Osmiorisbendi
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OSMIORISBENDI Is just a freelancer using the 23-EDD as musical interval system. Special Thanks to:…
OSMIORISBENDI Is just a freelancer using the 23-EDD as musical interval system. Special Thanks to: PALÄSTINA, Kevin Schilder, Ralph Jarzombek, Igliashon Jones, Fabrizio Fulvio Fausto Fiale, Neil Haverstick, Chris Vaisvil, Xenwolf, Norbert Oldani, Dante Rosati, Jeff Harrington, Andrew Heathwaite, Jon Catler, UND, MicrotonalMetalMaster !!!!!
{Palaestinensische Gemeinde.png} {22.png}
Tútim Dennsuul Wafiil
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... Dennsuul Wafiil Is a Xentonalist of Arica. Enthusiastic of all evolutible thing, works with …
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Dennsuul Wafiil Is a Xentonalist of Arica. Enthusiastic of all evolutible thing, works with the Armodue system on 23-EDD (3;1 relation).
{Icositriphonic Gitar 1.png} '6#' = 409.6 Hz Frecuence Tuning for all 23-EDD instruments.
TDW Notes:
Actually I stopped my works for a indefinable time, because i am working myself transmutation intensely, being my adherence here in this Xentonal avast a great causality for a new start in the existential designs of my life: The reconnection with the Great Mind of the Cosmos and its Universal Love Law. Really thanks so much all you guys for develop all that with magnificence, I love all you.
TDW Terminology:
EDD: Means Equal Divisions per Ditave. Ditave is the same thing as 2/1 ratio, like Tritave for 3/1 ratio or Pentave for 5/1 ratio.
Ketradektríatoh: Name coined to the MOSscale 11L3s for design a 14-tone scale with 3 intervals that aren't 'tones'.
Pluscuamtonality: Means the EDDs 'frettable' for play in common strings instruments like the guitars and Basses, considering a extense range since 13 ~ 57 EDDs.
Purdal: Is the name of a interval unit measure consistent by the division of the ditave in 9900 equal parts, designed and suggested by himself.
Superdiatonic: Name coined to the MOSscale 7L2s for design a 9-tone scale with the diatonic mold LLsLLLs expanded to the mold LLLsLLLLs, denomining to the ditave interval like "Decave" and not "Octave".
!!'!!!'!!!!'!!!'!!!!'!!!'!!!!'!!!'!!!!'!!!'!!!!'!!!'!!!!'!!!'!!!!'!!!'!!IS NOT EXIST YET.
Gentle region (extended version)
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... region article.
Margo Schulter, in a tuning list posting, defined the "gentle region…
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region article.
Margo Schulter, in a tuning list posting, defined the "gentle region" of temperaments with a fifth as generator as that of fifths about 1.49 to 2.65 cents sharp; later amending that to from 1.49 to 3.04 cents sharp. We can consider the first region to extend from fifths of size 17\29 to 64\109, and the extended region to reach 47\80. If we remove the restriction to tempering based on chains of fifths, we find that notable equal divisions in the smaller gentle region include multiples of 29edo, 46edo, 75edo, 104edo, 109edo, 121edo, 133edo, 155edo, 162edo, 167edo, 179edo, 191edo, 201edo, 213edo, 225edo and 237edo, plus 63edo and 80edo in the extended region.
Generator
Gentle region (extended version)
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This is an extended version of the Gentle region article.
Margo Schulter, in a tuning list postin…
This is an extended version of the Gentle region article.
Margo Schulter, in a tuning list posting, defined the "gentle region" of temperaments with a fifth as generator as that of fifths about 1.49 to 2.65 cents sharp; later amending that to from 1.49 to 3.04 cents sharp. We can consider the first region to extend from fifths of size 17\29 to 64\109, and the extended region to reach 47\80. If we remove the restriction to tempering based on chains of fifths, we find that notable equal divisions in the smaller gentle region include multiples of 29edo, 46edo, 75edo, 104edo, 109edo, 121edo, 133edo, 155edo, 162edo, 167edo, 179edo, 191edo, 201edo, 213edo, 225edo and 237edo, plus 63edo and 80edo in the extended region. GeneatorGenerator
Cents
2-3-7(b)-11-13(b)
pergen
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... Sometimes the temperament implies an enharmonic that isn't even a 2nd. For example, liese (2.3…
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Sometimes the temperament implies an enharmonic that isn't even a 2nd. For example, liese (2.3.5.7 with 81/80 and 1029/1000) is (P8, P11/3), with G = 7/5 = vd5, and E = 3·vd5 - P11 = v3dd3. The genchain is P1 -- vd5 -- ^M7 -- P11, or C -- Gbv -- B^ -- F.
This is all for single-comma temperaments. Each comma of a multiple-comma temperament also implies an enharmonic, and they may conflict. True double pergens, which are always multi-comma, have multiple notations. For example, the half-everything pergen has three possible notations, all equally valid. Even single-split pergens can have multiple commas that imply different enharmonics.
Chord names, scale names,names and staff notation
Using pergens, all rank-2 chords can be named using ups and downs, and if needed highs and lows as well. See the ups and downs page for chord naming conventions. The genchain and/or the perchain creates a lattice in which each note and each interval has its own name. The many enharmonic equivalents allow proper chord spelling.
In certain pergens, one spelling isn't always clearly better. For example, in half-4th, C E G A^ and C E G Bbv are the same chord, and either spelling might be used. This exact same issue occurs in 24-edo.
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Scales can be named similar to Meantone[7], as (P8, P5) [7] = unsplit heptatonic, or (P8, P5/2) [5] = half-fifth pentatonic, etc. The number of notes in the scale tend to be a multiple of m, e.g. half-octave pergens tend to have scales with an even number of notes.
Chord progressions can be written out by applying ups and downs to the chord roots as needed, e.g. I.v -- vIII.v -- vVI.^m -- I.v. A porcupine (third-4th) comma pump can be written out like so: C.v -- Av.^m -- Dv.v -- [Bvv=Bb^]^m -- Eb^.v -- G.^m -- G.v -- C.v. Brackets are used to show that Bvv and Bb^ are enharmonically equivalent. The equivalence is shown roughly half-way through the pump. Bvv is written first to show that this root is a vM6 above the previous root, Dv. Bb^ is second to show the P4 relationship to the next root, Eb^. Such an equivalence of course couldn't be used on the staff, where the chord would be written as either Bvv.^m or Bb^.^m, or possibly Bb^.vvM = Bb^ Dv F^.
Highs and lows can be added to the score justlows, like ups and downs can. Theydowns, precede the
Mizarian Porcupine Overture by Herman Miller (P8, P4/3)
{Mizarian Porcupine Overture.png}
pergen
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... Sometimes the temperament implies an enharmonic that isn't even a 2nd. For example, liese (2.3…
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Sometimes the temperament implies an enharmonic that isn't even a 2nd. For example, liese (2.3.5.7 with 81/80 and 1029/1000) is (P8, P11/3), with G = 7/5 = vd5, and E = 3·vd5 - P11 = v3dd3. The genchain is P1 -- vd5 -- ^M7 -- P11, or C -- Gbv -- B^ -- F.
This is all for single-comma temperaments. Each comma of a multiple-comma temperament also implies an enharmonic, and they may conflict. True double pergens, which are always multi-comma, have multiple notations. For example, the half-everything pergen has three possible notations, all equally valid. Even single-split pergens can have multiple commas that imply different enharmonics.
Chord names andnames, scale namesnames, staff notation
Using pergens, all rank-2 chords can be named using ups and downs, and if needed highs and lows as well. See the ups and downs page for chord naming conventions. The genchain and/or the perchain creates a lattice in which each note and each interval has its own name. The many enharmonic equivalents allow proper chord spelling.
In certain pergens, one spelling isn't always clearly better. For example, in half-4th, C E G A^ and C E G Bbv are the same chord, and either spelling might be used. This exact same issue occurs in 24-edo.
Given a specific temperament, the full period/generator mapping gives the notation of higher primes, and thus of any ratio. Thus JI chords can be named. For example, pajara (2.3.5.7 with 50/49 and 64/63) is (P8/2, P5), half-8ve, with P = vA4 or ^d5, G = P5, and E = ^^d2. The full mapping is [(2 2 7 8) (0 1 -2 -2)], which can also be written [(2 0) (2 1) (7 -2) (8 -2)]. This tells us 7/1 = 8·P - 2·G = 4·P8 - 2·P5 = WWm7, and 7/4 = m7. Likewise 5/1 = 7·P - 2·G = 7/1 minus a half-octave. From this it follows that 5/4 = m7 - ^d5 = vM3. A 4:5:6:7 chord is written C Ev G Bb = C7(v3).
A different temperament may result in the same pergen with the same enharmonic, but may still produce a different name for the same chord. For example, injera (2.3.5.7 with 81/80 and 50/49) is also half-8ve. However, the tipping point for the d2 enharmonic is at 700¢, and while pajara favors a fifth wider than that, injera favors a fifth narrower than that. Hence ups and downs are exchanged, and E = vvd2, and P = ^A4 = vd5. The mapping is [(2 2 0 1) (0 1 4 4)] = [(2 0) (2 1) (0 4) (1 4)]. Because the square mapping (the first two columns) are the same, the pergen is the same. Because the other columns are different, the higher primes are mapped differently. 5/4 = M3 and 7/4 = M3 + vd5 = vm7, and 4:5:6:7 = C E G Bbv = C,v7.
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(P8, P5/2) [7][5] = half-fifth
Chord progressions can be written out by applying ups and downs to the chord roots as needed, e.g. I.v -- vIII.v -- vVI.^m -- I.v. A porcupine (third-4th) comma pump can be written out like so: C.v -- Av.^m -- Dv.v -- [Bvv=Bb^]^m -- Eb^.v -- G.^m -- G.v -- C.v. Brackets are used to show that Bvv and Bb^ are enharmonically equivalent. The equivalence is shown roughly half-way through the pump. Bvv is written first to show that this root is a vM6 above the previous root, Dv. Bb^ is second to show the P4 relationship to the next root, Eb^. Such an equivalence of course couldn't be used on the staff, where the chord would be written as either Bvv.^m or Bb^.^m, or possibly Bb^.vvM = Bb^ Dv F^.
Highs and lows can be added to the score just like ups and downs can. They precede the note head and any sharps or flats. Scores for melody instruments can optionally have them above or below the staff. This score uses ups and downs, and has chord names.
Mizarian Porcupine Overture by Herman Miller (P8, P4/3)
{Mizarian Porcupine Overture.png}
Tipping points and sweet spots
The tipping point for half-octave with a d2 enharmonic is 700¢, 12-edo's 5th. As noted above, the 5th of pajara (half-8ve) tends to be sharp, thus it has E = ^^d2. But injera, also half-8ve, has a flat 5th, and thus E = vvd2. It is fine for two temperaments with the same pergen to be on opposite sides of the tipping point. But if a single temperament "tips over", either the up symbol sometimes means down in pitch, or even worse, the direction of ups and downs for a piece would reverse if the tuning is adjusted slightly. Fortunately, the temperament's "sweet spot", where the damage to those JI ratios likely to occur in chords is minimized, rarely contains the tipping point.
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Every rank-3 pergen can also be identified by its prime subgroup and its pergen number. A similar table could be made for all rank-3 pergens. The 2.3.5 and 2.3.7 subgroups are listed in the section on rank-3 pergens. The 2.5.7 subgroup's unsplit pergen is (P8, M3, ^M2), with ^M2 = 8/7 and ^1 = √256/245 = √(81/80 * 64/63 * 64/63) = about 38¢. The 3.5.7 subgroup's unsplit pergen is (P12, M6, ^M3), with ^M3 = 9/7 and ^1 = √6561/6125 = √[(81/80)3 * (64/63)2] = about 60¢.
Pergen squares
Pergen squaressquares, which were discovered by Praveen Venkataramana, are a
For (P8, P5), the pergen square has 4 notes, shown here with octave numbers (ignore the periods).
C2 -- G2
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gedra
edomapping Staff notation
Highs and lows can be added to the score just like ups and downs can. They precede the note head and any sharps or flats. Scores for melody instruments can optionally have them above or below the staff.
Combining pergens
Tempering out 250/243 creates third-4th, and 49/48 creates half-4th, and tempering out both commas creates sixth-4th. Therefore (P8, P4/3) + (P8, P4/2) = (P8, P4/6). If adding a comma to a temperament doesn't change the pergen, it's a strong extension, otherwise it's a weak extension.
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The LCM of the pergen's two splitting fractions could be called the height of the pergen. For example, (P8, P5) has height 1, and (P8/2, M2/4) has height 4. In single-pair notation, the enharmonic interval's number of ups or downs is equal to the height. The minimum number of ups or downs needed to notate the temperament is half the height, rounded down. If the height is 4 or 5, double-ups and double-downs will be needed.
Credits
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Praveen Venkataramana. Pergen squares are Praveen's creation.