This may help. https://www.mathsisfun.com/combinatorics/combinations-permutations-calculator.html
I don't know this math so a calculator in English helps me!
- Types to choose from? 7
- Number Chosen? 3
- Is Order important? No
- Is Repetition allowed? No
Combinations without repetition (n=7, r=3)
Using Items: a,b,c,d,e,f,g
List has 35 entries.
abc abd abe abf abg acd ace acf acg ade adf adg aef aeg afg bcd bce bcf bcg bde bdf bdg bef beg bfg cde cdf cdg cef ceg cfg def deg dfg efg
But that included tertian triads
ace & seventh chords
ade, tone clusters
But in any case, is there more than seven chords in a scale?
Yes. Even in fairly basic classical harmony, the seventh chords on
V are so common that 9 would be the bare minimum of major diatonic chords. Sequential harmony using all 7 diatonic seventh chords is a common pattern so that would be 14 chords. If you are looking for a rough number of chords for diatonic, functional harmony in a single key (not scale) 9-14 seems reasonable.
double harmonic major... diatonic chords... random notes from a scale... combinatorics
I think a lot of concepts are packed together.
With exotic scales like the double harmonic I think the typical approach is not building triads on each scale degree. It's more likely to be a folk music style or some eclectic modern style. 'Chords of the double harmonic scale' or the 'diatonic chords of double harmonic' isn't really a standard thing. Of course you can do it, but there aren't necessarily conventions to follow.
Combinatorics comes up in classical styles, but it seems to be mostly about permuting melodic material. Chords and harmony don't get treated with interchangeable reordering.
ii6 V7 I doesn't get permuted into
ii6 I V7 and
V7 ii6 I, etc. Also, I think a lot of the melodic combinations are constrained to harmony. So,
CEG can become
GCE, etc. Permutations of the notes of the chord. But the scale set
CDEFGABC is not just randomly permuted. So while there is the idea of Ars Combinatoria in the common practice era it wasn't random. In those styles harmony followed a kind of grammar that could not be arbitrarily re-ordered. Chord construction as well wasn't random sets of tones from a scale. You can get dozens or hundreds of 'chords' from a seven tone scale, but they won't be common practice chords.
Contrast that with 12 tone music where the whole point was to treat the 12 chromatic tones equally and un-do the harmonic grammar of common practice. When the harmony is freed from that constrained grammar you can create all kinds of tone sets (chords.) All possible permutations and sets are free to be used. Harmonically you are not constrained by convention.
When you look at music and combinatorics these contrasting aspects of Ars Combinatoria and 12-tone or other eclectic modern styles.