Easy Scales by Interpolating between Harmonic Series
A very easy way to construct a scale that's instantly recognizable, even without repeated listening/priming in the absence of listening the music in 12EDO, is to interpolate between harmonic series.
I'm using the notation <<harmonic series numbers from the root>> to denote what harmonic series certain scales contain. If this seems unclear or conflicts with an existing notation, please let me know.
Some of the most prominent scales in existence can be very quickly derived from just a few interlocked/interpolated harmonic series.
Take, for example, the diatonic major scale in 12EDO, where notes are approximately equal to
C | D | E | F | G | A | B |
1/1 | 9/8 or 10/9 | 5/4 | 4/3 | 3/2 | 5/3 | 15/8 or 17/9 |
This can be derived from the following harmonic series
(x/9) - 1/1, 10/9, 12/9 (4/3), 15/9 (5/3), 17/9
which is the same as the notes C D F A B and contains the subdominant major chord F A C
(x/8) - 1/1,9/8,10/8 (5/4), 12/8 (3/2), 15/8
which is the same as the notes C D E G B and contains the tonic major chord C E G along with the dominant major chord G B D
(x/12) - 1/1 5/4 4/3 3/2 5/3
the same as the notes C E F G A.
The x/12 and x/9 harmonic series become particularly stressed in the (Maqam) Rast, also known as the "Blues" scale, of
C | D | D#-E | F | G | A | A#-B |
1/1 | 9/8 or 10/9 | 11/9 | 4/3 | 3/2 | 5/3 or 27/16 | 11/6 |
Here the x/9 series uses the "blue tone" of 11/9 and grows into
(x/9) - 1/1 10/9 11/9 12/9 15/9 17/9
Meanwhile the addition of the "blue tone" of 11/6 and removal of 5/4 changes the x/12 series to
(x/12) - 1/1 4/3 3/2 5/3 11/6
And the removal of the 5/4 and 15/8 shrinks the x/8 series into
(x/8) - 1/1 9/8 3/2
If you don't like the small/compromised x/8 series, you can modify that scale by deleting the 11/9 and adding back the 5/4 and removing the 11/9 to make the scale more balanced between the size of the different series e.g.
(x/8) - 1/1 9/8 5/4 3/2
and
(x/9) - 1/1 10/9 4/3 5/3 17/9
But why limit yourself to/by x/8, x/9, and x/12 harmonic extensions of the root?
Part 2- Attempting to create scales from scratch given harmonic series segments from a root tone
A common pattern in the above scales is use of and interpolation between the x/9 and x/12 harmonic series
It turns out the x/7 harmonic series has many notes in common with or close to the x/8, x/9, and x/12 harmonic series.
We can take advantage of this to construct a more advanced scale in order the create the series
(x/7) = 7:9:11:12:13 (AKA 1/1 9/7 11/7 12/7 13/7)
(x/8) = 8:9:10:12:14:15
(x/9) = 9:10:12:14:15
(x/12) = 12:15:16:18:20
Some of these ratios are near each other, but not equal, such as 9/8 in the (x/8) and 10/9 in the (x/9) series or 14/9 in the (x/9) series and 11/7 in the x/7 series.
One way to evenly distribute error is to take the harmonic/geometric median between such fractions e.g. square-root-of ((11/7)/(14/9)) * 14/9 = about 1.56.
Doing such gives us the scale
1/1 | 11181/10000 | 5/4 | 9/7 | 4/3 | 3/2 | 156341/100000 | 5/3 | 26/15 | 28/15 |
between 10/9 and 9/8 | between 14/9 and 11/7 | between 12/7 and 7/4 | between 13/7 and 15/8 |
In addition, adding 16/15 to the above scale can yield an additional 5 note harmonic series, making the above a larger <<7-8-9-12-15>> scale
(x/15) = 15:16:20:26:28
And adding 6/5,8/5, and 9/5 gives
<<7-8-9-10-12-15>> scale
(x/10) = 10:12:15:16:18
Feel free to add comments or find/punch loopholes in this theory, but this is what I'd consider an idea "extended diatonic" scale in that it stresses likely the most important harmonic series in the diatonic scale while adding an extra (x/7) into the mix, giving the scale a wider range of tonal color.
More scales to come later...
Appendix
Above calculations such as the interpolation of (11/7)/(14/9) can also be expressed as commas e.g. 99/98, which can be plugged into Graham Breed's Temperament Finder on http://x31eq.com/temper/uv.html to reveal temperaments and ultimately scales likely to contain the above harmonic series segments.
. However, most of the more advanced scales on my method would require tempering out several different commas and, presumably, also countless related commas indirectly. I am afraid this would often result in huge and much more challenging to play in (think: well over 15 notes) scales needed to contain, say, the x/7,x/8,x/9,and x/12 harmonic series from the root tone with reasonable accuracy in regular temperament theory.
If any experts on Xenharmonic math, including related lists, can find a way to related the input of harmonic series segments to, say, MOS scales guaranteed to have them I would really appreciate it.