Hemipyth: Difference between revisions

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A '''hemipyth''' (or '''"hemipythagorean"''') interval is an [[interval]] in the √2.√3 [[subgroup]] i.e. intervals that can be constructed by multiplying half-integer powers of 2 and 3.

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 √(3/2) = √3/√2, semioctave √2 and the semifourth √(4/3) = 2/√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 ~3/2, ~2, or ~4/3 exactly in half and can thus be interpreted as neutral thirds, semioctaves, or semifourths within the temperament.

== Equal temperaments ==

An important property of edos > 1 is that they must by necessity include at least one of the notable hemipyth intervals:

An important property of edos > 1 is that they must by necessity include at least one of the notable hemipyth intervals:

* Either the edo is even and it features at least √2 (which is tuned "pure" when the octave is tuned pure).

* 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 √(3/2))

** 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 4/3 spans an even number of edosteps (leading to an approximation to √(4/3))

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

{| class="wikitable"

|+ style="font-size: 105%;" | List of edo mappings with full or partial hemipyth support

|-

! Edo (warts) !! Has √2 !! Has √(3/2) !! Has √(4/3)

! Edo (warts) !! Has <math>\sqrt{2}</math> !! Has <math>\sqrt{\frac{3}{2}}</math> !! Has <math>\sqrt{\frac{4}{3}}</math>

|-

| 2 || yes || no || no

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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 2/1, chain of fifths denotes multiples of 3/2, the sharp sign is equal to 2187/2048 etc.

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

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

Simple otonal chords can be plucked out of the harmonic segment 1:2:3:4:6:8:9:12:16:18:24:27:32:36:48:54:64:72:81:96:108:128:... e.g. 6:8:9 is a sus4 chord.

=== Neutral thirds ===

The 2.√(3/2) part can be notated using [[neutral chain-of-fifths notation]]. This introduces a neutral interval quality between major and minor, semisharps (a.k.a. demisharps) etc.

The <math>2\,.\sqrt{\frac{3}{2}}</math> part can be notated using [[neutral chain-of-fifths notation]]. This introduces a neutral interval quality between major and minor, semisharps (a.k.a. demisharps) etc.

A representative 3L 4s 4|2 (kleeth) scale would be spelled C, D, E{{demiflat2}}, F, G, A{{demiflat2}}, B{{demiflat2}}, (C).

A representative 3L 4s 4|2 (kleeth) scale would be spelled {{nowrap|C, D, E{{demiflat2}}, F, G, A{{demiflat2}}, B{{demiflat2}}, (C)}}

=== Semioctaves ===

In traditional notation the octave spans 7 diasteps which means that it splits into two interordinal 3½ diasteps or two perfect 4.5ths 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 {{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.

Intervals retain their quality when the frequency ratio is multiplied by the perfect semioctave √2 e.g. {{nowrap|M6 − P4.5 {{=}} M2.5 {{=}} (9/8)^(3/2)}}.

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>}}.

Relative interordinal intervals are either called by their double i.e. M2.5 is a major semifourth due to being exactly the half of an augmented fourth (Aug4), or by simply adding the suffix "-and-a-halfth" i.e. "major second-and-a-halfth". The semisecond gets the special nickname "sesquith".

The nominals for absolute pitches are denoted using lowercase Greek nominals (uppercase often looks identical to pre-existing Latin nominals). The logic being that Latin and Greek notes differ by a multiple of √2 when paired up alphabetically. The direction is determined by octaves starting from the middle C.

The nominals for absolute pitches are denoted using lowercase Greek nominals (uppercase often looks identical to pre-existing Latin nominals). The logic being that Latin and Greek notes differ by a multiple of <math>\sqrt{2}</math> when paired up alphabetically. The direction is determined by octaves starting from the middle C.

{| class="wikitable"

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! Nominal !! Pronuciation !! Meaning !! Ratio with middle C !! Cents

|-

| γ || gam || C + P4.5 || √2 || 600.000

| γ || gam || C + P4.5 || <math>\sqrt{2}</math> || 600.000

|-

| δ || del || D + P4.5 || √(81/~~32)~~ || 803.910

| δ || del || D + P4.5 || <math>\sqrt{\frac{81}{32}}</math> || 803.910

|-

| ε || eps || E + P4.5 || √(6561/~~2048)~~ || 1007.820

| ε || eps || E + P4.5 || <math>\sqrt{\frac{6561}{2048}}</math> || 1007.820

|-

| ζ || zet || F + P4.5 || √(32/9) || 1098.045

| ζ || zet || F + P4.5 || <math>\sqrt{\frac{32}{9}}</math> || 1098.045

|-

| η || eta || G - P4.5 || √(9/8) || 101.955

| η || eta || G - P4.5 || <math>\sqrt{\frac{9}{8}}</math> || 101.955

|-

| α || alp || A - P4.5 || (9/8)^(3/2) || 305.865

| α || alp || A - P4.5 || <math>\left(\frac{9}{8}\right)^{\frac{3}{2}}</math> || 305.865

|-

| β || bet || B - P4.5 || (9/8)^(5/2) || 509.775

| β || bet || B - P4.5 || <math>\left(\frac{9}{8}\right)^{\frac{5}{2}}</math> || 509.775

|}

Where to put the greek notes on a staff is still being decided. Probably on the same lines as traditional notes but with distinct noteheads. E.g. a middle η would look like a middle C, but with an upwards pointing triangular notehead.

A representative 10L 2s 10|0(2) scale would be spelled C, η, D, α, E, β, γ, G, δ, A, ε, B, (C).

A representative 10L 2s 10|0(2) scale would be spelled C, η, D, α, E, β, γ, G, δ, A, ε, B, (C).

An alternative solution, although one which looses bijectivity, is to keep only the traditional nominals while having a dedicated accidental pair for √(256/~~243)~~ (this was proposed by [[User:CompactStar|CompactStar]]).

An alternative solution, although one which looses bijectivity, is to keep only the traditional nominals while having a dedicated accidental pair for <math>\sqrt{\frac{256}{243}}</math> (this was proposed by [[User:CompactStar|CompactStar]]).

=== Semifourths ===

Luckily we don't need to introduce any more generalizations to the notation to indicate √(4/3). It's a neutral 2½ 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 {{frac|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 √(4/3) against the octave.

~~Nicknames are still assigned to make it easier to talk about the [[5L 4s]] scale generated by √(4/3) against the octave.~~

{| class="wikitable"

|+ style="font-size: 105%;" | Semifourth nominals

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! Nominal !! Pronunciation !! Meaning !! Ratio with middle C !! Cents

|-

| φ || phi || α{{demiflat2}} || √(4/3) || 249.022

| φ || phi || α{{demiflat2}} || <math>\sqrt{\frac{4}{3}}</math> || 249.022

|-

| χ || chi || β{{demiflat2}} || √(27/~~16)~~ || 452.933

| χ || chi || β{{demiflat2}} || <math>\sqrt{\frac{27}{16}}</math> || 452.933

|-

| ψ || psi || ε{{demiflat2}} || √3 || 950.978

| ψ || psi || ε{{demiflat2}} || <math>\sqrt{3}</math> || 950.978

|-

| ω || ome || ζ{{demisharp2}} || √(243/~~64)~~ || 1154.888

| ω || ome || ζ{{demisharp2}} || <math>\sqrt{\frac{243}{64}}</math> || 1154.888

|}

These particular definitions were chosen so that C, D, φ, χ, F, G, A, ψ, ω, (C) becomes the 6|2 (Stellerian) mode, all notated without accidentals.

These particular definitions were chosen so that {{nowrap|C, D, φ, χ, F, G, A, ψ, ω, (C)}} becomes the 6|2 (Stellerian) mode, all notated without accidentals.

=== Hemipyth ===

Putting it all together we can now spell a squashed Ionian scale, 10L 4s 10|2(2):

Putting it all together we can now spell a squashed Ionian scale, 10L 4s 10|2(2):

C, η, D, α{{demiflat2}}, E{{demiflat2}}, β{{demiflat2}}, F{{demisharp2}}, γ, G, δ, A{{demiflat2}}, ε{{demiflat2}}, B{{demiflat2}}, ζ{{demisharp2}}, (C)

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C, η, D, φ, E{{demiflat2}}, χ, F{{demisharp2}}, γ, G, δ, A{{demiflat2}}, ψ, B{{demiflat2}}, ω, (C)

The 4L 6s 4|4(2) scale (called Pacific), can be spelled like so:

The 4L 6s 4|4(2) scale (called Pacific), can be spelled like so:

C, η, α{{demiflat2}}, E{{demiflat2}}, F, γ, G, A{{demiflat2}}, ε{{demiflat2}}, ζ, C

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D, α, β{{demiflat2}}, F{{demisharp2}}, G, δ, A, B{{demiflat2}}, ζ{{demisharp2}}, η, D

Simple hemipyth chords can be plucked out of the square root of the Pythagorean segment 1:√2:√3:2:√6:√8:3:~~√12~~:4:~~√18~~:~~√24~~:~~√27~~:~~√32~~:6:~~√48~~:~~√54~~:8:~~√72~~:9:~~√96~~:~~√108~~:~~√128~~:~~...~~ e.g. 2:√6:3 is a neutral chord where spicy tension can be added by including the semioctave for 2:√6:√8:3 with no increase in complexity as far as the generator of the subgroup is concerned.

Simple hemipyth chords can be plucked out of the square root of the Pythagorean segment <math>1:</math> <math>\sqrt{2}:</math> <math>\sqrt{3}:2:</math> <math>\sqrt{6}:</math> <math>\sqrt{8}:3:</math> <math>\sqrt{12}:4:</math> <math>\sqrt{18}:</math> <math>\sqrt{24}:</math> <math>\sqrt{27}:</math> <math>\sqrt{32}:6:</math> <math>\sqrt{48}:</math> <math>\sqrt{54}:8:</math> <math>\sqrt{72}:9:</math> <math>\sqrt{96}:</math> <math>\sqrt{108}:</math> <math>\sqrt{128}:</math> <math>\ldots</math> e.g. <math>2:\sqrt{6}:3</math> is a neutral chord where spicy tension can be added by including the semioctave for <math>2:\sqrt{6}:\sqrt{8}:3</math> with no increase in complexity as far as the generator of the subgroup is concerned.

Here is a [https://xenpaper.com/#%7B58edo%7D%0A%23_5L_2s_5%7C1_(Ionian)%0A0_10_20_24_34_44_54_58_54_44_34_24_20_10_0%0A....%0A%23_3L_4s_4%7C2_(kleeth)%0A0_10_17_24_34_41_51_58_51_41_34_24_17_10_0%0A....%0A%23_10L_2s_10%7C0(2)%0A0_5_10_15_20_25_29_34_39_44_49_54_58_54_49_44_39_34_29_25_20_15_10_5_0%0A....%0A%23_5L_4s_6%7C2_(Stellerian)%0A0_10_12_22_24_34_44_46_56_58_56_46_44_34_24_22_12_10_0%0A....%0A%23_10L_4s_10%7C2(2)_(Squashed_Ionian)%0A0_5_10_12_17_22_27_29_34_39_41_46_51_56_58_56_51_46_41_39_34_29_27_22_17_12_10_5_0%0A....%0A%23_4L_6s_4%7C4(2)_(Pacific)%0A0_5_12_17_24_29_34_41_46_53_58_53_46_41_34_29_24_17_12_5_0 Xenpaper demo] of all five representative scales listed above.

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The semioctave is always tuned pure when the octave is tuned pure.

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 √(3/2) also makes it more tolerant of imprecise tuning.

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 ~√(4/3) (assuming it's featured in the tuning in the first place).

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).

=== Signposts ===

Due to their low damage in supporting temperaments octave 2/1, semioctave √2, fifth 3/2, fourth 4/3, neutral third √(3/2), ~~neural~~ sixth √(8/3), semifourth √(4/3), semitwelfth √3, "hemitone" √(9/8) and "contrahemitone" √(32/9) all provide good signposts for navigating around otherwise unfamiliar scales.

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.

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.

== Temperament interpretations ==

Under [[ploidacot]] classification diploid temperaments feature ~√2, dicot temperaments have ~√(3/2) and alpha-dicot temperaments feature ~√(4/3) (by virtue of having a ~√3).

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>).

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 !! ~√2 !! ~√(3/2) !! ~√(4/3) !! contorted !! rank-2

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

|-

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

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

|-

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

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

|-

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

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

|-

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

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

|-

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

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

|}

Above contorted tunings don't have a ~√2 period with a ~√3 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>~\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.

Some possible interpretations for ~√2 are:

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

{| class="wikitable"

|+ style="font-size: 105%;" | Higher-prime interpretations of √2

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

|-

! Temperament !! ~√2 !! contorted !! rank-2

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

|-

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

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|}

Some possible interpretations for ~√3 are:

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

{| class="wikitable"

|+ style="font-size: 105%;" | Higher-prime interpretations of √3

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

|-

! Temperament !! ~√3 !! contorted !! rank-2

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

|-

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

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{| class="wikitable"

|+ style="font-size: 105%;" | Higher-prime interpretations of √(3/2)

|+ style="font-size: 105%;" | Higher-prime interpretations of <math>\sqrt{\frac{3}{2}}</math>

|-

! Temperament !! ~√(3/2) !! contorted !! rank-2

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

|-

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

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[[File:The_Hymn_of_Pergele.mp3]]

The Hymn of Pergele, a short piece in [[Hemipyth]][10] 4|4(2) (Pacific mode of [[4L 6s]]), written by [[User:2^67-1|Cole]].

The Hymn of Pergele, a short piece in [[Hemipyth]][10] 4|4(2) (Pacific mode of [[4L 6s]]), written by [[User:2^67-1|Cole]].

[[Category:Rank 2]]

[[Category:Subgroup]]

@@ Line 1: / Line 1: @@
-A '''hemipyth''' (or '''"hemipythagorean"''') interval is an [[interval]] in the √2.√3 [[subgroup]] i.e. intervals that can be constructed by multiplying half-integer powers of 2 and 3.
+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 √(3/2) = √3/√2, semioctave √2 and the semifourth √(4/3) = 2/√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 ~3/2, ~2, or ~4/3 exactly in half and can thus be interpreted as neutral thirds, semioctaves, or semifourths within the temperament.
 == Equal temperaments ==
-An important property of edos > 1 is that they must by necessity include at least one of the notable hemipyth intervals:
+An important property of edos &gt;&nbsp;1 is that they must by necessity include at least one of the notable hemipyth intervals:
-* Either the edo is even and it features at least √2 (which is tuned "pure" when the octave is tuned pure).
+* 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 √(3/2))
+** 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 4/3 spans an even number of edosteps (leading to an approximation to √(4/3))
+** The closest approximation to 4/3 spans an even number of edosteps (leading to an approximation to <math>\sqrt{\frac{4}{3}}</math>)
 {| class="wikitable"
 |+ style="font-size: 105%;" | List of edo mappings with full or partial hemipyth support
 |-
-! Edo (warts) !! Has √2 !! Has √(3/2) !! Has √(4/3)
+! Edo (warts) !! Has <math>\sqrt{2}</math> !! Has <math>\sqrt{\frac{3}{2}}</math> !! Has <math>\sqrt{\frac{4}{3}}</math>
 |-
 | 2 || yes || no || no
@@ Line 77: / Line 77: @@
 == 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 2/1, chain of fifths denotes multiples of 3/2, the sharp sign is equal to 2187/2048 etc.
-A prototypical 5L 2s 5|1 (Ionian) scale would be spelled C, D, E, F, G, A, B, (C).
+A prototypical 5L&nbsp;2s 5|1 (Ionian) scale would be spelled C, D, E, F, G, A, B, (C).
 Simple otonal chords can be plucked out of the harmonic segment 1:2:3:4:6:8:9:12:16:18:24:27:32:36:48:54:64:72:81:96:108:128:... e.g. 6:8:9 is a sus4 chord.
 === Neutral thirds ===
-The 2.√(3/2) part can be notated using [[neutral chain-of-fifths notation]]. This introduces a neutral interval quality between major and minor, semisharps (a.k.a. demisharps) etc.
+The <math>2\,.\sqrt{\frac{3}{2}}</math> part can be notated using [[neutral chain-of-fifths notation]]. This introduces a neutral interval quality between major and minor, semisharps (a.k.a. demisharps) etc.
-A representative 3L 4s 4|2 (kleeth) scale would be spelled C, D, E{{demiflat2}}, F, G, A{{demiflat2}}, B{{demiflat2}}, (C).
+A representative 3L&nbsp;4s 4|2 (kleeth) scale would be spelled {{nowrap|C, D, E{{demiflat2}}, F, G, A{{demiflat2}}, B{{demiflat2}}, (C)}}
 === Semioctaves ===
-In traditional notation the octave spans 7 diasteps which means that it splits into two interordinal 3½ diasteps or two perfect 4.5ths 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 {{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.
-Intervals retain their quality when the frequency ratio is multiplied by the perfect semioctave √2 e.g. {{nowrap|M6 &minus; P4.5 {{=}} M2.5 {{=}} (9/8)^(3/2)}}.
+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>}}.
 Relative interordinal intervals are either called by their double i.e. M2.5 is a major semifourth due to being exactly the half of an augmented fourth (Aug4), or by simply adding the suffix "-and-a-halfth" i.e. "major second-and-a-halfth". The semisecond gets the special nickname "sesquith".
-The nominals for absolute pitches are denoted using lowercase Greek nominals (uppercase often looks identical to pre-existing Latin nominals). The logic being that Latin and Greek notes differ by a multiple of √2 when paired up alphabetically. The direction is determined by octaves starting from the middle C.
+The nominals for absolute pitches are denoted using lowercase Greek nominals (uppercase often looks identical to pre-existing Latin nominals). The logic being that Latin and Greek notes differ by a multiple of <math>\sqrt{2}</math> when paired up alphabetically. The direction is determined by octaves starting from the middle C.
 {| class="wikitable"
@@ Line 102: / Line 102: @@
 ! Nominal !! Pronuciation !! Meaning !! Ratio with middle C !! Cents
 |-
-| γ  || gam || C + P4.5 || √2 || 600.000
+| γ  || gam || C + P4.5 || <math>\sqrt{2}</math> || 600.000
 |-
-| δ || del || D + P4.5 || √(81/32) || 803.910
+| δ || del || D + P4.5 || <math>\sqrt{\frac{81}{32}}</math> || 803.910
 |-
-| ε || eps || E + P4.5 || √(6561/2048) || 1007.820
+| ε || eps || E + P4.5 || <math>\sqrt{\frac{6561}{2048}}</math> || 1007.820
 |-
-| ζ || zet || F + P4.5 || √(32/9) || 1098.045
+| ζ || zet || F + P4.5 || <math>\sqrt{\frac{32}{9}}</math> || 1098.045
 |-
-| η || eta || G - P4.5 || √(9/8) || 101.955
+| η || eta || G - P4.5 || <math>\sqrt{\frac{9}{8}}</math> || 101.955
 |-
-| α || alp || A - P4.5 || (9/8)^(3/2) || 305.865
+| α || alp || A - P4.5 || <math>\left(\frac{9}{8}\right)^{\frac{3}{2}}</math> || 305.865
 |-
-| β || bet || B - P4.5 || (9/8)^(5/2) || 509.775
+| β || bet || B - P4.5 || <math>\left(\frac{9}{8}\right)^{\frac{5}{2}}</math> || 509.775
 |}
 Where to put the greek notes on a staff is still being decided. Probably on the same lines as traditional notes but with distinct noteheads. E.g. a middle η would look like a middle C, but with an upwards pointing triangular notehead.
-A representative 10L 2s 10|0(2) scale would be spelled C, η, D, α, E, β, γ, G, δ, A, ε, B, (C).
+A representative 10L&nbsp;2s 10|0(2) scale would be spelled C, η, D, α, E, β, γ, G, δ, A, ε, B, (C).
-An alternative solution, although one which looses bijectivity, is to keep only the traditional nominals while having a dedicated accidental pair for √(256/243) (this was proposed by [[User:CompactStar|CompactStar]]).
+An alternative solution, although one which looses bijectivity, is to keep only the traditional nominals while having a dedicated accidental pair for <math>\sqrt{\frac{256}{243}}</math> (this was proposed by [[User:CompactStar|CompactStar]]).
 === Semifourths ===
-Luckily we don't need to introduce any more generalizations to the notation to indicate √(4/3). It's a neutral 2½ 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 {{frac|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 √(4/3) against the octave.
-Nicknames are still assigned to make it easier to talk about the [[5L 4s]] scale generated by √(4/3) against the octave.
 {| class="wikitable"
 |+ style="font-size: 105%;" | Semifourth nominals
@@ Line 132: / Line 133: @@
 ! Nominal !! Pronunciation !! Meaning !! Ratio with middle C !! Cents
 |-
-| φ || phi || α{{demiflat2}} || √(4/3) || 249.022
+| φ || phi || α{{demiflat2}} || <math>\sqrt{\frac{4}{3}}</math> || 249.022
 |-
-| χ || chi || β{{demiflat2}} || √(27/16) || 452.933
+| χ || chi || β{{demiflat2}} || <math>\sqrt{\frac{27}{16}}</math> || 452.933
 |-
-| ψ || psi || ε{{demiflat2}} || √3 || 950.978
+| ψ || psi || ε{{demiflat2}} || <math>\sqrt{3}</math> || 950.978
 |-
-| ω || ome || ζ{{demisharp2}} || √(243/64) || 1154.888
+| ω || ome || ζ{{demisharp2}} || <math>\sqrt{\frac{243}{64}}</math> || 1154.888
 |}
-These particular definitions were chosen so that C, D, φ, χ, F, G, A, ψ, ω, (C) becomes the 6|2 (Stellerian) mode, all notated without accidentals.
+These particular definitions were chosen so that {{nowrap|C, D, φ, χ, F, G, A, ψ, ω, (C)}} becomes the 6|2 (Stellerian) mode, all notated without accidentals.
 === Hemipyth ===
-Putting it all together we can now spell a squashed Ionian scale, 10L 4s 10|2(2):
+Putting it all together we can now spell a squashed Ionian scale, 10L&nbsp;4s 10|2(2):
 C, η, D, α{{demiflat2}}, E{{demiflat2}}, β{{demiflat2}}, F{{demisharp2}}, γ, G, δ, A{{demiflat2}}, ε{{demiflat2}}, B{{demiflat2}}, ζ{{demisharp2}}, (C)
@@ Line 151: / Line 152: @@
 C, η, D, φ, E{{demiflat2}}, χ, F{{demisharp2}}, γ, G, δ, A{{demiflat2}}, ψ, B{{demiflat2}}, ω, (C)
-The 4L 6s 4|4(2) scale (called Pacific), can be spelled like so:
+The 4L&nbsp;6s 4|4(2) scale (called Pacific), can be spelled like so:
 C, η, α{{demiflat2}}, E{{demiflat2}}, F, γ, G, A{{demiflat2}}, ε{{demiflat2}}, ζ, C
@@ Line 159: / Line 160: @@
 D, α, β{{demiflat2}}, F{{demisharp2}}, G, δ, A, B{{demiflat2}}, ζ{{demisharp2}}, η, D
-Simple hemipyth chords can be plucked out of the square root of the Pythagorean segment 1:√2:√3:2:√6:√8:3:√12:4:√18:√24:√27:√32:6:√48:√54:8:√72:9:√96:√108:√128:... e.g. 2:√6:3 is a neutral chord where spicy tension can be added by including the semioctave for 2:√6:√8:3 with no increase in complexity as far as the generator of the subgroup is concerned.
+Simple hemipyth chords can be plucked out of the square root of the Pythagorean segment <math>1:</math> <math>\sqrt{2}:</math> <math>\sqrt{3}:2:</math> <math>\sqrt{6}:</math> <math>\sqrt{8}:3:</math> <math>\sqrt{12}:4:</math> <math>\sqrt{18}:</math> <math>\sqrt{24}:</math> <math>\sqrt{27}:</math> <math>\sqrt{32}:6:</math> <math>\sqrt{48}:</math> <math>\sqrt{54}:8:</math> <math>\sqrt{72}:9:</math> <math>\sqrt{96}:</math> <math>\sqrt{108}:</math> <math>\sqrt{128}:</math> <math>\ldots</math> e.g. <math>2:\sqrt{6}:3</math> is a neutral chord where spicy tension can be added by including the semioctave for <math>2:\sqrt{6}:\sqrt{8}:3</math> with no increase in complexity as far as the generator of the subgroup is concerned.
 Here is a [https://xenpaper.com/#%7B58edo%7D%0A%23_5L_2s_5%7C1_(Ionian)%0A0_10_20_24_34_44_54_58_54_44_34_24_20_10_0%0A....%0A%23_3L_4s_4%7C2_(kleeth)%0A0_10_17_24_34_41_51_58_51_41_34_24_17_10_0%0A....%0A%23_10L_2s_10%7C0(2)%0A0_5_10_15_20_25_29_34_39_44_49_54_58_54_49_44_39_34_29_25_20_15_10_5_0%0A....%0A%23_5L_4s_6%7C2_(Stellerian)%0A0_10_12_22_24_34_44_46_56_58_56_46_44_34_24_22_12_10_0%0A....%0A%23_10L_4s_10%7C2(2)_(Squashed_Ionian)%0A0_5_10_12_17_22_27_29_34_39_41_46_51_56_58_56_51_46_41_39_34_29_27_22_17_12_10_5_0%0A....%0A%23_4L_6s_4%7C4(2)_(Pacific)%0A0_5_12_17_24_29_34_41_46_53_58_53_46_41_34_29_24_17_12_5_0 Xenpaper demo] of all five representative scales listed above.
@@ Line 175: / Line 176: @@
 The semioctave is always tuned pure when the octave is tuned pure.
-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 √(3/2) also makes it more tolerant of imprecise tuning.
+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 ~√(4/3) (assuming it's featured in the tuning in the first place).
+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).
 === Signposts ===
-Due to their low damage in supporting temperaments octave 2/1, semioctave √2, fifth 3/2, fourth 4/3, neutral third √(3/2), neural sixth √(8/3), semifourth √(4/3), semitwelfth √3, "hemitone" √(9/8) and "contrahemitone" √(32/9) all provide good signposts for navigating around otherwise unfamiliar scales.
+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.
 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.
 == Temperament interpretations ==
-Under [[ploidacot]] classification diploid temperaments feature ~√2, dicot temperaments have ~√(3/2) and alpha-dicot temperaments feature ~√(4/3) (by virtue of having a ~√3).
+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>).
 Full hemipyth support is indicated by at least "diploid dicot". Examples include:
@@ Line 191: / Line 192: @@
 |+ style="font-size: 105%;" | Higher-prime interpretations of hemipyth intervals
 |-
-! Temperament !! ~√2 !! ~√(3/2) !! ~√(4/3) !! contorted !! rank-2
+! Temperament !! <math>~\sqrt{2}</math> !! <math>~\sqrt{\frac{3}{2}}</math> !! <math>~\sqrt{\frac{4}{3}}</math> !! contorted !! rank-2
 |-
 | [[decimal]] || ~7/5 || ~5/4 || ~7/6 || no || yes
@@ Line 199: / Line 200: @@
 | [[sruti]] || ~45/32 || ~175/144 || ~81/70 || no || yes
 |-
-| [[Subgroup_temperaments#Pakkanian hemipyth|pakkanian hemipyth]] || ~17/12 || ~11/9 || ~15/13 || no || yes
+| [[Subgroup temperaments#Pakkanian hemipyth|pakkanian hemipyth]] || ~17/12 || ~11/9 || ~15/13 || no || yes
 |-
 | [[harry]] || ~17/12 || ~11/9 || ~15/13 || yes || yes
@@ Line 211: / Line 212: @@
 | [[semisema]] || ~108/77 || ~11/9 || ~7/6 || no || yes
 |-
-|| [[quadritikleismic]] || ~625/441 || ~49/40 || ~125/108 || yes || yes
+| [[quadritikleismic]] || ~625/441 || ~49/40 || ~125/108 || yes || yes
 |-
-|| [[decoid]] || ~99/70 || ~49/40 || ~4725/4096 || yes || yes
+| [[decoid]] || ~99/70 || ~49/40 || ~4725/4096 || yes || yes
 |}
-Above contorted tunings don't have a ~√2 period with a ~√3 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>~\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.
-Some possible interpretations for ~√2 are:
+Some possible interpretations for <math>~\sqrt{2}</math> are:
 {| class="wikitable"
-|+ style="font-size: 105%;" | Higher-prime interpretations of √2
+|+ style="font-size: 105%;" | Higher-prime interpretations of <math>\sqrt{2}</math>
 |-
-! Temperament !! ~√2 !! contorted !! rank-2
+! Temperament !! <math>~\sqrt{2}</math> !! contorted !! rank-2
 |-
 | [[jubilic]] || ~7/5 || no || yes (2.5.7)
@@ Line 234: / Line 235: @@
 |}
-Some possible interpretations for ~√3 are:
+Some possible interpretations for <math>~\sqrt{3}</math> are:
 {| class="wikitable"
-|+ style="font-size: 105%;" | Higher-prime interpretations of √3
+|+ style="font-size: 105%;" | Higher-prime interpretations of <math>\sqrt{3}</math>
 |-
-! Temperament !! ~√3 !! contorted !! rank-2
+! Temperament !! <math>~\sqrt{3}</math> !! contorted !! rank-2
 |-
 | [[semaphore]] || ~7/4 || no || yes (2.3.7)
@@ Line 249: / Line 250: @@
 {| class="wikitable"
-|+ style="font-size: 105%;" | Higher-prime interpretations of √(3/2)
+|+ style="font-size: 105%;" | Higher-prime interpretations of <math>\sqrt{\frac{3}{2}}</math>
 |-
-! Temperament !! ~√(3/2) !! contorted !! rank-2
+! Temperament !! <math>\sqrt{~\frac{3}{2}}</math> !! contorted !! rank-2
 |-
 | [[dicot]] || ~5/4 || no || yes (2.3.5)
@@ Line 284: / Line 285: @@
 [[File:The_Hymn_of_Pergele.mp3]]
-The Hymn of Pergele, a short piece in [[Hemipyth]][10] 4|4(2) (Pacific mode of [[4L 6s]]), written by [[User:2^67-1|Cole]].
+The Hymn of Pergele, a short piece in [[Hemipyth]][10] 4|4(2) (Pacific mode of [[4L&nbsp;6s]]), written by [[User:2^67-1|Cole]].
 [[Category:Rank 2]]
 [[Category:Subgroup]]