Hemipyth: Difference between revisions
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A '''hemipyth''' 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''' 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. | ||
Notable hemipyth intervals include the neutral third √(3/2) = √3/√2, semioctave √2 and the semifourth √(4/3) = | Notable hemipyth intervals include the neutral third √(3/2) = √3/√2, semioctave √2 and the semifourth √(4/3) = 2/√3. | ||
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 == | == 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 √2 (which is tuned "pure" when the octave is tuned pure). | ||
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{| class="wikitable" | {| class="wikitable" | ||
|+ 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 √2 !! Has √(3/2) !! Has √(4/3) | ! Edo (warts) !! Has √2 !! Has √(3/2) !! Has √(4/3) | ||
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|} | |} | ||
{{asterisk}} 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 hemipyth 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. You need to go all the way to 82edo in order to get an improvement in terms of relative error. | 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. You need to go all the way to 82edo in order to get an improvement in terms of relative error. | ||
== Notation == | == Notation == | ||
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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. | ||
=== Neutral thirds === | === 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 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. | ||
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 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 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 3½ diasteps or two perfect 4.5ths 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. M6 | 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)}}. | ||
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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{| class="wikitable" | {| class="wikitable" | ||
|+ Semioctave nominals | |+ style="font-size: 105%;" | Semioctave nominals | ||
|- | |- | ||
! Nominal !! Pronuciation !! Meaning !! Ratio with middle C !! Cents | ! Nominal !! Pronuciation !! Meaning !! Ratio with middle C !! Cents | ||
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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 √(256/243) (this was proposed by [[User:CompactStar|CompactStar]]). | ||
=== Semifourths === | === 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 √(4/3). It's a neutral 2½ or a α{{demiflat2}} (alp semiflat) w.r.t middle C. | ||
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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" | {| class="wikitable" | ||
|+ Semifourth nominals | |+ style="font-size: 105%;" | Semifourth nominals | ||
|- | |- | ||
! Nominal !! Pronunciation !! Meaning !! Ratio with middle C !! Cents | ! Nominal !! Pronunciation !! Meaning !! Ratio with middle C !! Cents | ||
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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 C, D, φ, χ, F, G, A, ψ, ω, (C) becomes the 6|2 (Stellerian) mode, all notated without accidentals. | ||
=== Hemipyth === | === 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): | ||
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Full hemipyth support is indicated by at least "diploid dicot". Examples include: | Full hemipyth support is indicated by at least "diploid dicot". Examples include: | ||
{| class="wikitable" | {| class="wikitable" | ||
|+ Higher-prime interpretations of hemipyth intervals | |+ style="font-size: 105%;" | Higher-prime interpretations of hemipyth intervals | ||
|- | |- | ||
! Temperament !! ~√2 !! ~√(3/2) !! ~√(4/3) !! contorted !! rank-2 | ! Temperament !! ~√2 !! ~√(3/2) !! ~√(4/3) !! contorted !! rank-2 | ||
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{| class="wikitable" | {| class="wikitable" | ||
|+ Higher-prime interpretations of √2 | |+ style="font-size: 105%;" | Higher-prime interpretations of √2 | ||
|- | |- | ||
! Temperament !! ~√2 !! contorted !! rank-2 | ! Temperament !! ~√2 !! contorted !! rank-2 | ||
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{| class="wikitable" | {| class="wikitable" | ||
|+ Higher-prime interpretations of √3 | |+ style="font-size: 105%;" | Higher-prime interpretations of √3 | ||
|- | |- | ||
! Temperament !! ~√3 !! contorted !! rank-2 | ! Temperament !! ~√3 !! contorted !! rank-2 | ||
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{| class="wikitable" | {| class="wikitable" | ||
|+ Higher-prime interpretations of √(3/2) | |+ style="font-size: 105%;" | Higher-prime interpretations of √(3/2) | ||
|- | |- | ||
! Temperament !! ~√(3/2) !! contorted !! rank-2 | ! Temperament !! ~√(3/2) !! contorted !! rank-2 | ||
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{| class="wikitable" | {| class="wikitable" | ||
|+ MOS patterns of hemipyth | |+ style="font-size: 105%;" | MOS patterns of hemipyth | ||
|- | |- | ||
! hemipyth[n] !! MOS pattern !! hardness (untempered) | ! hemipyth[n] !! MOS pattern !! hardness (untempered) | ||
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== Music == | == Music == | ||
{{todo|inline-1| Make more music }} | |||
[[File:The_Hymn_of_Pergele.mp3]] | [[File:The_Hymn_of_Pergele.mp3]] | ||