## Xenharmonic stuff

miekko
Posts: 6
Joined: Thu Nov 21, 2013 6:43 pm

### Xenharmonic stuff

I have for a long time been interested in music tuned in ways that differ from the 12-tone equal temperament that has been dominant in the west for about 200 years (and tunings that have been conceptually rather close to it have been dominant for about a hundred years more).

Due to the paucity of actual theory works that explain how to apply music theory to tunings other than 12-tet (possibly with the exception of 'meantone in general' - any tuning where a stack of four fifths give a passable major third) and the relatively small number of works written in these tunings, anyone who wants to learn to compose in microtonal tunings will have to spend some extra work just studying the characteristics and properties of unusual tunings. I have during recent months spent some time studying two temperaments that both have the peculiar property that they lack the octave.

1. 88 cent equal tuning
If you divide the fifth in 8 instead of 7 equal parts, you obtain a tuning where there are no octaves, but we get an interesting set of cool things to play with. As a side note, dividing the fifth in 4 steps is quite close to one scale used in some Georgian music (Caucasus-Georgia, not banjo-Georgia), thus a subset of this scale has some actual use in ethnic music somewhere. If we use 'cents' as our pitch-measurement, the perfect fifth is roughly 702 cents (700 in 12-tone equal temperament), we notice that this results in a step size of 87.75 - so basically 88% of the usual semitone (which is a hundred cents). By multiplying this by different integers we get
87.75
175.5
263.25
351
438.75
526.5
614.25
702
(and goes on from there)
Now, our usual tuning is basically 100-200-300-400-500-... (in fact, the cents measurement is designed in order that each 12-tone equal temperament interval is an exact multiple of 100). We notice that some of our intervals are 'squashed' - 200c -> 175, 100 -> 88, 300 -> 260, some are widened (400 -> 438, although arguably 400->351 could also be considered).

I will explain 'cents' and such and the relevance of ratios for harmony in a later post.

Turns out some of these intervals are interesting- e.g. if our usual minor third usually "represents" 6/5, it now is very close to 7/6, if the major third usually is an approximation of 5/4, it now is clearly 9/7. 614 cents is closer to a "simple" ratio than 600c was before - now it's pretty close to 10/7, and 175c is close to 10/9 instead of 9/8. What meaning this has I will get into later. Further up the line we get 965c, which is very close to 7/4 - in fact, another tuning in the 88 cent equal family is defined as what you get if you divide 7/4 in 11 equal steps (then, it's slightly over 88). Another definition is 7/6 by 3, another is 9/7 by five, another is 11/7 by 4 (that produces the narrowest intervals of all that I mentioned).

So, your minor chords end up significantly 'darker' than your usual minor chord, your major chord is significantly 'lighter' and quite more tense (it's not really a good consonance any longer). You do get 'standard' major chords - but only in wide voicings (C-G-e is actually more in tune than in 12-tone equal temperament, and the opposite voicing for standard minor chords - C-A-e - is also very well in tune. Restricts how you can do voice leading a bit, but that's the kind of challenge that makes this kind of stuff fun.)

This is a version that goes a bit overboard, I should probably remove quite a portion the piano-like thing and get some more actual melodies going (and more high-register stuff in general). It is still very much a draft, and I still am doing a lot of work on it. Feedback is welcome.
http://miekko.infa.fi/c88c.mp3

monoben
Posts: 103
Joined: Wed Feb 26, 2014 11:09 am

### Re: Xenharmonic stuff

This is great. very interesting read. The example is a bit crazy for my ears. Its been a while since you posted this, maybe since then you have made another version?

Koekepan
Posts: 222
Joined: Thu Dec 05, 2013 4:56 am

### Re: Xenharmonic stuff

An interesting question which is related to scale design is the actual natural scale of the speaking voice. If one can analyse that, one should have a very natural singing scale and it might turn out to correspond well to an existing theory-based scale.

Perhaps if you know a friendly audiologist?

monoben
Posts: 103
Joined: Wed Feb 26, 2014 11:09 am

### Re: Xenharmonic stuff

I've never heard the term audiologist before. Maybe I should search for it.
If one can analyse that.
Any idea how to go about that?

Koekepan
Posts: 222
Joined: Thu Dec 05, 2013 4:56 am

### Re: Xenharmonic stuff

monoben wrote:I've never heard the term audiologist before. Maybe I should search for it.
If one can analyse that.
Any idea how to go about that?
Sure. Get some actors or other volunteers to read a series of texts to an audiologist's (or other's) microphones, then spectrally analyse the speech by phrase and syllable.

Time-consuming, but perfectly feasible.