These three papers illustrate three subtle variations in approaching the combination of two or more harmonic or subharmonic segments. From there they become the basis of extracting subsets beginning with both pentatonic and tetrachordal scales or any subset of interest. Wilson found them also useful in imitating other scales such as pelog for example as if it was made of two subharmoni sulings based on different series . While Wilson presented these papers, dependent on the instrument design, with either a harmonic or subharmonic context, he thought of all these scales as invertible.


The flute scales are derived from a single subharmonic series plus an exit hole tuned to a 3/2 or 4/3 from one of its subharmonics. This is a practice he noticed in his collection of flutes from South America.The possibility of combined flutes are shown to play some subharmonic versions of the HelixSong scales. Wilson gave one of these flutes to Harry Partch which he used in "Delusion of the Fury."


The Diaphonic Cycles are derived from two subharmonic series. Originally they were conceived to be tuned on a guitar with pairs of strings tuned a 3/2 apart and with equally-spaced frets. The simplest possibilities are on page 2 [which have been rotated on page 5 to read easier]. On page 12 of the Diaphonic Cycles, Wilson shows the underlying subharmonic series whose factors at the meeting points represent the 'fundamental' of the two series or divisible by 2 or 3. In the second example on that page, the point where 12 and 18 meet, the underlying series is 6, and where 8 and 12 intersect, it is 4.

John Chalmers’ The Division of the Tetrachord devotes a section to these scales  [page159]


The HelixSong is a simple instrument designed orginally for metal tubes combining two harmonic series. Each series starts and stops on different shared tones found in its middle row. The first instrument provides alternative ways to use different sections of each series together to form both pentatonic and tetrachordal scales. These represent another pattern, that of combined Diaphonic Cycle in their harmonic form.

The 11-limit HelixSong has embedded within it a 5 and 7 tone Diaphonic scale, while the 17-limit has a 7 and 10 tone scale, and the the next example has 12 and 17 tone scales. Todd Manley helped Wilson to construct the first 11-limit HelixSong out of 2” aluminum tubing for which he, Lydia Ayers, and David Rosenthal wrote compositions for. The HelixSongs offer a gateway into many tetrachoral scales. It also contains the harmonic inversions of many of the Diaphonic Cycles (see Diaphonic Cycles paper).


The overlaps and interactions of these three scale systems are interesting and musically useful. Each HelixSong contains at least two Diaphonic Cycles but oriented harmonically instead of subharmonically. The 11-limit HelixSong contains the first two found on page 1 of the Diaphonic Cycles and the second 17-limit one encompasses the third. These series have not been taken any further but are worthy of investigation. In one paper elsewhere Wilson takes the simplest Diaphonic and by going through the common-tone modulations constructs a Diamond with it to show just one possible point of departure. The HelixSong paper also provides an example of its expansion into a Diamond.

~Comments and Questions Welcome~

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