Gene's very own nonoctave temperaments

I wouldn't say this about Bohlen-Pierce, where there's a certain perverse logic in in ignoring octaves, but a lot of this nonoctave business seems to me to come down to salesmanship. You don't see no-threes or no-fives generating such interest, and what nonoctave scales people pick on is only partly due to the need for small generator sizes.

For example, who promotes the nonoctave scale whose step size is a secor? I've not heard of it, but it makes as much sense as anything. It just doesn't get that old sales push. For another example, in Wendy Carlos style, I'll start with a generator which is 1/20th of a fifth: (3/2)^(1/20). Eleven of these give a major third, ten a neutral third, and nine a minor third, all to microtempering accuracy. That's related to a 5-limit temperament no one seems to care about, which is the 5-limit temperament for Neptune, the 7-limit microtemperament. For another example, take a generator very close to Alpha, namely, (8/3)^(1/22). Now three generators gives 8/7, five gives 5/4, eight gives 10/7, nineteen gives 7/6, twenty-two gives 8/3, and twenty-seven gives 10/3, all to extreme accuracy. This is the generator chain for tertiaseptal temperament.

But why stop at rank one? A nonoctave temperament I'm fond of is the rank two temperament with generators 4/3 and 6/5. Or is that nonoctave?

Gene>"I'll start with a generator which is 1/20th of a fifth: (3/2)^(1/20).
Eleven of these give a major third, ten a neutral third, and nine a minor
third, all to microtempering accuracy."

That's very convenient...but then for the 12th you get about 1.275 or around
14/11ths for the 15th you get 1.355 or around 19/14.

Though for the fourth you could get 14th at about 1.328, which really isn't
that far off 4/3. And then for a second you can use (3/2)^(6/20) which gives
about 1.129, not too far from 9/8 or 1.125.

So (assuming you use steps 6,9,11, and 14 starting) it has some pretty good
properties. But when you start changing the root of a chord from "middle C" I
assume you have to keep very skillful track of which keys you use since 6,9,11,
and 14 all have very different interval spacing (that is, if you want to keep
using pure intervals in your chords).
------------------

>"For another example, take a generator very close to Alpha, namely,
>(8/3)^(1/22). Now three generators gives 8/7, five gives 5/4, eight gives 10/7,
>nineteen gives 7/6, twenty-two gives 8/3, and twenty-seven gives 10/3, all to
>extreme accuracy. This is the generator chain for tertiaseptal temperament."

It seems like a similar deal here. To note...even taking (4/3)^x or
(5/4)*(4/3)^x gives low-limit ratios. But try taking 10/7 * (4/3)^x or
8/7*(4/3)^x and the ratios don't look so beautiful.

Please feel free to give a brilliant counter-example...but the main issue I
see with non-octave scales in composition is you seem to need to worry about one
of the following

A) Keep switching the set of keys of "modes" you use within a tuning when you
want purity in a chord with a different root tone (composition-ally quite
challenging to do while keeping a consistent mood).
B) Face the fact that making chords that cover a lot of frequency space IE have
notes ranging from a root to 19/10 or more over that root can often give sour
ratios, even if the closer spaced ones are very pure.

Lord willing there may be a way to overcome these issues...but personally I
have enough trouble mathematically finding ways to get good "tall/high-range"
chords working within the 2/1 octave.

Hi Gene,

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> I wouldn't say this about Bohlen-Pierce, where there's a certain perverse logic in in ignoring octaves, but a lot of this nonoctave business seems to me to come down to salesmanship. You don't see no-threes or no-fives generating such interest, and what nonoctave scales people pick on is only partly due to the need for small generator sizes.
>

Wendy's argument for her tuning choices was based on the idea that octaves can be accomplished by orchestration (i.e. tune instruments octaves apart), so that it's not important to have them on each instrument. She wrote some kind of computer program to perform a deep search of all generators from 1-of-10-EDO to 1-of-40-EDO (i.e. 120 cents to 30 cents) to find the best approximations to a bunch of simple ratios, and Alpha, Beta, and Gamma, were the top three.

You can read her article here:
http://www.wendycarlos.com/resources/pitch.html

> For example, who promotes the nonoctave scale whose step size is a secor? I've not
> heard of it, but it makes as much sense as anything.

Except that it does not, in its first "octave", approximate very many common simple ratios...which of course means I think it's great (8/7, 11/9, and 21/16 all approximated nearly-perfectly? Sign me up!) but other musicians probably aren't too keen on it.

>
It just doesn't get that old sales push. For another example, in Wendy Carlos style, I'll start with a generator which is 1/20th of a fifth: (3/2)^(1/20).
>

This is Carlos Gamma.

>
For another example, take a generator very close to Alpha, namely, (8/3)^(1/22). Now three generators gives 8/7, five gives 5/4, eight gives 10/7, nineteen gives 7/6, twenty-two gives 8/3, and twenty-seven gives 10/3, all to extreme accuracy. This is the generator chain for tertiaseptal temperament.
>

How significant is the advantage over Alpha here in approximating common small-ratio JI intervals?

-Igs

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> I wouldn't say this about Bohlen-Pierce, where there's a certain
> perverse logic in in ignoring octaves, but a lot of this
> nonoctave business seems to me to come down to salesmanship.
> You don't see no-threes or no-fives generating such interest,

One of the problems people have with better-than-12 tunings of
the 5-limit, and even extended JI, is that audiences can't tell
the difference. Because the octave is the most familiar
interval, it ought to be the most noticeable one to omit.
That's my interpretation of much of the interest in nonoctave
tunings. Gary Morrison stated it explicitly, in fact, as a
reason he turned to 88CET.

-Carl

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:

> Wendy's argument for her tuning choices was based on the idea that octaves can be accomplished by orchestration (i.e. tune instruments octaves apart), so that it's not important to have them on each instrument.

And the advantage is? I don't see it.

> > For example, who promotes the nonoctave scale whose step size is a secor? I've not
> > heard of it, but it makes as much sense as anything.
>
> Except that it does not, in its first "octave", approximate very many common simple ratios...which of course means I think it's great (8/7, 11/9, and 21/16 all approximated nearly-perfectly? Sign me up!) but other musicians probably aren't too keen on it.

Not true. We've got 8/7 at 2, 11/9 at 3, 7/5 at 5, 3/2 at 6, 8/5 at 7, 12/7 at 8 and 11/6 at 9; continuing on we have 9/4 at 12, 12/5 at 13, 18/7 at 14, 11/4 at 15, 22/7 at 17, 18/5 at 19, 22/5 at 22.

> For another example, take a generator very close to Alpha, namely, (8/3)^(1/22). Now three generators gives 8/7, five gives 5/4, eight gives 10/7, nineteen gives 7/6, twenty-two gives 8/3, and twenty-seven gives 10/3, all to extreme accuracy. This is the generator chain for tertiaseptal temperament.
> >
>
> How significant is the advantage over Alpha here in approximating common small-ratio JI intervals?

8/7: 0.377 cents sharp
5/4: 0.394 cents flat
10/7: 0.017 cents flat
7/3: 0.377 cents flat
8/3: just
10/3: 0.394 cents flat

Alpha is a valentine/starling type of tuning, not a microtemperament. While its tuning accuracy is decent, it's not into the tuning overkill range like tertiaseptal or this no-octaves version of tertiaseptal, and you'd get errors an order of magnitude larger, more or less.

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:

> That's my interpretation of much of the interest in nonoctave
> tunings. Gary Morrison stated it explicitly, in fact, as a
> reason he turned to 88CET.

At yet with that very precise choice he seems to suggest we ought to put the octaves back in. By that I mean that 88/1200 = 11/150. We will eventually get multiples of the octave anyway, and what's worse, or better, 11/150 is a very good generator for a rank two temperament, one we've called octacot but maybe we should have called morrison, just out of perversity, since he's almost rubbing our nose in it.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> > That's my interpretation of much of the interest in nonoctave
> > tunings. Gary Morrison stated it explicitly, in fact, as a
> > reason he turned to 88CET.
>
> At yet with that very precise choice he seems to suggest we ought
> to put the octaves back in. By that I mean that 88/1200 = 11/150.
> We will eventually get multiples of the octave anyway,

A fact I've pointed out many times in the past. However like
all omit temperaments, this one offers potentially reduced
complexity if you don't go too far out.

> and what's worse, or better, 11/150 is a very good generator
> for a rank two temperament, one we've called octacot but maybe
> we should have called morrison, just out of perversity, since
> he's almost rubbing our nose in it.

Any good omit temperament will eventually strike a system
that divides the gap, as it were. That doesn't mean omit
systems shouldn't be considered more closely.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:

> Any good omit temperament will eventually strike a system
> that divides the gap, as it were. That doesn't mean omit
> systems shouldn't be considered more closely.

There are omit systems like Bohlen-Pierce where the non-omit version is not that terrific. And there are omit systems like Alpha and 88cet, where the omission borders on perversity, like dancing on one leg to create a new and different dance style. The trouble with these is, the generator is less than 100 cents. So you go quite a long ways before you even run into the question of omit versus non-omit. You can roll happily along adding notes, and it's only when you've reached 16 notes of Alpha/valentine that you are faced for the first time with the question of whether this is going to be a MOS or whether you keep going. Same to a slightly lesser extent with 88cet/octacot. Hence the wonderful melodic properties Carlos alleges for Alpha are going to be equally true of valentine, which gets the octaves but not the good press. That's backwards.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

> --- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@> wrote:
>
> > Wendy's argument for her tuning choices was based on the idea that octaves can be accomplished by orchestration (i.e. tune instruments octaves apart), so that it's not important to have them on each instrument.
>
> And the advantage is? I don't see it.

The advantage is only there for those of us performing live on physical instruments: fewer pitches to navigate! The more intervals one wishes to accurately approximate in a given equal tuning, the more pitches one needs (as you well know), so eliminating the octave lessens the number of pitches. Of course, only Alpha truly accomplishes this; Beta and especially Gamma are basically versions of 19-EDO and 34-EDO optimized for non-2 harmonies (i.e. where the octaves are heavily tempered to improve accuracy of other intervals).

Compare Alpha to 31-tET, and you'll see that Alpha approximates many of the same simple ratios (8/7, 6/5, 5/4, 11/8, 10/7, 3/2, 11/7, 12/7, 9/5, etc) with similar accuracy, and yet with a generator more than twice as wide as 31-tET, meaning half as many pitches to deal with in a given octave span. For those of us who must perform our music on physical instruments (rather than programming it), this is a significant advantage!

> Not true. We've got 8/7 at 2, 11/9 at 3, 7/5 at 5, 3/2 at 6, 8/5 at 7, 12/7 at 8 and 11/6 at 9; continuing on we have 9/4 at 12, 12/5 at 13, 18/7 at 14, 11/4 at 15, 22/7 at 17, 18/5 at 19, 22/5 at 22.
>

But no 6/5 and no 5/4, the thirds which form the back-bone of common-practice music. Thus it is not suited for the same stuff as the Carlos tunings, which all follow the same harmonic basis as common-practice music. Not to say I don't think it's grand! Personally, I'll take the secor tuning over the Carlos tunings any day.

> > How significant is the advantage over Alpha here in approximating common small-ratio JI intervals?
>
> 8/7: 0.377 cents sharp
> 5/4: 0.394 cents flat
> 10/7: 0.017 cents flat
> 7/3: 0.377 cents flat
> 8/3: just
> 10/3: 0.394 cents flat

Yet it loses accuracy in 6/5, 11/8, 3/2, 12/7, 9/5, 15/8, and 9/4, and it approximates 25/16 instead of 11/7 (more complex). So it's a trade-off, not an improvement per se.

-Igs

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
Of course, only Alpha truly accomplishes this; Beta and especially Gamma are basically versions of 19-EDO and 34-EDO optimized for non-2 harmonies (i.e. where the octaves are heavily tempered to improve accuracy of other intervals).

Describing Gamma as an optimized 34-edo makes it sound like the 34 note MOS of the rank two temperament, and not Gamma at all. My argument is, why not just use the freakin' MOS?

> Compare Alpha to 31-tET, and you'll see that Alpha approximates many of the same simple ratios (8/7, 6/5, 5/4, 11/8, 10/7, 3/2, 11/7, 12/7, 9/5, etc) with similar accuracy, and yet with a generator more than twice as wide as 31-tET, meaning half as many pitches to deal with in a given octave span.

You are making out a case for the 15 or 16 note MOS of valentine here much more than for Alpha. This subject seems to have a tendency to make people blind to the obvious.

> For those of us who must perform our music on physical instruments (rather than programming it), this is a significant advantage!

Go valentine! Thanks for pointing out the advantages of rank two over rank one.

> > Not true. We've got 8/7 at 2, 11/9 at 3, 7/5 at 5, 3/2 at 6, 8/5 at 7, 12/7 at 8 and 11/6 at 9; continuing on we have 9/4 at 12, 12/5 at 13, 18/7 at 14, 11/4 at 15, 22/7 at 17, 18/5 at 19, 22/5 at 22.
> >
>
> But no 6/5 and no 5/4, the thirds which form the back-bone of common-practice music.

You are arguing both sides of the issue now. Do we want to get the backbone of common practice, or do we want to be strange and exotic? If the former, then of course you need octaves.

My responses are interwoven below:

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

>
> Describing Gamma as an optimized 34-edo makes it sound like the 34 note MOS of the rank two temperament, and not Gamma at all. My argument is, why not just use the freakin' MOS?
>
> You are making out a case for the 15 or 16 note MOS of valentine here much more than for Alpha. This subject seems to have a tendency to make people blind to the obvious.
>

Okay, well here's one reason to use the non-octave scale: you get access to different intervals in each octave-range. If you use the 15 or 16 note MOS of Valentine instead of Alpha, you lose intervals like 9/4, 26/11, and 18/7 (though I'll admit, there aren't too many intervals past those in the 2nd octave of Alpha very close to anything simple/consonant). Using non-octave scales allows for a large number of intervals despite a wide generator/chromatic step. In other words, it's a more "space-efficient" way of getting high accuracy and large intervallic variety while avoiding close chromatic steps.

> You are arguing both sides of the issue now. Do we want to get the backbone of common practice, or do we want to be strange and exotic? If the former, then of course you need octaves.
>

Why does it have to be a dichotomy? What if we want a touch of common-practice AND strange and exotic?

There are advantages to both approaches, Gene. I'm not saying one is superior to the other (in fact, I'm not terribly interested in either, though that Secor tuning looks promising), but there are certain advantages to both sides. Yes, using the MOS of those temperaments allows you to maintain a low number of pitches and many of the useful properties of the non-octave scales, and are worth mentioning to people looking at the non-octave scales. The only reason that I can see that the non-octave scales are more popular is, well, because they were proposed by Wendy Carlos, and she's pretty gosh-darn popular. She probably doesn't know a darn thing about temperaments, and if you educated her about Valentine et al. I'm sure she'd be fascinated.

For what it's worth, I'd like to see more sales-pitches for non-fifth tunings. You are correct that they receive very little press, especially compared to the Carlos tunings. We all have our pet tunings, I guess.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
>
> > Any good omit temperament will eventually strike a system
> > that divides the gap, as it were. That doesn't mean omit
> > systems shouldn't be considered more closely.
>
> There are omit systems like Bohlen-Pierce where the non-omit
> version is not that terrific. And there are omit systems like
> Alpha and 88cet, where the omission borders on perversity,
> like dancing on one leg to create a new and different dance
> style. The trouble with these is, the generator is less than
> 100 cents. So you go quite a long ways before you even run
> into the question of omit versus non-omit. You can roll
> happily along adding notes, and it's only when you've reached
> 16 notes of Alpha/valentine that you are faced for the first
> time with the question of whether this is going to be a MOS
> or whether you keep going. Same to a slightly lesser extent
> with 88cet/octacot. Hence the wonderful melodic properties
> Carlos alleges for Alpha are going to be equally true of
> valentine, which gets the octaves but not the good press.
> That's backwards.

Huh? If you keep going past 15 notes you get something quite
different than if you wrap at 2/1. Even after 46 places we're
missing the octave by 12 cents. BP gets within 2 cents after
41 places. Except with BP we're not supposed to go beyond 13.
And similarly, we could argue that with Alpha we should not
go beyond 9.

-Carl

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:

> For what it's worth, I'd like to see more sales-pitches for non-fifth tunings. You are correct that they receive very little press, especially compared to the Carlos tunings. We all have our pet tunings, I guess.
>

You know, when I think "no fives" I immediately think of 36, and "no sevens" brings 24 to mind, but I had to think a while before "no threes" brought up 31--because of course it does have a 3, just not so good as the 5 and the 7. 6 is another obvious number, and when I stuck 6 and 31 together I came up with something I didn't know--six supports hemiwuerschmidt temperament. This has a complex 3 anyway, so deep-sixing it is not so big a loss. The no-threes comma being tempered out is 3136/3125, which is a pretty important one, I'm thinking, in the no-threes universe. For less complexity, there's 50/49, which you could use for a no-threes version of pajara.

If you tune the no-threes version of hemiwuerschmidt, you get very good tuning accuracy, nearly to the microtemperament range, coupled with low complexity. I'll think about it some more and see if I can figure out how to sell it. Good press seems to be important. The 6-note MOS and the very low error are a place to start, I suppose.

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:

> Huh? If you keep going past 15 notes you get something quite
> different than if you wrap at 2/1.

That's sort of the point; it's different, but not in a way which helps as much as what you get by wrapping.

Even after 46 places we're
> missing the octave by 12 cents. BP gets within 2 cents after
> 41 places. Except with BP we're not supposed to go beyond 13.
> And similarly, we could argue that with Alpha we should not
> go beyond 9.

If you only have nine notes, the question hardly arises. Who wants just nine notes?

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

> > Huh? If you keep going past 15 notes you get something quite
> > different than if you wrap at 2/1.
>
> That's sort of the point; it's different, but not in a way
> which helps as much as what you get by wrapping.

Helps?

> > Even after 46 places we're
> > missing the octave by 12 cents. BP gets within 2 cents after
> > 41 places. Except with BP we're not supposed to go beyond 13.
> > And similarly, we could argue that with Alpha we should not
> > go beyond 9.
>
> If you only have nine notes, the question hardly arises. Who
> wants just nine notes?

9 notes/period is more than 7, which is what most music uses.

-Carl