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 primes [[2/1|2]] and [[3/1|3]]. Hemipythagorean is the pure tuning of [[Ploidacot/Diploid dicot|diploid dicot]]. | ||
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 ~{{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. | 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. | ||
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|+ style="font-size: 105%;" | List of edo mappings with full or partial hemipyth support | |+ style="font-size: 105%;" | List of edo mappings with full or partial hemipyth support | ||
|- | |- | ||
! Edo (warts) !! Has <math>\sqrt{2}</math> !! Has <math>\sqrt{\frac{3}{2}}</math> !! Has <math>\sqrt{\frac{4}{3}}</math> | ! Edo ([[Wart notation|warts]]) !! Has <math>\sqrt{2}</math> !! Has <math>\sqrt{\frac{3}{2}}</math> !! Has <math>\sqrt{\frac{4}{3}}</math> | ||
|- | |- | ||
| 2 || yes || no || no | | [[2edo|2]] || yes || no || no | ||
|- | |- | ||
| 3 || no || yes || no | | [[3edo|3]] || no || yes || no | ||
|- | |- | ||
| 4 || yes || yes || yes | | [[4edo|4]] || yes || yes || yes | ||
|- | |- | ||
| 5 || no || no || yes | | [[5edo|5]] || no || no || yes | ||
|- | |- | ||
| 6 || yes || yes || yes | | [[6edo|6]] || yes || yes || yes | ||
|- | |- | ||
| 7 || no || yes || no | | [[7edo|7]] || no || yes || no | ||
|- | |- | ||
| 8 || yes || no || no | | [[8edo|8]] || yes || no || no | ||
|- | |- | ||
| 9 || no || no || yes | | [[9edo|9]] || no || no || yes | ||
|- | |- | ||
| 10 || yes || yes || yes | | [[10edo|10]] || yes || yes || yes | ||
|- | |- | ||
| 11 || no || yes || no | | [[11edo|11]] || no || yes || no | ||
|- | |- | ||
| 12 || yes || no || no | | [[12edo|12]] || yes || no || no | ||
|- | |- | ||
| 13 || no || yes || no | | [[13edo|13]] || no || yes || no | ||
|- | |- | ||
| 13b || no || no || yes | | 13b || no || no || yes | ||
|- | |- | ||
| 14 || yes || yes || yes | | [[14edo|14]] || yes || yes || yes | ||
|- | |- | ||
| 15 || no || no || yes | | [[15edo|15]] || no || no || yes | ||
|- | |- | ||
| 16 || yes || no || no | | [[16edo|16]] || yes || no || no | ||
|- | |- | ||
| 17 || no || yes || no | | [[17edo|17]] || no || yes || no | ||
|- | |- | ||
| 18 || yes || no || no | | [[18edo|18]] || yes || no || no | ||
|- | |- | ||
| 18b || yes || yes || yes | | 18b || yes || yes || yes | ||
|- | |- | ||
| 19 || no || no || yes | | [[19edo|19]] || no || no || yes | ||
|- | |- | ||
| 20* || yes || yes || yes | | [[20edo|20]]* || yes || yes || yes | ||
|- | |- | ||
| 20b || yes || no || no | | 20b || yes || no || no | ||
|- | |- | ||
| 21 || no || yes || no | | [[21edo|21]] || no || yes || no | ||
|- | |- | ||
| 22 || yes || no || no | | [[22edo|22]] || yes || no || no | ||
|- | |- | ||
| 23 || no || no || yes | | [[23edo|23]] || no || no || yes | ||
|- | |- | ||
| 24 || yes || yes || yes | | [[24edo|24]] || yes || yes || yes | ||
|} | |} | ||
<nowiki>*</nowiki> Above the patent val of 20edo results in the same tuning as the patent val of 10edo, so it adds nothing new. | |||
Note how in hemipyth the patent val of 24edo is not tuned the same as 12edo's patent val. In fact 24edo is arguably the smallest edo where all of the important hemipyth intervals are tuned reasonably accurately. | Note how in hemipyth the patent val of 24edo is not tuned the same as 12edo's patent val. In fact, 24edo is arguably the smallest edo where all of the important hemipyth intervals are tuned reasonably accurately. | ||
Other edos with hemipyth-supporting patent vals are 28, 30, 34, 38, 44, 48, 52, 54, 58, etc. 58edo is the first one to reduce the absolute error of the neutral third generator compared to 24edo | Other edos with hemipyth-supporting patent vals are {{edos|28, 30, 34, 38, 44, 48, 52, 54, 58}}, etc. 58edo is the first one to reduce the absolute error of the neutral third generator compared to 24edo, though one needs to go all the way to [[82edo]] in order to get an improvement in terms of relative error. | ||
== Notation == | == Notation == | ||
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. | 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 {{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. | 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]]). | ||
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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. | 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>\sim | Nicknames are still assigned to make it easier to talk about the [[5L 4s]] scale generated by <math>\sim\sqrt{\frac{4}{3}}</math> against the octave. | ||
{| class="wikitable" | {| class="wikitable" | ||
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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 ~{{sfrac|4|3}} still results in a decent <math>\sim | The same goes for the semifourth. A poorly tuned ~{{sfrac|4|3}} still results in a decent <math>\sim\sqrt{\frac{4}{3}}</math> (assuming it's featured in the tuning in the first place). | ||
=== Signposts === | === Signposts === | ||
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== Temperament interpretations == | == Temperament interpretations == | ||
Under [[ploidacot]] classification diploid temperaments feature <math>\sim | Under [[ploidacot]] classification diploid temperaments feature <math>\sim\sqrt{2}</math>, dicot temperaments have <math>\sim\sqrt{\frac{3}{2}}</math> and alpha-dicot temperaments feature <math>\sim\sqrt{\frac{4}{3}}</math> (by virtue of having a <math>\sim\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>\sim | ! Temperament !! <math>\sim\sqrt{2}</math> !! <math>\sim\sqrt{\frac{3}{2}}</math> !! <math>\sim\sqrt{\frac{4}{3}}</math> !! contorted !! rank-2 | ||
|- | |- | ||
| [[decimal]] || ~{{sfrac|7|5}} || ~{{sfrac|5|4}} || ~{{sfrac|7|6}} || no || yes | | [[decimal]] || ~{{sfrac|7|5}} || ~{{sfrac|5|4}} || ~{{sfrac|7|6}} || no || yes | ||
| Line 209: | Line 209: | ||
| [[hemidim]] || ~{{sfrac|36|25}} || ~{{sfrac|25|21}} || ~{{sfrac|7|6}} || yes || yes | | [[hemidim]] || ~{{sfrac|36|25}} || ~{{sfrac|25|21}} || ~{{sfrac|7|6}} || yes || yes | ||
|- | |- | ||
| [[ | | [[baldur]]|| ~{{sfrac|99|70}} || ~{{sfrac|49|40}} || ~{{sfrac|231|200}} || no || no | ||
|- | |- | ||
| [[semisema]] || ~{{sfrac|108|77}} || ~{{sfrac|11|9}} || ~{{sfrac|7|6}} || no || yes | | [[semisema]] || ~{{sfrac|108|77}} || ~{{sfrac|11|9}} || ~{{sfrac|7|6}} || no || yes | ||
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|} | |} | ||
Above contorted tunings don't have a <math>\sim | Above contorted tunings don't have a <math>\sim\sqrt{2}</math> period with a <math>\sim\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>\sim | Some possible interpretations for <math>\sim\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>\sim | ! Temperament !! <math>\sim\sqrt{2}</math> !! contorted !! rank-2 | ||
|- | |- | ||
| [[jubilic]] || ~{{sfrac|7|5}} || no || yes (2.5.7) | | [[jubilic]] || ~{{sfrac|7|5}} || no || yes (2.5.7) | ||
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| [[semitonic]] || ~{{sfrac|17|12}} || no || yes (2.3.17) | | [[semitonic]] || ~{{sfrac|17|12}} || no || yes (2.3.17) | ||
|- | |- | ||
| [[kalismic temperaments|kalismic]] || ~{{sfrac|99|70}} || no || no | | [[kalismic temperaments|kalismic]] || ~{{sfrac|99|70}} || no || no (2.3.5.7.11) | ||
|} | |} | ||
Some possible interpretations for <math>\sim | Some possible interpretations for <math>\sim\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>\sim | ! Temperament !! <math>\sim\sqrt{3}</math> !! contorted !! rank-2 | ||
|- | |- | ||
| [[semaphore]] || ~{{sfrac|7|4}} || no || yes (2.3.7) | | [[semaphore]] || ~{{sfrac|7|4}} || no || yes (2.3.7) | ||
|- | |- | ||
| [[barbados]] || ~{{sfrac|26|15}} || no || yes (2.3.{{sfrac|13|5}}) | | [[barbados]] || ~{{sfrac|26|15}} || no || yes (2.3.{{sfrac|13|5}}) | ||
|- | |||
|[[Catalog of rank-4 temperaments#Seascape (160083/160000)|seascape]] | |||
|~{{sfrac|400|231}} | |||
|no | |||
|no (2.3.5.7.11) | |||
|} | |} | ||
Some possible interpretations for <math>\sim\ | Some possible interpretations for <math>\sim\sqrt{\frac{3}{2}}</math> are: | ||
{| class="wikitable" | {| class="wikitable" | ||
|+ style="font-size: 105%;" | Higher-prime interpretations of <math>\sqrt{\frac{3}{2}}</math> | |+ style="font-size: 105%;" | Higher-prime interpretations of <math>\sqrt{\frac{3}{2}}</math> | ||
|- | |- | ||
! Temperament !! <math>\sqrt{ | ! Temperament !! <math>\sim\sqrt{\frac{3}{2}}</math> !! contorted !! rank-2 | ||
|- | |- | ||
| [[dicot]] || ~{{sfrac|5|4}} || no || yes (2.3.5) | | [[dicot]] || ~{{sfrac|5|4}} || no || yes (2.3.5) | ||
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| [[Rastmic clan#Neutral|neutral]] || ~{{sfrac|11|9}} || no || yes (2.3.11) | | [[Rastmic clan#Neutral|neutral]] || ~{{sfrac|11|9}} || no || yes (2.3.11) | ||
|- | |- | ||
| [[ | | [[Breed family#Breed|breed]]|| ~{{sfrac|49|40}} || no || no (2.3.5.7) | ||
|- | |||
|[[Very high accuracy temperaments#Euzenius|euzenius]] | |||
|~{{sfrac|6250|5103}} | |||
|no | |||
|no (2.3.5.7) | |||
|} | |} | ||
== MOS patterns == | == MOS patterns == | ||
By default hemipyth[n] refers to a [[MOS]] pattern of size n inside the octave i.e. there are always two periods per octave. The generator is √3 unless otherwise stated. | {{Idiosyncratic terms|The mos names for hemipyth[14], hemipyth[24], and hemipyth[34] are proposals described on [[TAMNAMS_Extension #Naming mos descendants]].}} | ||
By default hemipyth[n] refers to a [[MOS]] pattern of size n inside the octave i.e. there are always two √2 periods per octave. The generator is √3 unless otherwise stated. | |||
{| class="wikitable" | {| class="wikitable" | ||
|+ style="font-size: 105%;" | MOS patterns of hemipyth | |+ style="font-size: 105%;" | MOS patterns of hemipyth | ||
|- | |- | ||
! hemipyth[n] !! MOS pattern !! hardness (untempered) | ! hemipyth[n] | ||
! TAMNAMS name !! MOS pattern !! hardness (untempered) | |||
|- | |||
| hemipyth[4] | |||
| biwood || [[2L 2s]] || 1.4094 | |||
|- | |||
| hemipyth[6] | |||
| citric || [[4L 2s]] || 2.4424 | |||
|- | |||
| hemipyth[10] | |||
| lime || [[4L 6s]] || 1.4424 | |||
|- | |||
| hemipyth[14] | |||
| m-chro lime || [[10L 4s]] || 2.260 | |||
|- | |- | ||
| hemipyth[ | | hemipyth[24] | ||
| f-enhar lime || [[10L 14s]] || 1.260 | |||
|- | |- | ||
| hemipyth[ | | hemipyth[34] | ||
| paso-lime | |||
| [[24L 10s]] | |||
| 3.8459 | |||
|- | |- | ||
| hemipyth[ | | hemipyth[58] | ||
| | |||
| [[24L 34s]] | |||
| 2.8459 | |||
|- | |- | ||
| hemipyth[ | | hemipyth[82] | ||
| | |||
| [[24L 58s]] | |||
| 1.8459 | |||
|- | |- | ||
| hemipyth[ | | hemipyth[106] | ||
| | |||
| [[82L 24s]] | |||
| 1.1822 | |||
|} | |} | ||
== Music == | == Music == | ||
[[File:The_Hymn_of_Pergele.mp3]] | |||
[[ | 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]]. | ||
== See also == | |||
* [[Hemipent]] – √2.√3.√5 subgroup | |||
[[Category:Rank 2]] | [[Category:Rank 2]] | ||
[[Category:Subgroup]] | [[Category:Subgroup]] | ||
[[Category:Listen]] | |||