septimal error equalization question

What is the size of the superpythagorean fifth whose cycle yields a 6:7:9
with same amount of deviation from the target intervals? What equal
temperament best approximates it? Also, how is the calculation done?

Oz.

Ozan wrote:

"What is the size of the superpythagorean fifth whose cycle yields a 6:7:9 with same amount of deviation from the target intervals? What equal temperament best approximates it?"

If you want a pure 9/7, then choose a fifth of ~708.7710238 cents. If you want a pure 7/6, then go for ~711.0430315 cents. If you want both of these mistuned by the same amount, then it is ~709.7447414 cents.

"Also, how is the calculation done?"

It's actually just the same procedure as with meantones, only the comma used is different and the tempering goes in the opposite direction. The comma in question is the septimal comma of 64/63. So in fact, the resulting temperaments really are 1/4-comma superpyth, 1/3-comma superpyth, and 2/7-comma superpyth, respectively.

Petr

Excellent. Thank you for clarifying this point Petr.

Oz.

----- Original Message -----
From: "Petr Par�zek" <p.parizek@chello.cz>
To: <tuning@yahoogroups.com>
Sent: 11 Mart 2007 Pazar 9:37
Subject: Re: [tuning] septimal error equalization question

> Ozan wrote:
>
> "What is the size of the superpythagorean fifth whose cycle yields a 6:7:9
> with same amount of deviation from the target intervals? What equal
> temperament best approximates it?"
>
> If you want a pure 9/7, then choose a fifth of ~708.7710238 cents. If you
> want a pure 7/6, then go for ~711.0430315 cents. If you want both of these
> mistuned by the same amount, then it is ~709.7447414 cents.
>
> "Also, how is the calculation done?"
>
> It's actually just the same procedure as with meantones, only the comma
used
> is different and the tempering goes in the opposite direction. The comma
in
> question is the septimal comma of 64/63. So in fact, the resulting
> temperaments really are 1/4-comma superpyth, 1/3-comma superpyth, and
> 2/7-comma superpyth, respectively.
>
> Petr
>
>

I notice 71-equal is an interesting temperament that has a fifth closest to
optimal size and sounds delicious with 7-limit. Has anyone used it before?

Oz.

----- Original Message -----
From: "Petr Par�zek" <p.parizek@chello.cz>
To: <tuning@yahoogroups.com>
Sent: 11 Mart 2007 Pazar 9:37
Subject: Re: [tuning] septimal error equalization question

> Ozan wrote:
>
> "What is the size of the superpythagorean fifth whose cycle yields a 6:7:9
> with same amount of deviation from the target intervals? What equal
> temperament best approximates it?"
>
> If you want a pure 9/7, then choose a fifth of ~708.7710238 cents. If you
> want a pure 7/6, then go for ~711.0430315 cents. If you want both of these
> mistuned by the same amount, then it is ~709.7447414 cents.
>
> "Also, how is the calculation done?"
>
> It's actually just the same procedure as with meantones, only the comma
used
> is different and the tempering goes in the opposite direction. The comma
in
> question is the septimal comma of 64/63. So in fact, the resulting
> temperaments really are 1/4-comma superpyth, 1/3-comma superpyth, and
> 2/7-comma superpyth, respectively.
>
> Petr
>
>

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> I notice 71-equal is an interesting temperament that has a fifth
closest to
> optimal size and sounds delicious with 7-limit. Has anyone used it
before?

I would think in that area, people zoom in on 72-eq.

I notice that 7/4 in 71-eq is not all that accurate, either. (980.282
cents). Not bad, but not the best available, and if you're interested
in 5-limit accuracy being thrown in there for good measure, the RMS
error of 31 or 53 is still better.

If you are only thinking 3 and 7 limit (no fives), the best bang for
the buck < 72 is perhaps 41, which has a lower RMS error for 3 and 7
than 72 does. I would guess plenty of folks have used 41, but I don't
know who. I know Kraig mentioned Wilson was a big 41 advocate.

-A.

----- Original Message -----
From: "Aaron Krister Johnson" <aaron@dividebypi.com>
To: <tuning@yahoogroups.com>
Sent: 11 Mart 2007 Pazar 16:25
Subject: [tuning] Re: septimal error equalization question

> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
> >
> > I notice 71-equal is an interesting temperament that has a fifth
> closest to
> > optimal size and sounds delicious with 7-limit. Has anyone used it
> before?
>
> I would think in that area, people zoom in on 72-eq.
>

But the cycle does not yield accurate 7-limit intervals. And the generator
interval is not the fifth.

> I notice that 7/4 in 71-eq is not all that accurate, either. (980.282
> cents). Not bad, but not the best available, and if you're interested
> in 5-limit accuracy being thrown in there for good measure, the RMS
> error of 31 or 53 is still better.
>

RMS?

Anyway, these are not superpythagorean systems.

> If you are only thinking 3 and 7 limit (no fives), the best bang for
> the buck < 72 is perhaps 41, which has a lower RMS error for 3 and 7
> than 72 does. I would guess plenty of folks have used 41, but I don't
> know who. I know Kraig mentioned Wilson was a big 41 advocate.
>
> -A.
>
>

I was too once. How about 49 then?

Oz.

----- Original Message ----- From: "Ozan Yarman" <ozanyarman@ozanyarman.com>
To: <tuning@yahoogroups.com>
Sent: Sunday, March 11, 2007 7:59 AM
Subject: Re: [tuning] septimal error equalization question

>I notice 71-equal is an interesting temperament that has a fifth closest to
> optimal size and sounds delicious with 7-limit. Has anyone used it before?
>
> Oz.

You're familiar with 22-tet, right? It has a similar fifth (709.0909 cents), being superpythagorean with the fifth tempered upward by a little less than 1/3 septimal comma.

There's also 27-tet (fifth = 711.1111), which is practically identical to 1/3-septimal comma.

~D.

--- In tuning@yahoogroups.com, "Aaron Krister Johnson" <aaron@...>
wrote:

> I would guess plenty of folks have used 41, but I don't
> know who. I know Kraig mentioned Wilson was a big 41 advocate.

I have. It's self-recommneding for magic temperament, among other
things.

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> I notice 71-equal is an interesting temperament that has a fifth
closest to
> optimal size and sounds delicious with 7-limit. Has anyone used it
before?

71 actually has two 7-limit versions, depending on whether you tune the
7 sharp or flat, but you are thinking of the version which supports
superpyth. It is indeed a good superpyth tuning, and is "poptimal" for
9-limit superpyth. That means it falls inside of a range of tunings
which are unweighted optimal for the 9-limit tonality diamond.

--- In tuning@yahoogroups.com, "Danny Wier" <dawiertx@...> wrote:

> You're familiar with 22-tet, right? It has a similar fifth (709.0909
cents),
> being superpythagorean with the fifth tempered upward by a little
less than
> 1/3 septimal comma.
>
> There's also 27-tet (fifth = 711.1111), which is practically
identical to
> 1/3-septimal comma.

Ozan's suggestion of 49 is the smallest poptimal tuning, for what that
is worth. I think 71 would be better, really, but of course the
question is what the complexity gets you, and that depends on whether
you need to worry about complexity at all.

----- Original Message -----
From: "Danny Wier" <dawiertx@sbcglobal.net>
To: <tuning@yahoogroups.com>
Sent: 11 Mart 2007 Pazar 18:45
Subject: Re: [tuning] septimal error equalization question

> ----- Original Message -----
> From: "Ozan Yarman" <ozanyarman@ozanyarman.com>
> To: <tuning@yahoogroups.com>
> Sent: Sunday, March 11, 2007 7:59 AM
> Subject: Re: [tuning] septimal error equalization question
>
>
> >I notice 71-equal is an interesting temperament that has a fifth closest
to
> > optimal size and sounds delicious with 7-limit. Has anyone used it
before?
> >
> > Oz.
>
> You're familiar with 22-tet, right? It has a similar fifth (709.0909
cents),
> being superpythagorean with the fifth tempered upward by a little less
than
> 1/3 septimal comma.
>
> There's also 27-tet (fifth = 711.1111), which is practically identical to
> 1/3-septimal comma.
>
> ~D.
>
>
>

Yes, I have encountered these. But somehow, I think 49 is better. It has a
fifth 710.204 cents wide.

Oz.

I knew something was to be said about it.

Oz.

----- Original Message -----
From: "Gene Ward Smith" <genewardsmith@coolgoose.com>
To: <tuning@yahoogroups.com>
Sent: 11 Mart 2007 Pazar 21:33
Subject: [tuning] Re: septimal error equalization question

> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
> >
> > I notice 71-equal is an interesting temperament that has a fifth
> closest to
> > optimal size and sounds delicious with 7-limit. Has anyone used it
> before?
>
> 71 actually has two 7-limit versions, depending on whether you tune the
> 7 sharp or flat, but you are thinking of the version which supports
> superpyth. It is indeed a good superpyth tuning, and is "poptimal" for
> 9-limit superpyth. That means it falls inside of a range of tunings
> which are unweighted optimal for the 9-limit tonality diamond.
>
>

So, did anyone think of composing in 71? It has a lot to say for itself.

Oz.

----- Original Message -----
From: "Gene Ward Smith" <genewardsmith@coolgoose.com>
To: <tuning@yahoogroups.com>
Sent: 11 Mart 2007 Pazar 21:37
Subject: [tuning] Re: septimal error equalization question

> --- In tuning@yahoogroups.com, "Danny Wier" <dawiertx@...> wrote:
>
> > You're familiar with 22-tet, right? It has a similar fifth (709.0909
> cents),
> > being superpythagorean with the fifth tempered upward by a little
> less than
> > 1/3 septimal comma.
> >
> > There's also 27-tet (fifth = 711.1111), which is practically
> identical to
> > 1/3-septimal comma.
>
> Ozan's suggestion of 49 is the smallest poptimal tuning, for what that
> is worth. I think 71 would be better, really, but of course the
> question is what the complexity gets you, and that depends on whether
> you need to worry about complexity at all.
>
>

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> So, did anyone think of composing in 71? It has a lot to say for
itself.

That's almost like asking if anyone has composing aything in superpyth.
And there, as with most of these things, you've got a wide-open field
of opportunity. I tried to convert a piece to superpyth for you using
Scala's new LT notation system, but it didn't work, alas.

So what was the piece?

----- Original Message -----
From: "Gene Ward Smith" <genewardsmith@coolgoose.com>
To: <tuning@yahoogroups.com>
Sent: 12 Mart 2007 Pazartesi 0:54
Subject: [tuning] Re: septimal error equalization question

> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
> >
> > So, did anyone think of composing in 71? It has a lot to say for
> itself.
>
> That's almost like asking if anyone has composing aything in superpyth.
> And there, as with most of these things, you've got a wide-open field
> of opportunity. I tried to convert a piece to superpyth for you using
> Scala's new LT notation system, but it didn't work, alas.
>
>

> If you are only thinking 3 and 7 limit (no fives), the best bang for
> the buck < 72 is perhaps 41, which has a lower RMS error for 3 and 7
> than 72 does. I would guess plenty of folks have used 41, but I don't
> know who. I know Kraig mentioned Wilson was a big 41 advocate.
>
> -A.

In my way early days, I was a big proponent of 41.

-Carl

> > I would guess plenty of folks have used 41, but I don't
> > know who. I know Kraig mentioned Wilson was a big 41 advocate.
>
> I have. It's self-recommneding for magic temperament, among other
> things.

I like it for schismatic. 12 fewer tones than 53 and
decent accuracy. The 7:4 is better, anyway.

Like Kraig, I think the Achilles' heel of 31 is the 9:4.
31 is dominating my ET ranking lists at the moment, but I
suspect when I get a stronger error weighting (one that
converges rapidly enough to make the 17-limit the last
signficant limit) 31 not look so great.

-Carl

> Ozan's suggestion of 49 is the smallest poptimal tuning, for what
> that is worth. I think 71 would be better, really, but of course
> the question is what the complexity gets you, and that depends on
> whether you need to worry about complexity at all.

It's < 79 and you get an ET. I wonder if the 693-cent fifth
is good enough for the meantone portion of Ozan's requirements.

-Carl

> Yes, I have encountered these. But somehow, I think 49 is better.
> It has a fifth 710.204 cents wide.

But no meantone option. -Carl

It is a stretch, but it does appear to work admirably. However, I would not
recommend a transition right away.

Oz.

----- Original Message -----
From: "Carl Lumma" <clumma@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: 12 Mart 2007 Pazartesi 3:08
Subject: [tuning] Re: septimal error equalization question

> > Ozan's suggestion of 49 is the smallest poptimal tuning, for what
> > that is worth. I think 71 would be better, really, but of course
> > the question is what the complexity gets you, and that depends on
> > whether you need to worry about complexity at all.
>
> It's < 79 and you get an ET. I wonder if the 693-cent fifth
> is good enough for the meantone portion of Ozan's requirements.
>
> -Carl
>
>

If you want both, 71 is the choice.

----- Original Message -----
From: "Carl Lumma" <clumma@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: 12 Mart 2007 Pazartesi 3:09
Subject: [tuning] Re: septimal error equalization question

> > Yes, I have encountered these. But somehow, I think 49 is better.
> > It has a fifth 710.204 cents wide.
>
> But no meantone option. -Carl
>
>

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

> It's < 79 and you get an ET. I wonder if the 693-cent fifth
> is good enough for the meantone portion of Ozan's requirements.

The experiments with flattone tunings which have been made suggest it
is workable.

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> If you want both, 71 is the choice.

Sounds good to me. The "7/4" is very close to a precise mixture of 7/4
and 16/9 (which would be at 982.5 cents) and could probably be heard
as either/both.

--- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...> wrote:
>
> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@> wrote:
> >
> > If you want both, 71 is the choice.
>
> Sounds good to me. The "7/4" is very close to a precise mixture of
7/4
> and 16/9 (which would be at 982.5 cents) and could probably be heard
> as either/both.

That's one of the main ideas behind superpyth.