Music Theory for Conworlders: Mavila

So, I've discussed a fair bit about scales that are rather closely related to twelve-tone equal temperament. I've explained informally how we can, by detuning the fifth, make four fifths equal a major third. I might have implied that we can find other near-identities and 'temper' out the difference - i.e. we're kind of doing that with (5/4)^3 = 2, in a sense, in 12-tone equal temperament. In that case, we only temper 5/4 by raising it a bit, and we keep 2 intact. We could raise 5/4 less and alter our octave or something, but let's not go there quite yet.

Could we, perchance, temper out a bigger difference? Yes, we could!

If we were to decrease the size of the fifth, so that the following holds:

Four fifths give a third

Two thirds give a fifth

 then it seems clear we've created a third that is midway between major and minor - the upper and lower half of the chord are of equal size. In fact, there's but one scale where the above solution exists (while maintaining octaves): eight fifths obviously must equal a fifth then, which tells us that it is seven-tone equal temperament we are dealing with. Seven-tone equal temperament tempers out the difference between major and minor thirds (or between minor and major seconds). The fifth is fairly low (about 685 cents). Thus, the scale we've obtained is EEEEEEE (for 'equal' or 'element'), instead of LLsLLLs. It is attested in human cultures - the piphat and pinpeat styles of Thailand and Cambodia use it. However, this does not make it an end point! We can go further!

For comparison, though, here's our familiar composition, now in 7 tone equal temperament on percussive-melodic instruments.  

Let's pick a generator that is fairly easy to deal with, but is lower than 685 - so say 675. This gives us a series of [0, 675, 150, 825, 300, 975, 450]. As we did with the regular meantone structure, we rotate it (by subtracting 675 from each element) and we get

mavila : 0, 150, 300, 525, 675, 825, 975.

for comparison, the regular 12-et major and minor scales

major: 0, 200, 400, 500, 700, 900, 1100

minor: 0, 200, 300, 500, 700, 800, 1000

So, like major, we have equal steps from C to D', and from D' to E'. (Where ' just marks "intervals obtained by naming the tones by the same sequence of letters and adding an apostrophe to show we're not doing the major scale" ; like major we have equal steps ever from F to B; like minor, we have the a 300c third, roughly a 800 cent sixth and roughly a 1000 cent seventh. Where major has small steps, this has large steps - where major has small steps, this has large. (If we were to rotate it to the same 'starting note' as minor, we'd actually get the opposite result - large steps corresponding to small and small to large, etc - but major intervals instead of minor and vice versa)

We get a thing that sounds like this:

Clearly, there's some drastic changes in sound from the previous versions of this song we've had: major is minor, minor is major! How did that happen?

 What happens here is that L < s, so it makes sense to change our notation, s:=L', L:=s' (where ' marks 'what previously was'). We obtain ssLsssL. Bizarro-major! If we do some calculations, we find that for s = 1, L = 2, we get 9 steps in total, for s = 2, L=3 we get 16 steps in total. We can of course try other numbers - 3,5; 4,5; ... - these numbers of course giving the relationship between the sizes of the large and small steps. we'll get larger temperaments. 23 and 25 are fairly popular for this particular scale. (In fact, if you look at 16 tone equal temperament, you'll find that it has 675 cent fifths, and thus was the scale I started out with here.) Of course, we could take a major key composition in our regular temperament and remap it to minor (or vice versa), but we won't retain the distribution of step-sizes then: instead of LL giving major turning into ss giving minor, we'll have to replace some L with s, and some L will remain L for that transition to happen. This way, [L, s] := [s, L] is possible.

What's interesting with this scale is that it's actually attested in real life. Not only that, but it is attested in the music of wildly separate peoples - some Indonesian gamelan scales and some African scales. You might have noticed it did not sound all too good with the vaguely organ-like timbre I used in the sample above. So let's go and change that:

Because the chords have no sustain on the instruments used, repetition is used instead.

We have something a bit more un-western already, even if we fundamentally work with the same kind of building blocks. But let's try and see what else we can do with it:

The variations for how the chord instrument plays its chords are not particularly systematic here. The patterns only illustrate some of the endless variations that could be used.

Also, the slight 'vibrato'-like sound on the melody instrument comes from three instruments playing it - one is a bit too sharp, the other a bit too flat, and this produces 'difference waves' that give it a shimmering quality.

We're already getting somewhere a bit afield from western European music! We could finally try and change the kind of accompaniment a bit more drastically, and mostly avoid the western chord. There's options for it - we could use dyads ('two tone chords') - iirc, much Indonesian gamelan music uses dyads quite a lot, mainly focuses on those closest in size to our 'fifths' and 'sixths'. Thus, their accompaniment tends to stick to parallel simultaneous playing of those intervals. Here, I'm doing another thing instead: sequences of tones, along the lines of CDECDECDE or GAFGAFGAF or GFEGFEGFE or things like that. I vary the patterns a bit:

Any composition in meantone[5] (i.e. our pentatonic scale) or meantone[7] (i.e. our major and minor scales, and their modes) can be translated to mavila while retaining a lot of the structural properties: large steps will turn to small steps, small steps to large; this means the same applies to combinations of steps - minor thirds turn major, major thirds turn minor. Perfect intervals will keep being perfect, but augmented intervals now are diminished instead, and vice versa. Major sixths turn into minor sixths, same goes for sevenths. Thus, a composition in our western scales has a shadow composition in mavila, where major is minor, minor is major, augmented is dim, and so on. Every mavila composition has a shadow composition in western scales where the same relations apply!

As an illustration of this, I recommend Mike Battaglia's Mavila Experiments, where a variety of pieces from the western classical canon are given a mavila treatment.

We even can take and expand this to include our harmonic and melodic minors - the scales we get by replacing one or two of the minor triads in a minor scale with major triads - instead of these two:

melodic minor ascending: 

major: F A C
minor:         C Eb G
major:                   G B D
or harmonic minor:
minor: F Ab C
minor:           C Eb G
major:                     G B D

(so C DEb F G A Bc, and C DEb F GAb  Bc)
we now get melodic major ascending:

minor: F Ab C
major:           C E G
minor:                   G Bb D
or harmonic major:
major: F A C
major:         C E G
minor:                 G Bb D       

 (CDEb FGAb Bb C, CDEb FG ABb C)
Of course these scales are also possible in twelve-tone equal temperament, but they are, somehow, less off in mavila than they are in meantone:y systems (12, 19, 31, ...), and for some reason, melodic and harmonic minor seem more off in mavila than they do in meantone. I will probably go and get into these once I get around to talking a bit more about melodic and harmonic minor and what kind of relation they have to the MOS structure.