Yahoo Tuning Groups Ultimate Backup tuning Constructing a dual- or triple-minor scale

I was playing around with Canton, which Gene posted recently. I have a little unfinished experimental material to show for it:

MIDI: http://www.freivald.org/~jake/documents/cantonese-retuned.mid
PDF score: http://www.freivald.org/~jake/documents/cantonese.pdf
Lilypad source: http://www.freivald.org/~jake/documents/cantonese.ly

...but I found myself not finding this to be different enough from 12-TET to matter. It has a mildly different character from 12-TET, and I even retuned it in John O'Sullivan's Blue Just tuning to see a greater difference in character. But I didn't feel like I was really writing non-twelve material. (I say this knowing that the lousy MIDI rendering on my PC may have something to do with it.)

So I asked myself what I wanted in order to "feel like I was doing microtonality". Then I thought I should use this as an example of tempering, if anyone's willing to spend the time on it.

I like the three minor thirds I've been playing with recently: 7/6 (266 cents), 13/11 (289), and 6/5 (316). I wanted to be able to play around with more than one minor third per chord in the I, IV, and V chords of the scale. I also wanted a 12-note scale so I could play in Lilypad.

That says I should have these intervals:
I:
1/1
7/6
13/11
6/5
4/3
3/2
2/1

IV:
4/3 * 1/1 = 4/3
4/3 * 7/6 = 14/9
4/3 * 13/11 = 52/33
4/3 * 6/5 = 8/5
4/3 * 4/3 = 16/9
4/3 * 3/2 = 2/1

V:
3/2 * 1/1 = 3/2
3/2 * 7/6 = 7/4
3/2 * 13/11 = 39/22
3/2 * 6/5 = 9/5
3/2 * 4/3 = 2/1
3/2 * 3/2 = 9/4 = 9/8

Sort and de-duplicate to get:

1 / 1
9 / 8
7 / 6
13 / 11
6 / 5
4 / 3
3 / 2
14 / 9
52 / 33
8 / 5
7 / 4
39 / 22
16 / 9
9 / 5
2 / 1

If I understand the way this works, I've selected intervals from the 2.3.5.7.13/11 subgroup, which is a five-dimensional tuning space. Is that a bad idea? If so, why?

This scale's not proper, but I'm not sure why that matters, so I'm ignoring that for now.

I have too many intervals: I want a 12-note scale. So I'll probably have to eventually pull out one of the minor thirds. I lean toward leaving in the 7/6 and the 6/5 because they're farther apart and therefore easier to hear; also, by removing the 13/11 from the tuning space, I collapse one dimension. That said, I'm open to suggestions.

Now, I'm open to tempering this scale, but I'm not sure whether I should. What benefit would I gain?

Here are some of the possible commas to temper. There may well be more, since I'm just eyeballing this. I'm including all three minor thirds, just in case something interesting pops up. I'm also including the intervals that these commas equate, even though I know (as Mike said before) people may generally not care about that when they choose temperaments.

7093/7085, 2 cents
difference between 4 9/8s and one 8/5
difference between 5 9/8s and one 9/5

352/351, 4.9 cents
difference between 3 4/3s and one octave + 13/11
difference between 4 4/3s and one octave + 52/33

81/80, 21.5 cents
difference between 4 4/3s and one octave + 8/5

1029/1024, 8.4 cents
difference between 3 7/4s and two octaves + 4/3
difference between 4 7/4s and three octaves + 7/6

1410/1409, 1.2 cents
difference between 5 9/5s and four octaves + 13/11

But now I'm lost. Why would I choose any particular one of these commas to temper?

7093/7085 doesn't seem very interesting, because those are the same equivalences I get with 12-EDO (4 whole tones makes m6, C-Ab; 5 whole tones makes m7, C-Bb).

I'm guessing that by tempering 352/351, I could formally eliminate the 13/11 and 52/33 intervals, while still getting them functionally in the scale; that makes that comma interesting, if I want to keep the 13/11 intervals and drop the 7s.

1029/1024 seems to make it possible to do interesting things with stacks of harmonic sevenths, which sounds pretty cool.

But those are just guesses.

Is there a way to do the tempering without knowing Maple? The xenwiki EDO pages don't say what commas they temper, unfortunately, because if it did I could choose EDOs that temper those commas and then select the closest value for each note in the scale. I don't know how to do the TOP computations to see which one gives the smallest level of distortion, but I should at least be able to get something workable. Would that be a reasonable approach, if I knew which EDOs temper which commas?

Does anyone have any thoughts on whether / how to temper this? Or on interesting commas I missed?

Am I even close to doing this right? Is it remotely interesting? :)

Thanks,
Jake

🔗genewardsmith <genewardsmith@...>

--- In tuning@yahoogroups.com, Jake Freivald <jdfreivald@...> wrote:

> 7093/7085, 2 cents
> difference between 4 9/8s and one 8/5
> difference between 5 9/8s and one 9/5

This is actually 32805/32768, the schisma. Tempering it out is characteristic of a lot of temperaments.

> 352/351, 4.9 cents
> difference between 3 4/3s and one octave + 13/11
> difference between 4 4/3s and one octave + 52/33
>
> 81/80, 21.5 cents
> difference between 4 4/3s and one octave + 8/5
>
> 1029/1024, 8.4 cents
> difference between 3 7/4s and two octaves + 4/3
> difference between 4 7/4s and three octaves + 7/6
>
> 1410/1409, 1.2 cents
> difference between 5 9/5s and four octaves + 13/11

Again, this is a little off--it's actually 650000/649539. If you are doing these approximations deliberately, you shouldn't: when tempering, it's crucial you use the exact comma, and not an approximation.

> But now I'm lost. Why would I choose any particular one of these commas
> to temper?

That would depend on what you want to do with it all.

> 7093/7085 doesn't seem very interesting, because those are the same
> equivalences I get with 12-EDO (4 whole tones makes m6, C-Ab; 5 whole
> tones makes m7, C-Bb).
>
> I'm guessing that by tempering 352/351, I could formally eliminate the
> 13/11 and 52/33 intervals, while still getting them functionally in the
> scale; that makes that comma interesting, if I want to keep the 13/11
> intervals and drop the 7s.

This equates the Pythagorean minor third, 32/27, with 13/11, and makes a pretty good fit with the schisma. It tends towards the sharp fifth end of the spectrum, and temperaments where fifths are of low complexity.

> 1029/1024 seems to make it possible to do interesting things with stacks
> of harmonic sevenths, which sounds pretty cool.

It's one of the key 7-limit intervals to temper out. 7 is of low complexity, and 3 three times higher in complexity, so it prevents 3 from being at the bottom of the spectrum of complexities.

> Is there a way to do the tempering without knowing Maple?

Of course. The easiest way would be to go here:

http://x31eq.com/temper/

And in particular the "unison vector search" page.

The xenwiki
> EDO pages don't say what commas they temper, unfortunately, because if
> it did I could choose EDOs that temper those commas and then select the
> closest value for each note in the scale.

If you want to temper 352/351, 1029/1024 and 32805/32768, just feed these into the maw of Graham's machine. This cuts rank six down to rank three. If you wanted rank two, guiron, the 41&77 temperament, suggests itself. 118et is pretty much unbeatable for 13-limit guiron, so a quick answer to your question would be "temper in 118". However, I'm not sure why you are choosing the commas you do. 118et does not temper out 81/80, which I doubt you want to do anyway in the light of the rest of your commas, but also not 650000/649539. If you add the latter to 351/350, 1029/1024 and 32805/32768, you get a perverse variation of guiron, so I'd just stick to guiron and 118, by the way.

🔗genewardsmith <genewardsmith@...>

🔗Mike Battaglia <battaglia01@...>

On Tue, Mar 15, 2011 at 9:57 PM, Jake Freivald <jdfreivald@...> wrote:
>
> ...but I found myself not finding this to be different enough from
> 12-TET to matter. It has a mildly different character from 12-TET, and I
> even retuned it in John O'Sullivan's Blue Just tuning to see a greater
> difference in character. But I didn't feel like I was really writing
> non-twelve material. (I say this knowing that the lousy MIDI rendering
> on my PC may have something to do with it.)

The fact that Blue Just is also pretty close to 12-equal has a lot to
do with it as well :)

> So I asked myself what I wanted in order to "feel like I was doing
> microtonality". Then I thought I should use this as an example of
> tempering, if anyone's willing to spend the time on it.

It's an interesting scale, and Gene's given some good suggestions to
temper it further. Keep in mind that the goal of tempering is to
increase the consonance of a scale, so this is not a bad thing.

Here's an exercise to try that may prove useful: go to 22-tet or
27-tet and load up the superpyth aeolian scale. Superpyth refers to a
scale in which the fifths are sharp, such that four of them is
tempered to make an octave-equivalent 9/7, as opposed to the more
common 5/4. Three fourths makes an approximate 7/6. So in a sense,
it's kind of like a 7-limit version of meantone.

Now load up a superpyth aeolian scale and play around for a while and
see if it sounds radically different from meantone aeolian. Try
comparing the version in 27-tet, which has almost pure 7/6's, to the
version in 19-tet, which has almost pure 6/5's.

You can probably detect the difference in that the 6:7:9 chords are a
bit more serene and restful sounding (at least that's how I hear them,
anyway) - but in another sense the scale sounds "the same." That is,
it's microtonal, kind of, but it doesn't sound like some completely
new, original, xenharmonic, novel sound that breaks from tradition.
(And for the record, I don't find that the pajara scales sound that
way either).

Now load up machine[11] in 17-tet and play 4:7:9:11, and transpose it
up and down the scale. That probably sounds completely different. Yes
it does.

If you want something different, in my experience, the best way to
trigger that is to start by finding scales that produce different
"puns," as it were.

-Mike

🔗genewardsmith <genewardsmith@...>

🔗Chris Vaisvil <chrisvaisvil@...>

🔗Jacques Dudon <fotosonix@...>

🔗genewardsmith <genewardsmith@...>

🔗Jacques Dudon <fotosonix@...>

🔗Mike Battaglia <battaglia01@...>

On Thu, Mar 17, 2011 at 1:43 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, "Jacques Dudon" <fotosonix@...> wrote:
>
> > > The point of tempering out 364/363, if you recall, was to make a 14/11 third and a 13/11 third come to a fifth. Whether canton is real microtonality or not, the use of 14/11 and 13/11 in place of major and minor thirds makes it xenharmonic at any rate, and I like it.
> >
> > Sounds interesting !
> > Have you noticed that it could easily temper 484/483, 897/896, 1288/1287 (etc.), and extend to prime 23 ?
>
> Nope! I think the 15-limit suffices for most musical purposes, and am not inclined to go above the prime 19 except in theory. So I don't usually look at such issues.

I used the 19-limit extensively in my early experiments with 72-TET. I
think it's fantastic, personally. I would generally go up to 19, but
sometimes throw 23 in as a "reserve" note - it doesn't sound like much
by itself, but if you're constructing a giant chord like
4:5:7:9:11:13:15:17:19, 23 works well on top of it. Plus, after 23 it
isn't until 29 that you actually hit a new prime, and 29 is pretty far
out there.

-Mike

🔗Kalle Aho <kalleaho@...>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Thu, Mar 17, 2011 at 1:43 PM, genewardsmith
> <genewardsmith@...> wrote:
> >
> > --- In tuning@yahoogroups.com, "Jacques Dudon" <fotosonix@> wrote:
> >
> > > > The point of tempering out 364/363, if you recall, was to make a 14/11 third and a 13/11 third come to a fifth. Whether canton is real microtonality or not, the use of 14/11 and 13/11 in place of major and minor thirds makes it xenharmonic at any rate, and I like it.
> > >
> > > Sounds interesting !
> > > Have you noticed that it could easily temper 484/483, 897/896, 1288/1287 (etc.), and extend to prime 23 ?
> >
> > Nope! I think the 15-limit suffices for most musical purposes, and am not inclined to go above the prime 19 except in theory. So I don't usually look at such issues.
>
> I used the 19-limit extensively in my early experiments with 72-TET. I
> think it's fantastic, personally. I would generally go up to 19, but
> sometimes throw 23 in as a "reserve" note - it doesn't sound like much
> by itself, but if you're constructing a giant chord like
> 4:5:7:9:11:13:15:17:19, 23 works well on top of it. Plus, after 23 it
> isn't until 29 that you actually hit a new prime, and 29 is pretty far
> out there.

Outside those giant chords higher intervals start to sound like
variants of lower ones. 24:29 at 327.622 cents is very close to
22-equal minor third, for example. Look at 31-odd limit diamond with
Scala, it's interesting. I wonder if it's possible to evoke 23:29
(401.303 cents) with some 12-equal voicing, probably not.

Kalle

Constructing a dual- or triple-minor scale

🔗Jake Freivald <jdfreivald@...>

🔗genewardsmith <genewardsmith@...>

🔗genewardsmith <genewardsmith@...>

🔗Mike Battaglia <battaglia01@...>

🔗genewardsmith <genewardsmith@...>

🔗genewardsmith <genewardsmith@...>

🔗Chris Vaisvil <chrisvaisvil@...>

🔗Jacques Dudon <fotosonix@...>

🔗genewardsmith <genewardsmith@...>

🔗Jacques Dudon <fotosonix@...>

🔗Mike Battaglia <battaglia01@...>

🔗Kalle Aho <kalleaho@...>