Armodue Theory of 16EDO

(summary translation from the Italian language site Armodue )

Referring not only to the 16-tone equal temperament, but also to half-equal and Lou Harrison's Just intonation 16 note scale, the natural octave division by Andrián Pertout and the 16-to-31 overtone scale, Armodue has been proposed as a new notation and theory system.

Desiring to make the approach to Armodue as easy as possible, but conscious that they had to give new names to the notes that constitute the system, the Italian creators of the Armodue system named them numbering from 1 to 9:

1, 1#, 2, 2#, 3, 3#, 4, 5, 5#, 6, 6#, 7, 7#, 8, 8#, 9

Consequently, the interval between a note at frequency n and other at frequency 2n is called a tenth or decave.

The basic semitone of Armodue, whatever concrete temperament is used, is always called eka (from Sanskrit eka: one, unit). In the chromatic Armodue scale, one eka always corresponds to the interval between any two consecutive notes.

For composing in Armodue it's useful to use a tetragram (staff with 4 lines):

The notes without accidentals form a Mavila superdiatonic scale (7L 2s).

If for the execution of a musical piece we need to write on two or more tetragrams, the notes will be written in the same way for every tetragram.

In other words, the "1" note will be written immediately under the first line in every tenth.

In Armodue we have only a numeric clef, that shows us the tenth:

The clefs 1,2,3... refers to the tenths: first, second, third...

So, in the illustrated example above, the first tetragram (from top) refers to the 3rd tenth (central tenth, corresponding to the octave C3-C4), the second tetragram to the 5th tenth and the third to the 2nd. If we need to write simultaneously on several staves, we will draw normal braces.

The keyboard conceived by the Armodue authors has the same disposition as Goldsmith's one (except the curvature):

The white keys, corresponding to the notes without accidentals in the notation system, form again a Mavila superdiatonic scale (7L 2s).

Dot-Pattern inlays, by Armodue

Armodue just intonation - connection to Lou Harrison and Andrián Pertout

Armodue Just Intonation is nothing else than the 16-note scale by Lou Harrison based on simple ratios and pure intervals. In this scale there are practically the same twelve intervals of the "natural" or "Zarlinian" scale (the semitone 16/15, the minor wholetone 10/9, the minor third 6/5, the major third 5/4, the pure forth and fifth 4/3 and 3/2, the minor sixth 8/5, the major sixth 5/3, the minor seventh that is the complement of the minor wholetone 9/5, the major seventh 15/8) in addition to the tritone and four other intervals that are based on the harmonic seventh (the ratios 8/7, 7/6, 12/7, 7/4).

The composer Andrián Pertout used a very similar scale in his composition "Sonus dulcis" (article on archive.org), namely the following (according to Armodue):

Armodue note Interval Ratio Cents
1 unison 1/1 0
1# major half-tone 16/15 112
2 minor tone 10/9 182
2# major tone 9/8 204
3 minor third 6/5 316
3# major third 5/4 386
4 fourth 4/3 498
5 harmonic tritone 45/32 590
5# cyclic tritone 64/45 610
6 fifth 3/2 702
6# diminished sixth 8/5 814
7 harmonic sixth 5/3 884
7# harmonic minor seventh 7/4 969
8 minor seventh 16/9 996
8# minor seventh 9/5 1018
9 major seventh 15/8 1088

Semi-equalized Armodue

One step of 16edo (75 cents) is nearly equal to two steps (2\31) of 31edo (77.42 cents). If we take the latter as a base, we get semi-equalized Armodue. In this temperament there is inevitably a smaller microtone (eka between the notes '7#' and '8'). leading to the 16 note MOS Valentine[16] of valentine temperament. Similarly we might use three steps of 46edo, 3\46, 78.26 cents, or five steps of 77edo, 77.92 cents.

Semi-equalized Armodue provides a balance between the symmetry of the equalized system and the purity of natural intervals: intervals of semi-equalized Armodue are very pure, and at the same time it preserves the symmetry of the equalized system and its interval sizes almost unchanged.

Armodue note cents (16edo) cents (semi-equalized Armodue based on 31edo)
1 0 0
1# 75 77.42
2 150 154.84
2# 225 232.26
3 300 309.68
3# 375 387.10
4 450 464.52
5 525 541.94
5# 600 619.35
6 675 696.77
6# 750 774.19
7 825 851.61
7# 900 929.03
8 975 967.74
8# 1050 1045.16
9 1125 1122.58