Hemipyth: Difference between revisions

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A '''hemipyth''' (or '''"hemipythagorean"''') interval is an [[interval]] in the <math>\sqrt{2}\,.\sqrt{3}</math> [[subgroup]] i.e. intervals that can be constructed by multiplying half-integer powers of 2 and 3.

Notable hemipyth intervals include the neutral third <math>\sqrt{\frac{3}{2}} = \frac{\sqrt{3}}{\sqrt{2}}</math>, semioctave <math>\sqrt{2}</math>, and the semifourth <math>\sqrt{\frac{4}{3}} = \frac{2}{\sqrt{3}}</math>.

Many temperaments naturally produce intervals that split ~3/2, ~2, or ~4/3 exactly in half and can thus be interpreted as neutral thirds, semioctaves, or semifourths within the temperament.

Many temperaments naturally produce intervals that split ~{{sfrac|3|2}}, ~2, or ~{{sfrac|4|3}} exactly in half and can thus be interpreted as neutral thirds, semioctaves, or semifourths within the temperament.

== Equal temperaments ==

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* Either the edo is even and it features at least <math>\sqrt{2}</math> (which is tuned "pure" when the octave is tuned pure).

* Or one of the following is true:

** The closest approximation to 3/2 spans an even number of edosteps (leading to an approximation to <math>\sqrt{\frac{3}{2}}</math>)

** The closest approximation to {{sfrac|3|2}} spans an even number of edosteps (leading to an approximation to <math>\sqrt{\frac{3}{2}}</math>)

** The closest approximation to 4/3 spans an even number of edosteps (leading to an approximation to <math>\sqrt{\frac{4}{3}}</math>)

** The closest approximation to {{sfrac|4|3}} spans an even number of edosteps (leading to an approximation to <math>\sqrt{\frac{4}{3}}</math>)

{| class="wikitable"

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== Notation ==

The Pythagorean (2.3) part of hemipyth can be notated using traditional notation where octaves represent multiples of 2/1, chain of fifths denotes multiples of 3/2, the sharp sign is equal to 2187/2048 etc.

The Pythagorean (2.3) part of hemipyth can be notated using traditional notation where octaves represent multiples of {{sfrac|2|1}}, chain of fifths denotes multiples of {{sfrac|3|2}}, the sharp sign is equal to {{sfrac|2187|2048}} etc.

A prototypical 5L 2s 5|1 (Ionian) scale would be spelled C, D, E, F, G, A, B, (C).

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=== Semioctaves ===

In traditional notation the octave spans 7 diasteps which means that it splits into two interordinal {{~~frac~~|3|1|2}} diasteps or two perfect 4.5ths ("four-and-a-halves") if we wish to remain backwards compatible with the 1-indexed traditional notation.

In traditional notation the octave spans 7 diasteps which means that it splits into two interordinal {{sfrac|3|1|2}} diasteps or two perfect 4.5ths ("four-and-a-halves") if we wish to remain backwards compatible with the 1-indexed traditional notation.

Intervals retain their quality when the frequency ratio is multiplied by the perfect semioctave <math>\sqrt{2}</math>, e.g. {{nowrap|M6 − P4.5 {{=}} M2.5 {{=}} ({{frac|9|8}})<sup>3/2</sup>}}.

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=== Semifourths ===

Luckily we don't need to introduce any more generalizations to the notation to indicate <math>\sqrt{\frac{4}{3}}</math>. It's a neutral {{~~frac~~|2|1|2}} or a α{{demiflat2}} (alp semiflat) w.r.t middle C.

Luckily we don't need to introduce any more generalizations to the notation to indicate <math>\sqrt{\frac{4}{3}}</math>. It's a neutral {{sfrac|2|1|2}} or a α{{demiflat2}} (alp semiflat) w.r.t middle C.

Nicknames are still assigned to make it easier to talk about the [[5L 4s]] scale generated by <math>~\sqrt{\frac{4}{3}}</math> against the octave.

Nicknames are still assigned to make it easier to talk about the [[5L 4s]] scale generated by <math>\sim\vsp\sqrt{\frac{4}{3}}</math> against the octave.

{| class="wikitable"

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The neutral third receives only half of the tuning damage of the fifth so it has a strong character even if the fifth isn't tuned very pure. The irrational nature of <math>\sqrt{\frac{3}{2}}</math> also makes it more tolerant of imprecise tuning.

The same goes for the semifourth. A poorly tuned ~4/3 still results in a decent <math>~\sqrt{\frac{4}{3}}</math> (assuming it's featured in the tuning in the first place).

The same goes for the semifourth. A poorly tuned ~{{sfrac|4|3}} still results in a decent <math>\sim\vsp\sqrt{\frac{4}{3}}</math> (assuming it's featured in the tuning in the first place).

=== Signposts ===

Due to their low damage in supporting temperaments, the octave ({{~~frac~~|2|1}}), semioctave <math>\left(\sqrt{2}\right)</math>, perfect fifth ({{~~frac~~|3|2}}), perfect fourth ({{~~frac~~|4|3}}), neutral third <math>\left(\sqrt{\frac{3}{2}}\right)</math>, neutral sixth <math>\left(\sqrt{\frac{8}{3}}\right)</math>, semifourth <math>\left(\sqrt{\frac{4}{3}}\right)</math>, semitwelfth <math>\left(\sqrt{3}\right)</math>, "hemitone" <math>\left(\sqrt{\frac{9}{8}}\right)</math>, and "contrahemitone" <math>\left(\sqrt{\frac{32}{9}}\right)</math> all provide good signposts for navigating around otherwise unfamiliar scales.

Due to their low damage in supporting temperaments, the octave ({{sfrac|2|1}}), semioctave <math>\left(\sqrt{2}\right)</math>, perfect fifth ({{sfrac|3|2}}), perfect fourth ({{sfrac|4|3}}), neutral third <math>\left(\sqrt{\frac{3}{2}}\right)</math>, neutral sixth <math>\left(\sqrt{\frac{8}{3}}\right)</math>, semifourth <math>\left(\sqrt{\frac{4}{3}}\right)</math>, semitwelfth <math>\left(\sqrt{3}\right)</math>, "hemitone" <math>\left(\sqrt{\frac{9}{8}}\right)</math>, and "contrahemitone" <math>\left(\sqrt{\frac{32}{9}}\right)</math> all provide good signposts for navigating around otherwise unfamiliar scales.

While untempered semitones usually come as unequal pairs consisting of an augmented unison and a minor second, the "hemitone" is always exactly the geometric half of a 9/8 whole tone. The "contrahemitone" is its octave-complement.

While untempered semitones usually come as unequal pairs consisting of an augmented unison and a minor second, the "hemitone" is always exactly the geometric half of a {{sfrac|9|8}} whole tone. The "contrahemitone" is its octave-complement.

== Temperament interpretations ==

Under [[ploidacot]] classification diploid temperaments feature <math>~\sqrt{2}</math>, dicot temperaments have <math>~\sqrt{\frac{3}{2}}</math> and alpha-dicot temperaments feature <math>~\sqrt{\frac{4}{3}}</math> (by virtue of having a <math>~\sqrt{3}</math>).

Under [[ploidacot]] classification diploid temperaments feature <math>\sim\vsp\sqrt{2}</math>, dicot temperaments have <math>\sim\vsp\sqrt{\frac{3}{2}}</math> and alpha-dicot temperaments feature <math>\sim\vsp\sqrt{\frac{4}{3}}</math> (by virtue of having a <math>\sim\vsp\sqrt{3}</math>).

Full hemipyth support is indicated by at least "diploid dicot". Examples include:

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|+ style="font-size: 105%;" | Higher-prime interpretations of hemipyth intervals

|-

! Temperament !! <math>~\sqrt{2}</math> !! <math>~\sqrt{\frac{3}{2}}</math> !! <math>~\sqrt{\frac{4}{3}}</math> !! contorted !! rank-2

! Temperament !! <math>\sim\vsp\sqrt{2}</math> !! <math>\sim\vsp\sqrt{\frac{3}{2}}</math> !! <math>\sim\vsp\sqrt{\frac{4}{3}}</math> !! contorted !! rank-2

|-

| [[decimal]] || ~7/5 || ~5/4 || ~7/6 || no || yes

| [[decimal]] || ~{{sfrac|7|5}} || ~{{sfrac|5|4}} || ~{{sfrac|7|6}} || no || yes

|-

| [[anguirus]] || ~45/32 || ~56/45 || ~7/6 || no || yes

| [[anguirus]] || ~{{sfrac|45|32}} || ~{{sfrac|56|45}} || ~{{sfrac|7|6}} || no || yes

|-

| [[sruti]] || ~45/32 || ~175/144 || ~81/70 || no || yes

| [[sruti]] || ~{{sfrac|45|32}} || ~{{sfrac|175|144}} || ~{{sfrac|81|70}} || no || yes

|-

| [[Subgroup temperaments#Pakkanian hemipyth|pakkanian hemipyth]] || ~17/12 || ~11/9 || ~15/13 || no || yes

| [[Subgroup temperaments#Pakkanian hemipyth|pakkanian hemipyth]] || ~{{sfrac|17|12}} || ~{{sfrac|11|9}} || ~{{sfrac|15|13}} || no || yes

|-

| [[harry]] || ~17/12 || ~11/9 || ~15/13 || yes || yes

| [[harry]] || ~{{sfrac|17|12}} || ~{{sfrac|11|9}} || ~{{sfrac|15|13}} || yes || yes

|-

| [[semimiracle]] || ~91/64 || ~11/9 || ~15/13 || yes || yes

| [[semimiracle]] || ~{{sfrac|91|64}} || ~{{sfrac|11|9}} || ~{{sfrac|15|13}} || yes || yes

|-

| [[hemidim]] || ~36/25 || ~25/21 || ~7/6 || yes || yes

| [[hemidim]] || ~{{sfrac|36|25}} || ~{{sfrac|25|21}} || ~{{sfrac|7|6}} || yes || yes

|-

| [[greenland]] || ~99/70 || ~49/40 || ~15/13~231/200 || no || no

| [[greenland]] || ~{{sfrac|99|70}} || ~{{sfrac|49|40}} || ~{{sfrac|15|13}}~{{sfrac|231|200}} || no || no

|-

| [[semisema]] || ~108/77 || ~11/9 || ~7/6 || no || yes

| [[semisema]] || ~{{sfrac|108|77}} || ~{{sfrac|11|9}} || ~{{sfrac|7|6}} || no || yes

|-

| [[quadritikleismic]] || ~625/441 || ~49/40 || ~125/108 || yes || yes

| [[quadritikleismic]] || ~{{sfrac|625|441}} || ~{{sfrac|49|40}} || ~{{sfrac|125|108}} || yes || yes

|-

| [[decoid]] || ~99/70 || ~49/40 || ~4725/4096 || yes || yes

| [[decoid]] || ~{{sfrac|99|70}} || ~{{sfrac|49|40}} || ~{{sfrac|4725|4096}} || yes || yes

|}

Above contorted tunings don't have a <math>~\sqrt{2}</math> period with a <math>~\sqrt{3}</math> generator, but introduce further splits. Higher than rank-2 temperaments introduce further structure that goes beyond basic hemipyth.

Above contorted tunings don't have a <math>\sim\vsp\sqrt{2}</math> period with a <math>\sim\vsp\sqrt{3}</math> generator, but introduce further splits. Higher than rank-2 temperaments introduce further structure that goes beyond basic hemipyth.

Some possible interpretations for <math>~\sqrt{2}</math> are:

Some possible interpretations for <math>\sim\vsp\sqrt{2}</math> are:

{| class="wikitable"

|+ style="font-size: 105%;" | Higher-prime interpretations of <math>\sqrt{2}</math>

|-

! Temperament !! <math>~\sqrt{2}</math> !! contorted !! rank-2

! Temperament !! <math>\sim\vsp\sqrt{2}</math> !! contorted !! rank-2

|-

| [[jubilic]] || ~7/5 || no || yes (2.5.7)

| [[jubilic]] || ~{{sfrac|7|5}} || no || yes (2.5.7)

|-

| [[diaschismic]] || ~45/32 || no || yes (2.3.5)

| [[diaschismic]] || ~{{sfrac|45|32}} || no || yes (2.3.5)

|-

| [[semitonic]] || ~17/12 || no || yes (2.3.17)

| [[semitonic]] || ~{{sfrac|17|12}} || no || yes (2.3.17)

|-

| [[kalismic temperaments|kalismic]] || ~99/70 || no || no

| [[kalismic temperaments|kalismic]] || ~{{sfrac|99|70}} || no || no

|}

Some possible interpretations for <math>~\sqrt{3}</math> are:

Some possible interpretations for <math>\sim\vsp\sqrt{3}</math> are:

{| class="wikitable"

|+ style="font-size: 105%;" | Higher-prime interpretations of <math>\sqrt{3}</math>

|-

! Temperament !! <math>~\sqrt{3}</math> !! contorted !! rank-2

! Temperament !! <math>\sim\vsp\sqrt{3}</math> !! contorted !! rank-2

|-

| [[semaphore]] || ~7/4 || no || yes (2.3.7)

| [[semaphore]] || ~{{sfrac|7|4}} || no || yes (2.3.7)

|-

| [[barbados]] || ~26/15 || no || yes (2.3.13/5)

| [[barbados]] || ~{{sfrac|26|15}} || no || yes (2.3.{{sfrac|13|5}})

|}

Some possible interpretations for ~~~√(~~3/2) are:

Some possible interpretations for <math>\sim\vsp\frac{3}{2}</math> are:

{| class="wikitable"

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! Temperament !! <math>\sqrt{~\frac{3}{2}}</math> !! contorted !! rank-2

|-

| [[dicot]] || ~5/4 || no || yes (2.3.5)

| [[dicot]] || ~{{sfrac|5|4}} || no || yes (2.3.5)

|-

| [[Rastmic clan#Neutral|neutral]] || ~11/9 || no || yes (2.3.11)

| [[Rastmic clan#Neutral|neutral]] || ~{{sfrac|11|9}} || no || yes (2.3.11)

|-

| [[jove]] || ~11/9~49/40 || no || no

| [[jove]] || ~{{sfrac|11|9}}~{{sfrac|49|40}} || no || no

|}

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+{{texchars}}
 A '''hemipyth''' (or '''"hemipythagorean"''') interval is an [[interval]] in the <math>\sqrt{2}\,.\sqrt{3}</math> [[subgroup]] i.e. intervals that can be constructed by multiplying half-integer powers of 2 and 3.
 Notable hemipyth intervals include the neutral third <math>\sqrt{\frac{3}{2}} = \frac{\sqrt{3}}{\sqrt{2}}</math>, semioctave <math>\sqrt{2}</math>, and the semifourth <math>\sqrt{\frac{4}{3}} = \frac{2}{\sqrt{3}}</math>.
-Many temperaments naturally produce intervals that split ~3/2, ~2, or ~4/3 exactly in half and can thus be interpreted as neutral thirds, semioctaves, or semifourths within the temperament.
+Many temperaments naturally produce intervals that split ~{{sfrac|3|2}}, ~2, or ~{{sfrac|4|3}} exactly in half and can thus be interpreted as neutral thirds, semioctaves, or semifourths within the temperament.
 == Equal temperaments ==
@@ Line 9: / Line 10: @@
 * Either the edo is even and it features at least <math>\sqrt{2}</math> (which is tuned "pure" when the octave is tuned pure).
 * Or one of the following is true:
-** The closest approximation to 3/2 spans an even number of edosteps (leading to an approximation to <math>\sqrt{\frac{3}{2}}</math>)
+** The closest approximation to {{sfrac|3|2}} spans an even number of edosteps (leading to an approximation to <math>\sqrt{\frac{3}{2}}</math>)
-** The closest approximation to 4/3 spans an even number of edosteps (leading to an approximation to <math>\sqrt{\frac{4}{3}}</math>)
+** The closest approximation to {{sfrac|4|3}} spans an even number of edosteps (leading to an approximation to <math>\sqrt{\frac{4}{3}}</math>)
 {| class="wikitable"
@@ Line 77: / Line 78: @@
 == Notation ==
-The Pythagorean (2.3) part of hemipyth can be notated using traditional notation where octaves represent multiples of 2/1, chain of fifths denotes multiples of 3/2, the sharp sign is equal to 2187/2048 etc.
+The Pythagorean (2.3) part of hemipyth can be notated using traditional notation where octaves represent multiples of {{sfrac|2|1}}, chain of fifths denotes multiples of {{sfrac|3|2}}, the sharp sign is equal to {{sfrac|2187|2048}} etc.
 A prototypical 5L&nbsp;2s 5|1 (Ionian) scale would be spelled C, D, E, F, G, A, B, (C).
@@ Line 89: / Line 90: @@
 === Semioctaves ===
-In traditional notation the octave spans 7 diasteps which means that it splits into two interordinal {{frac|3|1|2}} diasteps or two perfect 4.5ths ("four-and-a-halves") if we wish to remain backwards compatible with the 1-indexed traditional notation.
+In traditional notation the octave spans 7 diasteps which means that it splits into two interordinal {{sfrac|3|1|2}} diasteps or two perfect 4.5ths ("four-and-a-halves") if we wish to remain backwards compatible with the 1-indexed traditional notation.
 Intervals retain their quality when the frequency ratio is multiplied by the perfect semioctave <math>\sqrt{2}</math>, e.g. {{nowrap|M6 &minus; P4.5 {{=}} M2.5 {{=}} ({{frac|9|8}})<sup>3/2</sup>}}.
@@ Line 124: / Line 125: @@
 === Semifourths ===
-Luckily we don't need to introduce any more generalizations to the notation to indicate <math>\sqrt{\frac{4}{3}}</math>. It's a neutral {{frac|2|1|2}} or a α{{demiflat2}} (alp semiflat) w.r.t middle C.
+Luckily we don't need to introduce any more generalizations to the notation to indicate <math>\sqrt{\frac{4}{3}}</math>. It's a neutral {{sfrac|2|1|2}} or a α{{demiflat2}} (alp semiflat) w.r.t middle C.
-Nicknames are still assigned to make it easier to talk about the [[5L&nbsp;4s]] scale generated by <math>~\sqrt{\frac{4}{3}}</math> against the octave.
+Nicknames are still assigned to make it easier to talk about the [[5L&nbsp;4s]] scale generated by <math>\sim\vsp\sqrt{\frac{4}{3}}</math> against the octave.
 {| class="wikitable"
@@ Line 178: / Line 179: @@
 The neutral third receives only half of the tuning damage of the fifth so it has a strong character even if the fifth isn't tuned very pure. The irrational nature of <math>\sqrt{\frac{3}{2}}</math> also makes it more tolerant of imprecise tuning.
-The same goes for the semifourth. A poorly tuned ~4/3 still results in a decent <math>~\sqrt{\frac{4}{3}}</math> (assuming it's featured in the tuning in the first place).
+The same goes for the semifourth. A poorly tuned ~{{sfrac|4|3}} still results in a decent <math>\sim\vsp\sqrt{\frac{4}{3}}</math> (assuming it's featured in the tuning in the first place).
 === Signposts ===
-Due to their low damage in supporting temperaments, the octave&nbsp;({{frac|2|1}}), semioctave&nbsp;<math>\left(\sqrt{2}\right)</math>, perfect&nbsp;fifth&nbsp;({{frac|3|2}}), perfect&nbsp;fourth&nbsp;({{frac|4|3}}), neutral&nbsp;third&nbsp;<math>\left(\sqrt{\frac{3}{2}}\right)</math>, neutral&nbsp;sixth&nbsp;<math>\left(\sqrt{\frac{8}{3}}\right)</math>, semifourth&nbsp;<math>\left(\sqrt{\frac{4}{3}}\right)</math>, semitwelfth&nbsp;<math>\left(\sqrt{3}\right)</math>, "hemitone"&nbsp;<math>\left(\sqrt{\frac{9}{8}}\right)</math>, and "contrahemitone"&nbsp;<math>\left(\sqrt{\frac{32}{9}}\right)</math> all provide good signposts for navigating around otherwise unfamiliar scales.
+Due to their low damage in supporting temperaments, the octave&nbsp;({{sfrac|2|1}}), semioctave&nbsp;<math>\left(\sqrt{2}\right)</math>, perfect&nbsp;fifth&nbsp;({{sfrac|3|2}}), perfect&nbsp;fourth&nbsp;({{sfrac|4|3}}), neutral&nbsp;third&nbsp;<math>\left(\sqrt{\frac{3}{2}}\right)</math>, neutral&nbsp;sixth&nbsp;<math>\left(\sqrt{\frac{8}{3}}\right)</math>, semifourth&nbsp;<math>\left(\sqrt{\frac{4}{3}}\right)</math>, semitwelfth&nbsp;<math>\left(\sqrt{3}\right)</math>, "hemitone"&nbsp;<math>\left(\sqrt{\frac{9}{8}}\right)</math>, and "contrahemitone"&nbsp;<math>\left(\sqrt{\frac{32}{9}}\right)</math> all provide good signposts for navigating around otherwise unfamiliar scales.
-While untempered semitones usually come as unequal pairs consisting of an augmented unison and a minor second, the "hemitone" is always exactly the geometric half of a 9/8 whole tone. The "contrahemitone" is its octave-complement.
+While untempered semitones usually come as unequal pairs consisting of an augmented unison and a minor second, the "hemitone" is always exactly the geometric half of a {{sfrac|9|8}} whole tone. The "contrahemitone" is its octave-complement.
 == Temperament interpretations ==
-Under [[ploidacot]] classification diploid temperaments feature <math>~\sqrt{2}</math>, dicot temperaments have <math>~\sqrt{\frac{3}{2}}</math> and alpha-dicot temperaments feature <math>~\sqrt{\frac{4}{3}}</math> (by virtue of having a <math>~\sqrt{3}</math>).
+Under [[ploidacot]] classification diploid temperaments feature <math>\sim\vsp\sqrt{2}</math>, dicot temperaments have <math>\sim\vsp\sqrt{\frac{3}{2}}</math> and alpha-dicot temperaments feature <math>\sim\vsp\sqrt{\frac{4}{3}}</math> (by virtue of having a <math>\sim\vsp\sqrt{3}</math>).
 Full hemipyth support is indicated by at least "diploid dicot". Examples include:
@@ Line 192: / Line 193: @@
 |+ style="font-size: 105%;" | Higher-prime interpretations of hemipyth intervals
 |-
-! Temperament !! <math>~\sqrt{2}</math> !! <math>~\sqrt{\frac{3}{2}}</math> !! <math>~\sqrt{\frac{4}{3}}</math> !! contorted !! rank-2
+! Temperament !! <math>\sim\vsp\sqrt{2}</math> !! <math>\sim\vsp\sqrt{\frac{3}{2}}</math> !! <math>\sim\vsp\sqrt{\frac{4}{3}}</math> !! contorted !! rank-2
 |-
-| [[decimal]] || ~7/5 || ~5/4 || ~7/6 || no || yes
+| [[decimal]] || ~{{sfrac|7|5}} || ~{{sfrac|5|4}} || ~{{sfrac|7|6}} || no || yes
 |-
-| [[anguirus]] || ~45/32 || ~56/45 || ~7/6 || no || yes
+| [[anguirus]] || ~{{sfrac|45|32}} || ~{{sfrac|56|45}} || ~{{sfrac|7|6}} || no || yes
 |-
-| [[sruti]] || ~45/32 || ~175/144 || ~81/70 || no || yes
+| [[sruti]] || ~{{sfrac|45|32}} || ~{{sfrac|175|144}} || ~{{sfrac|81|70}} || no || yes
 |-
-| [[Subgroup temperaments#Pakkanian hemipyth|pakkanian hemipyth]] || ~17/12 || ~11/9 || ~15/13 || no || yes
+| [[Subgroup temperaments#Pakkanian hemipyth|pakkanian hemipyth]] || ~{{sfrac|17|12}} || ~{{sfrac|11|9}} || ~{{sfrac|15|13}} || no || yes
 |-
-| [[harry]] || ~17/12 || ~11/9 || ~15/13 || yes || yes
+| [[harry]] || ~{{sfrac|17|12}} || ~{{sfrac|11|9}} || ~{{sfrac|15|13}} || yes || yes
 |-
-| [[semimiracle]] || ~91/64 || ~11/9 || ~15/13 || yes || yes
+| [[semimiracle]] || ~{{sfrac|91|64}} || ~{{sfrac|11|9}} || ~{{sfrac|15|13}} || yes || yes
 |-
-| [[hemidim]] || ~36/25 || ~25/21 || ~7/6 || yes || yes
+| [[hemidim]] || ~{{sfrac|36|25}} || ~{{sfrac|25|21}} || ~{{sfrac|7|6}} || yes || yes
 |-
-| [[greenland]] || ~99/70 || ~49/40 || ~15/13~231/200 || no || no
+| [[greenland]] || ~{{sfrac|99|70}} || ~{{sfrac|49|40}} || ~{{sfrac|15|13}}~{{sfrac|231|200}} || no || no
 |-
-| [[semisema]] || ~108/77 || ~11/9 || ~7/6 || no || yes
+| [[semisema]] || ~{{sfrac|108|77}} || ~{{sfrac|11|9}} || ~{{sfrac|7|6}} || no || yes
 |-
-| [[quadritikleismic]] || ~625/441 || ~49/40 || ~125/108 || yes || yes
+| [[quadritikleismic]] || ~{{sfrac|625|441}} || ~{{sfrac|49|40}} || ~{{sfrac|125|108}} || yes || yes
 |-
-| [[decoid]] || ~99/70 || ~49/40 || ~4725/4096 || yes || yes
+| [[decoid]] || ~{{sfrac|99|70}} || ~{{sfrac|49|40}} || ~{{sfrac|4725|4096}} || yes || yes
 |}
-Above contorted tunings don't have a <math>~\sqrt{2}</math> period with a <math>~\sqrt{3}</math> generator, but introduce further splits. Higher than rank-2 temperaments introduce further structure that goes beyond basic hemipyth.
+Above contorted tunings don't have a <math>\sim\vsp\sqrt{2}</math> period with a <math>\sim\vsp\sqrt{3}</math> generator, but introduce further splits. Higher than rank-2 temperaments introduce further structure that goes beyond basic hemipyth.
-Some possible interpretations for <math>~\sqrt{2}</math> are:
+Some possible interpretations for <math>\sim\vsp\sqrt{2}</math> are:
 {| class="wikitable"
 |+ style="font-size: 105%;" | Higher-prime interpretations of <math>\sqrt{2}</math>
 |-
-! Temperament !! <math>~\sqrt{2}</math> !! contorted !! rank-2
+! Temperament !! <math>\sim\vsp\sqrt{2}</math> !! contorted !! rank-2
 |-
-| [[jubilic]] || ~7/5 || no || yes (2.5.7)
+| [[jubilic]] || ~{{sfrac|7|5}} || no || yes (2.5.7)
 |-
-| [[diaschismic]] || ~45/32 || no || yes (2.3.5)
+| [[diaschismic]] || ~{{sfrac|45|32}} || no || yes (2.3.5)
 |-
-| [[semitonic]] || ~17/12 || no || yes (2.3.17)
+| [[semitonic]] || ~{{sfrac|17|12}} || no || yes (2.3.17)
 |-
-| [[kalismic temperaments|kalismic]] || ~99/70 || no || no
+| [[kalismic temperaments|kalismic]] || ~{{sfrac|99|70}} || no || no
 |}
-Some possible interpretations for <math>~\sqrt{3}</math> are:
+Some possible interpretations for <math>\sim\vsp\sqrt{3}</math> are:
 {| class="wikitable"
 |+ style="font-size: 105%;" | Higher-prime interpretations of <math>\sqrt{3}</math>
 |-
-! Temperament !! <math>~\sqrt{3}</math> !! contorted !! rank-2
+! Temperament !! <math>\sim\vsp\sqrt{3}</math> !! contorted !! rank-2
 |-
-| [[semaphore]] || ~7/4 || no || yes (2.3.7)
+| [[semaphore]] || ~{{sfrac|7|4}} || no || yes (2.3.7)
 |-
-| [[barbados]] || ~26/15 || no || yes (2.3.13/5)
+| [[barbados]] || ~{{sfrac|26|15}} || no || yes (2.3.{{sfrac|13|5}})
 |}
-Some possible interpretations for ~√(3/2) are:
+Some possible interpretations for <math>\sim\vsp\frac{3}{2}</math> are:
 {| class="wikitable"
@@ Line 254: / Line 255: @@
 ! Temperament !! <math>\sqrt{~\frac{3}{2}}</math> !! contorted !! rank-2
 |-
-| [[dicot]] || ~5/4 || no || yes (2.3.5)
+| [[dicot]] || ~{{sfrac|5|4}} || no || yes (2.3.5)
 |-
-| [[Rastmic clan#Neutral|neutral]] || ~11/9 || no || yes (2.3.11)
+| [[Rastmic clan#Neutral|neutral]] || ~{{sfrac|11|9}} || no || yes (2.3.11)
 |-
-| [[jove]] || ~11/9~49/40 || no || no
+| [[jove]] || ~{{sfrac|11|9}}~{{sfrac|49|40}} || no || no
 |}