Composing with tablets

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Preface

Perhaps the first thing to note about the use of tablets as an aid to composition is that while the tablet for a given note is always non-unique, it can become unique if we require all of the notes sounding in a particular chord to share the same chord identifier, and this is the most basic of ways to make use of tablets. In other words, if c is a chord identifier in a tablet, a chord can often be notated as the set of notes {note(n1, c), note(n2, c), ..., note(nk, c)}, all with the same chord identifier c. If a piece of music uses chords sharing the same note identifier exclusively, we can completely separate the numbers n, the note skeleton, from the sequence of chord identifers, and change each around at will completely independently.

4et tablets

Let us consider a simple 4edo piece as a starting point:

[0 3 6 9]

[0 3 5 10]

[0 2 5 11]

[0 2 5 12]

[0 3 6 13]

[0 3 5 14]

[0 2 5 15]

[0 2 5 16]

[0 3 6 9]

[0 3 5 10]

[0 2 5 11]

[0 2 5 12]

[0 3 6 13]

[0 3 5 14]

[0 2 5 15]

[0 2 5 16]

[0 3 6 9]

[0 3 5 10]

[0 2 5 11]

[0 2 5 12]

[0 3 6 9]

[0 3 5 10]

[0 2 5 11]

[0 2 5 12]

[0 3 6 9]

[0 3 5 10]

[0 2 5 11]

[0 2 5 12]

[0 3 6 9]

[0 3 5 10]

[0 2 5 11]

[0 2 5 12]

[0 2 5 11]

[0 2 5 11]

[0 3 6 9]

[0 3 6 9]

[0 2 5 11]

[0 2 5 11]

[0 3 6 9]

[0 3 6 9]

[0 2 5 11]

[0 2 5 11]

[0 3 6 9]

[0 3 6 9]

[0 2 5 11]

[0 2 5 11]

[0 3 6 9]

[0 3 6 9]

[0 2 5 16]

[0 2 5 15]

[0 3 5 14]

[0 3 6 13]

[0 2 5 12]

[0 2 5 11]

[0 3 5 10]

[0 3 6 9]

[0 3 6 8]

[0 3 6 8]

[0 3 6 8]

[0 3 6 8]

[0 2 5 8]

[0 2 5 8]

[0 2 5 8]

[0 2 5 8]

This has sixteen measures of four notes each. If to each two measures we attach the chord identifier of a 7-limit 4et tablet, then we may convert all of the 4et notes into 7-limit JI notes by means of the associated chord identifier and therefore by the assoicated tablet. For instance, from the sequence of chords [-1 -1 -1], [0 -1 -1], [0 0 -1], [0 -1 0], [-1 -1 0], [-1 0 -1], [-1 0 0], [0 0 0] we obtain this piece.

This traverses all eight tetrads of the 7-limit tonality diamond, a scale of thirteen notes. But since we have the piece in tablet form, it is easy to transform it in a variety of ways. For instance, we may add [1 1 1] to each chord identifier. This makes for a piece which traverses the stellated hexany, a scale of fourteen notes. The rather surprising increase from thirteen to fourteen notes is possible since each scale has 32 tablets to each octave, but more than one tablet corresponds to each note. The 32 tablets of the diamond, corresponding to thirteen notes, transform to the 32 tablets of the stellated hexany, corresponding to fourteen notes.

This is not, of course, the only way to transform the original tonality diamond piece into one in the stellated hexany. More or less at random we could try converting the chord identifier [a b c] to [-a -c -b]; this sort of transformation will send two related chords to another pair of related chords. Doing this leads to the following piece. But we have perfect freedom to choose our chord identifiers however we like; we could for example use [0 0 0], [0 1 -1], [0 2 -1], [0 2 -2], [0 3 -2], [0 3 -3], [1 3 -3], [2 3 -3]. This modulates upwards by 2401/2400, and leads to the following piece.

Because the keenanismic tablet also makes use of 4et, we can take the same 4et melodic skeleton we started with and add chords for a progression of keenanismic tablets. If we use [1 1 1], [1 1 1], [0 3 1], [1 4 0], [0 5 -1], [1 5 -3], [3 4 -4], [4 4 -6], [5 4 -8], [4 5 -9], [5 5 -11], [4 7 -11], [5 8 -12], [4 9 -13], [5 9 -15], [5 9 -15] as the sixteen keenanismic tetrads for the sixteen measures of the 4et piece, a chord progressing where each chord shares a triad with each preceeding and following, we obtain the following piece. This in theory transposes up 2401/2400 again, but it does very little tuning damage to temper it out, leading to agni temperament. The tuning used here, 284et, tempers out both 2401/2400 and 385/384.

5et tablets

We start with a 5et piece, a simple-minded one being the 62 measures of

[3 0 4], [2 0 4], [2 4 3], [3 4 2], [4 3 2], [1 3 2], [3 2 0], [3 2 0], [0 3 4], [0 2 4], [4 1 3], [4 0 3], [3 1 4], [3 2 4], [2 3 0], [2 3 0], [3 2 0], [2 3 0], [1 4 0], [0 4 1], [1 4 2], [2 4 3], [3 2 4], [3 1 4], [2 3 0], [3 2 0], [4 2 0], [4 1 0], [4 1 0], [4 1 0], [2 3 0], [1 3 0], [3 4 0], [2 3 0], [2 3 0], [1 2 0], [0 2 3], [1 2 3], [3 2 4], [3 2 4], [2 3 4], [3 2 4], [3 2 4], [2 3 4], [2 4 1], [2 1 0], [2 3 0], [2 3 0], [1 0 3], [4 0 3], [4 3 1], [1 3 4], [3 1 4], [2 1 4], [1 4 0], [1 4 0], [0 1 3], [0 4 3], [3 2 1], [3 0 1], [1 2 3], [1 4 3], [4 1 0], [4 1 0], [1 4 0], [4 1 0], [2 3 0], [0 3 2], [2 3 4], [4 3 1], [1 4 3], [1 2 3], [4 1 0], [1 4 0], [3 4 0], [3 2 0], [3 2 0], [3 2 0], [4 1 0], [2 1 0], [1 3 0], [4 1 0], [4 1 0], [2 4 0], [0 4 1], [2 4 1], [1 4 3], [1 4 3], [4 1 3], [1 4 3], [1 4 3], [4 1 3], [4 3 2], [4 2 0], [4 1 0], [4 1 0], [1 0 3], [4 0 3], [4 3 1], [1 3 4], [3 1 4], [2 1 4], [1 4 0], [1 4 0], [0 1 3], [0 4 3], [3 2 1], [3 0 1], [1 2 3], [1 4 3], [4 1 0], [4 1 0], [1 4 0], [4 1 0], [2 3 0], [0 3 2], [2 3 4], [4 3 1], [1 4 3], [1 2 3], [4 1 0], [1 4 0], [3 4 0], [3 2 0], [3 2 0], [3 2 0], [4 1 0], [2 1 0], [1 3 0], [4 1 0], [4 1 0], [2 4 0], [0 4 1], [2 4 1], [1 4 3], [1 4 3], [4 1 3], [1 4 3], [1 4 3], [4 1 3], [4 3 2], [4 2 0], [4 1 0], [4 1 0], [2 0 1], [3 0 1], [3 1 2], [2 1 3], [1 2 3], [4 2 3], [2 3 0], [2 3 0], [0 2 1], [0 3 1], [1 4 2], [1 0 2], [2 4 1], [2 3 1], [3 2 0], [3 2 0], [2 3 0], [3 2 0], [4 1 0], [0 1 4], [4 1 3], [3 1 2], [2 3 1], [2 4 1], [3 2 0], [2 3 0], [1 3 0], [1 4 0], [1 4 0], [1 4 0], [3 2 0], [4 2 0], [2 1 0], [3 2 0], [3 2 0], [4 3 0], [0 3 2], [4 3 2], [2 3 1], [2 3 1], [3 2 1], [2 3 1], [2 3 1], [3 2 1], [3 1 4], [3 4 0], [3 2 0], [3 2 0], [3 0 4], [2 0 4], [2 4 3], [3 4 2], [4 3 2], [1 3 2], [3 2 0], [3 2 0], [0 3 4], [0 2 4], [4 1 3], [4 0 3], [3 1 4], [3 2 4], [2 3 0], [2 3 0], [3 2 0], [2 3 0], [1 4 0], [0 4 1], [1 4 2], [2 4 3], [3 2 4], [3 1 4], [2 3 0], [3 2 0], [4 2 0], [4 1 0], [4 1 0], [4 1 0], [2 3 0], [1 3 0], [3 4 0], [2 3 0], [2 3 0], [1 2 0], [0 2 3], [1 2 3], [3 2 4], [3 2 4], [2 3 4], [3 2 4], [3 2 4], [2 3 4], [2 4 1], [2 1 0], [2 3 0], [2 3 0], [2 3 0], [2 3 0], [2 3 0], [2 3 0], [2 3 0], [2 3 0], [2 3 0], [2 3 0]

This can be rendered thusly. We can give this more much needed pizazz by turning into a meantone piece, for instance by using chord identifiers which change every two measures from identifier 30 down to 0, tuned in 31et. This results in this meantone piece. On the other hand, we can convert it to a 7-limit JI piece by chord identifiers

[0 0 0], [-1 0 0], [-2 0 0], [-3 0 0], [-4 0 0], [-5 0 0], [-5 1 0], [-5 2 0], [-5 3 0], [-5 4 0], [-5 5 0], [-5 6 0], [-5 6 1], [-5 6 2], [-5 6 3], [-5 6 3]

The chord [-5 6 3] has root 4374/4375 rather than 1/1, but note 0 no longer corresponds to it so we end the piece by moving the bass down 9/8.