Circle of tempered 1.4 (7/5), 3/2's, and 1.6 (8/5)'s

I did a blind test on intervals around 3/2 and I learned something. That is just how sensitive 3/2 is as an interval. For example, move from 6/5 (1.2) to 11/9 (1.22222) and it sounds worse but not breathtakingly so. Move from 11/10 to 12/11 covering intervals in-between and there's barely a difference.

But move from 3/2 to 34/23 = 1.4782 and...wow that bites! Wolf indeed.... Turns out (at least to my ear) even the fairly low numbered 16/11 = 1.454545 doesn't work too well as an "alternative 5th" either.
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So it seems we have two fairly obvious options. Hit the 3/2 almost directly so the root and partials "match" ALA mean-tone and JI diatonic....or stay far far away from it to avoid conflict. The problem with using exclusively the first option, of course, is you end up with a bunch of scales that IMVHO sound exactly like mean-tone (and the ones that are closest to using the exact 3/2 as a generator ARE mean-tone). :-D

Firstly, I wonder what solutions people have come up with in history to deal with the "fussy" 5ths issue....what ideas are known about and/or do you have concerning this issue?

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Secondly, I found that by using intervals within about 10 cents of 7/5 (1.4....stays far from fifth), 3/2....hits fifth directly, and 8/5 (1.6....stays far from fifth) as alternating generators, I can obtain scales like the below (note...this is not a JI scale hence fractions are not used)

1
1.1012882
1.2075
1.3766
1.5
1.65193224
1.8233

Note many other scales are of course possible via this method...I just like this one because the smallest interval is not far from 12/11 and doesn't drop into the "13/12 or smaller" area of rapidly increasing dissonance (in both the critical band dissonance and harmonic entropy theories). I also wonder if there's an existing name/theory for such a method...although so far I have yet to hear of one.

Any thoughts or suggestions?

Ah, tempered fifths...

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> I did a blind test on intervals around 3/2 and I learned something. That is just how sensitive 3/2 is as an interval. For example, move from 6/5 (1.2) to 11/9 (1.22222) and it sounds worse but not breathtakingly so. Move from 11/10 to 12/11 covering intervals in-between and there's barely a difference.
>
> But move from 3/2 to 34/23 = 1.4782 and...wow that bites! Wolf indeed.... Turns out (at least to my ear) even the fairly low numbered 16/11 = 1.454545 doesn't work too well as an "alternative 5th" either.
>

Try 32/21 or 22/15. Better yet, try either of those in a triad with a 7/6 or a 9/8. There seems to be a little "zone" around 3/2 where if you're between 15 and 25 cents sharp or flat of a perfect 3/2, you get significant roughness, but when you get close to a 30-cent deviation, you start landing on some more stable intervals. These intervals may not sound great on their own (though they sound better an octave up), but when you make them part of a triad they can sound actually perfectly stable.

Don't believe it? Listen for yourself: I recently put up a guitar piece on the xenharmonic ning called "Into the Gloam" in 18-EDO, which hits both of those intervals (32/21 and 22/15) very close to Just, and I think you'll find the chords in that song sound remarkably stable.

OTOH, I notice that 16-EDO hits very close to that 34/23 at 675 cents, and I find that interval "unusable" as a fifth (though it's good for different purposes!), even though it's "closer" in pitch to a 3/2 than is 22/15 or 32/21.

16/11 I would NEVER suggest as an alternative to the fifth...despite its low complexity, it packs a decidedly-dissonant bite more akin to a species of tritone. When I used to play a Catler 12-tone Ultra Plus guitar, which has (among other things) frets for a Just 11/8 (which can also be used to make a 16/11), I could rarely distinguish 11/8 from sqrt(2)/1 (600 cents) unless played in a high register. Ditto 16/11. 50 cents from a 3/2 is too far to maintain the functionality of a fifth.

An interval of 3/5 of an octave (720 cents) lands almost precisely at a 50/33, you could give that a try too as it seems to be on some sort of psychoacoustic threshold. It beats, but if you put it in a triad with a good major or minor third, it seems to work just fine. Easley Blackwood considered the triads of 15-EDO stable enough to serve as a final sonority in a piece, and I've found the triads of 25-EDO (which feature either a 9/7 or 5/4 which are very near Just) work pretty darn well also.

Bizarre, the 22/15 (1.46666 works quite well for me) despite being in the middle of an area of sour tones. 50/33 actually works quite beautifully as well...surprisingly relaxed feel.

Igs>"An interval of 3/5 of an octave (720 cents) lands almost precisely at a
50/33"
Ah mirroring around simple intervals "strikes again"...that "mirroring" has to be one of the most useful theories I've seen, along with harmonic entropy and critical band dissonance. I believe Cameron used that technique to optimize one of my scales and, again, here it's working beautifully. Helping me break past this "5ths issue" is infinitely helpful...thank you! :-)

As such the latest series of "alternative 5ths" I'm going to be messing with are the following of your suggestions
22/15 = 1.46666 and
50/33 = 1.51515

Hopefully this time around I can get those "magically not-so-dissonant" 12/11 and 11/10 ratios into my scale system (for sake of "impossible" clustered chords) and avoid the exponentially increasing in dissonance 13/12 and closer ratios while keeping plenty of good intervals around good old 3/2 (and not having to, say, chop my 7 note scales into, say, 6 note ones to eliminate the issue). :-)

BTW, Igs, here's a result of those "alternative 5ths" at work. I'm working on more but consider this an appetizer. :-D

Modified/"Tempered" Infinity Scale using Igs's alternative 5ths as generators (very slightly rounded)

1/1
1.11111
1.22222
1.34444
1.4666
1.63
1.8333333
2/1

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:

> So it seems we have two fairly obvious options. Hit the 3/2 almost directly so the root and partials "match" ALA mean-tone and JI diatonic....or stay far far away from it to avoid conflict. The problem with using exclusively the first option, of course, is you end up with a bunch of scales that IMVHO sound exactly like mean-tone (and the ones that are closest to using the exact 3/2 as a generator ARE mean-tone). :-D

Not really--there's the schismic family to consider. Those are the temperaments, identical at the five limit, which use Fb as the major third. But why does the generator need to be a fifth?

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:

> 1
> 1.1012882
> 1.2075
> 1.3766
> 1.5
> 1.65193224
> 1.8233

This looks kind of like a warped version of Porcupine[7], the 7 note MOS of porcupine temperament. What do you think of this scale:

10/9, 6/5, 4/3, 16/11, 8/5, 7/4, 2

It's a JI equivalent of Porcupine[7]; if you tune it in 22 equal, you get Porcupine[7].

Gene>"But why does the generator need to be a fifth?"
Actually it doesn't, but I have not found a good way to get intervals in the right range to "spiral" around to 2 without using estimates of the 5th as at least one of the generators.
What I figure I do need is to have any dyads the scale system avoid hitting the "dead area" in between 7/5,3/2, and 8/5 where very few intervals work. Given that I can't find a way to make just, say, 7/5 and 8/5 generators create an "MOS" at the octave while making a scale that also nears 7TET (to maximize root-tone critical band consonance for the root tones)....I threw the 3/2 back in there as one of the generators.

And, hence I tried alternating between 7/5, 3/2, and 9/5...I've just made so many scales that seem to work very well for seconds, thirds, and odd intervals like 11/9, 11/10, and 12/11...only to find an occasional "wolf" fifth that indirectly comes about by generating optimizing such "other" intervals.
Admittedly, I have a bit of an obsession with getting a fifth or some sort of alternative to it to work well because I think it's safe to assume that's one of the first things musicians will try to do in a scale is fine an interval that can work in that area between 7/5 and 8/5...and again I'm trying to push toward my mission of an "easy for anyone to use" micro-tonal scale that does not sound such at all like 12TET yet does not pose a steep learning curve to the public.

So far one of the better (IMVHO) results I've gotten from using 3/2,7/5, and 9/5 approximated ratios (IE within 13 cents of each of these) as generators is
1
1.11111111 (10/9)
1.22222222 (11/9)
1.3444444 (???)
1.5 (3/2)
1.6666666666 (5/3)
1.8333333 (11/6)
2

It avoids the nasty-sounding (40/33) and tense 1.375 (11/8) ratios for the most part and doesn't hit any off-sounding "alternative 5th" ratios such as 16/11.
But I'm more than up for suggestions about other generation methods or improvements. Again I don't "need" to have 5ths...but I do (in general) figure I need to either hit them near dead-on or avoid them to avoid "Wolves"....

Gene>"This looks kind of like a warped version of Porcupine[7] , the 7 note
MOS of porcupine temperament. What do you think of this scale:
10/9, 6/5, 4/3, 16/11, 8/5, 7/4, 2"

That porcupine temperament subset looks pretty good.
I would have to compose with it to know for sure but just at a glance (note I'm a very picky critic when it comes to certain dyads which I've found, over time, much harder to use in composition)....

Good:
No really narrowly spaced intervals IE nothing under around 13/12.
A good selection of minor thirds in general IE 6/5, 4/3 over 10/9 = about 6/5, 8/5 over 4/3 = 6/5, 7/4 over 16/11 = 6/5.
Seems to have some good major thirds as well.

Bad:
16/11 and 6/5 form the sour 40/33 interval...locked between the IMVHO much cleaner sounding 6/5 and 11/9.

16/11 itself feels sour to me (one of those "bad imitation 5ths" I was discussing/worrying about).
8/5 and 10/9 also form a fairly sour 13/9 (another "bad imitation 5th").
7/4 and 6/5 form another sour 16/11-ish interval.

In general...ratios between 7/5 and 8/5 which form from certain dyads in the scale go very sour.
Not that it's a special problem by any means, I have the same thing pop up in a large percentage of my scales...my "better" ones only have 2-3 sour 5th-ish intervals out of 7. I've seen the same issue pop up in Ptolemy's Homalon scales, which I also like a lot in general, as well.