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Old 8th April 2013, 05:48 AM   #97
calebprime
Penultimate Amazing
 
Join Date: Jul 2006
Posts: 13,001
Why I like 171edo as a master tuning system

My background has been Just Intonation and of course 12 notes per octave.

I've been looking for a system of equal divisions of the octave (EDO) that would bridge the gap between a system of ratios and a system of equal divisions.

Such a system has to be very accurate, but defining accuracy turns out to be tricky.

For one thing, a little inaccuracy results in a little waver (beating) in a typical chord played by instruments with harmonic overtones -- instruments like plucked strings, bowed strings, woodwinds, organs, voice, etc. A little very slow waver actually sounds better to me than none at all.

It turns out that the tuning app I use called Lil' Miss Scale Oven for the Mac has a feature that gives you a clue about what EDOs are most accurate. But the answers it gives depend completely on what information you give it.

You enter some target ratios in a window.

You define how much accuracy in another window. (I like to enter this as n/50 -- so that I can get lots of detail and not miss anything.)

It figures out that so-many-steps will give you x number of cents (1200 per octave) mean deviation given the ratios you have entered.

So, you have to choose approximately how big, how accurate an EDO you're looking for, for what purpose.

My purpose is mainly Pythagorean chains of 3/2 approximations, and 13-limit JI, but without the chains. In other words, I want to hear fairly accurate "overtone series" chords, such as 8,9,10,11,12,13,14,15. I want to be able to have a chain of 3/2's of the same size as large as possible. I want to have somewhere between 50 and 300 notes as my master tuning system.

The whole idea of an EDO is to simplify your thinking and get rid of certain distinctions that you don't want to make.

For a long time I was happy enough with 87EDO, but I've found that the 3/2 approximation of 703.448 cents was a little too wide or big for chains of 3/2's to be accurate from one end of the chain to the other.

A master tuning system isn't something you try to sing. It's a system of the finest distinctions that you'll need to make for your music. Generally, the smaller the number, the more it simplifies -- so the more convenient -- but the less accurate it is.

I asked some of the tuning experts on a Yahoo tuning group.

I asked Lil' Miss Scale Oven, also, as described above.

Here are the answers that LMSO gives with different target ratios entered. As the size grows, the tuning is getting more accurate. The initial accuracy is arbitrary.

1/1, 3/2 -- 29,41,53,200,253,306 EDOs. (Very accurate "5ths" or 3/2s.)

1/1, 5/4, 3/2 -- 19,22,31,34,53,118,289,323,441 EDOs (a major triad.)

1/1, 5/4, 3/2, 7/4 -- 31,53,68,72,84,99,130,140,171,270,441 EDOs (triad plus harmonic seventh)

1/1, 9/8, 5/4, 11/8, 3/2, 7/4 -- 41,72,87,94,118,130,176,183,224,270,494 EDOs (add another 3-ratio and an 11-ratio)

1/1, 16/15, 9/8, 6/5, 5/4, 11/8, 3/2, 13/8, 27/16, 7/4, 15/8 -- (filling in the gaps, starting to approach my usual keyboard tuning) 41,53,72,77,87,118,130,183,224,270,407 EDOs


chain of approximate "4ths" or 4/3s: 9/5, 6/5, 8/5, 16/15, 64/45, 243/128, 81/64, 27/16, 9/8, 3/2, 1/1, 4/3, 16,9, 32/27, 128/81, 256/243, 45/32,15/8, 5/4 -- 53,118,171,323,335,376 EDOs



So, for me, 171EDO is in the sweet zone (The Goldilocks zone) between not too big and not too inaccurate, and does ratios of 3,5,7 and 15 very well, and 11s and 13s ok.

The approximation of a 3/2 in 171EDO is 701.754 cents. That is exactly 100 steps in 171EDO, so that's convenient.

It sounds a little brighter and a lot more accurate than the familiar "5th" or 3/2 approximation of 12 or 72edo at 700 cents. I like that.

171 is divisible by 3, so that means that a system of 171 pitches will have increments at exactly 400 and 800 cents -- an equal-tempered "augmented" triad with 1/1. That's a sound and a way of thinking that I like.

What I hear is that I can barely tell it from 13-limit Just Intonation, but it still simplifies things considerably, and brings all those ratios into a closed, additive system.

In the 11 ratios, it's as much as 4 cents off, but I can live with that.

It was very hard to choose. 224EDO is the most accurate. 159EDO (3x53) is excellent and is used by tuning expert Ozan Yarman. 89 is a good smaller system. 87 is good but for the wide 5ths.

72EDO is the choice of some of the Boston Microtonalists. I don't make the same assumptions they do, or have the same goals.

94EDO is an excellent all-around master system.

130EDO shows up on every list. I forget as I write this why I didn't choose it. (Maybe the harmonic 7ths weren't so good?)

But 171 is my choice for the next few years.

Last edited by calebprime; 8th April 2013 at 06:18 AM.
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