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. | 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>. | 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 | 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 == | == 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). | * 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: | * Or one of the following is true: | ||
** The closest approximation to 3 | ** 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 | ** 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" | {| class="wikitable" | ||
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== Notation == | == Notation == | ||
The Pythagorean (2.3) part of hemipyth can be notated using traditional notation where octaves represent multiples of 2 | 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 | A prototypical {{nowrap|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. | 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. | ||
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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. | 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 | A representative {{nowrap|3L 4s 4{{!}}2}} (kleeth) scale would be spelled {{nowrap|C, D, E{{demiflat2}}, F, G, A{{demiflat2}}, B{{demiflat2}}, (C)}} | ||
=== Semioctaves === | === Semioctaves === | ||
In traditional notation the octave spans 7 diasteps which means that it splits into two interordinal {{ | 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 | 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". | 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". | ||
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! Nominal !! Pronuciation !! Meaning !! Ratio with middle C !! Cents | ! Nominal !! Pronuciation !! Meaning !! Ratio with middle C !! Cents | ||
|- | |- | ||
| γ | | γ || gam || C + P4.5 || <math>\sqrt{2}</math> || 600.000 | ||
|- | |- | ||
| δ || del || D + P4.5 || <math>\sqrt{\frac{81}{32}}</math> || 803.910 | | δ || del || D + P4.5 || <math>\sqrt{\frac{81}{32}}</math> || 803.910 | ||
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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. | 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 | A representative {{nowrap|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 <math>\sqrt{\frac{256}{243}}</math> (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 === | === 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 {{ | 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> | 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" | {| class="wikitable" | ||
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! Nominal !! Pronunciation !! Meaning !! Ratio with middle C !! Cents | ! Nominal !! Pronunciation !! Meaning !! Ratio with middle C !! Cents | ||
|- | |- | ||
| φ || phi || α{{demiflat2}} || <math>\sqrt{\frac{4}{3}}</math> || 249.022 | | φ || phi || α{{nbhsp}}{{demiflat2}} || <math>\sqrt{\frac{4}{3}}</math> || 249.022 | ||
|- | |- | ||
| χ || chi || β{{demiflat2}} || <math>\sqrt{\frac{27}{16}}</math> || 452.933 | | χ || chi || β{{nbhsp}}{{demiflat2}} || <math>\sqrt{\frac{27}{16}}</math> || 452.933 | ||
|- | |- | ||
| ψ || psi || ε{{demiflat2}} || <math>\sqrt{3}</math> || 950.978 | | ψ || psi || ε{{nbhsp}}{{demiflat2}} || <math>\sqrt{3}</math> || 950.978 | ||
|- | |- | ||
| ω || ome || ζ{{demisharp2}} || <math>\sqrt{\frac{243}{64}}</math> || 1154.888 | | ω || ome || ζ{{nbhsp}}{{demisharp2}} || <math>\sqrt{\frac{243}{64}}</math> || 1154.888 | ||
|} | |} | ||
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=== Hemipyth === | === Hemipyth === | ||
Putting it all together we can now spell a squashed Ionian scale, 10L | Putting it all together we can now spell a squashed Ionian scale, {{nowrap|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) | 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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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. | 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:// | Here is a [https://luphoria.com/xenpaper/#%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. | ||
== Musical significance == | == Musical significance == | ||
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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 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 | 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 === | === 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 ({{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 | 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 == | == Temperament interpretations == | ||
Under [[ploidacot]] classification diploid temperaments feature <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: | 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 | |+ style="font-size: 105%;" | Higher-prime interpretations of hemipyth intervals | ||
|- | |- | ||
! Temperament !! <math> | ! 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 | | [[decimal]] || ~{{sfrac|7|5}} || ~{{sfrac|5|4}} || ~{{sfrac|7|6}} || no || yes | ||
|- | |- | ||
| [[anguirus]] || ~45 | | [[anguirus]] || ~{{sfrac|45|32}} || ~{{sfrac|56|45}} || ~{{sfrac|7|6}} || no || yes | ||
|- | |- | ||
| [[sruti]] || ~45 | | [[sruti]] || ~{{sfrac|45|32}} || ~{{sfrac|175|144}} || ~{{sfrac|81|70}} || no || yes | ||
|- | |- | ||
| [[Subgroup temperaments#Pakkanian hemipyth|pakkanian hemipyth]] || ~17 | | [[Subgroup temperaments#Pakkanian hemipyth|pakkanian hemipyth]] || ~{{sfrac|17|12}} || ~{{sfrac|11|9}} || ~{{sfrac|15|13}} || no || yes | ||
|- | |- | ||
| [[harry]] || ~17 | | [[harry]] || ~{{sfrac|17|12}} || ~{{sfrac|11|9}} || ~{{sfrac|15|13}} || yes || yes | ||
|- | |- | ||
| [[semimiracle]] || ~91 | | [[semimiracle]] || ~{{sfrac|91|64}} || ~{{sfrac|11|9}} || ~{{sfrac|15|13}} || yes || yes | ||
|- | |- | ||
| [[hemidim]] || ~36 | | [[hemidim]] || ~{{sfrac|36|25}} || ~{{sfrac|25|21}} || ~{{sfrac|7|6}} || yes || yes | ||
|- | |- | ||
| [[greenland]] || ~99 | | [[greenland]] || ~{{sfrac|99|70}} || ~{{sfrac|49|40}} || ~{{sfrac|15|13}}~{{sfrac|231|200}} || no || no | ||
|- | |- | ||
| [[semisema]] || ~108 | | [[semisema]] || ~{{sfrac|108|77}} || ~{{sfrac|11|9}} || ~{{sfrac|7|6}} || no || yes | ||
|- | |- | ||
| [[quadritikleismic]] || ~625 | | [[quadritikleismic]] || ~{{sfrac|625|441}} || ~{{sfrac|49|40}} || ~{{sfrac|125|108}} || yes || yes | ||
|- | |- | ||
| [[decoid]] || ~99 | | [[decoid]] || ~{{sfrac|99|70}} || ~{{sfrac|49|40}} || ~{{sfrac|4725|4096}} || yes || yes | ||
|} | |} | ||
Above contorted tunings don't have a <math> | 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> | Some possible interpretations for <math>\sim\vsp\sqrt{2}</math> are: | ||
{| class="wikitable" | {| class="wikitable" | ||
|+ style="font-size: 105%;" | Higher-prime interpretations of <math>\sqrt{2}</math> | |+ style="font-size: 105%;" | Higher-prime interpretations of <math>\sqrt{2}</math> | ||
|- | |- | ||
! Temperament !! <math> | ! Temperament !! <math>\sim\vsp\sqrt{2}</math> !! contorted !! rank-2 | ||
|- | |- | ||
| [[jubilic]] || ~7 | | [[jubilic]] || ~{{sfrac|7|5}} || no || yes (2.5.7) | ||
|- | |- | ||
| [[diaschismic]] || ~45 | | [[diaschismic]] || ~{{sfrac|45|32}} || no || yes (2.3.5) | ||
|- | |- | ||
| [[semitonic]] || ~17 | | [[semitonic]] || ~{{sfrac|17|12}} || no || yes (2.3.17) | ||
|- | |- | ||
| [[kalismic temperaments|kalismic]] || ~99 | | [[kalismic temperaments|kalismic]] || ~{{sfrac|99|70}} || no || no | ||
|} | |} | ||
Some possible interpretations for <math> | Some possible interpretations for <math>\sim\vsp\sqrt{3}</math> are: | ||
{| class="wikitable" | {| class="wikitable" | ||
|+ style="font-size: 105%;" | Higher-prime interpretations of <math>\sqrt{3}</math> | |+ style="font-size: 105%;" | Higher-prime interpretations of <math>\sqrt{3}</math> | ||
|- | |- | ||
! Temperament !! <math> | ! Temperament !! <math>\sim\vsp\sqrt{3}</math> !! contorted !! rank-2 | ||
|- | |- | ||
| [[semaphore]] || ~7 | | [[semaphore]] || ~{{sfrac|7|4}} || no || yes (2.3.7) | ||
|- | |- | ||
| [[barbados]] || ~26 | | [[barbados]] || ~{{sfrac|26|15}} || no || yes (2.3.{{sfrac|13|5}}) | ||
|} | |} | ||
Some possible interpretations for | Some possible interpretations for <math>\sim\vsp\frac{3}{2}</math> are: | ||
{| class="wikitable" | {| class="wikitable" | ||
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! Temperament !! <math>\sqrt{~\frac{3}{2}}</math> !! contorted !! rank-2 | ! Temperament !! <math>\sqrt{~\frac{3}{2}}</math> !! contorted !! rank-2 | ||
|- | |- | ||
| [[dicot]] || ~5 | | [[dicot]] || ~{{sfrac|5|4}} || no || yes (2.3.5) | ||
|- | |- | ||
| [[Rastmic clan#Neutral|neutral]] || ~11 | | [[Rastmic clan#Neutral|neutral]] || ~{{sfrac|11|9}} || no || yes (2.3.11) | ||
|- | |- | ||
| [[jove]] || ~11 | | [[jove]] || ~{{sfrac|11|9}}~{{sfrac|49|40}} || no || no | ||
|} | |} | ||
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[[File:The_Hymn_of_Pergele.mp3]] | [[File:The_Hymn_of_Pergele.mp3]] | ||
The Hymn of Pergele, a short piece in [[Hemipyth]][10] 4 | The Hymn of Pergele, a short piece in {{nowrap|[[Hemipyth]][10] 4{{!}}4(2)}} (Pacific mode of [[4L 6s]]), written by [[User:2^67-1|Cole]]. | ||
[[Category:Rank 2]] | [[Category:Rank 2]] | ||
[[Category:Subgroup]] | [[Category:Subgroup]] | ||
[[Category:Listen]] | |||