Hemipyth: Difference between revisions
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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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=== 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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[[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]] | ||