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So I was watching this video and he goes to show that there's 48 (!) chords in the double harmonic major. Why so many? I thought there were only seven diatonic chords. Should one take just 3 random notes from a scale and call each a chord?

In math it's under combinatorics. so combinations with no regards to order:

n!/(r!(n-r)!)

where n is number of things to choose from, and r is how much we take from it. If I plug it in it's 7!/(3!(7-3)!) = 35 different types of 3-note chords per scale. He says 48 because he's adding some four note chords as well. But in any case, is there more than seven chords in a scale?

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  • I think you meant to ask "...seven diatonic tertian triads..." otherwise you could have lots of different counts depending on what set of chords you want in the set... and what topo morto said about heptatonic scales. – Michael Curtis Jun 5 at 16:36
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    Consider the chromatic scale. How many chords then? And we may only be talking triads... – Tim Jun 5 at 16:44
  • You don't need an exotic scale to get more than seven 3-note chords. Consider e.g. sus2 and sus4 chords. – Your Uncle Bob Jun 5 at 16:47
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    Readers of this question may be interested in Jay Hook's article "Why Are There Twenty-Nine Tetrachords? A Tutorial on Combinatorics and Enumeration in Music Theory." – Richard Jun 5 at 17:25
  • If you include 7ths, 9th, 11ths and 13ths there are quite a bit. But their function from a harmony perspective may not all be independent. – ggcg Jun 5 at 18:16
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Does a scale have more than seven chords?

Yes, any scale of at least 4 or 5 notes (depending on whether you're defining a chord as having at least 2, or at least 3, notes) can generate more than 7 chords.

Should one take just 3 random notes from a scale and call each a chord?

Depending on how you are defining 'chord', you can take any group of 2/3 or more notes and call it a chord.

Of course not all of these chords will be triads.

I thought there were only seven diatonic chords.

A diatonic scale has only seven diatonic triads. But we aren't limited to thinking in terms of triadic harmony (nor are we limited to the diatonic scale, for that matter).

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7

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 acg, quartal ade, tone clusters abc etc.

But in any case, is there more than seven chords in a scale?

Yes. Even in fairly basic classical harmony, the seventh chords on ii and 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 CGE, GEC, 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.

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Well, if you harmonize the scale to get all the chords that belongs to the given key you will end up with exactly the same amount of chords as notes in the scale. To harmonize the major scale for triads you get the chords with following notes from the scale: first chord 1,3 and 5, second chord 2,4 and 6, third chord 3,5,7, fourth chord 4,6 and 1, etc. to the 7th chord, 7,2,4.

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    This runs into problems with scales like the harmonic minor scale when you introduce enharmonic spellings. Even restricted to triads the C harmonic minor scale gives both Fm and Fdim, both G and Gaug, all three of Ab, Abm, and Abdim, and both Bdim and Baug, yielding a total of 12 triads when you add the other triads. – David Bowling Jun 28 at 13:59

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