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 |

One may additionally describe a PC set with a corresponding circulant matrix. Note that the 'fifthiest' hexatonic scale (221223) has - like most hexatonic scales - a non-invertable circulant matrix (with determinant of zero). In fact the only symmetric 6-sets with non-zero determinants thereof are the pair 122133, 121323 (interval vector <2,2,4,3,2,2>).

ReplyDeleteOf the 12 pairs of dual=inverse asymmetrics, 4 have non-zero circulant determinants - 131223/221313, 122223/222213, 112134/312114 and 211224/221124.

Of the 8 quads of dual≠inverse asymmetrics, 3 have non-zero circulant determinants - 121224/221214:221133/112233, 113124/213114:132114/112314 and 121125/211215:211134 311124.

The six note oriental scale (131142) is one of those final 3 in that 4-homometry group, but the six note blues scale (321132) - also in that group - is not and has an uninvertible circulant.

In total, then, only 8 out of the 35 hexatonic co-homometries have invertible circulants.