The Pentatonic Family pt.2/expanded

Here is a paper where we return to look at another cycle of Pentatonics within a 12 tone context in order for others to apply to the scale and subsets of ones choice. It is better to start with part one if missed it here.
I have revised this from the other day.

A family of 9 tone Scales with some 14 ones on the side.

We have to thank John Chalmers for his Tritriadic Scales. It causes us to keep focus on the material generated by the combination of tonic, dominant and subdominant and how wide those parameters can be extended. It seems to be an idea that has been in the air recently. Warren Burt used in an installation, which you can listen to a bit here 3 11-limit harmonic series fifths apart. Such a structure while embedded in a Partch diamond (harmonic hexads on 1/1, 3/2, and 9/8) seems to stand on its own with it own life. The piece I am working on now for solo vibraphone that I will be playing in Brisbane and in Helenburgh explores these relationships within 7 tone scales much in the way Indonesians will use pentatonic within their 7 tone pelog. I mention some ideas about this here in relation to the idea of tempering to accommodate these three scales.
It appears it a similar idea caught the interest of Steve Grainger who took a classic 7-limit Pentatonic and extended it likewise up and down. This he mailed to me.
I made a mistake right off and assumed it was not a “constant structure”, that is a scale that where each occurrence of a ratio is always subtended by the same number of steps. It is a quality that gives a good melodic flow that otherwise only has harmonic relationships holding it together. Having both is worthwhile to pursue.
But I moved ahead and found 2 solutions right off for 14 tone scales. It is an unusual number to have as a scale and I seem to have come up with quite a few over the years. I am not sure why. These are illustrated and one could pick out of either set of alternatives depending on what one might like.

There is an important feature here though of Grainger’s 9-tone scale that as far as I know has been overlooked. It appears that there is a whole family of pentatonic based on 3 fifths plus 2 notes in between the fourths that will all produce 9 tone constant structures. These tones have to be larger than a 9/8 but smaller than a 32/27, most of your smaller minor thirds.

This scale would have been of interest to Rod Poole and note sure if La Monte Young has toyed with the idea, since it develops in the direction of 3 and 7s. It is in that territory

Nonogonal Music

While others have noticed that Pelog fourths form a good cycle at 9 tones, it was Wilson’s work along these lines that seem to represent it the best. There were three forms that he found had been used historically as the basis of pentatonics, made by rotating around the cycle 5 at a time. Since 2 tones are not tuned they skip to the next tone which gives us 3 different forms. These can be seen in the three rings in this diagram. If one uses all 9 tones one can characterize these as 5 fourths in a row, 4 in a row and skipping two, or the last can be seen as 3 in a row skipping two and then two more or these last 2 first. While 12 ET is a bad way to represent pelog, one can form a 9 tone cycle by using tritones in 3 places, in stead of perfect fourths for those who refuse to concede to mesotones. One will have to figure these out for oneself, and the same idea can be transferred to any other scale, of equal steps or not. I have even applied in use to the meta Slendro I use. A larger form of this diagram can be downloaded here. While not as precise geometrically as one could do, I still find it visually interesting.