Tablet: Difference between revisions
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=What is a tablet?= | =What is a tablet?= | ||
By a ''tablet'' (the name by analogy with tablature) is meant a pair [n, c] consisting of an approximate note-number n (an integer) and a chord denoting element c, typically a [http://en.wikipedia.org/wiki/Tuple tuple] of integers, which defines a type of chord up to octave equivalence. Together they define a note in a just intonation group or regular temperament. The representation of the note is non-unique, as for any note there will be a variety of tablets for it depending on the specified type of chord and chord element, but by means of a val or val-like mapping, the number n in the tablet is definable from the note. For a discussion of how tablets can be used as a compositional tool, see [[ | By a ''tablet'' (the name by analogy with tablature) is meant a pair [n, c] consisting of an approximate note-number n (an integer) and a chord denoting element c, typically a [http://en.wikipedia.org/wiki/Tuple tuple] of integers, which defines a type of chord up to octave equivalence. Together they define a note in a just intonation group or regular temperament. The representation of the note is non-unique, as for any note there will be a variety of tablets for it depending on the specified type of chord and chord element, but by means of a val or val-like mapping, the number n in the tablet is definable from the note. For a discussion of how tablets can be used as a compositional tool, see [[Composing with tablets]]. | ||
There is really no better way of defining more precisely what a tablet is than by giving examples, which are considered below. | There is really no better way of defining more precisely what a tablet is than by giving examples, which are considered below. | ||
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==The 7-limit 4et tablet== | ==The 7-limit 4et tablet== | ||
Suppose m0, m1, m2 and m3 are four [[ | Suppose m0, m1, m2 and m3 are four [[monzo]]s denoting four notes of a 7-limit tetrad, either otonal or utonal, with distinct pitch classes, so that all four chord elements are represented. The product of the four notes is represented by the monzo m = m0+m1+m2+m3, and the pitch class for m, which ignores the first coefficient m[1] defining octaves, is represented by the other three, m[2], m[3] and m[4]. This pitch class is uniquely associated to the tetrad, and can be used to name it. If the tetrad is otonal with the root being |* e3 e5 e7>, where the asterisk can be any integer value, then m = |* 4e3+1 4e5+1 4e7+1>. on the other hand, if it is utonal, with the fifth of the chord |* e3 e5 e7> then m = |* 4e3-1 4e5-1 4e7-1>. If we denote the 3-tuple [(m[3]+m[4]-2)/2 (m[2]+m[4]-2)/2 (m[2]+m[3]-2)/2] by [a b c], then if the tetrad is otonal, [a b c] = [e5+e7 e3+e7 e3+e5], whereas if it is utonal [a b c] = [e5+e7-1 e3+e7-1 e3+e5-1]. From this it follows that [a b c] is a triple of integers, and that a+b+c is even if the tetrad is otonal, and odd if the tetrad is utonal. Hence every triple of integers refers uniquely to a 7-limit tetrad; this is the [[The Seven Limit Symmetrical Lattices|cubic lattice of 7-limit tetrads]]. | ||
If t = [n, [a b c]] is a tablet, define u = n - 7a - 4b - 2c. If a+b+c is even, then | If t = [n, [a b c]] is a tablet, define u = n - 7a - 4b - 2c. If a+b+c is even, then | ||
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==The keenanismic tablet== | ==The keenanismic tablet== | ||
This is based on the five [[ | This is based on the five [[keenanismic tetrads]], which are essentially tempered chords tempering out 385/384, plus the otonal and utonal tetrads. If we take the intervals for an otonal tetrad in close position, 5/4-6/5-7/6-8/7, we can permute them in 24 ways. The number of circular permutations, leading to chords which are not simply transpositions, is six. Keenanismic tempering makes all six chords [[Dyadic chord|dyadic]]. in addition the steps 8/7-5/4-8/7-14/11 lead to a keenanismic tempered 35/32-5/4-3/2-12/7 chord. | ||
Taking the product of all four notes of the tetrad and ignoring octaves. we can uniquely associate these chords to no-twos monzos |* w x y>. If the chord is transposed by some quantity with monzo M, the chord product is changed to |* x y z> + 4M, so that the no-twos monzos modulo four represent the types of chords. It's convenient to modify this slightly, and represent the chords by [2-x 2-y 2-z]. We then have the following for the mod four reductions: | Taking the product of all four notes of the tetrad and ignoring octaves. we can uniquely associate these chords to no-twos monzos |* w x y>. If the chord is transposed by some quantity with monzo M, the chord product is changed to |* x y z> + 4M, so that the no-twos monzos modulo four represent the types of chords. It's convenient to modify this slightly, and represent the chords by [2-x 2-y 2-z]. We then have the following for the mod four reductions: | ||
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==The pele tablet== | ==The pele tablet== | ||
This is a tablet for the rank 3 13-limit temperament [[ | This is a tablet for the rank 3 13-limit temperament [[Hemifamity family#Pele|pele]]. It is based on the following 71 chords, in the 5-limit transversal: | ||
chords = [ | chords = [ | ||
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==The meantone add6/9 tablet== | ==The meantone add6/9 tablet== | ||
The meantone add6/9 tablet is based on the [[ | The meantone add6/9 tablet is based on the [[Meantone add6-9 pentad|meantone add6/9 pentad]], which can also be called the add2/9 pentad, the meantone pentatonic scale or Meantone[5]. The tablet is extremely simple, consiting of an ordered pair [n, c], where we have a meantone transversal for the notes defined by u = n-8c, where | ||
<ul><li>If u mod 5 = 0 then</li></ul>note(n, c) = |u/5 c> | <ul><li>If u mod 5 = 0 then</li></ul>note(n, c) = |u/5 c> | ||
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<ul><li>If u mod 5 = 4 then</li></ul>note(n, c) = |(u-4)/5-4 c+3> | <ul><li>If u mod 5 = 4 then</li></ul>note(n, c) = |(u-4)/5-4 c+3> | ||
In all cases <5 8|note(n, c) = n. Tempering | In all cases <5 8|note(n, c) = n. Tempering the Pythgorean transversal by flattening 3 gives, as usual, a meantone tuning. | ||
==The 5et portent tablet== | ==The 5et portent tablet== | ||
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Once again, <5 8 12 14 17|note(n, c) = c. | Once again, <5 8 12 14 17|note(n, c) = c. | ||
The selection of these particular representatives for each of the twelve types of chords is based on each of them having a common | The selection of these particular representatives for each of the twelve types of chords is based on each of them having a common triad—three common notes—in common with the utonal pentad, the first chord in the chords list. The chord representatives do not need to be given in terms of a 2.5.7 subgroup transversal, and a more perspicuous way of expressing the same portent temperament pentads is | ||
<ul><li>JI pentads</li></ul>1-9/8-5/4-3/2-7/4 | <ul><li>JI pentads</li></ul>1-9/8-5/4-3/2-7/4 | ||
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=The 6et tutone tutonic tablet= | =The 6et tutone tutonic tablet= | ||
This tablet is based on the [[ | This tablet is based on the [[tutonic sextad]], which in terms of the 99/98 (Huygens) version of 11-limit meantone consists of a chain of five tones, followed by an augmented second; in other words a {81/80, 126/125, 99/98}-tempered version of 9/8-9/8-9/8-9/8-9/8-8/7, which in terms of notes rather than steps is a tempered 1-9/8-5/4-7/5-11/7-7/4. Using this chord as the basis for harmony puts one in [[Chromatic pairs#Tutone|tutone temperament]], a 2.9.7.11 subgroup temperament, and the sextad can be called Tutone[6], the tutone haplotonic scale. | ||
If the tablet is the ordered pair [n, c] and if u = n-19c, then if i = u mod 6, define note(n, c) = |(u-i)/6-3i 2c+2i>. This gives a 3-limit interval which tempers to a note of tutone satisfying the identity <12 19|note(n, c) = 2n. We can also express this in terms of a subgroup monzo as <6 19|note(n, c) = n, where note(n, c) in subgroup monzo terms is |(u-i)/6-3i c+i>. | If the tablet is the ordered pair [n, c] and if u = n-19c, then if i = u mod 6, define note(n, c) = |(u-i)/6-3i 2c+2i>. This gives a 3-limit interval which tempers to a note of tutone satisfying the identity <12 19|note(n, c) = 2n. We can also express this in terms of a subgroup monzo as <6 19|note(n, c) = n, where note(n, c) in subgroup monzo terms is |(u-i)/6-3i c+i>. | ||
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<ul><li>If u mod 7 = 6, then</li></ul>note(n, [r e3 e5 e7 e11 e13]) = |(u-20)/7 e3 e5 e7+1 e11 e13> | <ul><li>If u mod 7 = 6, then</li></ul>note(n, [r e3 e5 e7 e11 e13]) = |(u-20)/7 e3 e5 e7+1 e11 e13> | ||
On the other hand if r is odd: | On the other hand, if r is odd: | ||
<ul><li>If u mod 7 = 0, then</li></ul>note(n, [r e3 e5 e7 e11 e13]) = |u/7 e3 e5 e7 e11 e13> | <ul><li>If u mod 7 = 0, then</li></ul>note(n, [r e3 e5 e7 e11 e13]) = |u/7 e3 e5 e7 e11 e13> | ||
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=The orwell nonad tablet= | =The orwell nonad tablet= | ||
The [[ | The [[orwell tetrad|orwell nonad]] is the Orwell[9] MOS, considered as an essentially tempered dyadic chord (which it is, in the 15-limit.) While nine notes is a lot of notes for a chord, obviously in practice subchords could be used. | ||
If the tablet is the ordered pair [n, c] and if u = n-11c, then setting i = u mod 9, define a transversal for note(n, c) by determining if i is even or odd, and setting | If the tablet is the ordered pair [n, c] and if u = n-11c, then setting i = u mod 9, define a transversal for note(n, c) by determining if i is even or odd, and setting | ||
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if i is odd. We then have <9 14 21 25|note(n, c) = n. | if i is odd. We then have <9 14 21 25|note(n, c) = n. | ||
[[Category: | [[Category:Chords]] | ||
[[Category: | [[Category:Composition]] | ||
[[Category: | [[Category:Notation]] | ||