbeetle'' (2023) [https://xotla.bandcamp.com/track/mollusc-merchant-80edo Bandcamp] | [https://www.youtube.com/watch?v=5cb0WHAwVuM YouTube]
+* "Mollusc Merchant" from ''Jazzbeetle'' (2023) [https://xotla.bandcamp.com/track/mollusc-merchant-80edo Bandcamp] | [https://www.youtube.com/watch?v=5cb0WHAwVuM Original YouTube video] [https://www.youtube.com/watch?v=LnWJzffO7dY YouTube video without AI visuals] (2025)
 ==Instruments==