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Author Topic: Friendly chromatics  (Read 82818 times)

Gedankenwelt

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Re: Friendly chromatics
« Reply #15 on: January 22, 2012, 04:58:41 PM »

Well, seems I guessed wrong, I guess I should have asked before writing such a long text.

so heres an example of a recurrent sequence for
mavila:        7 9 16 25 41
and heres
meantone:  5 7 12 19 31
father:        5 8 13 21 34

I first didn't find any explanation why 5 7 12 ... generates meantone EDOs. Like the sequence of fibonacci ratios 5/8, 8/13, 13/21 converges to the golden ratio, I tried 7/12, 12/19, 19/31, ... , which seems to converge to a value slightly above the perfect fifth*, which doesn't seem to have any specific relation to meantone temperament. Then I realized there are two different recurrent sequences, from which ratios can be built that seem to converge to some kind of meantone fifth at ~ 696.2 Cent:

Code: [Select]
EDO sequence:    2   5   7  12    19    31    50    81 ...
MT 5th sequence: 1   3   4   7    11    18    29    47 ...
Ratios:         1/2 3/5 4/7 7/12 11/19 18/31 29/50 47/81 ...
(again, ratios in octave = 1/1200 Cent)

This seems to work for any such sequence of EDO tunings, and with any pair of scale degrees (like 7th degree in 12-EDO and 11th degree in 19-EDO) as a starting point for another sequence. It has something to do with a/b < c/d being equivalent to a/b < (a+c)/(b+d) < c/d (unless certain values equal zero), for example:

11/19 < 7/12 <=> 11/19 < 18/31 < 7/12,
11/19 < 18/31 <=> 11/19 < 29/50 < 18/31, and so on ...

I still prefer the fifth from 31-EDO, I think it's almost perfect for meantone.


*19/31 can be interpreted as 19/31 octave = 1200 * 19/31 Cent = 2^(19/31) as a frequency ratio, which is slightly greater than 3/2, the ratio of the perfect fifth.
« Last Edit: January 22, 2012, 05:04:35 PM by Gedankenwelt »
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Easy Listening

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Re: Friendly chromatics
« Reply #16 on: January 22, 2012, 05:11:37 PM »

Thank you Gedankenwelt, you just blew my mind.
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Gedankenwelt

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Re: Friendly chromatics
« Reply #17 on: January 27, 2012, 07:19:44 PM »

Wow, there really seems to be a close relation to the golden ratio. Take a look at the following intervals:

Code: [Select]
EDO 12 19 31 50 81 131 212 343 555 ...
Fifth 7 11 18 29 47 76 123 199 322 ...
Interval
----------------------------------------------------------------------------------------------------------------------------------
...
#1 (-4, 7) 1 1 2 3 5 8 13 21 34 ...
b2 (3, -5) 1 2 3 5 8 13 21 34 55 ...
2  (-1, 2) 2 3 5 8 13 21 34 55 89 ...
b3 (2, -3) 3 5 8 13 21 34 55 89 144 ...
4  (1, -1) 5 8 13 21 34 55 89 144 233 ...
b6 (3, -4) 8 13 21 34 55 89 144 233 377 ...
b9 (4, -5) 13 21 34 55 89 144 233 377 610 ...
...
(Intervals in standard notation, and harmonic coordinates (#octaves, #fifths), then the degree in the respective EDO-tuning)

If the sequence is continued for an infinite number of steps, the fifth converges to (15 - sqrt(5) )/22 octave, which is about 0.5802 octave, or 696.2 Cent. Along with the fifth, the ratio between adjacent intervals in this list (which are consecutive fibonacci numbers) converges to the golden ratio.

This means if a fifth with ~ 696.2 Cent is used as a generator, then the following interval pairs (among others) differ by the golden ratio (in terms of logarithmic frequency ratios): b2/#1, 2/b2, b3/2, 4/b3, b6/4, b9/b6. So harmonically, it's a meantone tuning, and melodically there's a golden ratio between many intervals that are used in classical western music.

The name for such a tuning was rather obvious: Golden meantone tuning ("golden mean" is synonymous to "golden ratio"). And when I searched for it, there was obviously someone else who had the same idea before:

http://www.rev.net/~aloe/music/golden.html
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twistedblues

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Re: Friendly chromatics
« Reply #18 on: October 29, 2012, 06:20:18 PM »

Keep an eye on http://ubertar.com/, Paul Rubenstein is a member here and is working on a guitar with full-width movable frets, i believe he may be patenting so we have to wait for details ... might be worth contacting him to discuss?
I'm not sure i could choose one EDO over another, i'm such an experimenter i know i would need movable frets as i would change my mind the next day ...


THIS would be amazing!!
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All_Your_Bass

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Re: Friendly chromatics
« Reply #19 on: January 28, 2013, 08:35:44 AM »

Double sharps/flats on a single tone do not map* correctly* between 19 and 12 but single accidentals do.

So one can map many things between them as long as double sharps/flats are not used.
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