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  1. page pergen edited ... Finding an example temperament To find an example of a temperament with a specific pergen, we…
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    Finding an example temperament
    To find an example of a temperament with a specific pergen, we must find the comma(s) the temperament tempers out. To construct a comma that creates a single-split pergen, find a ratio for P or G that contains only one higher prime, with color depth of 1 (i.e. exponent of ±1), of appropriate cents to add up to approximately the octave or the multigen. The comma is the difference between the stacked ratios and the larger interval. For example, (P8/4, P5) requires a P of about 300¢. The comma is the difference between 4⋅P and P8. If P is 6/5, the comma is 4⋅P - P8 = (6/5)4 ÷ (2/1) = 648/625, the diminished temperament. If P is 7/6, the comma is P8 - 4⋅P = (2/1) · (7/6)-4, the quadruple red temperament. Neither 13/11 nor 32/27 would work for P, too many and too few higher primes respectively.
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    interval be X.x. Then n⋅Xn⋅x gens = n⋅I = X⋅M,x⋅M, where M
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    The comma iscan be found from this equation.equation, if n and x are coprime. For example,
    If the pergen's notation is known, an even easier method is to simply assume that the up symbol equals a comma that maps to P1, such as 81/80 or 64/63 (see mapping commas in the next section). Thus for (P8/4, P5), if P = vm3, and ^1 = 64/63, P is 32/27 ÷ 64/63 = 7/6. This method is notation-dependent: (P8/2, P5) with P = vA4 and ^1 = 81/80 gives P = 45/32, but if P = ^4, then P = 27/20.
    Finding the comma(s) for a double-split pergen is trickier. As previously noted, if a pergen's multigen is (a,b), the octave is split into at least |b| parts. Therefore if a pergen (P8/m, (a,b)/n) has m = |b|, it is explicitly false. If so, proceed as if the octave were unsplit: (P8/2, M2/4) requires G ~ 50¢, perhaps 33/32, and the comma is 4⋅G - M2 = (33/32)^4 / (9/8) = (-17, 2, 0, 0, 4).
    (view changes)
    11:33 pm
  2. page pergen edited ... Finding an example temperament To find an example of a temperament with a specific pergen, we…
    ...
    Finding an example temperament
    To find an example of a temperament with a specific pergen, we must find the comma(s) the temperament tempers out. To construct a comma that creates a single-split pergen, find a ratio for P or G that contains only one higher prime, with color depth of 1 (i.e. exponent of ±1), of appropriate cents to add up to approximately the octave or the multigen. The comma is the difference between the stacked ratios and the larger interval. For example, (P8/4, P5) requires a P of about 300¢. The comma is the difference between 4⋅P and P8. If P is 6/5, the comma is 4⋅P - P8 = (6/5)4 ÷ (2/1) = 648/625, the diminished temperament. If P is 7/6, the comma is P8 - 4⋅P = (2/1) · (7/6)-4, the quadruple red temperament. Neither 13/11 nor 32/27 would work for P, too many and too few higher primes respectively.
    Another method: if the generator's cents are known, look on the genchain for an interval that approximates a ratio of color depth ±1. Let the interval be I, and the genspan of this interval be X. Then n⋅X gens = n⋅I = X⋅M, where M is the multigen and M/n is the generator. The comma is found from this equation. For example, suppose (P8, P5/5) has G = 140¢. The genchain is all multiples of 140¢. Looking at the cents, 280¢ is about 7/6. Thus 2G = 7/6, and 10G = 5⋅(7/6) = 2⋅P5. Thus 2⋅P5 - 5⋅(7/6) = 0G = 0¢, and the comma is (3, 7, 0, -5). If the period is split and the generator isn't, use the perchain instead of the genchain. For example, (P8/7, P5) has a period of 141¢. 2 gens = 343¢, about 11/9. Thus 7⋅(11/9) = 2⋅P8, and the comma is (-2, -14, 0, 0, 7).
    If the pergen's notation is known, an even easier method is to simply assume that the up symbol equals a comma that maps to P1, such as 81/80 or 64/63 (see mapping commas in the next section). Thus for (P8/4, P5), if P = vm3, and ^1 = 64/63, P is 32/27 ÷ 64/63 = 7/6. This method is notation-dependent: (P8/2, P5) with P = vA4 and ^1 = 81/80 gives P = 45/32, but if P = ^4, then P = 27/20.
    Finding the comma(s) for a double-split pergen is trickier. As previously noted, if a pergen's multigen is (a,b), the octave is split into at least |b| parts. Therefore if a pergen (P8/m, (a,b)/n) has m = |b|, it is explicitly false. If so, proceed as if the octave were unsplit: (P8/2, M2/4) requires G ~ 50¢, perhaps 33/32, and the comma is 4⋅G - M2 = (33/32)^4 / (9/8) = (-17, 2, 0, 0, 4).
    (view changes)
    11:25 pm

Saturday, June 9

  1. page 23edo edited ... {23edoMavilaMOS.jpg} 23-EDO was proposed by ethnomusicologist Erich von Hornbostel as the re…
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    {23edoMavilaMOS.jpg}
    23-EDO was proposed by ethnomusicologist Erich von Hornbostel as the result of continuing a circle of "blown" fifths of ~678-cent fifths that (he argued) resulted from "overblowing" a bamboo pipe.
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    more details. AlsoAlso note that
    As with 9-EDO, 16-EDO, and 25-EDO, one way to treat 23-EDO is as a Pelogic temperament, tempering out the "comma" of 135/128 and equating three 'acute 4/3's with 5/1 (related to the Armodue system). This means mapping '3/2' to 13 degrees of 23, and results in a 7 notes Anti-diatonic scale of 3 3 4 3 3 3 4 (in steps of 23-EDO), which extends to 9 notes Superdiatonic scale (3 3 3 1 3 3 3 3 1). One can notate 23-EDO using the Armodue system, but just like notating 17-EDO with familiar diatonic notation, flats will be lower in pitch than enharmonic sharps, because in 23-EDO, the "Armodue 6th" is sharper than it is in 16-EDO, just like the Diatonic 5th in 17-EDO is sharper than in 12-EDO. In other words, 2b is lower in pitch than 1#, just like how in 17-EDO, Eb is lower than D#.
    However, one can also map 3/2 to 14 degrees of 23-EDO without significantly increasing the error, taking us to a 7-limit temperament where two 'broad 3/2's equals 7/3, meaning 28/27 is tempered out, and six 4/3's octave-reduced equals 5/4, meaning 4096/3645 is tempered out. Both of these are very large commas, so this is not at all an accurate temperament, but it is related to 13-EDO and 18-EDO and produces MOS scales of 5 and 8 notes: 5 5 4 5 4 (the "anti-pentatonic") and 4 1 4 1 4 4 1 4 (the "quarter-tone" version of the Blackwood/Rapoport/Wilson 13-EDO "subminor" scale). Alternatively we can treat this temperament as a 2.9.21 subgroup, and instead of calling 9 degrees of 23-EDO a Sub-"4/3", we can call it 21/16. Here three 21/16's gets us to 9/4, meaning 1029/1024 is tempered out. This allows us to treat a triad of 0-4-9 degrees of 23-EDO as an approximation to 16:18:21, and 0-5-9 as 1/(16:18:21); both of these triads are abundant in the 8-note MOS scale.
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    Hod
    3 1 3 1 3 1 3 1 3 1 3
    Palestine 11
    2 2 2 2 1 2 2 2 1 2 2 2 1
    Mode Tishrei
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    2 2 1 2 2 1 2 2 1 2 2 1 2 1
    2 1 2 2 1 2 2 1 2 2 1 2 2 1
    Palestine 14
    1 1 1 4 1 1 1 1 4 1 1 1 1 4
    2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
    2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1
    Palestine 17
    2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 2 1 1 1
    Books
    ...
    Instruments
    {Icositriphonic_Guitar.PNG}
    An Icositriphonic23-EDD 8-string Guitar. 23-ED2 Guitar by Ron Sword.
    {Icositriphonic_Bass.JPG}
    An Icositriphonic P-Bass. 23-ED2

    {Bajo 23-EDD.jpg}
    23-EDD
    Bass by Tútim Dennsuul Wafiil.
    {Icositriphonic Gitar 1.png}
    An Icositriphonic 'Hendrix' TravelGuitar. 23-ED2 'Gitar'
    Osmiorisbendi.
    {Baritarra 23-EDD.jpg}
    23-EDD Baritar
    by Tútim Dennsuul Wafiil.Osmiorisbendi.
    {Icositritar 1.png}
    The 'Icositritar', 23-ED223-EDD 5-string Acoustic
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    Tútim Dennsuul Wafiil.Wafiil (RIP).
    {Teclado Icositrifónico.PNG}
    An illustrative image of a 23-ED2Illustrative 23-EDD Keyboard
    Chris Vaisvil made a do it yourself 23 edo electric guitar out of less than $50 of material. Here he is playing it.
    {playing.jpg}
    (view changes)
    6:59 pm
  2. file Bajo 23-EDD.jpg uploaded
    6:50 pm
  3. page OrientacionGeneral edited ... La música suena muy bien monofónicamente o con armonías simples, pero ahora tenemos un problem…
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    La música suena muy bien monofónicamente o con armonías simples, pero ahora tenemos un problema: en un sistema de afinación justa podemos tocar en Do mayor y sonará de maravilla, pero si tocamos en Solb(por ejemplo), los tonos no están acomodados a la serie armónica, entonces será necesario afinar estas fracciones para Solb; es por este detalle que se tempera un instrumento, osea se mueve un poco la vibración resultante de las fracciones para que las tonalidades tengan más acercamiento a parciales en común, esto debe entenderse como una renuncia obligatoria a los parciales exactos, pero los cambios suelen ser tan sutiles que apenas se notan, y se notan más o menos en diferentes tonalidades. El proceso de estudio es largo y escabroso, pues los sonidos se pierden en el aire y es difícil poder entender cómo afinaban exactamente los griegos o romanos, en fin, para la cultura occidental se considera un punto de inflexión la época de Bach, por haber podido sintetizar un temperamento y una música que se acomodaba para cada una de las tonalidades del teclado (con el Clave Bien Temperado, efectivamente), eso sí hay que recordar que la afinación de Bach no se sabe hoy con exactitud, él no dejó prosa al respecto y lo más parecido que conocemos es la afinación Werckmeister III, y justamente sabemos esto porque la obra de Bach es una especie de apoteosis de los avances en el contrapunto, la construcción de claves, el nacer del piano, y las matemáticas simples que facilitaron las maneras de temperar. Tengamos en cuenta también que a Bach no lo interpretamos hoy con esta afinación.
    Temperamentos Equitativos (INGLÉS FALTA CREAR PÁGINA EN ESPAÑOL): cada uno de ellos con una única colección de intervalos totalmente transportables y modulables. Pueden ser tratados como Temperamentos o no.
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    mandolinas, charangos, bandurrias, ukeleles, etc., tienen
    En la práctica común global e histórica, el olvidado uso de temperamentos históricos (INGLÉS FALTA CREAR PÁGINA EN ESPAÑOL):
    Música Turka; desarrollo del Makam (INGLÉS FALTA CREAR PÁGINA EN ESPAÑOL)
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    Música Gamelán; Cultura Indonesia;
    Griego antiguo, bizantino (INGLÉS FALTA CREAR PÁGINA EN ESPAÑOL); notación Bizantina (INGLÉS FALTA CREAR PÁGINA EN ESPAÑOL)
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    (Griego ξένος) "extrangero"."extranjero".
    Temperamentos regulares (INGLÉS FALTA CREAR PÁGINA EN ESPAÑOL): (incluyendo Temperamentos Lineales): una práctica de siglos de antigüedad que ha experimentado un levantón matemático, en la que la Entonación Justa es selectiva y periódicamente desafinada de diversas maneras, para satisfacer mejor una variedad de los deseos de composición, como la modulación armónica o construcción de ciertas propiedades en las escalas.
    Momento de Simetría(MOS) (INGLÉS FALTA CREAR PÁGINA EN ESPAÑOL): un medio de iteración de un solo intervalo que resulta generativa, con un módulo un intervalo periódico, para producir escamas de dos paso a los tamaños. Creación de Erv Wilson.
    (view changes)
    5:12 pm
  4. msg My well temperament message posted My well temperament 1 1.0534979424 1.1203511866 1.1851851852 1.25 1.3333333333 1.40625 1.5 1.5802469136 1.6735…
    My well temperament
    1
    1.0534979424
    1.1203511866
    1.1851851852
    1.25
    1.3333333333
    1.40625
    1.5
    1.5802469136
    1.6735823752
    1.7777777778
    1.875

    My C tuning is 264, which would also assume a G tuning of 396.

    Fifths:

    Ab–Eb: just
    Eb–Bb: just
    Bb–F: just
    F–C: just
    C–G: just
    G–D: -1÷3 comma
    D–A: -1÷3 comma
    A–E: -1÷3 comma
    E–B: just
    B–F#: just
    F#–C#: -schisma
    C#–G#: just

    The C–E and G–B major thirds are also just, as well as the E–G minor third.
    4:06 am
  5. msg I like this. message posted I like this. It's a nice system in my opinion. It allows to do pitch slides more smoothly than in any 12–tone sy…
    I like this.
    It's a nice system in my opinion. It allows to do pitch slides more smoothly than in any 12–tone system, such as 12–edo or well temperament. It's quite natural.
    3:46 am

Thursday, June 7

  1. 2:43 am

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