A.D. FOKKER
The lattice. - Denomination of notes by lettered symbols and additional signs. - Unison vectors. - The use of unison vectors for simplifying notation. - Periodicity blocks. - Complementary scales of twelve semitones. - Complementary scales of twenty-two sruti's. - Scales of nineteen tritotones. - Scales of thirty-one diëses. - Scale of fourty-one supracommas. - Seales of fifty-three commas. - Conversion of vectors into numerical factors - Table of unison vectors, microsons and minisons.
The lattice
Starting from an original pitch level, by adding and subtracting fundamental
pitch intervals, an infinite variety of musical notes will be produced.
For fundamental intervals I shall take the octave, representing the
harmonic frequency ratio 2/1, the perfect fifth with the ratio 3/2, the
perfect major third, ratio 5/4, and the concordant perfect seventh, ratio 7/4.
By common general agreement all notes differing by an arbitrary number
of octaves only, are considered as unison, and as one and the same note.
Thus we are left with only a threefold variety of notes. A note will be
defined by the numbers, positive or negative, of the respective intervals
constituting the note. It is quite natural to visualize these three numbers
by coordinates in a lattice, thus making a harmonic note-lattice. The
nearest part around the origin (0,0,0) will comprise 3^{3} = 27 notes,
defined by
-1 1 -1 0 1 -1 1 1 -1 -1 0 -1 0 0 -1 1 0 -1 -1 -1 -1 0 -1 -1 1 -1 -1 -1 1 0 0 1 0 1 1 0 -1 0 0 0 0 0 1 0 0 -1 -1 0 0 -1 0 1 -1 0 -1 1 1 0 1 1 1 1 1 -1 0 1 0 0 1 0 0 1 -1 -1 1 0 -1 1 1 -1 1The axis of the fifths is from left to right. The axis of the major thirds is in the plane of the paper, from below to above. The axis of the sevenths runs from back to front.
Denomination of notes by lettered symbols and additional signs. , , /, \, , , , .
In music the usual notation is by simple letters for notes, one fifth apart. Thus we have a correspondence as follows.
-3, 0, 0 | -2, 0, 0 | -1, 0, 0 | 0, 0, 0 | 1, 0, 0 | 2, 0, 0 | 3, 0, 0 |
f | c | g | D | a | e | b |
The note D obviously is a note of symmetry. I shall always put it as such. For denomination of perfect major and minor thirds additional signs, the so-called sharps and flats are in use. Thus we have
-3,-1, 0 | -2,-1, 0 | -1,-1, 0 | 0,-1, 0 | 0, 1, 0 | 1, 1, 0 | 2, 1, 0 | 3, 1, 0 |
d-flat | a | e | b | f-sharp | c | g | d |
A column of successive major thirds is written as shown.
0, 3, 0 | c |
0, 2, 0 | a |
0, 1, 0 | f |
0, 0, 0 | D |
0,-1, 0 | b |
0,-2, 0 | g |
0,-3, 0 | e |
Care must be taken to avoid an ambiguity, in that not only 0,1,0, one major
third above D, bears the name f-sharp, but the fifth 4,0,0 too,
one place beyond b (3,0,0) is commonly called f-sharp. The
latter, however, is sharper than the former, by one syntonic comma, the
fraction 81/80. Therefore it is appropriate for 4,0,0 to use an additional
commatic sign, a stroke sloping upward, /, and to name it
f-sharp-comma-up, /f. Likewise
there is a difference of a syntonic comma between b-flat for 0,-1,0 and
b-flat-comma-down, -4,0,0, or \b.
Similar discriminations are to be made between \e and
e, or between
/c and c.
Again, discrimination is necessary between \e and e, or c
and
/c, \f and f. The rule will be that the commatic stroke is
added to that
homonym of a pair, which in the lattice is farther away from the centre D,
reckoned by the number of steps required for coming from D. Because
3 + 2 is more than 1 + 3, therefore /f =
(3,2,0) has the commatic sign,
and f=(-1,3,0) is without stroke.
The concordant perfect sevenths necessitate the adoption of some more
additional signs. The seventh above D is a trifle flatter than both
c and /c.
It is 63/64 flatter than c and 35/36 flatter than /c.
We choose the sign
invented by Giuseppe Tartini (1752), a hook pointing down, and symbolising
something like a half flat, c. We call the note
c-half-flat. Again, the
seventh of c (-2,0,0) is a trifle flatter than both
b-flats (-4,0,0) and (0,-1,0).
We call it b-flat-and-half. The name of the seventh of
c-half-flat, i.e. the
second seventh of D, consequently must be
b-flat-and-half-and-half.
The subseventh below D is a trifle sharper than both e's.
We propose
to call it e-half-sharp and to use an additional sign suggesting a sharp
with one stroke only, e. The subseventh of e is
a trifle
sharper than f.
It will be called f-sharp-and-half. Consequently the second subseventh
of D must bear the name f-sharp-and-half-and-half. Such
considerations lead to the following correspondences.
-3, 0,-1 | -2, 0,-1 | -1, 0,-1 | 0, 0,-1 | 0, 0, 1 | 1, 0, 1 | 2, 0, 1 | 3, 0, 1 |
g | d | a | e | c | g | d | a |
g-half-sharp | d-half-sharp | a-half-sharp | e-half-sharp | c-half-flat | g-half-flat | d-half-flat | a-half-flat |
Again, a row of consecutive sevenths:
0, 0, 3 | 0, 0, 2 | 0, 0, 1 | 0, 0, 0 | 0, 0,-1 | 0, 0,-2 | 0, 0,-3 |
a | b | c | D | e | f | g |
Thus there is a row of six sevenths from a-flat-and-half-and-half-and-half
to g-sharp-and-half-and-half-and-half.
Obviously here the symbols by numbers offer some ease of definition.
By the combination of thirds and sevenths curious mixtures of sharps and flats
may appear. The note (0,2,0) must be called a-sharp. The seventh (0,2,1)
consequently will be g-sharp-half-flat. Another seventh added makes
(0,2,2) to be f-sharp-half-flat-and-half, f. One would like to substitute simply f for
that.
We shall presently return to similar questions after having seen what
unison vectors are.
Unison vectors
All pairs of notes differing by octaves only are considered unisons.
There are other pairs of notes which in musical practice are considered
to be unisonous.
Three major thirds, from the origin 0,0,0 lead to a note 0,3,0. On the
ordinary key-instruments this note is taken to be identical with the
octave 0,0,0. It should be 125/128 flat.
As a rule no attention is paid to the difference of c (-2,0,0) and
/c (2,-1,0).
But the difference is a syntonic comma, vid. 81/80.
Again, twelve consecutive fifths are supposed to lead to a note, seven
octaves up. Thus 12,0,0 is taken to be unison with 0,0,0. Yet the difference
is 531441/524288, or 74/73 sharp, the pythagorean comma. We incidentally
met a-flat-and-half-and-half-and-half, 0,0,3, which practically is
unison with g (-1,0,0). The difference is (1,0,3), that is 1029/1024,
or 206/205 sharp.
Again, there is b-sharp (2,2,0) which most often is identified with
c-half-flat
(0,0,1), because the difference (2,2,-1) is no more than 225/224 sharp.
The vectors, which in the harmonic note lattice connect such notes
taken for unisons will be called unison vectors. In the last section quite
a number of unison vectors have been tabulated.
Given two vectors, accepted as unison vectors, it is obvious that their
sum too must be accepted to be a unison vector. Thus, adding (1,0,3) +
(2,2,-1), their sum (3,2,2) must be a unison vector too. It represents the
residual fraction 108/107. Subtracting (12,0,0) - (4,-1,0) = (8,1,0) makes
the well known schisma, with a residual fraction 800/799.
Having a unison vector, by successive multiplying one gets a series of
unison notes, and thus a periodicity in the harmonic note lattice.
Accordingly, having three different unison vectors, by a three-dimensional
periodicity, blocks will be formed. The whole lattice will be divided in
countless periodicity blocks, each of them representing in a way the whole
of the lattice. Obviously there is a considerable economy in handling the
finite number of notes within a periodicity block instead of the numberless
notes of the complete lattice.
The use of unison vectors to simplify notation in practice
Giuseppe Tartini invented already a special sign for contracting one
half-flat with a flat, making one flat-and-half, .
That is a matter of definition. Accordingly we also contract a half-sharp
with a sharp, writing .
In practice some unison vectors may be useful to simplify the notation
by substituting for the sevenths other notes which require flats and sharps
only.
For substituting one seventh we have
(2,2,-1) = 225/224, (1,5,1) = 738/737, and (-7,4,1) = 4375/4374.
For substituting two sevenths we have
(1,-3,-2) = 323/322, and (0,-5,2) = 285/284.
For substituting three sevenths we have
(1,0,3) = 206/205, and (-7,-1,3) = 268/267.
The shifts produced by these substitutions amount to 2/5 comma at
most. Here are some examples.
For eliminating one seventh:
( 2, 2,-1) | + ( 0, 0, 1) | = ( 2, 2, 0), | or 225/224 | · c = b, |
(-2,-2, 1) | + ( 0, 0,-1) | = (-2,-2, 0), | or 224/225 | · e = f, |
( 2, 2,-1) | + (-2,-1, 1) | = ( 0, 1, 0), | or 225/224 | · g = f, |
(-2,-2, 1) | + ( 2, 0,-1) | = ( 0,-2, 0), | or 224/225 | · f = g, |
( 2, 2,-1) | + ( 0, 2, 1) | = ( 2, 4, 0), | or 225/224 | · g = f, |
( 1, 5, 1) | + ( 0, 0,-1) | = ( 1, 5, 0), | or 738/737 | · e = d, |
(-1,-5,-1) | + ( 1, 0, 1) | = ( 0,-5, 0), | or 737/738 | · g = a, |
(-1,-5,-1) | + ( 0, 2, 1) | = (-1,-3, 0), | or 737/738 | · g = a, |
(-7, 4, 1) | + ( 0, 0,-1) | = (-7, 4, 0), | or 4375/4374 | · e = \\e, |
( 7,-4,-1) | + ( 0, 2, 1) | = ( 7,-2, 0), | or 4374/4375 | · g = //g, |
( 1,-3,-2) | + ( 0, 1, 2) | = ( 1,-2, 0), | or 323/322 | · d = d, |
(-1, 3, 2) | + ( 0, 0,-2) | = (-1, 3, 0), | or 322/323 | · f = f, |
( 1,-3,-2) | + ( 0, 2, 2) | = ( 1,-1, 0), | or 323/322 | · f = f, |
( 0,-5, 2) | + ( 0, 0,-2) | = ( 0,-5, 0), | or 285/284 | · f = a, |
( 0, 5,-2) | + ( 0, 2, 2) | = ( 0, 7, 0), | or 284/285 | · f = d, |
( 1, 0, 3) | + ( 0, 0,-3) | = ( 1, 0, 0), | or 206/205 | · g = a, |
( 1, 0, 3) | + (-2,-1,-3) | = (-1,-1, 0), | or 206/205 | · d = e, |
(-7,-1, 3) | + ( 0, 0,-3) | = (-7,-1, 0), | or 268/267 | · g = \\b, |
(-7,-1, 3) | + (-2,-1,-3) | = (-9,-2, 0), | or 268/267 | · d = \\f, |
Periodicity blocks
In a previous paper^{*)} I dealt with the periodicity meshes, formed by unison vectors in the two-dimensional plane lattice of perfect fifths and major thirds. I dwelt on five kinds of selections by various choices of these vectors.
| 0 -3 | | 4 -1 | | 4 -1 | | 4 -1 | 12 = | | = | | 19 = | | = | | | 4 -1 | | 8 1 | | -1 5 | | -5 6 | | 4 2 | | 4 2 | | 4 -1 | | 1 -8 | 22 = | | = | | 31 = | | = | | | -1 5 | | 3 7 | | 3 7 | | 4 -1 | | 8 1 | | 8 1 | | 8 1 | | 8 1 | 41 = | | = | | 53 = | | = | | | -1 5 | | 7 6 | | -5 6 | | 3 7 |The investigation will now be extended to an equal number of periodicity blocks in the three-dimensional note lattice, including perfect sevenths. These blocks too will contain 12, 19, 22, 31, 41, 53 notes respectively, by various choices of unison vectors. Here is the survey
| 4 2 0 | 12 = | 4 -3 2 | | 2 2 -1 | | 4 -1 0 | | 4 -1 0 | | -7 -1 3 | 19 = | 2 2 -1 | = | 2 2 -1 | = | 2 2 -1 | | -3 3 1 | | -7 4 1 | | 5 -6 0 | | -1 3 2 | 22 = | -7 -1 3 | | 2 2 -1 | | 4 -1 0 | | 4 -1 0 | | 1 5 1 | 31 = | 2 2 -1 | = | 0 3 5 | = | -1 -2 4 | | 1 0 3 | | 1 0 3 | | 1 -3 -2 | | -2 -2 1 | 41 = | -7 -1 3 | | 4 -3 2 | | 4 -3 2 | | -7 4 1 | | -7 4 1 | 53 = | 2 2 -1 | = | 1 -3 -2 | = | 1 5 1 | | -1 3 2 | | 3 0 4 | | 1 -3 -2 |For investigating the contents of a periodicity block I project it on a plane of fifths and major thirds, adjusting it in such a way, that the centre of the block lies in one of the notes of the lattice. That note will be chosen as the origin, 0,0,0. It is to become the centre of symmetry, and consequently will be called D. It is not difficult to find the projections of the sections of consecutive planes parallel to the plane of fifths and major thirds, and to see what notes have their projections inside these projected sections.
| 4 2 0 | | -1 3 2 | 12 = | 4 -3 2 | 22 = | -7 -1 3 | | 2 2 -1 | | 2 2 -1 |Fig. 1. Three-dimensional scale of semitones. Among the 15 notes shown, there are three unison pairs. These have been indicated by similar markings.
Fig. 2. Three-dimensional scale of sruti's. Vertices of the periodicity block have been marked by small circles, and the edges of the block are shown. There are unison pairs. Where both partners in such a pair are visible, they have got similar markings.
Complementary scales of twelve semitones
The periodicity block is defined by unison vectors (4,2,0), residual fraction 88/89, (4,-3,2), residual fraction 128/129, and (2,2,-1), residual fraction 225/224. We find a collection of 15 notes, as shown in the diagram. Among them there are two unison pairs, linked up by (4,-3,-2) and one unison pair, linked up by (4,2,0). Accordingly, there are two complementary scales, which together have a centre of symmetry which is not a centre of symmetry of any of the two scales.
0 | 0, 0, 0 | D | D | 0, 0, 0 | 0 | |||
18/19 | 18/19 | |||||||
1 | 3, 1, 0 | /d | /d | 3, 1, 0 | 1 | |||
16/17 | 16/17 | |||||||
2 | 0,-2, 1 | f | f | 0,-2, 1 | 2 | |||
18/19 | 16/17 | |||||||
3 | 3,-1, 1 | /f | (128/129) | e | -1, 2,-1 | 3 | ||
13/14 | 15/16 | |||||||
4 | -2, 1,-1 | \f | f | -2, 1,-1 | 4 | |||
20/21 | 20/21 | |||||||
5 | -1, 0, 0 | g | g | -1, 0, 0 | 5 | |||
18/19 | 15/16 | |||||||
6 | 2, 1, 0 | g | (88/89) | a | -2,-1, 0 | 6 | ||
15/16 | 18/19 | |||||||
7 | 1, 0, 0 | a | a | 1, 0, 0 | 7 | |||
20/21 | 20/21 | |||||||
8 | 2,-1, 1 | b | /b | 2,-1, 1 | 8 | |||
15/16 | 13/14 | |||||||
9 | 1,-2, 1 | c | (128/129) | \b | -3, 1,-1 | 9 | ||
16/17 | 18/19 | |||||||
10 | 0, 2,-1 | b | b | 0, 2,-1 | 10 | |||
16/17 | 16/17 | |||||||
11 | -3,-1, 0 | \d | \d | -3,-1, 0 | 11 | |||
18/19 | 18/19 | |||||||
12 | 0, 0, 0 | D | D | 0, 0, 0 | 12 |
Complementary scales of twenty-two sruti's
I do not claim that there is a closer correspondence of these notes with the sruti's of India beyond the number of twenty-two in an octave. The unison vectors chosen are (-1,3,2), fraction 322/323, (7,1,-3), fraction 267/268, and (2,2,-1), fraction 225/224. If the centre of the periodicity block is chosen in a note, D, there are eight unison notes in the vertices, and there are ten unison pairs in two opposite faces of the block. In these pairs the unison ratio is 224/225. Here follow the scales.
If we try to make a primary tetrad 4 : 5 : 6 : 7 : 8, we do so with intervals with 7 + 6 + 5 + 4 sruti's, e.g. D : f : a : c : D, then we find that f and a belong to different scales, and instead of c the scales have in common /b, which is 133/132 sharp, 4/5 of a comma.
Scales of nineteen tritotones
Three variants will be shown. Three periodicity blocks have been formed by combination of the unison vectors here shown, with the implied fractions.
H: 4,-1, 0 81/80 K: 4,-1, 0 L: -7,-1, 3 268/267 2, 2,-1 225/224 2, 2,-1 2, 2,-1 -3, 3, 1 80/79 -7, 4, 1 4375/4374 5,-6, 0 213/267
Fig. 3. Three scales of 19 tritotones. Two of them lie in a single layer. The notes unison with the centre D have been indicated with dummy cubes.
The scales are not complementary. Each scale has a proper centre of symmetry. For comparison's sake they have been tabulated together.
Looking at the diagrams, we see that all variants show a considerable number
of perfect minor thirds, where two cubes share a common edge.
Both the variants H and L have 12 perfect minor thirds, and variant K
has even 16 perfect minor thirds. That is why tritotone equal temperament
is especially good in minor thirds.
Again, we see that variant H shows 19 × 2 + 4 × 5 = 58 bare cube
faces. For variant K this number is 19 × 2 + 7 × 2 + 9 × 2
= 70. In the variant L only 8 faces out of a total of 19 × 6 = 114 are
shared by neighbouring cubes. Therefore 53 intervals are not perfect
representations of the fundamental intervals, in L.
None of the three variants possesses a single perfect seventh.
Scales of thirty-one diëses
I have taken three variants, from three periodicity blocks, which have been formed by the unison vectors mentioned below with their respective representative fractions.
M: 4,-1, 0 81/80 N: 4,-1, 0 P: 1, 5, 1 738/737 2, 2,-1 225/224 0, 3, 5 565/564 1,-3,-2 323/322 1, 0, 3 206/205 1, 0, 3 -1,-2, 4 2401/2400
Fig. 4. Three scales of 31 diëses. In scale M the central cube is not visible. In the scales N and P the centres have heen marked with D.
The scales are symmetrical in themselves. Therefore it will be enough to give the notes and intervals from 0 to 16 only. The sixteenth note being the inverse of the fifteenth, the following notes are the symmetrical inverse notes of the preceding ones, and the steps follow in the reverse order.
In the scale N (central column) the steps alternate quite regularly between
one seventh of a tone (63/64) and one quarter tone (30/31 or 35/36)
One were tempted to say: between major and minor diëses, were it not,
that the name minor diësis is already in use for 128/125, very near 43/42.
In the scale P the steps evenly are all near one-fifth of a tone, between
40/41 and 49/50.
In the scale M the steps also range from one quarter tone to one seventh
of a tone, but irregularly and with intermediate values.
Looking at the diagrams we find that in diagram M we see, from front
and from back, 2 × 11 = 22 bare cube faces. That means that we shall
find 31 - 11 = 20 perfect sevenths.
From left and from right we find 2 × 9 = 18 bare cube faces. Thus there
will be 31 - 9 = 22 perfect fifths.
From above and from below we see another 2 × 15 = 30 bare cube faces.
That means that there are 31 - 15 = 16 perfect major thirds.
In the diagram of scale N we see, from front and from back, 2 × 15 = 30
bare faces. There will be found 31 - 15 = 16 perfect sevenths. From left
and from right we see 2 × 13 bare faces. There will be 31 - 13 = 18
perfect fifths. From above and from below we see 2 × 25 bare faces.
Consequently there will be found 31 - 25 = 6 perfect major thirds only.
In much the same way we conclude, that in the scale P (the right column)
there are 31 - 10 = 21 perfect sevenths, 31 - 27 = 4 perfect fifths only, and
31 - 8 = 23 perfect major thirds.
Thus the numbers of available perfect fundamental intervals are 22
fifths, 16 major thirds, 21 perfect sevenths for scale M, 18, 6 and 16 for
scale N, and 4, 23, 21 for scale P. The comparison shows that scale M
is the best, all round. Scale N has a considerable deficiency of major
thirds, and scale P is very poor in perfect fifths.
It would take us too far to investigate and to compare the numbers
and kinds of concordant triads and tetrads.
Scale of forty-one supracommas
We take the scale formed by the contents of the periodicity block made by the unison vectors as given here with their defining fractions.
-7,-1, 3 268/267 2, 2,-1 225/224 4,-3, 2 128/129The scale is quite symmetrical, numbers n and (41 - n) being inverse one to the other. Here is the scale.
0 | D | 0, 0, 0 | 41 | D | 0, 0, 0 | |||
80/81 | 81/80 | |||||||
1 | /d | 4,-1, 0 | 40 | \d | -4, 1, 0 | |||
41/42 | 42/41 | |||||||
2 | \e | -3, 0, 1 | 39 | /c | 3, 0,-1 | |||
80/81 | 81/80 | |||||||
3 | e | 1,-1, 1 | 38 | c | -1, 1,-1 | |||
49/50 | 50/49 | |||||||
4 | d | 1, 1,-1 | 37 | d | -1,-1, 1 | |||
80/81 | 81/80 | |||||||
5 | /d | 5, 0,-1 | 36 | \d | -5, 0, 1 | |||
41/42 | 42/41 | |||||||
6 | \e | -2, 1, 0 | 35 | /c | 2,-1, 0 | |||
80/81 | 81/80 | |||||||
7 | e | 2, 0, 0 | 34 | c | -2, 0, 0 | |||
49/50 | 50/49 | |||||||
8 | d | 2, 2,-2 | 33 | d | -2,-2, 2 | |||
61/62 | 62/61 | |||||||
9 | f | -1, 0, 1 | 32 | b | 1, 0,-1 | |||
49/50 | 50/49 | |||||||
10 | e | -1, 2,-1 | 31 | c | 1,-2, 1 | |||
125/126 | 126/125 | |||||||
11 | f | 1,-1, 0 | 30 | b | -1, 1, 0 | |||
41/42 | 42/41 | |||||||
12 | \g | -6, 0, 1 | 29 | /a | 6, 0,-1 | |||
80/81 | 81/80 | |||||||
13 | g | -2,-1, 1 | 28 | a | 2, 1,-1 | |||
80/81 | 81/80 | |||||||
14 | /g | 2,-2, 1 | 27 | \a | -2, 2,-1 | |||
49/50 | 50/49 | |||||||
15 | f | 2, 0,-1 | 26 | b | -2, 0, 1 | |||
61/62 | 62/61 | |||||||
16 | a | -1,-2, 2 | 25 | g | 1, 2,-2 | |||
49/50 | 50/49 | |||||||
17 | g | -1, 0, 0 | 24 | a | 1, 0, 0 | |||
80/81 | 81/80 | |||||||
18 | /g | 3,-1, 0 | 23 | \a | -3, 1, 0 | |||
41/42 | 42/41 | |||||||
19 | \a | -4, 0, 1 | 22 | /g | 4, 0,-1 | |||
80/81 | 81/80 | |||||||
20 | a | 0,-1, 1 | 21 | g | 0, 1,-1 | |||
49/50 | 50/49 | |||||||
21 | g | 0, 1,-1 | 20 | a | 0,-1, 1 |
Fig. 5. A scale of 41 supracommas in a periodicity block in the three-dimensional harmonic note lattice. The central cube has been marked D.
Very often commas 80/81 alternate with steps twice as large, 41/42,
or nearly so (49/50). Two steps are only 125/126, hardly a bit larger than
one of the unison vectors constituting the periodicity block (128/129),
about 5/8 of a comma.
On inspection of the diagram, and counting the bare faces of cubes,
where no perfect intervals are available in the scale, we find that in this
scale there are 41 - 11 = 30 perfect fifths, 41 - 31 = 10 perfect major thirds,
and 41 - 35 = 6 perfect sevenths only.
Scales of fifty-three commas
We shall avail ourselves of three sets of unison vectors.
Q: 2, 2,-1 225/224 R: -7, 4, 1 4375/4374 S: -7, 4, 1 -4, 3,-2 129/128 1, 5, 1 738/737 1,-3,-2 1,-3,-2 323/322 1,-3,-2 3, 0, 4 92/93
Fig. 6. Three scales of 53 commas located in the lattice. The central note of scale Q is not visible. In scales R and S they have been marked D.
For reasons of space economy, I shall give half the scales only. They are symmetrical. The notes nrs. n and (53 - n) are inverse one to the other, and the steps are the same in the two halves, following in reverse order. Here are the notes from 0 to 27 inclusive.
The scale R shows a rather uniform distribution of the notes over the
octave, The majority of the steps, 34 out of 53, measure <= 1 in 80. The
smallest step is 1 in 90 (0.9 comma), the biggest is 1 in 64 (1.25 comma).
The scale Q has a much wider variety of steps. Not only do we find two
steps of 107/108 (0.75 comma) but 12 steps 125/126 (0.64 comma) and
even two steps of 205/206 (2/5 comma). To make good for these losses,
we find bigger steps, as big as 49/50 (8/5 comma).
The variant S has the most licentious scale. Four of the steps, 564/565
may even be considered as unison. Again, three other steps are of the
order of 2/5 of a comma. There is no step bigger than 63/64 (5/4 comma).
To make up for the losses owing to the unison intervals, the number of
supracommas (63/64 and 64/65) amounts to 30. This variant could be
taken as a scale of 53 - 4 - 3 = 46 notes, because of the nearly unison steps.
The number of perfect intervals realised in the scales again are to be
found by inspection of the diagrams, in the way explained before. We find:
in scale Q: 37 perfect fifths, 35 perfect major thirds, 26 perfect sevenths;
in scale R: 43 perfect fifths, 30 perfect major thirds, 6 perfect sevenths;
in scale S: 60 perfect fifths, 22 perfect major thirds, 12 perfect sevenths;
The scores of the totality of perfect intervals are 98 for Q, 80 for R, and 44 for S.
Conversion of vectors into numerical factors
In the course of the investigation time and again we wanted to know
the numerical fraction value of intervals, expressed by an interval vector.
These will be found in the table below.
The greater part of the figures given have been taken from the Just
Intonation Tables, made by mr John Chalmers, Ph. D., by means of a
computer in 1966, and kindly put at my disposal. My thanks are due to
him for this very valuable tool in numerical work.
Table of Unison Vectors, Microsons and Minisons | ||||
harmonic vector symbols | numerical interval | approximation | cents | |
-7, 4, 1 | 4375/4374 | 4375/4374 | 0.396 | |
-1,-2, 4 | 2401/2400 | 2401/2400 | 0.721 | |
-8, 2, 5 | 420175/419904 | 1550/1549 | 1.117 | |
9, 3,-4 | 2460375/2458624 | 1405/1404 | 1.233 | |
8, 1, 0 | 32805/32768 | 800/799 | 1.954 | |
1, 5, 1 | 65625/65536 | 738/737 | 2.349 | |
0, 3, 5 | 2100875/2097152 | 565/564 | 3.071 | |
-8,-6, 2 | 102760448/102515625 | 420/419 | 4.130 | |
1,-3,-2 | 6144/6125 | 323/322 | 5.362 | |
0,-5, 2 | 3136/3125 | 285/284 | 6.083 | |
-7,-1, 3 | 10976/10935 | 268/267 | 6.479 | |
2, 2,-1 | 225/224 | 225/224 | 7.712 | |
-5, 6, 0 | 15625/15552 | 216/215 | 8.107 | |
8,-4, 2 | 321489/320000 | 214/213 | 8.037 | |
1, 0, 3 | 1029/1024 | 206/205 | 8.433 | |
3, 7, 0 | 2109375/2097152 | 173/172 | 10.061 | |
-5,-2,-3 | 2097152/2083725 | 157/156 | 11.120 | |
3,-1,-3 | 1728/1715 | 133/132 | 13.074 | |
-4, 3,-2 | 4000/3969 | 129/128 | 13.469 | |
2,-3, 1 | 126/125 | 126/125 | 13.795 | |
-5, 1, 2 | 245/243 | 123/122 | 14.191 | |
10,-2, 1 | 413343/409600 | 110/109 | 15.748 | |
3, 2, 2 | 33075/32768 | 108/107 | 16.144 | |
-3, 0,-4 | 65536/64827 | 93/92 | 18.831 | |
3,-6,-1 | 110592/109375 | 91/90 | 19.157 | |
-4,-2, 0 | 2048/2025 | 89/88 | 19.553 | |
5, 1,-4 | 2430/2401 | 84/83 | 20.785 | |
4,-1, 0 | 81/80 | 81/80 | 21.506 | |
-3, 3, 1 | 875/864 | 80/79 | 21.902 | |
12, 0, 0 | 531441/524288 | 74/73 | 23.460 | |
5, 4, 1 | 1063125/1048576 | 73/72 | 23.856 | |
4, 2, 5 | 34034175/33554432 | 71/70 | 24.577 | |
-3,-5,-2 | 4194304/4134375 | 70/69 | 24.915 | |
-10,-1,-1 | 2097152/2066715 | 69/68 | 25.310 | |
5,-4,-2 | 31104/30625 | 65/64 | 26.868 | |
-2, 0,-1 | 64/63 | 64/63 | 27.264 | |
-3,-2, 3 | 686/675 | 62/61 | 27.985 | |
-1, 5, 0 | 3125/3072 | 59/58 | 29.614 | |
-2, 3, 4 | 300125/294912 | 58/57 | 30.335 | |
-1,-3,-3 | 131072/128625 | 54/53 | 32.626 | |
-8, 1,-2 | 327680/321489 | 53/52 | 33.022 | |
-9,-1, 2 | 100352/98415 | 52/51 | 33.743 | |
0, 2,-2 | 50/49 | 50/49 | 34.976 | |
-1, 0, 2 | 49/48 | 49/48 | 35.697 | |
1, 7,-1 | 234375/229376 | 47/46 | 37.325 | |
7, 1, 2 | 535815/524288 | 46/45 | 37.651 | |
0, 5, 3 | 1071875/1048576 | 46/45 | 38.046 | |
1,-1,-4 | 12288/12005 | 45/44 | 40.338 | |
0,-3, 0 | 128/125 | 43/42 | 41.059 | |
-7, 1, 1 | 2240/2187 | 42/41 | 41.455 | |
2, 4,-3 | 5625/5488 | 41/40 | 42.687 | |
1, 2, 1 | 525/512 | 40/39 | 43.408 | |
0, 0, 5 | 16807/16384 | 40/39 | 44.130 | |
1,-6,-2 | 786432/765625 | 38/37 | 46.421 | |
-6,-2,-1 | 131072/127575 | 37/36 | 46.817 | |
2,-1,-1 | 36/35 | 36/35 | 48.770 | |
-6, 1, 4 | 12005/11664 | 35/34 | 49.887 | |
2, 2, 4 | 540225/524288 | 34/33 | 51.841 | |
2,-6, 1 | 16128/15625 | 32/31 | 54.854 | |
-5,-2, 2 | 6272/6075 | 32/31 | 55.249 | |
4, 1,-2 | 405/392 | 31/30 | 56.482 | |
3,-1, 2 | 1323/1280 | 31/30 | 57.203 | |
-4, 3, 3 | 42875/41472 | 31/30 | 57.599 | |
4,-4, 0 | 648/625 | 28/27 | 62.565 | |
-3, 0, 1 | 28/27 | 28/27 | 62.961 | |
-1, 2, 0 | 25/24 | 25/24 | 70.672 | |
1,-1, 1 | 21/20 | 21/20 | 84.467 | |
3, 1, 0 | 135/128 | 19/18 | 92.179 | |
-3,-3, 1 | 3584/3375 | 17/16 | 104.020 | |
-1, 4,-2 | 625/588 | 17/16 | 105.648 |