The interval given by Ruland for a pentatonic based on the "Ancient Indian Cycle of Sevenths" is not so precise in practice. I was pretty crestfallen when I got to hear my first attempt (based on a 4/7 interval). I knew that I wanted the equivalent of 2.4 modern semitones but I wasn't sure what this would sound like or how to get there in a more precise way. However today I figured that the fifth root of 0.5 (=0.870551) would give me a precise ratio to work with. Actually the 4/7 was not so far off in the second octave, but precision really helps in approaching unusual tones.
Thing still sounds weird, but I am confident in its weirdness!
Thankfully, after my students worked so hard on all that math, the ratio given for Persia of 8/13 is much more precise. This gives ten equal divisions from which seven are selected to form the scale.
which is a variation of the GiQuan based around the interval of 1/7 and leaving enough room between frets for football team stickers!
Green: Persian Scale = Equal intervals based on Cycle of Sixths = 8/13
The above would be a nice project - four fretboards based on different mathematical intervals (with possibly and "even-tempered" one alongside for comparison).
But they also show the following as easily achievable using "natural harmonics":
Diatonic:
Do = open
Re = 8/9
Mi = 4/5
Fa = 3/4
So = 2/3
La = 3/5
Ta = 5/9
Do = 1/2
Mixolidyan but with two octaves gives all modes as previously:
. . . . also "chromatic", with only the flat ninth and major seventh elusive
(what do you mean "Quite right too!"?)
Do = open
Ra = ?
RE = 8/9
Ma = 5/6
Mi = 4/5
Fa = 3/4
Fi/Sa = 7/10
So = 2/3
Le = 5/8
La = 3/5
Ta = 5/9
Ti = ?
Do = 1/2
Here's a chromatic fretboard without those pesky semitones "Ra" (flat nine) and "Ti" (Major seventh) - all you need are two strings tuned a fifth apart.
This is really quite lovely - a fretboard with all the chromatic options (bar changing key!) and no fractional intervals in the double digits . . . . if it works!
