An interesting idea. Following from the statement:
Our previous work demonstrated that chords with frequency ratios 3:5:7: and 5:7:9: have many perceptual harmonic properties of major triads (4:5:6 ratios). The traditional diatonic scale can be constructed from three major chords, the tonic, the dominant, and the subdominant chords.
I took some liberties with names and numbers to simplify the results.
Build a 'C major triad' , C E G. ratio 4:5:6
The 'C' is the 5th of the subdominant chord, F, and G is the root of the dominant.
Using this ratio to build the F and G triads, and using foldover octave equivalence, the C 'major' scale, [starting on the arbitrary frequency of 400 Hz], seems to come out as follows:
This scale may already have a name. The three primary major triads would be beat free. I think that melodies constructed from here might be quite interesting for the general population.
By extension up and down, F# and Bb would be introduced. Built on the D, the notes would be D, 450Hz, F# 562Hz and A 675. This A is higher than that in the F major triad. Maybe the solution would be to split the difference and tune A to 670Hz. It would be about equally inharmonic with both triads . . .
On 2012, Jun 11, at 3:32 PM, Richard F. Lyon wrote: