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 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]].

=== Significance of echidna ===

As an equal temperament, it is well-tuned for the important 11-limit and 17-limit half-octave-period temperament [[echidna]] (the 22&58 temperament), which affords great freedom in a 36-note MOS and still many choices in a 22-note MOS, offering a high-accuracy rank 2 detemper of [[22edo]], which in comparison conflates many important distinctions of the 11-limit. This is not insignificant as many abundant intervals of echidna, such as (especially) [[11/10]], [[9/7]] and [[17/16]], are tuned so accurately that they [[#Consistent circles|form 80-note consistent circles]]. Echidna supports [[srutal archagall]], which is also tuned near-optimally for [[fiventeen]] — specifically, for the characteristic fiventeen pentad, 30:34:40:45:51:60, consisting of steps of [[20/17]] and [[9/8]]~[[17/15]], and is the smallest edo to improve on the tuning of srutal archagall + fiventeen after [[34edo]]. In its representation of echidna, the least accurate tuning is that of [[7/4]], which is (relatively) very sharp in 80edo, for which [[58edo]] does better as a tuning of echidna (though much worse as a tuning for srutal archagall/diaschismic and especially fiventeen); one can reason this makes the 80edo tuning of echidna feel more like a detemper of 22edo (especially given the smaller step size between adjacent notes equated in 22edo).

As an equal temperament, it is well-tuned for the important 11-limit and 17-limit half-octave-period temperament [[echidna]] (the {{nowrap|22 & 58}} temperament), which affords great freedom in a 36-note MOS and still many choices in a 22-note MOS, offering a high-accuracy rank 2 detemper of [[22edo]], which in comparison conflates many important distinctions of the 11-limit. This is not insignificant as many abundant intervals of echidna, such as (especially) [[11/10]], [[9/7]] and [[17/16]], are tuned so accurately that they [[#Consistent circles|form 80-note consistent circles]]. Echidna supports [[srutal archagall]], which is also tuned near-optimally for [[fiventeen]] — specifically, for the characteristic fiventeen pentad, 30:34:40:45:51:60, consisting of steps of [[20/17]] and [[9/8]]~[[17/15]], and is the smallest edo to improve on the tuning of srutal archagall + fiventeen after [[34edo]]. In its representation of echidna, the least accurate tuning is that of [[7/4]], which is (relatively) very sharp in 80edo, for which [[58edo]] does better as a tuning of echidna (though much worse as a tuning for srutal archagall/diaschismic and especially fiventeen); one can reason this makes the 80edo tuning of echidna feel more like a detemper of 22edo (especially given the smaller step size between adjacent notes equated in 22edo).

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

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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..{{cent}}, 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).

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 27e&53 temperament [[quartonic]].. Even the [[#Significance of echidna|aforementioned]] 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 80&231 temperament [[superlimmal]], where it represents [[27/25]]~[[40/37]], implying a slightly sharp tuning for 27/25, which is characteristic.

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 [[#Significance of echidna|aforementioned]] 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.

=== Commas ===

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| [[Bidia]]

| [[Pentorwell]]

| 80 & 104

| [[Linus]] retraction

| ?

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

|+ style="font-size: 105%;" | 20-note circles by gen. ~~–~~ '''all are related to [[Degrees]]'''

|+ style="font-size: 105%;" | 20-note circles by gen. – '''all are related to [[Degrees]]'''

|-

! [[Interval]]

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| {{monzo| 127 -80 }}

| {{mapping| 80 127 }}

| ~~−0~~.961

| −0.961

| 0.960

| 6.40

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| 2048/2025, 390625000/387420489

| {{mapping| 80 127 186}}

| ~~−1~~.169

| −1.169

| 0.837

| 5.59

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

| {{mapping| 80 127 186 225 }}

| ~~−1~~.426

| −1.426

| 0.851

| 5.68

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

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

| ~~−1~~.353

| −1.353

| 0.775

| 5.17

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

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

| ~~−1~~.105

| −1.105

| 0.901

| 6.01

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

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

| ~~−0~~.949

| −0.949

| 0.917

| 6.12

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

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

| ~~−0~~.903

| −0.903

| 0.867

| 5.78

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

* 31&80 {{multival| 7 6 15 27 -24 -23 -20 … }}

* {{nowrap|31 & 80}} {{multival| 7 6 15 27 -24 -23 -20 … }}

* 72&80 {{multival| 24 30 40 24 32 24 0 … }}

* {{nowrap|72 & 80}} {{multival| 24 30 40 24 32 24 0 … }}

* 34&80 {{multival| 2 -4 -50 22 16 2 -40 … }}

* {{nowrap|34 & 80}} {{multival| 2 -4 -50 22 16 2 -40 … }}

* 46&80 {{multival| 2 -4 30 22 16 2 40 … }}

* {{nowrap|46 & 80}} {{multival| 2 -4 30 22 16 2 40 … }}

* 29&80 {{multival| 3 34 45 33 24 -37 20 … }}

* {{nowrap|29 & 80}} {{multival| 3 34 45 33 24 -37 20 … }}

* 12&80 {{multival| 4 -8 -20 -36 32 4 0 … }}

* {{nowrap|12 & 80}} {{multival| 4 -8 -20 -36 32 4 0 … }}

* 22&80 {{multival| 6 -10 12 -14 -32 6 -40 … }}

* {{nowrap|22 & 80}} {{multival| 6 -10 12 -14 -32 6 -40 … }}

* 58&80 {{multival| 6 -10 12 -14 -32 6 40 … }}

* {{nowrap|58 & 80}} {{multival| 6 -10 12 -14 -32 6 40 … }}

* 41&80 {{multival| 7 26 25 -3 -24 -33 20 … }}

* {{nowrap|41 & 80}} {{multival| 7 26 25 -3 -24 -33 20 … }}

In each case, the numbers joined by an ampersand represent 19-limit [[patent val]]s (meaning obtained by rounding to the nearest integer) and the first and most important part of the wedgie is given.

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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 ~~it is~~ distinct

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

== Detemperaments ==

@@ 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 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]].
 === Significance of echidna ===
-As an equal temperament, it is well-tuned for the important 11-limit and 17-limit half-octave-period temperament [[echidna]] (the 22&58 temperament), which affords great freedom in a 36-note MOS and still many choices in a 22-note MOS, offering a high-accuracy rank 2 detemper of [[22edo]], which in comparison conflates many important distinctions of the 11-limit. This is not insignificant as many abundant intervals of echidna, such as (especially) [[11/10]], [[9/7]] and [[17/16]], are tuned so accurately that they [[#Consistent circles|form 80-note consistent circles]]. Echidna supports [[srutal archagall]], which is also tuned near-optimally for [[fiventeen]] — specifically, for the characteristic fiventeen pentad, 30:34:40:45:51:60, consisting of steps of [[20/17]] and [[9/8]]~[[17/15]], and is the smallest edo to improve on the tuning of srutal archagall + fiventeen after [[34edo]]. In its representation of echidna, the least accurate tuning is that of [[7/4]], which is (relatively) very sharp in 80edo, for which [[58edo]] does better as a tuning of echidna (though much worse as a tuning for srutal archagall/diaschismic and especially fiventeen); one can reason this makes the 80edo tuning of echidna feel more like a detemper of 22edo (especially given the smaller step size between adjacent notes equated in 22edo).
+As an equal temperament, it is well-tuned for the important 11-limit and 17-limit half-octave-period temperament [[echidna]] (the {{nowrap|22 &amp; 58}} temperament), which affords great freedom in a 36-note MOS and still many choices in a 22-note MOS, offering a high-accuracy rank 2 detemper of [[22edo]], which in comparison conflates many important distinctions of the 11-limit. This is not insignificant as many abundant intervals of echidna, such as (especially) [[11/10]], [[9/7]] and [[17/16]], are tuned so accurately that they [[#Consistent circles|form 80-note consistent circles]]. Echidna supports [[srutal archagall]], which is also tuned near-optimally for [[fiventeen]] — specifically, for the characteristic fiventeen pentad, 30:34:40:45:51:60, consisting of steps of [[20/17]] and [[9/8]]~[[17/15]], and is the smallest edo to improve on the tuning of srutal archagall + fiventeen after [[34edo]]. In its representation of echidna, the least accurate tuning is that of [[7/4]], which is (relatively) very sharp in 80edo, for which [[58edo]] does better as a tuning of echidna (though much worse as a tuning for srutal archagall/diaschismic and especially fiventeen); one can reason this makes the 80edo tuning of echidna feel more like a detemper of 22edo (especially given the smaller step size between adjacent notes equated in 22edo).
 === Potential for a general-purpose system ===
@@ Line 16: / Line 16: @@
 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..{{cent}}, 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).
-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 27e&53 temperament [[quartonic]].. Even the [[#Significance of echidna|aforementioned]] 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 80&231 temperament [[superlimmal]], where it represents [[27/25]]~[[40/37]], implying a slightly sharp tuning for 27/25, which is characteristic.
+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 &amp; 53}} temperament [[quartonic]].. Even the [[#Significance of echidna|aforementioned]] 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.
 === Commas ===
@@ Line 243: / Line 243: @@
 | [[Bidia]]
 | [[Pentorwell]]
-| 80 & 104
+| 80 &amp; 104
 | [[Linus]] retraction
 | ?
@@ Line 274: / Line 274: @@
 {| class="wikitable center-all left-5"
-|+ style="font-size: 105%;" | 20-note circles by gen. &ndash; '''all are related to [[Degrees]]'''
+|+ style="font-size: 105%;" | 20-note circles by gen. – '''all are related to [[Degrees]]'''
 |-
 ! [[Interval]]
@@ Line 509: / Line 509: @@
 | {{monzo| 127 -80 }}
 | {{mapping| 80 127 }}
-| &minus;0.961
+| −0.961
 | 0.960
 | 6.40
@@ Line 516: / Line 516: @@
 | 2048/2025, 390625000/387420489
 | {{mapping| 80 127 186}}
-| &minus;1.169
+| −1.169
 | 0.837
 | 5.59
@@ Line 523: / Line 523: @@
 | 1728/1715, 2048/2025, 3136/3125
 | {{mapping| 80 127 186 225 }}
-| &minus;1.426
+| −1.426
 | 0.851
 | 5.68
@@ Line 530: / Line 530: @@
 | 176/175, 540/539, 896/891, 1331/1323
 | {{mapping| 80 127 186 225 277 }}
-| &minus;1.353
+| −1.353
 | 0.775
 | 5.17
@@ Line 537: / Line 537: @@
 | 169/168, 176/175, 325/324, 364/363, 540/539
 | {{mapping| 80 127 186 225 277 296 }}
-| &minus;1.105
+| −1.105
 | 0.901
 | 6.01
@@ Line 544: / Line 544: @@
 | 136/135, 169/168, 176/175, 221/220, 364/363, 540/539
 | {{mapping| 80 127 186 225 277 296 327 }}
-| &minus;0.949
+| −0.949
 | 0.917
 | 6.12
@@ Line 551: / Line 551: @@
 | 136/135, 169/168, 176/175, 190/189, 221/220, 364/363, 400/399
 | {{mapping| 80 127 186 225 277 296 327 340 }}
-| &minus;0.903
+| −0.903
 | 0.867
 | 5.78
@@ Line 559: / Line 559: @@
 et [[support]]s a profusion of 19-limit (and lower) rank-2 temperaments which have mostly not been explored. We might mention:
-* 31&amp;80 {{multival| 7 6 15 27 -24 -23 -20 … }}
+* {{nowrap|31 &amp; 80}} {{multival| 7 6 15 27 -24 -23 -20 … }}
-* 72&amp;80 {{multival| 24 30 40 24 32 24 0 … }}
+* {{nowrap|72 &amp; 80}} {{multival| 24 30 40 24 32 24 0 … }}
-* 34&amp;80 {{multival| 2 -4 -50 22 16 2 -40 … }}
+* {{nowrap|34 &amp; 80}} {{multival| 2 -4 -50 22 16 2 -40 … }}
-* 46&amp;80 {{multival| 2 -4 30 22 16 2 40 … }}
+* {{nowrap|46 &amp; 80}} {{multival| 2 -4 30 22 16 2 40 … }}
-* 29&amp;80 {{multival| 3 34 45 33 24 -37 20 … }}
+* {{nowrap|29 &amp; 80}} {{multival| 3 34 45 33 24 -37 20 … }}
-* 12&amp;80 {{multival| 4 -8 -20 -36 32 4 0 … }}
+* {{nowrap|12 &amp; 80}} {{multival| 4 -8 -20 -36 32 4 0 … }}
-* 22&amp;80 {{multival| 6 -10 12 -14 -32 6 -40 … }}
+* {{nowrap|22 &amp; 80}} {{multival| 6 -10 12 -14 -32 6 -40 … }}
-* 58&amp;80 {{multival| 6 -10 12 -14 -32 6 40 … }}
+* {{nowrap|58 &amp; 80}} {{multival| 6 -10 12 -14 -32 6 40 … }}
-* 41&amp;80 {{multival| 7 26 25 -3 -24 -33 20 … }}
+* {{nowrap|41 &amp; 80}} {{multival| 7 26 25 -3 -24 -33 20 … }}
 In each case, the numbers joined by an ampersand represent 19-limit [[patent val]]s (meaning obtained by rounding to the nearest integer) and the first and most important part of the wedgie is given.
@@ Line 676: / Line 676: @@
 | [[Degrees]]
 |}
-<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 == Detemperaments ==