Tablet: Difference between revisions

There is really no better way of defining more precisely what a tablet is than by giving examples, which are considered below.

If we want a way to uniquely denote the octave-equivalent class of a 5-limit major triad, we can simply give the root of the chord, ignoring octaves. In terms of monzos that means a major triad would be denoted |* e3 e5>, where the asterisk can be anything. Since the asterisk is not being used to impart information, we can use it to distinguish major from minor triads. So we might decide that [r e3 e5] defines a major triad with root given by |* e3 e5> when r is even, and a minor triad when r is odd.

Then if c is a tablet, note(n, c) gives the note belonging to the triad denoted by c, satisfying <3 5 7|note(n, c) = n.

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 a+b+c is odd, then define note(t) as -note(-n, [-1-a -1-b -1-c]). Then note(t) is the note defined by the tablet t. If t = [n, w], where w is a 3-tuple, then [note([n, w)), note(n+1, w), note(n+2), w), note(n+3, w)] is a 7-limit tetrad, and <4 6 9 11|note(n, t) = n.

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.

If a and b are two allowable triples of integers representing tetrads (so that their reduction modulo four is one of the seven allowed types) then we may derive a [http://mathworld.wolfram.com/PositiveDefiniteQuadraticForm.html positive definite quaratic form] on 3-tuples [x y z] by Q(x, y, z) = 4x^2 + 3y^2 + 3z^2 + 4xy + 4xz + 2xy, such that Q(a-b) = 8 if a and b are 3-tuples representing tetrads with a common triad. The square root √Q(a-b) is a Eulidean measure of distance, and √8 is the minimum distance between tetrad 3-tuples. Such 3-tuple pairs a minimum distance apart share at least an interval in common, and usually a triad, and are therefore closely related.

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:

14/11-13/11-13/11-9/8]]:

If we define u = n - 9a - 5b - 3c then supposing a+b+c is even,

Once again, if a+b+c is odd, then define note(t) as -note(-n, [-1-a -1-b -1-c]). If t = [n, w], where w is a 3-tuple, then [note([n, w)), note(n+1, w), note(n+2), w), note(n+3, w), note(n+4), w)] is a complete 9-odd-limit pentad, where <5 8 12 14|note(n, t) = n.

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

In all cases <5 8|note(n, c) = n. Tempering the Pythgorean transversal by flattening 3 gives, as usual, a meantone tuning.

This is based on the following twelve chords, which are expressed in terms of the 2.5.7 transversal of the 11-limit rank three temperament portent, which tempers out 385/384, 441/440 and hence also 1029/1024 and 3025/3024.

Each of the other eleven chords shares a triad with the first, otonal, pentad, and all of the chords can be related by an infinite but locally finite graph by drawing an edge between chords with a common triad.

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>.

Let <r e3 e5 e7 e11 e13| denote an otonal 13-limit septad with root given by |* e3 e5 e7 e11 e13> when r is even, which in close position is 9/8-5/4-11/8-3/2-13/8-7/4-2. If r is odd, let it denote a utonal pentad which in close position is 12/11-6/5-4/3-3/2-12/7-24/13-2

<7 11 16 20 24 26|note(n, [r e3 e5 e7 e11 e13]) = n.

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 i is odd. We then have <9 14 21 25|note(n, c) = n.

[[Category:Chords]]

[[Category:Composition]]

[[Category:Notation]]