20191110

Harmonic Minor - Seventh Haven

First, Count your Sevenths

By a 'seventh' we just mean a tetrad in the 12-tone 'universe' built from stacked thirds (either major or minor) each third being constructed by skipping exactly alternate scale notes. Further, the whole tetrad is constructively (i.e. with no inversions) contained strictly within the span of one octave. The commonest seventh chords encountered are the minor 7th, the major 7th and - perhaps the most well known of all - the dominant 7th, often known simply as the 7th.

The diatonic scale - with its (major, or Ionian Mode) semitone skip pattern of 2212221 - can carry four such stacks, the aformentioned three, plus the one beginning on the subtonic, the 'half-diminished' or 7♭5 chord. Naturally this capability obtains for all seven modes (Ionian, Dorian, Phrygian, Lydian, Mixolydian, Aeolian, Locrian).

The Melodic Minor scale - with its semitone skip pattern of 2122221 - can carry five such stacks. Although it can no longer support a 'standard' major 7th, this loss is amply replaced by two others, the more unusual major augmented seventh +M7 rooted on the scale's tonic, and the 'psycho chord', the minor major seventh mM7 - rooted on the scale's submediant. Again, naturally, this capability obtains for all seven of this scale's modes - more usually regarded as scales in their own right (e.g. Locrian Super, Altered, Melodic Minor, Javanese, Lydian Augmented, Overtone, Hindi, Locrian Natural, etc.).

But there are two more 7ths. All seven may be counted systematically. Starting from a root note labelled 0 you add either 3 or 4 semitones (i.e. the minor or major third), then those two 'waypoints' each offer two further choices by adding 3 or 4 semitone skips, finalising to 8 choices with the concluding 3 or 4 semitone skip to finish the tetrad. The 8th choice is excluded, however, as 4+4+4 tops it off with the root (albeit an octave above) and is not a seventh chord.

In order of 'thirds-stacking' then,we have:

  • 333 - diminished
  • 334 - half-diminished (or minor 7th flat 5)
  • 343 - minor 7th
  • 344 - minor major 7th
  • 433 - 7th
  • 434 - major 7th
  • 443 - augmented major 7th (or major 7th sharp 5)

As 4-gons embedded within the chromatic 12-gon, they may be represented thus (tonic, or root, note at the top, scale ascent clockwise):

Seven Sevenths

Note that the tetrads are generally 'closed' with interval skips of less than 3 - this is simply a fourth skip necessary to complete the tetragonal embedding within the 12-tone dodecagon and should not be seen as part of the seventh's construction. Also, blue indicates the chords represented are their own inversions whereas pink pairs are mutual inverses (e.g. half-diminished is an inversion of 7th, and vice versa).

In the key of F (chosen mainly to confine the tetrads behind a treble clef fence) these chords are:

Effy Sevenths

No heptatonic scale constructable from just half steps or whole steps (interval strings comprising only 1s and 2s) is capable of supporting all of these sevenths. It's necessary to include a 3 step, and - once included - it must be flanked on each side by 1 steps otherwise a two note jump in either direction would exceed 4 semitones and break the minor/major third requirement. This leaves four skips of 1s and 2s - needless to say, consecutive 1 skips are also prohibited since they'd lead to two note jumps incapable of providing any kind of a third (naturally we regard a diminished third as out of bounds).

And the scale known commonly known as the Harmonic Minor, with its interval string of 2122131, is the 'only' heptatonic scale able to accommodate all seven tetrads. The word 'only' is in scare quotes because, as with all heptatonic scales, it has seven modes and these, too, have acquired their own names (often several) as scales, e.g. Locrian Ultra, Harmonic Minor, Mohammedan, Locrian Natural 6, Major Augmented, Harmonic Major, Lydian Diminished, Romanian, Phrygian Dominant, Spanish, Jewish, Aeolian Harmonic etc..

Harmonic minor scale, interval string 2122131, as 7-gon in 12-gon

Tonic at top, scale-note skipping clockwise.

In fact, as the interval string is reversible (say, to the Ethiopian scale, interval-strung as 2212131) in a way that the diatonic and melodic minor 'modes' are not (both of which are their own inverses), a further seven scales or modes are available to support the seven sevenths. There are thus up to fourteen in all.

As an illustration, here's a descent through the scale degrees of an Indian scale (also known as Makam Huzzam, Maqam Saba Zamzam, Phrygian flat 4, according to this catalogue of scale names - thanks to Paul Erlich for pointing me there) showing each of the seven sevenths:

An Indian Descent

An Odd Connection to Block Designs

We have seven 'shapes' (see the above tetragons) embeddable within a seven-sided polygon. Perhaps the simplest example found in every introduction to the subject matter of block designs is one generated by quadratic residues in a field of integers modulo a prime number, itself congruent to the number 3 modulo 4, the first 'interesting' such prime being 7 (3 itself being too trivial). More often than not, the Fano Plane turns up as a diagram:

5 6 3 4 0 1 2

There are seven 'lines' in this figure (if one allows the inscribed circle as being a line), each going through three points. Each such line is a distinct block of the design shown below as a set of seven blocks of three integers each.

The quadratic residues used to construct this simple block design are the squares of the integers modulo 7. Thus 12 = 1, 22 = 4, 32 = 2 (9/7 leaves remainder 2), etc.. We can say etc. here because the next square, i.e. 42 = 16 also leaves remainder 2 after division by 7, and no results other than 1, 2 and 4 will ever turn up.

This set, { 1, 4, 2 }, constitutes the first block (it is, in fact, a representation of the multiplicative group ℤ*7) of the design and subsequent blocks are generated simply by adding 1 (modulo 7, as always) to each of its elements. Thus the second block is { 2, 5, 3 }, the third is { 3, 6, 4 }, the fourth { 4, 0, 5 }, the fifth, sixth and seventh { 5, 1, 6 }, { 6, 2, 0 } and { 0, 3, 1 } - after which further generations would repeat from the first block.

1234560
4560123
2345601

The (symmetric) block design (v=7, k=3, λ=1) displayed as b=7 vertically oriented blocks, in which each variety (integers 0 … 6) turns up r=3 times each and in which each of the 7×6/2=21 pairs of varieties (e.g. {0,1}, {2,5}, {3,4} …) turns up once (λ=1).

Such designs are symmetric in that the number, v, of varieties being distributed is the same as the number, b, of blocks where each block holds k distinct varieties. The varieties are to have equal representation - r of each - throughout the whole design. Seen as a rectangular arrangement of b blocks it's easy to see that vr = bk.

A less obviously visualised property of the block design - in this case a 2-design - is that each pair (hence the 2) of varieties is also equally represented. There are v(v-1)/2 possible pairs and if appearing λ times amongst the b blocks (of k(k-1)/2 pairs in each block), we must have that λv(v-1)=bk(k-1) or, using the earlier equation, λ(v-1)=r(k-1).

It's important to remember that block designs are a way to disperse varieties of objects in a way that - on the face of it - looks somewhat random or unpredictable but which is in fact engineered to give each variety equal exposure in several positions (blocks, the location or sequence of which is unimportant). In the case of 2-designs, each possible pair of varieties will also turn up the same number of times. In the more general t-design case, each t-sized subset of the v varieties (of which there are v(v-1)(v-2)…(v-t+1) distinct possibilities) have equal representation.

Although the designs themselves may be conveniently be built from integers - using the heavy lifting machinery of addition, multiplication, exponentiation etc. - once the design has been generated the numbers may be replaced by something more abstract (the integers’ responsibilities then being delegated, relegated even, to mere labelhood). These objects or varieties can be anything (well, anything distinguishable) and need not be integers. They may be breeds of plant, species of bacteria,

colours: … and even music:
Fano Triads

where, in the above sequence of triads based on the seven blocks, we have mapped the integers 0 … 6 to the pitches B, C, D, E, F, G# and A.

It may seem perverse to sharpen the G in what would normally look like a perfectly ordinary chord sequence in the C major scale, but there's another, seveny, musical reason for it - and it's not the (implied) 7/4 time signature, but that the fact that each triad might be seen as a tetrad missing its fifth. Respectively those missing fifths would be A, B, C, D, E, F and G#. We shall put them in the bass (labelled with the integers 6, 0, 1, 2, 3, 4 and 5) and bung in an extra root note below. Notice that we would then have the complete sequence of all seven sevenths:

Plesio-Fano Tetrads

I want to show you something. It may mean something to you, it may not. I don't know. I don't know anymore.

Ricky Roma

G# A E F B C D

20190726

All Interval Sets revisited

Recently, I spotted a five year old comment by David Feldman, which reminded me of my earlier post on All Interval Sets.

The brute force method, in brief, is to look for 6 term polynomials of the form

s(x) = 1 + x + xp + xq + xr + xs, (2 < p < q < r < s < 31)

such that

s(x) · s(x-1)
= 6 + x + x2 + x3 + x4 + … + x-3 + x-2 + x-1
= 6 + x + x2 + x3 + x4 + … + x28 + x29 + x30 modulo (x31-1)

since that product, fully written out, will look like the 36 term polynomial

1 + x + xp + xq + xr + xs +
x-1 + 1 + xp-1 + xq-1 + xr-1 + xs-1 +
x-p + x1-p + 1 + xq-p + xr-p + xs-p +
x-q + x1-q + xp-q + 1 + xr-q + xs-q +
x-r + x1-r + xp-r + xq-r + 1 + xs-r +
x-s + x1-s + xp-s + xq-s + xr-s + 1

where 6 of the 36 terms are unavoidably collapsed into units (from, e.g. xr · x-r ≡ 1) and the remaining 30 terms - products of 6 × 5 = 30 terms with unequal exponents - are present exactly once. The four 'unknowns' p, q, r and s must be chosen so that the nonzero differences { 0-1, 0-p, 0-q, 0-r, 0-s, 1-0, 1-p, 1-q, 1-r, 1-s, p-0, p-1, p-q, p-r, p-s, q-0, q-1, q-p, q-r, q-s, r-0, r-1, r-p, r-q, r-s, s-0, s-1, s-p, s-q, s-r } must equate to {±1, ±2, ±3, ±4, ±5, ±6, ±7, ±8, ±9, ±10, ±11, ±12, ±13, ±14, ±15}. I.e. where the 28 member set

{ -p, -q, -r, -s, 1-p, 1-q, 1-r, 1-s, p, p-1, p-q, p-r, p-s, q, q-1, q-p, q-r, q-s, r, r-1, r-p, r-q, r-s, s, s-1, s-p, s-q, s-r }
must be the same as the set
{±2, ±3, ±4, ±5, ±6, ±7, ±8, ±9, ±10, ±11, ±12, ±13, ±14, ±15}
- we already know where the ±1s are. Bear in mind that the first, variable, set is by no means presented in any kind of numerical value order. We know things like q-p > 0, and s-p > s-r, and q-r < 0, but not much else.

Whilst the brute force search, finding p, q, r and s - as 10 solution quartets -

pqrs
381218
3101426
461321
4101217
6182229
8111317
11192628
14202429
15192124
15202228

does not take very long for 31 EDO, one can appreciate that for the larger octave division sets where the k(k-1) distinct differences between k distinct pitch classes must all occur exactly once, higher values of k represents a combinatorial explosion in search times (or large systems of Diophantine simultaneous equations).

The Feldman Observation

Prof Feldman finds a rather quicker way to get the all interval sets for 31 EDO, and begins with the fact that the number 3 can generate the complete set of integers 1 … 30 when repeatedly multiplied by itself, 29 times, modulo 31.

This is not as weirdly out of left-field as it sounds. All it takes is a little messing around with multiplicative group theory. Here are the integers we're looking at:

30,31,32,33,34,35, …, 327,328,329
1,3,9,27,19,26,16,17,20,29,25,13,8,24,10,30,28,22,4,12,5,15,14,11,2,6,18,23,7,21

Now this set - which is to say the set of all integers between 1 and 30 inclusive - forms a group, G, with 30 elements. Insofar as

  • the multiplicative identity, 1, is an element
  • any pair, multiplied together modulo 31, is an element (e.g. 25×11 = 275 = (31×8) + 27 ≡ 27)
  • every element has exactly one multiplicative inverse (e.g. 27-1 = 23, since 27×23 = 621 = (31×20) + 1)
  • … and of course integer multiplication in modulo arithmetic is associative

All except the third property is plain to see. You may wish to verify the less obvious one with 28×27 multiplications, or read up on multiplicative groups.

Now this order 30 group has a particular subgroup, H of order 3 (Sylow says there's only one), comprising elements { 3^0, 3^10, 3^20 }, i.e. { 1, 25, 5 }, as the following (rather simple) multiplication table shows:

×311525
11525
55251
252515

Because this subgroup is of order 3, G's elements may be partitioned into 30/3 = 10 cosets of 3 members each. One of them being H itself, and the other 9 being non-groups - since the triples perforce lack the identity.

Here they are with the cosets in columns (with the first column being the actual subgroup H)

H×3kk
0369121518212427
 1271629830415223
 25242812146731917
 511182192620131022

He then combines the first two cosets (columns) to produce the 6-member set H×30H×33 = {1, 5, 11, 24, 25, 27}.

That this set is the 'same' as one found by brute-force is not hard to see. It's basically x24s(x), as the following argument should show.

Differences are completely unaffected by the subtraction of the same constant from each of those six numbers, and it's easy to see that a subtraction of 24 from each will be appropriate since it takes 24 and 25 to 0 and 1 respectively.

{1-24, 5-24, 11-24, 24-24, 25-24, 27-24} = {-23, -19, -13, 0, 1, 3} ≡31 {8, 12, 18, 0, 1, 3} = {0, 1, 3, 8, 12, 18}

since the 24-24, 25-24 gives us a 0, 1 (the 1 + x of our general s(x) 'all interval' polynomial). We can now see that we have recovered the first row of our brutishly forced table, i.e. where p=3, q=8, r=12, s=18.

Were this 'instant solution' not remarkable enough, we may recover the others - again pretty much instantly - by pairing up the remaining 8 cosets:

6∪9161821122829
12∪15914630826
18∪21207315413
24∪2719223221017

The first column entries ab label the rows as being the union of cosets H×3a and H×3b.

All 10 cosets of H (including, as usual, subgroup H itself) have now been accounted for. The group G's 30 elements have thus been partitioned into 5 hexads.

These 5 hexads look like plausible all-interval sets for 31 EDO (we already know the first one is precisely one such) since in each one of them we can see two elements differing by 1 (giving us our 0, 1 exponents for the polynomial s(x)). If we perform the appropriate row subtractions, then, we get

24-23-19-13013x24s0(x)
28-12-10-7-1601x28s1(x)
816-222018x8s2(x)
3174012110x3s3(x)
22-3-2010-12-5x22s4(x)

where the first column shows the value subtracted from each element in that row, and the last column shows what the original row was, written as a pitch class polynomial, where sk(x) is some six term polynomial beginning with 1 + x. We can now present (after modulo 31 normalisation) the p, q, r, s sets in the following table's right hand column (ignoring the 0s and 1s now common to each row).

812180133,8,12,18
192124150115,19,21,24
1629220186,18,22,29
1740121104,10,12,17
281110192611,19,26,28

reproducing five rows (respectively 1st, 9th, 5th, 4th and 7th) of the 'brute' table. Which is basically the full set of all interval patterns since the remaining 5 are simply inversions of the 5 found here (all-interval sets come in pairs of mutually inverted pitch class sets).

It's not clear to me what directed him towards the idea of constructing a set from the union of an order 3 subgroup and its 'first' non-group coset. What is clear is that a more obviously direct approach using an order 6 subgroup would not have worked. For whilst G's subgroup K = {1, 5, 6, 25, 26, 30} has the right number of elements for a 31 EDO all-interval set, it already has two differences of 1 (26-25 and 6-5) in it and so is a complete non-starter.

Is 73 the Best Number?

Some may know that Sheldon Cooper believes it is. But even he may be unaware of a musically interesting property of this particular prime.

First of all, being 9×8+1 it brings us to 73 EDO, just like 6×5+1 brings us to 31 EDO, both of which allow for the possibility of all-interval sets in which all possible (nonzero) differences between pitch classes in pitch class subsets of size 9 and 6 respectively occur exactly once. In polynomial terms we're looking for polynomials such as

s(x) = 1 + x + xp3 + xp4 + xp5 + xp6 + xp7 + xp8 + xp9

where 2 < p3 < p4 < p5 < p6 < p7 < p8 < p9 < 73) such that

s(x) · s(x-1)
= 9 + x + x2 + x3 + x4 + … + x-3 + x-2 + x-1
= 9 + x + x2 + x3 + x4 + … + x70 + x71 + x72 modulo (x73-1)

Of course we've already computed these sets with our brute-force methodology and where, as predicted, the search for the 8 exponent sets found took somewhat longer than it did for 31 EDO - about an hour of computation with a C program. By the time we got to 91 EDO, the computation was taking about a day to complete.

As it happens, life is particularly easy with 73 EDO. First of all, we need a multiplicative group of order 72 to hold our 72 difference counting exponents, and the integer 5 generates it quite nicely, in that the set {50, 51, 52, 53, …, 570, 571} all calculated modulo 73, contains every integer in the range 1 … 72 exactly once. This is a well-known fact about this group, so no work is required here. Although even though you don't need to do it, a computer algebra system such as GAP will take mere seconds to assure you that this is the case.

gap> c:=List([0..71],i->(5^i) mod 73);
[ 1, 5, 25, 52, 41, 59, 3, 15, 2, 10, 50, 31, 9, 45, 6, 30, 4, 20, 27, 62,
18, 17, 12, 60, 8, 40, 54, 51, 36, 34, 24, 47, 16, 7, 35, 29, 72, 68, 48,
21, 32, 14, 70, 58, 71, 63, 23, 42, 64, 28, 67, 43, 69, 53, 46, 11, 55, 56,
61, 13, 65, 33, 19, 22, 37, 39, 49, 26, 57, 66, 38, 44 ]
gap> Sort(c);
gap> c=List([1..72]);
true

And GAP will generate the appropriate group for you, pretty much instantly.

gap> G:=Units(Integers mod 73);
<group of size 72 with 1 generators>
gap> Int(GeneratorsOfGroup(G)[1]);
5

Now we need to find a subgroup H, ideally of order 9, of G. It's certainly possible since 9 divides 72 exactly, and such a subgroup would have an index of 8 = 72 ÷ 9. GAP provides us with a function which gives us subgroups of such an index:

gap> K:=LowIndexSubgroups(G,8);
[ <group of size 72 with 1 generators>, <group of size 36 with 4 generators>,
<group of size 24 with 4 generators>, <group of size 18 with 3 generators>,
<group of size 12 with 3 generators>, <group of size 9 with 2 generators> ]

And we see that the 6th entry is exactly what we're looking for. We can list its elements:

gap> List(K[6]);
[ Z(73)^0, Z(73)^64, Z(73)^48, Z(73)^56, Z(73)^40, Z(73)^24, Z(73)^32, Z(73)^16, Z(73)^8 ]
gap> H:=K[6];
<group of size 9 with 2 generators>
gap> List(H,i->Int(i));
[ 1, 37, 64, 55, 32, 8, 16, 4, 2 ]

And here is H's multiplication table:

×7313764553281642
113764553281642
3737645532816421
6464553281642137
5555328164213764
3232816421376455
881642137645532
1616421376455328
442137645532816
221376455328164

We can see that there's only one element-difference of 1 (from 2 - 1), which is encouraging. We can sort H's elements and subtract 1 from each (remember that subtraction of the same constant from each element has no impact upon their differences)

gap> S:=List(H,i->Int(i));
[ 1, 37, 64, 55, 32, 8, 16, 4, 2 ]
gap> Sort(S);
gap> S-1;
[ 0, 1, 3, 7, 15, 31, 36, 54, 63 ]

Inspection of the n=73 table in that earlier post shows that the set {0, 1, 3, 7, 15, 31, 36, 54, 63} we have here is precisely the first row of that table, under the heading p1 p2 p3 p4 p5 p6 p7 p8 p9.

But of course there's more. Being of index 8, this order 9 subgroup H is associated with another 7 cosets. We already know that 73 EDO provides 4 pairs of mutually inverse all-interval sets. Can it be that the remaining 7 cosets here provide exactly what we need, in what is essentially no time at all (compared to a daysworth of computational searching)?

Since we were careful to make S the sorted list of the group H's operations, rather than overwrite H as a list, the GAP variable H, for H, remains available to us and we can list all 8 cosets in integer form with a one-liner:

gap> C:=List(RightCosets(G,H),i->List(i,j->Int(j)));
[ [ 1, 37, 64, 55, 32, 8, 16, 4, 2 ], [ 22, 11, 21, 42, 47, 30, 60, 15, 44 ],
[ 46, 23, 24, 48, 12, 3, 6, 38, 19 ], [ 72, 36, 9, 18, 41, 65, 57, 69, 71 ],
[ 63, 68, 17, 34, 45, 66, 59, 33, 53 ], [ 51, 62, 52, 31, 26, 43, 13, 58, 29 ],
[ 27, 50, 49, 25, 61, 70, 67, 35, 54 ], [ 10, 5, 56, 39, 28, 7, 14, 40, 20 ] ]

And we know what to do now - although there is likely a better way to do this in GAP if you're going to do it a lot. First we will construct an array of 8 elements to be subtracted, modulo 73, from each of the 8 integer lists (in order to rebase each of them with a 0, 1 pair of elements). Then we will perform that subtraction in a loop, sort each of the 8 lists, and finally re-present:

gap> b:= [1,21,23,71,33,51,49,39];;
gap> for i in [1..8] do; C[i] := (C[i] - b[i]) mod 73; od;
gap> for c in C do; Sort(c); od;
gap> Sort(C);
gap> Display(C);
[[ 0, 1,  3,  7, 15, 31, 36, 54, 63 ],
 [ 0, 1,  5, 12, 18, 21, 49, 51, 59 ],
 [ 0, 1,  7, 11, 35, 48, 51, 53, 65 ],
 [ 0, 1,  9, 21, 23, 26, 39, 63, 67 ],
 [ 0, 1, 11, 20, 38, 43, 59, 67, 71 ],
 [ 0, 1, 12, 20, 26, 30, 33, 35, 57 ],
 [ 0, 1, 15, 23, 25, 53, 56, 62, 69 ],
 [ 0, 1, 17, 39, 41, 44, 48, 54, 62 ]]

And indeed these 8 sets correspond exactly, row for row, to the previously calculated all-interval sets for 73 EDO. And it took only a few minutes rather than an hour or so. The longest part of the process was visually inspecting the cosets in order to construct the 'subtractions array', b.

In any event, for investigations like these, we will find a 'set difference' function useful. We may define it in GAP as:

setdifferences:=function(n, s)
local x, r;
 x := Indeterminate(Integers, "x");
 s := s - Minimum(s);
 r := Sum(List(s, i -> x^i));
 s := Sum(List(s, i -> x^(n-i)));
 return CoefficientsOfUnivariatePolynomial((r*s) mod (x^n-1));
end;

Such a function will essentially perform the calculation of s(x) · s(x-1) and return the coefficients of the polynomial. For example:

gap> setdifferences(73, [22, 11, 21, 42, 47, 30, 60, 15, 44]);
[ 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]

which reveals to a human onlooker that the supplied pitch class set does indeed form an-all interval set within 73 EDO. For a non-human tester, the following function may be more useful:

isalldifferenceset:=function(s)
local x, k, n;
  k := Size(s);
  n := k * (k - 1) + 1;
  s := s - Minimum(s);
  if Maximum(s) < n then
    x := ShallowCopy(setdifferences(n, s));
    if (Size(x) = n) and (x[1] = k) then
      Remove(x, 1);
      return (Minimum(x) = 1) and (Maximum(x) = 1);
    fi;
    return false;
  fi;
  return false;
end;

133 EDO

We already mentioned that it took about a day's worth of computation to find the 12 all-interval sets within 91 EDO (the next suitable microtonality beyond 73 EDO). How long would brute force technique take to search through 111 EDO to find all-interval sets of 11 pitch classes? I don't know because I've not tried it. And the next one up is the sets of size 12 in 133 EDO - how long would that take?

As a teaser, let's try our GAP function out with something daring:

gap> setdifferences(133,[ 0,1,10,58,60,64,82,87,98,101,113,126 ]);
[ 12, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1 ]

Using group theory it doesn't actually take all that much effort to research 12×11+1 = 133 EDO. Here is a list of the 36 (normal-form) all-interval sets of 12 pitch classes therefrom:

0110586064828798101113126
01253040465396100114122131
0115254552586163808492
018143045475666106109129
0116212449515862688094
0131721586573100105111124
019192431525658697298
015121531333956768598
012633394453616384118130
01521243949617592125127
018213943485473105117131
01233757627583869092102
01622334050596388119131
016183968798298102124126
0173537506689108113122130
0152444717480105112120122
01151820243152608595107
0142751577989100118120125
 
0182133364752707476124
01312203438818894104109
014250547173768289109119
015252868788789104120126
0140546672768385110113118
0110232934616976113117131
01366265767882103110115125
013649587895101103119122129
01416507173819095101108
01794259738595110113129
01317296180869195113126
01324244485159727797111
013154671758494101112128
01810323652556695116128
0141221264568849799127
011214222954606390110129
012739497482103110114116119
01914163445557783107130

Some hints about this - unlike 31 and 73, the number 133 is not prime but has divisors 7 and 19. This means the associated multiplicative group (G) has order (7-1)×(19-1) = 108. Also, 108 is divisible by 36, 12, and 3.

20190509

The Polygony And The Octasy

We return to the matter of all interval sets, as described in general in a previous post, but in particular of those in the 12 tone universe inhabited by the usual musics.

Tetrad

We have, in this dodecaphonic universe, four all-interval tetrads. Some may better know these as, respectively, PC Sets 4-Z29A, 4-Z15A, 4-Z29B and 4-Z15B in the Fortean bestiary. There are only two distinct ‘shapes’, however, as each set can be paired with its mirror image - its musical inversion (the A and B forms).

0 1 2 3 4 5 6 7 8 9 A B 4-Z29A 4-Z15A 4-Z29B 4-Z15B
The Tetrarchy

Octad = Tetrad + Tetrad

Each of these tetrads may add a transposition of one of the others to form an octad, provided that no pitch class takes up a space occupied by the other. For example we may add 4 semitones to the second (which - so transposed - no longer collides with the first) and add this to the first to produce

A pair of non-interfering tetrads

Due to the non-colliding pitch class limitation, it turns out that - of the 66 possible ways to combine two of them - there are only 14 non-colliders. Even then, we find that two turn out to be the well known ‘octatonic scale’ shape (covering both minor and major versions of those Jazz/Stravinsky/Messiaen/etc scales) built from four consecutive semitone+wholetone steps. These shapes result by combining one tetrad shape with the inversion of the other.

So finally, due to similar symmetries, we end up with congruences (by which we mean only modal equivalences, where the shapes are rotatable into each other) in only 7 distinct octads, all of them symmetric (i.e. inversionally identical).

These are the Fortean PC Sets known as 8-28, 8-25, 8-26, 8-9, 8-17, 8-10, 8-3.

8-28 8-25 8-26 8-9 8-17 8-10 8-3
The Octarchy

If we were to take (as seems usual but it's really not compulsory) that pitch class 0 represents the note C, then an application of this particular ‘tetradic addition’ would be the addition of 4-Z29A rendered as the set of pitches C, C♯, E♭, G and the set 4-Z15A (prime-form rendered as C, C♯, E, F♯) transposed (by the aforementioned four semitones) up to E, F, A♭, B♭. This produces the eight pitches (in order) C, C♯, E♭, E, F, G, A♭, B♭ (corresponding to pitch classes 0, 1, 3, 4, 5, 7, 8, 10).

Forte-wise, this might be expressed as something like 8-26 = 4-Z29A + 4-Z15A.T4, where the .T4 operator applied to a PC set indicates its transposition (up) by four semitones.

The following image shows the seven distinct constructions in an arrangement where the prime-form 4-29A is fixed at the top of a ‘pitch class clock’. Addends and sums are oriented appropriately with respect to it. Each tetrad pair (in pink, indicating their inherent inversional asymmetry) is in one clock and its summed octad (blue, indicating inversional symmetry) is in its own clock to its right.

It's OK to fix one tetrad (we've chosen the first) at the top in this way because any other possible tetrad pairing will be rotationally or inversionally identical to one of these shapes.

Labels are Fortean PC Set names located at PC element 0 positions. Consequently some may almost be upside down.

The Octacy

So a second example of such a ‘Fortean operational notation’ is demonstrated by 8-25's ‘11 o'clock’ orientation showing the image 4-Z29A + 4-Z15A.T5 = 8-25.TB where .TB is a transposition 11 semitones up (clockwise rotation by ‘one hour’), or 1 semitone down (an ‘hour’ anticlockwise). Shifting this expression clockwise 1 semitone (to ‘right’ the 8-25 to prime form's ‘midnight’) would require an application of .T1, and, bearing in mind that .TB.T1 ≡ .T0 is effectively a no-op, the equality could be reversed and rewritten as 8-25 = 4-Z29A.T1 + 4-Z15A.T6 (as the operational composition .T5.T1 is, of course, .T6).

All That Bebop

There are several bebop scales, all of them - by design - octatonic.

Major and Minor

One of the prime forms above (Forte 8-26) gives us (in two modes of the same sequence) both the Bebop Major and Bebop Harmonic Minor.

Forte 8-26.T4 as Major Bebop C, D, E, F, G, A♭, A, B (PC 0 = C)
C C♯ D E♭ E F F♯ G A♭ A B♭ B 4-Z29A.T4
→ { 4, 5, 7, 11 }
4-Z15A.T8
→ { 8, 9, 0, 2 }
{ E, F, G, B } ∪ { A♭, A, C, D }
Forte 8-26.T7 as Harmonic Minor Bebop C, D, E♭, F, G, A♭, B♭, B (PC 0 = C)
4-Z29A.T7
→ { 7, 8, 10, 2 }
4-Z15A.TB
→ { 11, 0, 3, 5 }
{ G, A♭, B♭, D } ∪ { B, C, E♭, F }
Major and Minor Beboppery from All Interval Sets

Operationallywise, one might also say that BebopMajor.T3 = BebopHarmonicMinor (or, alternatively, BebopHarmonicMinor.T9 = BebopMajor), were one so seduced by operational notations.

Dominant and Dorian

The Bebop Dominant and Bebop Dorian scales are, like the preceding Major and Minor Harmonic, modal variations of the same PC Set, known in Forte-speak as 8-23. It's not one of our all-interval tetradic composites, but is nevertheless a symmetric set - its inversion is the same set. The figure below exhibits the rotations needed to recover the scales from the prime form - the Fortean 8-23 label appearing as usual at its 0 pitch class vertex.

And, just to draw attention to the fact that musical applications (instantiations) of pitch class sets do not require that pitch class zero be eternally attached to the note C, this time we'll exemplify the Dominant scale in G and the Dorian in D - they should go nicely with the above bebop major.

G A♭ A B♭ B C C♯ D E♭ E F G♭ D E♭ E F G♭ G G♯ A B♭ B C D♭ 8-23 8-23.T9 (with PC 0 = G)
G, A, B, C, D, E, F, G♭
Bebop Dominant in G
8-23.T2 (with PC 0 = D)
D, E, F, G♭, G, A, B, C
Bebop Dorian in D
Dominant and Dorian Beboppery modes of 8-23

Dominant Flat Nine

Another Bebop scale related to an all interval set is the Bebop Dominant Flat Nine. As this scale is not self-inverting, it can't be one of the combined tetrads. Nonetheless, as we shall soon see, it is yet related to one.

In the scale of C, it would be C, D♭ E, F, G, A, B♭ B - a mode of the PC Set { 0, 1, 3, 5, 6, 7, 8, 9 } - aka Forte 8-Z15B - or operationally 8-Z15B.T9.

This set's ‘unused’ pitch classes, viz. { 2, 4, 10, 11 }, can be operationally written as 4-Z15A.TA. This is because { 0 + 10, 1 + 10, 4 + 10, 6 + 10 } = { 10, 11, 14, 16 }, the same as (on our 12 hour clock) { 10, 11, 2, 4 } and the irrelevancy of set element presentation order finishes it off (as { 0, 1, 4, 6 } + 10). This means that the bebop dominant flat nine is (a transposition of) the PC set complementary to our second all interval set. Or, more formally, 8-Z15B + 4-Z15A.TA = 12-1 (where 12-1 is Fortean for the complete chromatic scale).

The transposition of the above by 4 semitones - to modally shift 8-Z15B into the actual bebop dominant flat nine scale - leaves this expression essentially unaltered, due to our modulo polynomial arithmetic. (But we might be tempted to say BebopDominantFlatNine is anti SecondAllIntervalTetrad.T2).

Bebop Dominant Flat Nine instantiated in C, D♭, E, F, G, A, B♭, B (PC 0 = C)
8-Z15B 4-Z15A.TA
{ 10, 11, 2, 4 }
≡ -8-Z15B
8-Z15B.T4 ≡
{ 0, 1, 4, 5, 7, 9, 10, 11 }
Bebop Δ♭9 + 4-Z15B.T2 = Chromatic

Forte: 8-Z15B.T4 + 4-Z15A.T2 = 12-1
Flat Nine Beboppery as an anti All-Interval Set

By the way, it's no coincidence that both of these sets share the same number 15 (in 8-Z15B and 4-Z15A) - Forte numbered his sets fully aware of complementarities.

Another pair of Bebops

A final pair of bebop scales in this collection are found as modes of the non-invertible PC sets categorised by Forte as 8-22A and 8-27A.

The sets (as scales) are known as the Altered Bebop Dorian and the Bebop Melodic Minor.

A B♭ B C D♭ D E♭ E F G♭ G A♭ 8-22A 8-27A 8-22A.TA (PC 0 = D)
D, E, F, G, A, B, C, D♭
Bebop Altered Dorian in D
8-27A.T7 (PC 0 = A)
A, B, C, D, E, F, G♭, A♭
Bebop Melodic Minor in A
Bebopperies unrelated to All-Interval Sets

Of course when we say that these octatonic sets are unrelated to all-interval sets, this does not mean that one cannot extract an all-interval subset from them. For example a 4-Z29A may be extracted from either of these scales, viz. (E, F, G, B) from the altered dorian and (B, C, D, G♭) from the melodic minor (both following the pattern {0, 1, 3, 7} from E and B respectively). It's simply that the four pitches remaining in each scale - respectively (A, C, D♭, D) and (E, F, A♭, A) - are not congruent with any all-interval set (inversions included).

It's also possible to pick out a 4-Z15A as (D♭, D, F, G) - from the altered dorian. We leave it as an exercise for the student to spot any other possible extractions.

In any event, certainly neither octatonic set's complement is congruent to such a tetrad.