80 HexaSets


221223114114211125131313111117222222122214113223113115111414213123132123212124211233112116111144122133121323121215111333212223222123113214123114111225221115131223221313131124211314111315131115122223222213112224222114121233212133123123213213112134312114111126211116211224221124121134311214122124212214211323112323121224221214221133112233112125212115321114111234112215122115231114111324113124213114132114112314121314131214131133113133121125211215211134311124111216121116311115111135

This is a complete catalogue of all 80 Pitch Class Sets containing exactly half - i.e 6 - of the 12 available pitch classes in conventional, i.e. non-microtonal, music. These sets carry particular weight within the dodecatonic universe because their duals (i.e. the pitch class sets comprising those 6 pitch classes omitted from each such set) are automatically included in this collection.

Any twelve tone row, therefore, can be considered a sequence of two of these sets - an observation that was not lost on either Arnold Schoenberg or Josef Matthias Hauer who used this as the basis for a quite different non-serial methodology for constructing 12-tone music. Very little is heard of this modus operandi these days, but it's there to investigate and play with.

Although this collection has been presented before, we're re-presenting the material in a different light since the set of 80 distinct polygonal shapes presented there are partitioned into 35 subsets of various singletons, pairs and quadruples, each bound by a common consonance or interval vector, and sorted by consonance (descending alphabetically).

The following presentation alternatively partitions the 80 sets in the same 35 ways - i.e. according to their common (descending) consonance - but this time in groups of singletons, pairs and quadruples used as classifiers within the major categories of (a) symmetric pitch class sets (of which there are 20) and (b) non-symmetric pitch class sets (the remaining 60).

The columns

The consonance column is - basically - the interval vector without the commas or the angled brackets, and crucially with the first and fifth components swapped. Within each sub-table, rows are ordered by descending value of this marker. Thus the more fifths the set has, the higher up in the list it appears. As a major second is just a fifth of a fifth, such an ordering secondarily descending-sorts next on those, and so on. The highest possible value of the consonance amongst hexachords, or hexatonic scales, is 543210 (no tritones!) and this 'fifthiest' of hexatonic sets thus appears first.

The second column carries the interval vector for reference. It's functionally redundant because it can be recovered directly from the consonance by the simple swap mentioned above, but we imagine folk would like to see it.

The third column presents the Forte name(s) of the polygons presented in the row, being laid out with the same 'geography' as the 1, 2 or 4 objects shown.

The fourth column presents, with the same layout, the interval path clockwise around the polygon, or semitone-skipping up the corresponding hexatonic scale or chord (including the final wraparound to the start). Pitch class sets so represented are in Forte's prime form with the largest skips at the end and larger-packed to the right where still ambiguous. This is, essentially, the interval path reversed and descendingly sorted (arguably the most bothersome sort-ordering you could possibly present to a computer!)

Note that where more than one prime form is presented, i.e. where there are non-equal inverses (with reverse interval paths) or non-equal duals (with complementary and reverse complementary interval paths) the one with the 'primest' of the pair, or quartet, of prime forms (the one which reverse-descend-sorts first) is taken to be the representative of the whole animal (unified as they are by their common consonance). We suppose you might call it the prime-prime form, or perhaps the tropic-form as it uniquely identifies each of the 35 (Carter, Gerhard, Perle) reductions of Hauer's 44 tropes.

The fifth column presents our various hexagonal representations of the sets. Each runs, as usual, clockwise from the top (at pitch class 0) and its 6 sides are annotated by the corresponding digit from its 'key-free-mode-free' interval path (from column 4).

The final column (dual transposition) shows how many semitones to transpose the set's dual if you need it transposed for use in a twelve-tone serial (tropic or otherwise) composition. Where only two polygons are shown in a row, the dual is the one that is transposed by that number. Hover over it with your mouse and it should demonstrate what we mean.

For example, the 'least fifthiest quadratrope' appears as the last row of the whole table. Its representative is interval-pathed as 111216 and might represent (let's be conventional and employ C as pitch class 0) the set of pitches {C, C#, D, E♭, F, F#}. Its dual (or anti-scale) is directly underneath, interval-pathed as 311115. As presented it would also be rooted on pitch class 0 - exactly what we want for a canonical 'shape' identifier. But it's no use if we want the 6 notes excluded from the first set. So the '4', sitting at the bottom of the next column, tells you how many semitones to offset your application of the dual's interval path. I.e. start at pitch class 4 (rather than 0) and add your interval steps from that pitch class origin to recover the desired {E, G, A♭, A, B♭, B} from the 311115. If the transposition is shown as negative (because it represents an anticlockwise rotation in the mouse-hovering demonstrations) then just add 12, as usual in clockland.

20 Symmetric Hexatonics (identical to their inverses)
6 Symmetric Hexatonics also identical to their duals
consonance interval
vector
Forte
name
interval
path
hexagon dual
transposition

543210
<1,4,3,2,5,0> 6-32 221223 6

420243
<4,2,0,2,4,3> 6-7 114114 -3

343230
<3,4,3,2,3,0> 6-8 211125 6

303630
<3,0,3,6,3,0> 6-20 131313 2

143250
<5,4,3,2,1,0> 6-1 111117 6

060603
<0,6,0,6,0,3> 6-35 222222 1
14 Symmetric Hexatonics but different to their duals
consonance interval
vector
Forte
names
interval
paths
hexagon and dual hexagon dual
transposition

432321
<2,3,2,3,4,1> 6-Z26 6-Z48 122214 113223 -3

421242
<4,2,1,2,4,2> 6-Z6 6-Z38 113115 111414 -4

324222
<2,2,4,2,3,2> 6-Z29 6-Z50 213123 132123 4

234222
<2,3,4,2,2,2> 6-Z23 6-Z45 212124 211233 -5

232341
<4,3,2,3,2,1> 6-Z4 6-Z37 112116 111144 -5

224322
<2,2,4,3,2,2> 6-Z28 6-Z49 122133 121323 -5

224232
<3,2,4,2,2,2> 6-Z13 6-Z42 121215 111333 -4
60 Hexatonics with no Reflectional Symmetries
2 Cyclicly Symmetric Hexatonics whose inverses are their duals
consonance interval
vector
Forte
names
interval
paths
hexagon and dual hexagon dual
transposition

224223
<2,2,4,2,2,3> 6-30A 6-30B 123123 213213 -4
24 Nonsymmetric Hexatonics whose inverses are their duals
consonance interval
vector
Forte
names
interval
paths
hexagon and dual hexagon dual
transposition

443211
<1,4,3,2,4,1> 6-Z33A 6-Z33B 212223 222123 4

422232
<3,2,2,2,4,2> 6-18A 6-18B 113214 123114 3

342231
<3,4,2,2,3,1> 6-9A 6-9B 111225 221115 4

323421
<2,2,3,4,3,1> 6-31A 6-31B 131223 221313 6

322431
<3,2,2,4,3,1> 6-16A 6-16B 131124 211314 -5

322242
<4,2,2,2,3,2> 6-5A 6-5B 111315 131115 4

242412
<1,4,2,4,2,2> 6-34A 6-34B 122223 222213 2

241422
<2,4,1,4,2,2> 6-22A 6-22B 112224 222114 3

225222
<2,2,5,2,2,2> 6-27A 6-27B 121233 212133 5

223431
<3,2,3,4,2,1> 6-15A 6-15B 112134 312114 3

143241
<4,4,3,2,1,1> 6-2A 6-2B 111126 211116 5

142422
<2,4,2,4,1,2> 6-21A 6-21B 211224 221124 5
2 Nonsymmetric Hexatonics (mutual inverses) identical to their duals
consonance interval
vector
Forte
names
interval
paths
hexagon and inverse hexagon dual
transpositions

323430
<3,2,3,4,3,0> 6-14A 6-14B 121134 311214 6 6
32 Nonsymmetric Hexatonics different to their duals
consonance interval
vector
Forte
names
interval
paths
hexagon and inverse hexagon
dual hexagon and inverse dual hexagon
dual
transpositions

433221
<2,3,3,2,4,1> 6-Z25A 6-Z25B




6-Z47A 6-Z47B
122124 212214




211323 112323
-5 -3



333321
<2,3,3,3,3,1> 6-Z24A 6-Z24B




6-Z46A 6-Z46B
121224 221214




221133 112233
5 -3



333231
<3,3,3,2,3,1> 6-Z11A 6-Z11B




6-Z40A 6-Z40B
112125 212115




321114 111234
3 -4



332232
<3,3,2,2,3,2> 6-Z12A 6-Z12B




6-Z41A 6-Z41B
112215 122115




231114 111324
3 -4



322332
<3,2,2,3,3,2> 6-Z43A 6-Z43B




6-Z17A 6-Z17B
113124 213114




132114 112314
3 -3



313431
<3,1,3,4,3,1> 6-Z19A 6-Z19B




6-Z44A 6-Z44B
121314 131214




131133 113133
5 -3



233331
<3,3,3,3,2,1> 6-Z10A 6-Z10B




6-Z39A 6-Z39B
121125 211215




211134 311124
6 5



233241
<4,3,3,2,2,1> 6-Z3A 6-Z3B




6-Z36A 6-Z36B
111216 121116




311115 111135
4 -5


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