34edo: Difference between revisions

Line 21:

! Cents

! Approx. ratios<ref group="note">{{sg|limit=2.3.5.11.13.17.23 [[subgroup]]}}</ref>

! Ratios of 7 Using the 34 Val

! Ratios of 7 Using the 34 Val

! Ratios of 7 Using the 34d Val

! Ratios of 7 Using the 34d Val

! colspan="3" | [[Ups and Downs Notation]]

! colspan="2" | [[Solfege|Solfeges]]

Line 65:

| [[15/14]], [[21/20]]

| vA1, ^m2

| downaug 1sn, upminor 2nd

| downaug 1sn, upminor 2nd

| vD#, ^Eb

| fru

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|

| ^M2, vm3

| upmajor 2nd, downminor 3rd

| upmajor 2nd, downminor 3rd

| ^E, vF

| ru/no

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|

| ^M6, vm7

| upmajor 6th, downminor 7th

| upmajor 6th, downminor 7th

| ^B, vC

| lu/tho

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== Notation ==

=== Ups and downs notation ===

34edo can be notated with [[ups and downs]], spoken as up, dup, downsharp, sharp, upsharp etc. and down, dud, upflat etc. Note that dup is equivalent to dudsharp and dud is equivalent to dupflat.

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Like [[17edo]], 34edo contains good approximations of just intervals involving 3, 11, and 13 – specifically, 13/8, 13/12, 13/11, 13/9, 11/9 and their inversions – while failing to closely approximate ratios of 7 given its step size. 34edo adds ratios of 5 into the mix – including 5/4, 6/5, 9/5, 15/8, 13/10, 15/13, and their inversions – as well as 17 – including 17/16, 18/17, 17/12, 17/11, 17/10, 17/13, 17/15 and their inversions. Since it distinguishes between 9/8 and 10/9 (exaggerating the difference between them, the [[syntonic comma]] of 81/80, from 21.5 cents to 35.3 cents), it is suitable for quasi-5-limit JI but is not a [[meantone]] system. While no number of fifths (3/2) land on major or minor thirds, an even number of major or minor thirds will be the same pitch as a pitch somewhere in the circle of seventeen fifths.

The sharpening of ~13 ~~cents~~ of 11/8 can fit with the 9/8 and 13/8 which both are about 7 cents sharp. This is the basis of a subtle trick: the guitarist tunes the high 'E' string flat by several cents, enough to be imperceptible in many contexts, but which makes chords/harmonies against those several intervals tuned more justly.

The sharpening of ~13{{c}} of 11/8 can fit with the 9/8 and 13/8 which both are about 7 cents sharp. This is the basis of a subtle trick: the guitarist tunes the high 'E' string flat by several cents, enough to be imperceptible in many contexts, but which makes chords/harmonies against those several intervals tuned more justly.

Likewise the 16~~-cent~~ flat 27\34 approximate 7/4 can be musically useful especially in [[kleismic]] or [[4L 3s]] contexts (with generator a 9\34 minor third). On the other hand, the slightly worse and sharper 7/4, 28\34, sounds more like the "dominant seventh" found in blues and jazz – which some listeners are accustomed to. ([[68edo]] contains a copy of 34edo and has the intervals 7/4 and 11/8 tuned nearly just.)

Likewise the 16{{c}} flat 27\34 approximate 7/4 can be musically useful especially in [[kleismic]] or [[4L 3s]] contexts (with generator a 9\34 minor third). On the other hand, the slightly worse and sharper 7/4, 28\34, sounds more like the "dominant seventh" found in blues and jazz – which some listeners are accustomed to. ([[68edo]] contains a copy of 34edo and has the intervals 7/4 and 11/8 tuned nearly just.)

=== Interval mappings ===

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== Tuning by ear ==

In principle, one can approximate 34edo by ear using only 5-limit intervals, using the fact that 17edo is very close to a circle of seventeen [[25/24]] chromatic semitones to within 1.5 ~~cents~~, and using a pure 5/4 which is less than 2 ~~cents~~ off for the second chain. The overall tuning error, assuming everything is tuned perfectly, will be less than 3.5 ~~cents~~, or a relative error of less than 10%.

In principle, one can approximate 34edo by ear using only 5-limit intervals, using the fact that 17edo is very close to a circle of seventeen [[25/24]] chromatic semitones to within 1.5{{c}}, and using a pure 5/4 which is less than 2{{c}} off for the second chain. The overall tuning error, assuming everything is tuned perfectly, will be less than 3.5{{c}}, or a relative error of less than 10%.

== Approximation to irrational intervals ==

As a Fibonacci number, 34edo contains a fraction of an octave which is a close approximation to the [[logarithmic phi]] – 21 degrees of 34edo, approximately 741.2 ~~cents~~. Repeated iterations of this interval generates [[moment of symmetry]] scales with near-phi relationships between the step sizes. As a 2.3.5.13 temperament, the 21\34 generator is an approximate 20/13, and the temperament tempers out 512/507 and {{monzo| -6 2 6 0 0 -13 }}. From the tempering of 512/507, two 16/13 neutral thirds are an approximate 3/2, defining an essentially tempered neutral triad with a sharp rather than a flat fifth. (On the other hand, the frequency ratio phi is ~ 833 ~~cents~~, and the equal divisions of octave approximating this interval closely are 13edo and [[36edo]].)

As a Fibonacci number, 34edo contains a fraction of an octave which is a close approximation to the [[logarithmic phi]] – 21 degrees of 34edo, approximately 741.2{{c}}. Repeated iterations of this interval generates [[moment of symmetry]] scales with near-phi relationships between the step sizes. As a 2.3.5.13 temperament, the 21\34 generator is an approximate 20/13, and the temperament tempers out 512/507 and {{monzo| -6 2 6 0 0 -13 }}. From the tempering of 512/507, two 16/13 neutral thirds are an approximate 3/2, defining an essentially tempered neutral triad with a sharp rather than a flat fifth. (On the other hand, the frequency ratio phi is ~ 833{{c}}, and the equal divisions of octave approximating this interval closely are 13edo and [[36edo]].)

=== Counterpoint ===

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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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|}

In the 5-limit, 34edo [[support]]s [[hanson]], [[srutal]], [[tetracot]], [[würschmidt]] and [[vishnu]] temperaments. It does less well in the [[7-limit]], with two mappings possible for [[7/4]]: a flat one from the [[patent val]], and a sharp one from the 34d val. By way of the patent val 34 supports [[keemun]] temperament, and 34d is an excellent alternative to [[22edo]] for 7-limit [[pajara]] temperament. In the [[11-limit]], 34de supports 11-limit [[pajaric]], and in fact is quite close to the [[POTE tuning]]; it adds [[4375/4374]] to the commas of 11-limit pajaric. On the other hand, the 34d val supports pajara, vishnu and würschmidt, adding 4375/4374 to the commas of pajara. Among subgroup temperaments, the patent val supports [[semaphore]] on the 2.3.7 subgroup.

In the 5-limit, 34edo [[support]]s [[hanson]], [[srutal]], [[tetracot]], [[würschmidt]], and [[vishnu]] temperaments. It does less well in the [[7-limit]], with two mappings possible for [[7/4]]: a flat one from the [[patent val]], and a sharp one from the 34d val. By way of the patent val 34 supports [[keemun]] temperament, and 34d is an excellent alternative to [[22edo]] for 7-limit [[pajara]] temperament. In the [[11-limit]], 34de supports 11-limit [[pajaric]], and in fact is quite close to the [[POTE tuning]]; it adds [[4375/4374]] to the commas of 11-limit pajaric. On the other hand, the 34d val supports pajara, vishnu and würschmidt, adding 4375/4374 to the commas of pajara. Among subgroup temperaments, the patent val supports [[semaphore]] on the 2.3.7 subgroup.

=== Rank-2 temperaments ===

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{| class="wikitable"

|+ Rank-2 temperaments by period and generator

|+ style="font-size: 105%;" | Rank-2 temperaments by period and generator

|-

! Periods per 8ve

! Periods per 8ve

! Generator

! Cents

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| 3\34

| 105.88

| [[11L 1s]] [[11L 12s]]

| [[11L 1s]] [[11L 12s]]

|

|-

| 5\34

| 176.471

| [[6L 1s]] [[7L 6s]] [[7L 13s]] 7L 20s

| [[6L 1s]] [[7L 6s]] [[7L 13s]]<br />[[7L 20s]]

| [[Tetracot]], [[bunya]] (34d), [[modus]] (34d), [[monkey]] (34), [[wollemia]] (34)

|-

| 7\34

| 247.059

| [[5L 4s]] [[5L 9s]] [[5L 14s]] [[5L 19s]] Pathological 5L 24s

| [[5L 4s]] [[5L 9s]] [[5L 14s]] [[5L 19s]] Pathological 5L 24s

| [[Immunity]] (34), [[immunized]] (34d)

|-

| 9\34

| 317.647

| [[4L 3s]] [[4L 7s]] [[4L 11s]] [[15L 4s]]

| [[4L 3s]] [[4L 7s]] [[4L 11s]] [[15L 4s]]

| [[Hanson]], [[keemun]] (34), [[catalan]] (34d), [[catakleismic]] (34d)

|-

| 11\34

| 388.235

| [[3L 7s]] [[3L 10s]] [[3L 13s]] [[3L 16s]] [[3L 19s]] [[3L 22s]] Pathological [[3L 25s]] Pathological 3L 28s

| [[3L 7s]] [[3L 10s]] [[3L 13s]] [[3L 16s]] [[3L 19s]] [[3L 22s]] Pathological [[3L 25s]] Pathological 3L 28s

| [[Würschmidt]] (34d), [[worschmidt]] (34)

|-

| 13\34

| 458.824

| [[3L 2s]] [[5L 3s]] [[8L 5s]] [[13L 8s]]

| [[3L 2s]] [[5L 3s]] [[8L 5s]] [[13L 8s]]

| [[Petrtri]]

|-

| 15\34

| 529.412

| [[2L 3s]] [[2L 5s]] [[7L 2s]] [[9L 7s]] 9L 16s

| [[2L 3s]] [[2L 5s]] [[7L 2s]] [[9L 7s]] 9L 16s

| [[Mabila]]

|-

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| 2\34

| 70.588

| [[16L 2s]]

| [[16L 2s]]

| [[Vishnu]]

|-

| 3\34

| 105.882

| [[2L 6s]] [[2L 8s]] [[10L 2s]] [[12L 10s]]

| [[2L 6s]] [[2L 8s]] [[10L 2s]] [[12L 10s]]

| [[Srutal]] (34d), [[pajara]] (34d), [[diaschismic]] (34)

|-

| 4\34

| 141.176

| [[2L 6s]] [[8L 2s]] [[8L 10s]] 8L 16s

| [[2L 6s]] [[8L 2s]] [[8L 10s]] 8L 16s

| [[Fifive]], [[crepuscular]] (34d), [[fifives]] (34)

|-

| 5\34

| 176.471

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

| [[6L 2s]] [[6L 8s]] [[14L 6s]]

| [[Stratosphere]]

|-

| 6\34

| 211.765

| [[4L 2s]] [[6L 4s]] [[6L 10s]] [[6L 16s]] Pathological 6L 22s

| [[4L 2s]] [[6L 4s]] [[6L 10s]] [[6L 16s]] Pathological 6L 22s

| [[Antikythera]]

|-

| 7\34

| 247.059

| [[4L 2s]] [[4L 6s]] [[10L 4s]] [[10L 14s]]

| [[4L 2s]] [[4L 6s]] [[10L 4s]] [[10L 14s]]

| [[Tobago]]

|-

| 8\34

| 282.353

| [[2L 2s]] [[4L 2s]] [[4L 6s]] [[4L 10s]] [[4L 14s]] 4L 18s 4L 22s Pathological 4L 26s

| [[2L 2s]] [[4L 2s]] [[4L 6s]] [[4L 10s]] [[4L 14s]] 4L 18s 4L 22s Pathological 4L 26s

| [[Bikleismic]]

|}

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{| class="commatable wikitable center-all left-3 right-4 left-6"

|-

! [[Harmonic limit|Prime limit]]

! [[Harmonic limit|Prime limit]]

! [[Ratio]]<ref group="note">{{rd}}</ref>

! [[Monzo]]

@@ Line 21: / Line 21: @@
 ! Cents
 ! Approx. ratios<ref group="note">{{sg|limit=2.3.5.11.13.17.23 [[subgroup]]}}</ref>
-! Ratios of 7 <br />Using the 34 Val
+! Ratios of 7<br />Using the 34 Val
-! Ratios of 7 <br />Using the 34d Val
+! Ratios of 7<br />Using the 34d Val
 ! colspan="3" | [[Ups and Downs Notation]]
 ! colspan="2" | [[Solfege|Solfeges]]
@@ Line 65: / Line 65: @@
 | [[15/14]], [[21/20]]
 | vA1, ^m2
-| downaug 1sn, <br />upminor 2nd
+| downaug 1sn,<br />upminor 2nd
 | vD#, ^Eb
 | fru
@@ Line 109: / Line 109: @@
 |
 | ^M2, vm3
-| upmajor 2nd, <br />downminor 3rd
+| upmajor 2nd,<br />downminor 3rd
 | ^E, vF
 | ru/no
@@ Line 329: / Line 329: @@
 |
 | ^M6, vm7
-| upmajor 6th, <br />downminor 7th
+| upmajor 6th,<br />downminor 7th
 | ^B, vC
 | lu/tho
@@ Line 415: / Line 415: @@
 == Notation ==
 === Ups and downs notation ===
 edo can be notated with [[ups and downs]], spoken as up, dup, downsharp, sharp, upsharp etc. and down, dud, upflat etc. Note that dup is equivalent to dudsharp and dud is equivalent to dupflat.
@@ Line 466: / Line 465: @@
 Like [[17edo]], 34edo contains good approximations of just intervals involving 3, 11, and 13 – specifically, 13/8, 13/12, 13/11, 13/9, 11/9 and their inversions – while failing to closely approximate ratios of 7 given its step size. 34edo adds ratios of 5 into the mix – including 5/4, 6/5, 9/5, 15/8, 13/10, 15/13, and their inversions – as well as 17 – including 17/16, 18/17, 17/12, 17/11, 17/10, 17/13, 17/15 and their inversions. Since it distinguishes between 9/8 and 10/9 (exaggerating the difference between them, the [[syntonic comma]] of 81/80, from 21.5 cents to 35.3 cents), it is suitable for quasi-5-limit JI but is not a [[meantone]] system. While no number of fifths (3/2) land on major or minor thirds, an even number of major or minor thirds will be the same pitch as a pitch somewhere in the circle of seventeen fifths.
-The sharpening of ~13 cents of 11/8 can fit with the 9/8 and 13/8 which both are about 7 cents sharp. This is the basis of a subtle trick: the guitarist tunes the high 'E' string flat by several cents, enough to be imperceptible in many contexts, but which makes chords/harmonies against those several intervals tuned more justly.
+The sharpening of ~13{{c}} of 11/8 can fit with the 9/8 and 13/8 which both are about 7 cents sharp. This is the basis of a subtle trick: the guitarist tunes the high 'E' string flat by several cents, enough to be imperceptible in many contexts, but which makes chords/harmonies against those several intervals tuned more justly.
-Likewise the 16-cent flat 27\34 approximate 7/4 can be musically useful especially in [[kleismic]] or [[4L 3s]] contexts (with generator a 9\34 minor third). On the other hand, the slightly worse and sharper 7/4, 28\34, sounds more like the "dominant seventh" found in blues and jazz – which some listeners are accustomed to. ([[68edo]] contains a copy of 34edo and has the intervals 7/4 and 11/8 tuned nearly just.)
+Likewise the 16{{c}} flat 27\34 approximate 7/4 can be musically useful especially in [[kleismic]] or [[4L&nbsp;3s]] contexts (with generator a 9\34 minor third). On the other hand, the slightly worse and sharper 7/4, 28\34, sounds more like the "dominant seventh" found in blues and jazz – which some listeners are accustomed to. ([[68edo]] contains a copy of 34edo and has the intervals 7/4 and 11/8 tuned nearly just.)
 === Interval mappings ===
@@ Line 489: / Line 488: @@
 == Tuning by ear ==
-In principle, one can approximate 34edo by ear using only 5-limit intervals, using the fact that 17edo is very close to a circle of seventeen [[25/24]] chromatic semitones to within 1.5 cents, and using a pure 5/4 which is less than 2 cents off for the second chain. The overall tuning error, assuming everything is tuned perfectly, will be less than 3.5 cents, or a relative error of less than 10%.
+In principle, one can approximate 34edo by ear using only 5-limit intervals, using the fact that 17edo is very close to a circle of seventeen [[25/24]] chromatic semitones to within 1.5{{c}}, and using a pure 5/4 which is less than 2{{c}} off for the second chain. The overall tuning error, assuming everything is tuned perfectly, will be less than 3.5{{c}}, or a relative error of less than 10%.
 == Approximation to irrational intervals ==
-As a Fibonacci number, 34edo contains a fraction of an octave which is a close approximation to the [[logarithmic phi]] – 21 degrees of 34edo, approximately 741.2 cents. Repeated iterations of this interval generates [[moment of symmetry]] scales with near-phi relationships between the step sizes. As a 2.3.5.13 temperament, the 21\34 generator is an approximate 20/13, and the temperament tempers out 512/507 and {{monzo| -6 2 6 0 0 -13 }}. From the tempering of 512/507, two 16/13 neutral thirds are an approximate 3/2, defining an essentially tempered neutral triad with a sharp rather than a flat fifth. (On the other hand, the frequency ratio phi is ~ 833 cents, and the equal divisions of octave approximating this interval closely are 13edo and [[36edo]].)
+As a Fibonacci number, 34edo contains a fraction of an octave which is a close approximation to the [[logarithmic phi]] – 21 degrees of 34edo, approximately 741.2{{c}}. Repeated iterations of this interval generates [[moment of symmetry]] scales with near-phi relationships between the step sizes. As a 2.3.5.13 temperament, the 21\34 generator is an approximate 20/13, and the temperament tempers out 512/507 and {{monzo| -6 2 6 0 0 -13 }}. From the tempering of 512/507, two 16/13 neutral thirds are an approximate 3/2, defining an essentially tempered neutral triad with a sharp rather than a flat fifth. (On the other hand, the frequency ratio phi is ~ 833{{c}}, and the equal divisions of octave approximating this interval closely are 13edo and [[36edo]].)
 === Counterpoint ===
@@ Line 503: / Line 502: @@
 ! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal <br />8ve stretch (¢)
+! rowspan="2" | Optimal<br />8ve stretch (¢)
 ! colspan="2" | Tuning error
 |-
@@ Line 545: / Line 544: @@
 |}
-In the 5-limit, 34edo [[support]]s [[hanson]], [[srutal]], [[tetracot]], [[würschmidt]] and [[vishnu]] temperaments. It does less well in the [[7-limit]], with two mappings possible for [[7/4]]: a flat one from the [[patent val]], and a sharp one from the 34d val. By way of the patent val 34 supports [[keemun]] temperament, and 34d is an excellent alternative to [[22edo]] for 7-limit [[pajara]] temperament. In the [[11-limit]], 34de supports 11-limit [[pajaric]], and in fact is quite close to the [[POTE tuning]]; it adds [[4375/4374]] to the commas of 11-limit pajaric. On the other hand, the 34d val supports pajara, vishnu and würschmidt, adding 4375/4374 to the commas of pajara. Among subgroup temperaments, the patent val supports [[semaphore]] on the 2.3.7 subgroup.
+In the 5-limit, 34edo [[support]]s [[hanson]], [[srutal]], [[tetracot]], [[würschmidt]], and [[vishnu]] temperaments. It does less well in the [[7-limit]], with two mappings possible for [[7/4]]: a flat one from the [[patent val]], and a sharp one from the 34d val. By way of the patent val 34 supports [[keemun]] temperament, and 34d is an excellent alternative to [[22edo]] for 7-limit [[pajara]] temperament. In the [[11-limit]], 34de supports 11-limit [[pajaric]], and in fact is quite close to the [[POTE tuning]]; it adds [[4375/4374]] to the commas of 11-limit pajaric. On the other hand, the 34d val supports pajara, vishnu and würschmidt, adding 4375/4374 to the commas of pajara. Among subgroup temperaments, the patent val supports [[semaphore]] on the 2.3.7 subgroup.
 === Rank-2 temperaments ===
@@ Line 552: / Line 551: @@
 {| class="wikitable"
-|+ Rank-2 temperaments by period and generator
+|+ style="font-size: 105%;" | Rank-2 temperaments by period and generator
 |-
-! Periods <br />per 8ve
+! Periods<br />per 8ve
 ! Generator
 ! Cents
@@ Line 568: / Line 567: @@
 | 3\34
 | 105.88
-| [[11L 1s]] <br />[[11L 12s]]
+| [[11L&nbsp;1s]]<br />[[11L&nbsp;12s]]
 |
 |-
 | 5\34
 | 176.471
-| [[6L 1s]] <br />[[7L 6s]] <br />[[7L 13s]] <br />7L 20s
+| [[6L&nbsp;1s]]<br />[[7L&nbsp;6s]]<br />[[7L&nbsp;13s]]<br&nbsp;/>[[7L&nbsp;20s]]
 | [[Tetracot]], [[bunya]] (34d), [[modus]] (34d), [[monkey]] (34), [[wollemia]] (34)
 |-
 | 7\34
 | 247.059
-| [[5L 4s]] <br />[[5L 9s]] <br />[[5L 14s]] <br />[[5L 19s]] <br />Pathological 5L 24s
+| [[5L&nbsp;4s]]<br />[[5L&nbsp;9s]]<br />[[5L&nbsp;14s]]<br />[[5L&nbsp;19s]]<br />Pathological 5L&nbsp;24s
 | [[Immunity]] (34), [[immunized]] (34d)
 |-
 | 9\34
 | 317.647
-| [[4L 3s]] <br />[[4L 7s]] <br />[[4L 11s]] <br />[[15L 4s]]
+| [[4L&nbsp;3s]]<br />[[4L&nbsp;7s]]<br />[[4L&nbsp;11s]]<br />[[15L&nbsp;4s]]
 | [[Hanson]], [[keemun]] (34), [[catalan]] (34d), [[catakleismic]] (34d)
 |-
 | 11\34
 | 388.235
-| [[3L 7s]] <br />[[3L 10s]] <br />[[3L 13s]] <br />[[3L 16s]] <br />[[3L 19s]] <br />[[3L 22s]] <br />Pathological [[3L 25s]] <br />Pathological 3L 28s
+| [[3L&nbsp;7s]]<br />[[3L&nbsp;10s]]<br />[[3L&nbsp;13s]]<br />[[3L&nbsp;16s]]<br />[[3L&nbsp;19s]]<br />[[3L&nbsp;22s]]<br />Pathological [[3L&nbsp;25s]]<br />Pathological 3L&nbsp;28s
 | [[Würschmidt]] (34d), [[worschmidt]] (34)
 |-
 | 13\34
 | 458.824
-| [[3L 2s]] <br />[[5L 3s]] <br />[[8L 5s]] <br />[[13L 8s]]
+| [[3L&nbsp;2s]]<br />[[5L&nbsp;3s]]<br />[[8L&nbsp;5s]]<br />[[13L&nbsp;8s]]
 | [[Petrtri]]
 |-
 | 15\34
 | 529.412
-| [[2L 3s]] <br />[[2L 5s]] <br />[[7L 2s]] <br />[[9L 7s]] <br />9L 16s
+| [[2L&nbsp;3s]]<br />[[2L&nbsp;5s]]<br />[[7L&nbsp;2s]]<br />[[9L&nbsp;7s]]<br />9L&nbsp;16s
 | [[Mabila]]
 |-
@@ Line 604: / Line 603: @@
 | 2\34
 | 70.588
-| [[16L 2s]]
+| [[16L&nbsp;2s]]
 | [[Vishnu]]
 |-
 | 3\34
 | 105.882
-| [[2L 6s]] <br />[[2L 8s]] <br />[[10L 2s]] <br />[[12L 10s]]
+| [[2L&nbsp;6s]]<br />[[2L&nbsp;8s]]<br />[[10L&nbsp;2s]]<br />[[12L&nbsp;10s]]
 | [[Srutal]] (34d), [[pajara]] (34d), [[diaschismic]] (34)
 |-
 | 4\34
 | 141.176
-| [[2L 6s]] <br />[[8L 2s]] <br />[[8L 10s]] <br />8L 16s
+| [[2L&nbsp;6s]]<br />[[8L&nbsp;2s]]<br />[[8L&nbsp;10s]]<br />8L&nbsp;16s
 | [[Fifive]], [[crepuscular]] (34d), [[fifives]] (34)
 |-
 | 5\34
 | 176.471
-| [[6L 2s]] <br />[[6L 8s]] <br />[[14L 6s]]
+| [[6L&nbsp;2s]]<br />[[6L&nbsp;8s]]<br />[[14L&nbsp;6s]]
 | [[Stratosphere]]
 |-
 | 6\34
 | 211.765
-| [[4L 2s]] <br />[[6L 4s]] <br />[[6L 10s]] <br />[[6L 16s]] <br />Pathological 6L 22s
+| [[4L&nbsp;2s]]<br />[[6L&nbsp;4s]]<br />[[6L&nbsp;10s]]<br />[[6L&nbsp;16s]]<br />Pathological 6L&nbsp;22s
 | [[Antikythera]]
 |-
 | 7\34
 | 247.059
-| [[4L 2s]] <br />[[4L 6s]] <br />[[10L 4s]] <br />[[10L 14s]]
+| [[4L&nbsp;2s]]<br />[[4L&nbsp;6s]]<br />[[10L&nbsp;4s]]<br />[[10L&nbsp;14s]]
 | [[Tobago]]
 |-
 | 8\34
 | 282.353
-| [[2L 2s]] <br />[[4L 2s]] <br />[[4L 6s]] <br />[[4L 10s]] <br />[[4L 14s]] <br />4L 18s <br />4L 22s <br />Pathological 4L 26s
+| [[2L&nbsp;2s]]<br />[[4L&nbsp;2s]]<br />[[4L&nbsp;6s]]<br />[[4L&nbsp;10s]]<br />[[4L&nbsp;14s]]<br />4L&nbsp;18s<br />4L&nbsp;22s<br />Pathological 4L&nbsp;26s
 | [[Bikleismic]]
 |}
@@ Line 643: / Line 642: @@
 {| class="commatable wikitable center-all left-3 right-4 left-6"
 |-
-! [[Harmonic limit|Prime <br />limit]]
+! [[Harmonic limit|Prime<br />limit]]
 ! [[Ratio]]<ref group="note">{{rd}}</ref>
 ! [[Monzo]]