36edo: Difference between revisions

Line 9:

=== Harmonics ===

In the 5-limit, 36edo offers no improvement over 12edo, since its nearest approximation to 5:4 is the overly-familiar 400-cent major third. However, it excels at the 7th harmonic and intervals involving 7. As a 3 and 7 tuning, or in other words as a tuning for the 2.3.7 [[~~just intonation~~ subgroup]], 36edo's single degree of around 33 cents serves a double function as 49:48, the so-called [https://en.wikipedia.org/wiki/Septimal_diesis Slendro diesis] of around 36 cents, and as 64:63, the so-called [https://en.wikipedia.org/wiki/Septimal_comma septimal comma] of around 27 cents. Meanwhile, its second degree functions as 28:27, the so-called [https://en.wikipedia.org/wiki/Septimal_third-tone Septimal third-tone] (~~which~~ = 49:48 x 64:63). The 2.3.7 subgroup can be extended to the [[k*N_subgroups|2*36 subgroup]] 2.3.25.7.55.13.17, and on this subgroup it tempers out the same commas as [[72edo]] does in the full [[17-limit]].

In the 5-limit, 36edo offers no improvement over 12edo, since its nearest approximation to 5:4 is the overly-familiar 400-cent major third. However, it excels at the 7th harmonic and intervals involving 7. As a 3 and 7 tuning, or in other words as a tuning for the 2.3.7 [[subgroup]], 36edo's single degree of around 33 cents serves a double function as 49:48, the so-called [https://en.wikipedia.org/wiki/Septimal_diesis Slendro diesis] of around 36 cents, and as 64:63, the so-called [https://en.wikipedia.org/wiki/Septimal_comma septimal comma] of around 27 cents. Meanwhile, its second degree functions as 28:27, the so-called [https://en.wikipedia.org/wiki/Septimal_third-tone Septimal third-tone] (since {{nowrap|28/27 {{=}} 49:48 × 64:63}}). The 2.3.7 subgroup can be extended to the [[k*N_subgroups|2*36 subgroup]] 2.3.25.7.55.13.17, and on this subgroup it tempers out the same commas as [[72edo]] does in the full [[17-limit]].

=== Mappings ===

The 36edo patent val, like 12, tempers out 81/80, 128/125 and 648/625 in the 5-limit. It departs from 12 in the 7-limit, tempering out 686/675, and as a no-fives temperament, 1029/1024 and 118098/117649. The no-fives temperament tempering out 1029/1024, [[slendric]], is well supported by 36edo, its generator of ~8/7 represented by 7 steps of 36edo. In the 11-limit, the patent val tempers out 56/55, 245/242 and 540/539, and is the [[optimal patent val]] for the rank four temperament tempering out 56/55, as well as the rank three temperament [[Didymus rank three family|melpomene]] tempering out 81/80 and 56/55. In the 13-limit, it tempers out 78/77, in the 17-limit 51/50, and in the 19-limit 76/75~~, 91/90~~ and 96/95.

The 36edo patent val, like 12, tempers out 81/80, 128/125 and 648/625 in the 5-limit. It departs from 12 in the 7-limit, tempering out 686/675, and as a no-fives temperament, 1029/1024 and 118098/117649. The no-fives temperament tempering out 1029/1024, [[slendric]], is well supported by 36edo, its generator of ~8/7 represented by 7 steps of 36edo. In the 11-limit, the patent val tempers out 56/55, 245/242 and 540/539, and is the [[optimal patent val]] for the rank four temperament tempering out 56/55, as well as the rank three temperament [[Didymus rank three family|melpomene]] tempering out 81/80 and 56/55. In the 13-limit, it tempers out 78/77 and 91/90, in the 17-limit 51/50, and in the 19-limit 76/75 and 96/95.

As a 5-limit temperament, the patent val for 36edo is [[Wedgies and Multivals|contorted]], meaning there are notes of it which cannot be reached from the unison using only 5-limit intervals. A curious alternative val for the 5-limit is {{val| 36 65 116 }}, which is not contorted. It is also a meantone val, in the sense that 81/80 is tempered out. However, the "comma" {{monzo|29 0 -9}} is also tempered out, and the "fifth", 29\36, is actually approximately 7/4, whereas the "major third", 44\36, is actually approximately 7/3. Any 5-limit musical piece or scale which is a [[transversal]] for a meantone piece or scale will be converted to a no-fives piece tempering out 1029/1024 in place of 81/80 by applying this val.

Line 386:

If the arrows are taken to have their own layer of enharmonic spellings, then in some cases notes may be best denoted using double arrows.

===Sagittal notation===

=== Sagittal notation ===

This notation uses the same sagittal sequence as [[43edo#Sagittal notation|43-EDO]], is a subset of the notation for [[72edo#Sagittal notation|72-EDO]], and is a superset of the notations for EDOs [[18edo#Sagittal notation|18]], [[12edo#Sagittal notation|12]], and [[6edo#Sagittal notation|6]].

~~====Evo flavor====~~

==== Evo flavor ====

File:36-EDO_Evo_Sagittal.svg

Line 399:

</imagemap>

====Revo flavor====

==== Revo flavor ====

File:36-EDO_Revo_Sagittal.svg

Line 413:

Line 412:

For people accustomed to 12edo, 36edo is one of the easiest (if not ''the'' easiest) higher edo to become accustomed to. This is because one way to envision it is as an extended 12edo to which [https://en.wikipedia.org/wiki/Blue_note blue notes] (which are a sixth-tone lower than normal) and "red notes" (a sixth-tone higher) have been added.

The intervals in 36edo are all either the familiar 12edo intervals, or else "red" and "blue" versions of them. ~~Unlike~~ [[24edo]], ~~which has genuinely foreign~~ intervals such as 250 cents (halfway between a tone and a third) and 450 cents (halfway between a fourth and a third), the new intervals in 36edo all variations on existing ones. Unlike 24edo, 36edo is also relatively free of what Easley Blackwood called "discordant" intervals. The 5th and 11th harmonics fall almost halfway in between scale degrees of 36edo, and thus intervals containing them can be approximated two different ways, one of which is significantly sharp and the other significantly flat. The 333.333-cent interval (the "red minor third") sharply approximates 6/5 and flatly approximates 11/9, for instance, whereas the sharp 11/9 is 366.667 cents and the flat 6/5 is 300 cents. However, 11/10, 20/11, 15/11, and 22/15 all have accurate and consistent approximations since the errors on the 5th and 11th harmonics cancel out with both tending sharp.

The intervals in 36edo are all either the familiar 12edo intervals, or else "red" and "blue" versions of them. In [[24edo]], intervals such as 250 cents (halfway between a tone and a third) and 450 cents (halfway between a fourth and a third) tend to sound genuinely foreign, the new intervals in 36edo are all variations on existing ones. Unlike 24edo, 36edo is also relatively free of what Easley Blackwood called "discordant" intervals. The 5th and 11th harmonics fall almost halfway in between scale degrees of 36edo, and thus intervals containing them can be approximated two different ways, one of which is significantly sharp and the other significantly flat. The 333.333-cent interval (the "red minor third") sharply approximates 6/5 and flatly approximates 11/9, for instance, whereas the sharp 11/9 is 366.667 cents and the flat 6/5 is 300 cents. However, 11/10, 20/11, 15/11, and 22/15 all have accurate and consistent approximations since the errors on the 5th and 11th harmonics cancel out with both tending sharp.

36edo is fairly cosmopolitan because many genres of world music can be played in it. Because of the presence of blue notes, and the closeness with which the 7th harmonic and its intervals are matched, 36edo is an ideal scale to use for African-American genres of music such as blues and jazz, in which septimal intervals are frequently encountered. Indonesian gamelan music using pelog easily adapts to it as well, since 9edo is a subset and can be notated as every fourth note, and Slendro can be approximated in several different ways as well. 36edo can therefore function as a "bridge" between these genres and Western music. Arabic and Persian music do not adapt as well, however, since their microtonal intervals consist of mostly quarter tones.

Line 419:

Line 418:

The "red unison" and "blue unison" are in fact the same interval (33.333 cents), which is actually fairly consonant as a result of being so narrow (it is perceived as a unison, albeit noticeably "out of tune", but still not overly unpleasant). In contrast, most people consider 24edo's 50 cent step to sound much more discordant when used as a subminor second.

People with perfect (absolute) pitch often have a difficult time listening to xenharmonic and non-12edo scales, ~~which is due to~~ their ability to memorize and become accustomed to the pitches and intervals of 12edo ~~resulting~~ in other pitches and intervals sounding out of tune. This is not as much of a problem with 36edo, due to its similarity to 12. With practice, it might even be possible to extend one's perfect pitch to be able to recognize blue and red notes.

People with perfect (absolute) pitch often have a difficult time listening to xenharmonic and non-12edo scales,since their ability to memorize and become accustomed to the pitches and intervals of 12edo results in other pitches and intervals sounding out of tune. This is not as much of a problem with 36edo, due to its similarity to 12. With practice, it might even be possible to extend one's perfect pitch to be able to recognize blue and red notes.

=== "Quark" ===

Line 902:

Line 901:

| 10/9

| [[Squirrel]] (36), [[coendou]] (36c)

| [[1L 6s]], [[7L 1s]], [[7L 8s]], [[7L 15s]], [[7L 22s]]

| [[1L 6s]], [[7L 1s]], [[7L 8s]], [[7L 15s]], [[7L 22s]]

|-

| 1

Line 909:

Line 908:

| 8/7

| [[Slendric]] / [[mothra]] / [[guiron]]

| [[1L 4s]], [[1L 5s]], [[5L 1s]], [[5L 6s]], [[5L 11s]], [[5L 16s]], [[5L 21s]], [[5L 26s]]

| [[1L 4s]], [[1L 5s]], [[5L 1s]], [[5L 6s]], [[5L 11s]], [[5L 16s]], [[5L 21s]], [[5L 26s]]

|-

| 1

Line 916:

Line 915:

|

| [[3L 1s]], [[3L 4s]], [[3L 7s]], [[10L 3s]], [[13L 10s]]

| [[3L 1s]], [[3L 4s]], [[3L 7s]], [[10L 3s]], [[13L 10s]]

|-

| 1

Line 923:

Line 922:

|

| [[2L 1s]], [[3L 2s]], [[3L 5s]], [[3L 8s]], [[11L 3s]], [[11L 14s]]

| [[2L 1s]], [[3L 2s]], [[3L 5s]], [[3L 8s]], [[11L 3s]], [[11L 14s]]

|-

| 1

Line 930:

Line 929:

| 7/5

| [[Liese]], [[pycnic]] (36c)

| [[2L 1s]], [[2L 3s]], [[2L 5s]], [[2L 7s]], [[2L 9s]], …, [[2L 15s]], [[17L 2s]]

| [[2L 1s]], [[2L 3s]], [[2L 5s]], [[2L 7s]], [[2L 9s]], …, [[2L 15s]], [[17L 2s]]

|-

| 2

Line 937:

Line 936:

| 10/9

| [[Hedgehog]] (36ceff), [[echidna]] (36)

| [[2L 4s]], [[6L 2s]], [[8L 6s]], [[14L 8s]]

| [[2L 4s]], [[6L 2s]], [[8L 6s]], [[14L 8s]]

|-

| 2

Line 944:

Line 943:

| 8/7

| [[Baladic]] / [[echidnic]]

| [[4L 2s]], [[6L 4s]], [[10L 6s]], [[10L 16s]]

| [[4L 2s]], [[6L 4s]], [[10L 6s]], [[10L 16s]]

|-

| 3

Line 951:

Line 950:

|

| [[6L 3s]], [[6L 9s]], [[15L 6s]]

| [[6L 3s]], [[6L 9s]], [[15L 6s]]

|-

| 4

Line 958:

Line 957:

|

| [[4L 4s]], [[4L 8s]], [[4L 12s]], [[16L 4s]]

| [[4L 4s]], [[4L 8s]], [[4L 12s]], [[16L 4s]]

|-

| 4

Line 965:

Line 964:

|

| [[4L 4s]], [[8L 4s]], [[8L 12s]], [[8L 20s]]

| [[4L 4s]], [[8L 4s]], [[8L 12s]], [[8L 20s]]

|-

| 6

Line 979:

Line 978:

| 4/3<br>(36/35)

| [[Niner]]

| [[9L 9s]]

| [[9L 9s]]

|-

| 12

Line 986:

Line 985:

| 8/7<br>(64/63)

| [[Catler]]

| [[12L 12s]]

| [[12L 12s]]

|-

| 18

@@ Line 9: / Line 9: @@
 === Harmonics ===
-In the 5-limit, 36edo offers no improvement over 12edo, since its nearest approximation to 5:4 is the overly-familiar 400-cent major third. However, it excels at the 7th harmonic and intervals involving 7. As a 3 and 7 tuning, or in other words as a tuning for the 2.3.7 [[just intonation subgroup]], 36edo's single degree of around 33 cents serves a double function as 49:48, the so-called [https://en.wikipedia.org/wiki/Septimal_diesis Slendro diesis] of around 36 cents, and as 64:63, the so-called [https://en.wikipedia.org/wiki/Septimal_comma septimal comma] of around 27 cents. Meanwhile, its second degree functions as 28:27, the so-called [https://en.wikipedia.org/wiki/Septimal_third-tone Septimal third-tone] (which = 49:48 x 64:63). The 2.3.7 subgroup can be extended to the [[k*N_subgroups|2*36 subgroup]] 2.3.25.7.55.13.17, and on this subgroup it tempers out the same commas as [[72edo]] does in the full [[17-limit]].
+In the 5-limit, 36edo offers no improvement over 12edo, since its nearest approximation to 5:4 is the overly-familiar 400-cent major third. However, it excels at the 7th harmonic and intervals involving 7. As a 3 and 7 tuning, or in other words as a tuning for the 2.3.7 [[subgroup]], 36edo's single degree of around 33 cents serves a double function as 49:48, the so-called [https://en.wikipedia.org/wiki/Septimal_diesis Slendro diesis] of around 36 cents, and as 64:63, the so-called [https://en.wikipedia.org/wiki/Septimal_comma septimal comma] of around 27 cents. Meanwhile, its second degree functions as 28:27, the so-called [https://en.wikipedia.org/wiki/Septimal_third-tone Septimal third-tone] (since {{nowrap|28/27 {{=}} 49:48 × 64:63}}). The 2.3.7 subgroup can be extended to the [[k*N_subgroups|2*36 subgroup]] 2.3.25.7.55.13.17, and on this subgroup it tempers out the same commas as [[72edo]] does in the full [[17-limit]].
 {{harmonics in equal|36|prec=2}}
 === Mappings ===
-The 36edo patent val, like 12, tempers out 81/80, 128/125 and 648/625 in the 5-limit. It departs from 12 in the 7-limit, tempering out 686/675, and as a no-fives temperament, 1029/1024 and 118098/117649. The no-fives temperament tempering out 1029/1024, [[slendric]], is well supported by 36edo, its generator of ~8/7 represented by 7 steps of 36edo. In the 11-limit, the patent val tempers out 56/55, 245/242 and 540/539, and is the [[optimal patent val]] for the rank four temperament tempering out 56/55, as well as the rank three temperament [[Didymus rank three family|melpomene]] tempering out 81/80 and 56/55. In the 13-limit, it tempers out 78/77, in the 17-limit 51/50, and in the 19-limit 76/75, 91/90 and 96/95.
+The 36edo patent val, like 12, tempers out 81/80, 128/125 and 648/625 in the 5-limit. It departs from 12 in the 7-limit, tempering out 686/675, and as a no-fives temperament, 1029/1024 and 118098/117649. The no-fives temperament tempering out 1029/1024, [[slendric]], is well supported by 36edo, its generator of ~8/7 represented by 7 steps of 36edo. In the 11-limit, the patent val tempers out 56/55, 245/242 and 540/539, and is the [[optimal patent val]] for the rank four temperament tempering out 56/55, as well as the rank three temperament [[Didymus rank three family|melpomene]] tempering out 81/80 and 56/55. In the 13-limit, it tempers out 78/77 and 91/90, in the 17-limit 51/50, and in the 19-limit 76/75 and 96/95.
 As a 5-limit temperament, the patent val for 36edo is [[Wedgies and Multivals|contorted]], meaning there are notes of it which cannot be reached from the unison using only 5-limit intervals. A curious alternative val for the 5-limit is {{val| 36 65 116 }}, which is not contorted. It is also a meantone val, in the sense that 81/80 is tempered out. However, the "comma" {{monzo|29 0 -9}} is also tempered out, and the "fifth", 29\36, is actually approximately 7/4, whereas the "major third", 44\36, is actually approximately 7/3. Any 5-limit musical piece or scale which is a [[transversal]] for a meantone piece or scale will be converted to a no-fives piece tempering out 1029/1024 in place of 81/80 by applying this val.
@@ Line 386: / Line 386: @@
 If the arrows are taken to have their own layer of enharmonic spellings, then in some cases notes may be best denoted using double arrows.
-===Sagittal notation===
+=== Sagittal notation ===
 This notation uses the same sagittal sequence as [[43edo#Sagittal notation|43-EDO]], is a subset of the notation for [[72edo#Sagittal notation|72-EDO]], and is a superset of the notations for EDOs [[18edo#Sagittal notation|18]], [[12edo#Sagittal notation|12]], and [[6edo#Sagittal notation|6]].
-====Evo flavor====
+==== Evo flavor ====
 <imagemap>
 File:36-EDO_Evo_Sagittal.svg
@@ Line 399: / Line 399: @@
 </imagemap>
-====Revo flavor====
+==== Revo flavor ====
 <imagemap>
 File:36-EDO_Revo_Sagittal.svg
@@ Line 413: / Line 412: @@
 For people accustomed to 12edo, 36edo is one of the easiest (if not ''the'' easiest) higher edo to become accustomed to. This is because one way to envision it is as an extended 12edo to which [https://en.wikipedia.org/wiki/Blue_note blue notes] (which are a sixth-tone lower than normal) and "red notes" (a sixth-tone higher) have been added.
-The intervals in 36edo are all either the familiar 12edo intervals, or else "red" and "blue" versions of them. Unlike [[24edo]], which has genuinely foreign intervals such as 250 cents (halfway between a tone and a third) and 450 cents (halfway between a fourth and a third), the new intervals in 36edo all variations on existing ones. Unlike 24edo, 36edo is also relatively free of what Easley Blackwood called "discordant" intervals. The 5th and 11th harmonics fall almost halfway in between scale degrees of 36edo, and thus intervals containing them can be approximated two different ways, one of which is significantly sharp and the other significantly flat. The 333.333-cent interval (the "red minor third") sharply approximates 6/5 and flatly approximates 11/9, for instance, whereas the sharp 11/9 is 366.667 cents and the flat 6/5 is 300 cents. However, 11/10, 20/11, 15/11, and 22/15 all have accurate and consistent approximations since the errors on the 5th and 11th harmonics cancel out with both tending sharp.
+The intervals in 36edo are all either the familiar 12edo intervals, or else "red" and "blue" versions of them. In [[24edo]], intervals such as 250 cents (halfway between a tone and a third) and 450 cents (halfway between a fourth and a third) tend to sound genuinely foreign, the new intervals in 36edo are all variations on existing ones. Unlike 24edo, 36edo is also relatively free of what Easley Blackwood called "discordant" intervals. The 5th and 11th harmonics fall almost halfway in between scale degrees of 36edo, and thus intervals containing them can be approximated two different ways, one of which is significantly sharp and the other significantly flat. The 333.333-cent interval (the "red minor third") sharply approximates 6/5 and flatly approximates 11/9, for instance, whereas the sharp 11/9 is 366.667 cents and the flat 6/5 is 300 cents. However, 11/10, 20/11, 15/11, and 22/15 all have accurate and consistent approximations since the errors on the 5th and 11th harmonics cancel out with both tending sharp.
 edo is fairly cosmopolitan because many genres of world music can be played in it. Because of the presence of blue notes, and the closeness with which the 7th harmonic and its intervals are matched, 36edo is an ideal scale to use for African-American genres of music such as blues and jazz, in which septimal intervals are frequently encountered. Indonesian gamelan music using pelog easily adapts to it as well, since 9edo is a subset and can be notated as every fourth note, and Slendro can be approximated in several different ways as well. 36edo can therefore function as a "bridge" between these genres and Western music. Arabic and Persian music do not adapt as well, however, since their microtonal intervals consist of mostly quarter tones.
@@ Line 419: / Line 418: @@
 The "red unison" and "blue unison" are in fact the same interval (33.333 cents), which is actually fairly consonant as a result of being so narrow (it is perceived as a unison, albeit noticeably "out of tune", but still not overly unpleasant). In contrast, most people consider 24edo's 50 cent step to sound much more discordant when used as a subminor second.
-People with perfect (absolute) pitch often have a difficult time listening to xenharmonic and non-12edo scales, which is due to their ability to memorize and become accustomed to the pitches and intervals of 12edo resulting in other pitches and intervals sounding out of tune. This is not as much of a problem with 36edo, due to its similarity to 12. With practice, it might even be possible to extend one's perfect pitch to be able to recognize blue and red notes.
+People with perfect (absolute) pitch often have a difficult time listening to xenharmonic and non-12edo scales,since their ability to memorize and become accustomed to the pitches and intervals of 12edo results in other pitches and intervals sounding out of tune. This is not as much of a problem with 36edo, due to its similarity to 12. With practice, it might even be possible to extend one's perfect pitch to be able to recognize blue and red notes.
 === "Quark" ===
@@ Line 902: / Line 901: @@
 | 10/9
 | [[Squirrel]] (36), [[coendou]] (36c)
-| [[1L 6s]], [[7L 1s]], [[7L 8s]], [[7L 15s]], [[7L 22s]]
+| [[1L&nbsp;6s]], [[7L&nbsp;1s]], [[7L&nbsp;8s]], [[7L&nbsp;15s]], [[7L&nbsp;22s]]
 |-
 | 1
@@ Line 909: / Line 908: @@
 | 8/7
 | [[Slendric]] / [[mothra]] / [[guiron]]
-| [[1L 4s]], [[1L 5s]], [[5L 1s]], [[5L 6s]], [[5L 11s]], [[5L 16s]], [[5L 21s]], [[5L 26s]]
+| [[1L&nbsp;4s]], [[1L&nbsp;5s]], [[5L&nbsp;1s]], [[5L&nbsp;6s]], [[5L&nbsp;11s]], [[5L&nbsp;16s]], [[5L&nbsp;21s]], [[5L&nbsp;26s]]
 |-
 | 1
@@ Line 916: / Line 915: @@
 |
 |
-| [[3L 1s]], [[3L 4s]], [[3L 7s]], [[10L 3s]], [[13L 10s]]
+| [[3L&nbsp;1s]], [[3L&nbsp;4s]], [[3L&nbsp;7s]], [[10L&nbsp;3s]], [[13L&nbsp;10s]]
 |-
 | 1
@@ Line 923: / Line 922: @@
 |
 |
-| [[2L 1s]], [[3L 2s]], [[3L 5s]], [[3L 8s]], [[11L 3s]], [[11L 14s]]
+| [[2L&nbsp;1s]], [[3L&nbsp;2s]], [[3L&nbsp;5s]], [[3L&nbsp;8s]], [[11L&nbsp;3s]], [[11L&nbsp;14s]]
 |-
 | 1
@@ Line 930: / Line 929: @@
 | 7/5
 | [[Liese]], [[pycnic]] (36c)
-| [[2L 1s]], [[2L 3s]], [[2L 5s]], [[2L 7s]], [[2L 9s]], …, [[2L 15s]], [[17L 2s]]
+| [[2L&nbsp;1s]], [[2L&nbsp;3s]], [[2L&nbsp;5s]], [[2L&nbsp;7s]], [[2L&nbsp;9s]], …, [[2L&nbsp;15s]], [[17L&nbsp;2s]]
 |-
 | 2
@@ Line 937: / Line 936: @@
 | 10/9
 | [[Hedgehog]] (36ceff), [[echidna]] (36)
-| [[2L 4s]], [[6L 2s]], [[8L 6s]], [[14L 8s]]
+| [[2L&nbsp;4s]], [[6L&nbsp;2s]], [[8L&nbsp;6s]], [[14L&nbsp;8s]]
 |-
 | 2
@@ Line 944: / Line 943: @@
 | 8/7
 | [[Baladic]] / [[echidnic]]
-| [[4L 2s]], [[6L 4s]], [[10L 6s]], [[10L 16s]]
+| [[4L&nbsp;2s]], [[6L&nbsp;4s]], [[10L&nbsp;6s]], [[10L&nbsp;16s]]
 |-
 | 3
@@ Line 951: / Line 950: @@
 |
 |
-| [[6L 3s]], [[6L 9s]], [[15L 6s]]
+| [[6L&nbsp;3s]], [[6L&nbsp;9s]], [[15L&nbsp;6s]]
 |-
 | 4
@@ Line 958: / Line 957: @@
 |
 |
-| [[4L 4s]], [[4L 8s]], [[4L 12s]], [[16L 4s]]
+| [[4L&nbsp;4s]], [[4L&nbsp;8s]], [[4L&nbsp;12s]], [[16L&nbsp;4s]]
 |-
 | 4
@@ Line 965: / Line 964: @@
 |
 |
-| [[4L 4s]], [[8L 4s]], [[8L 12s]], [[8L 20s]]
+| [[4L&nbsp;4s]], [[8L&nbsp;4s]], [[8L&nbsp;12s]], [[8L&nbsp;20s]]
 |-
 | 6
@@ Line 979: / Line 978: @@
 | 4/3<br>(36/35)
 | [[Niner]]
-| [[9L 9s]]
+| [[9L&nbsp;9s]]
 |-
 | 12
@@ Line 986: / Line 985: @@
 | 8/7<br>(64/63)
 | [[Catler]]
-| [[12L 12s]]
+| [[12L&nbsp;12s]]
 |-
 | 18