A weird partial 13-limit linear temperament

Hi guys,

has any of you tried this? Period = 2/1, generator = 13th root of 130. First of all, the generator itself is close to 16/11. Then, 5 of them are close to 13/2. And finally, 8 of them are close to 20/1. Unfortunately, there's only a poor approximation to 3/2. In contrast, the chord of 1:2:5:11:13 is imitated so well that the intervals are just over half a cent away from JI!
Here's what it sounds like: http://download.yousendit.com/04C4C2D07E9D6FE9

Petr

--- In tuning@yahoogroups.com, Petr Pařízek <p.parizek@...> wrote:
>
> Hi guys,
>
> has any of you tried this? Period = 2/1, generator = 13th root
> of 130. First of all, the generator itself is close to 16/11.
> Then, 5 of them are close to 13/2. And finally, 8 of them are
> close to 20/1. Unfortunately, there's only a poor approximation
> to 3/2. In contrast, the chord of 1:2:5:11:13 is imitated so
> well that the intervals are just over half a cent away from JI!
> Here's what it sounds like:
> http://download.yousendit.com/04C4C2D07E9D6FE9

I don't think it likely anyone here's tried it, because
other than a few investigations into bases lacking 2, our
theory is entirely based on complete prime or odd limits.
It's a huge problem I've long complained about, but this
example does far more to convince.

-Carl

Carl wrote:

> It's a huge problem I've long complained about, but this
> example does far more to convince.

In what way?

Petr

Carl Lumma wrote:
> --- In tuning@yahoogroups.com, Petr Paří­zek <p.parizek@...> wrote:
>> Hi guys,
>>
>> has any of you tried this? Period = 2/1, generator = 13th root
>> of 130. First of all, the generator itself is close to 16/11.
>> Then, 5 of them are close to 13/2. And finally, 8 of them are
>> close to 20/1. Unfortunately, there's only a poor approximation
>> to 3/2. In contrast, the chord of 1:2:5:11:13 is imitated so
>> well that the intervals are just over half a cent away from JI!
>> Here's what it sounds like:
>> http://download.yousendit.com/04C4C2D07E9D6FE9
> > I don't think it likely anyone here's tried it, because
> other than a few investigations into bases lacking 2, our
> theory is entirely based on complete prime or odd limits.
> It's a huge problem I've long complained about, but this
> example does far more to convince.
> > -Carl

I agree. It's clear from this example that there are lots of potentially interesting temperaments that are being neglected. Fortunately, I think that much of the theory is still applicable to these kinds of temperaments with some adaptation.

This tuning really stands out in the list of the best temperaments using primes 2, 5, 11, 13 without 3 or 7. The next one with a slightly better error is more than twice as complex, and the next less complex one has 5 times the error. The generator mapping is

[<1, x, -2, x, 4, 1], <0, x, 8, x, -1, 5]>

where the x's indicate that prime factors 3 and 7 are skipped. I like to use generators less than half the period size, so I'll use a modified generator mapping:

[<1, x, 6, x, 3, 6], <0, x, -8, x, 1, -5]>

It can be identified as a combination of 11-ET and 13-ET, or "11&13".

TOP-MAX P = 1199.875918, G = 551.653738
TOP-RMS P = 1199.895944, G = 551.650860

Here's a summary of some of the best temperaments in this group, with the generator mappings, complexity, and error for each one.

[<3, 7, 10, 11], <0, 0, 1, 0]> c = 2.597928, e = 8.441030
[<2, 5, 7, 8], <0, -1, 0, -2]> c = 4.140159, e = 7.266490
[<1, 2, 3, 4], <0, 2, 3, -2]> c = 5.071741, e = 4.055217
[<1, 2, 5, 4], <0, 1, -5, -1]> c = 5.883697, e = 2.756537
[<1, 3, 3, 3], <0, -3, 2, 3]> c = 6.877559, e = 2.232620
[<1, 5, 5, 6], <0, -7, -4, -6]> c = 9.525147, e = 1.773663
[<1, 1, 1, 2], <0, 7, 13, 9]> c = 11.861587, e = 0.615011
[<1, 6, 3, 6], <0, -8, 1, -5]> c = 12.555849, e = 0.124082
[<1, 1, -5, 0], <0, 5, 32, 14]> c = 29.380455, e = 0.119306
[<1, 2, 3, 2], <0, 7, 10, 37]> c = 30.120391, e = 0.082695
[<1, 2, 7, 14], <0, 1, -11, -32]> c = 30.414513, e = 0.026798
[<3, 8, 9, 8], <0, -3, 4, 9]> c = 36.987094, e = 0.024900
[<37, 86, 128, 137], <0, -1, 0, -1]> c = 57.803545, e = 0.023404
[<1, 6, -5, -18], <0, -10, 23, 59]> c = 67.401607, e = 0.010724
[<1, -9, 0, -18], <0, 36, 11, 69]> c = 68.264791, e = 0.010453

This is really unexplored territory, so it's not likely that any of these have names. I only found two of them in a list of 13-limit temperaments.

[<1, 5, 2, 0, 5, 4], <0, -11, 1, 9, -5, -1]>
[<1, -2, 1, 3, 1, 2], <0, 19, 7, -1, 13, 9]>

Herman

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
>
> Carl wrote:
> > It's a huge problem I've long complained about, but this
> > example does far more to convince.
>
> In what way?
>
> Petr

It sounds good, and therefore shows that a theory
dependent on complete prime limits is limited/flawed.

-Carl

> I agree. It's clear from this example that there are lots of
> potentially interesting temperaments that are being neglected.
> Fortunately, I think that much of the theory is still
> applicable to these kinds of temperaments with some adaptation.

Absolutely. But certain problems arise. For one, how do you
manage the explosion of basis sets? For another, the terminology
gets impossible. Just look at the kind of trouble you can
get into:

[I wrote...]
>Give me an abelian group G*, where G is a collection of positive
>odd integers not exceeding 17 (with 2 <= |G| <= 9). A "block" B
>in G* is a finite subset of G* such that elements of U, another
>finite subset of G*, cannot be recovered from the factors of any
>element of B, with B n U = [null].
>
>Require every element of U be =< 2^( 1/|G| ).
>
>Now find all B such that 5 <= |B| <= 10 and all elements of the
>corresponding U are squares of G.
>
>Maybe somebody can tell me what problems I'll encounter by
>allowing G to contain composite odds. I still don't understand
>why adding a 'factor-backwards' rule won't preserve something
>equivalent to the fundamental theorem of arithmetic.
>
>Anyway, so far I'd call these untempered periodicity blocks of
>5-10 elements whose chroma are square rationals no bigger than
>240 cents. The whole point of the obtuse language is that the
>bases for defining JI and "square" are not typical prime or odd
>limits.

> This tuning really stands out in the list of the best
> temperaments

Ackg. :)

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> > I agree. It's clear from this example that there are lots of
> > potentially interesting temperaments that are being neglected.
> > Fortunately, I think that much of the theory is still
> > applicable to these kinds of temperaments with some adaptation.
>
> Absolutely. But certain problems arise. For one, how do you
> manage the explosion of basis sets?

I've suggested using chordal harmonic entropy. You would rank
all the chords of n-elements by consonance, then extract the
most common basis sets from the top 100 of them (say), or perhaps
even use the top 10 as basis sets directly.

Unfortunately, nobody knows how to calculate chordal h.e.

I've also suggested it could be used instead of cents error,
but Graham has pointed out that the behavior of the h.e.
function may make the optimization problem hard. But until
we can see the behavior of the chordal h.e. funtion, we won't
really know that.

-Carl

i am lazy so what is that cents,

,',',',Kraig Grady,',',',
'''''''North/Western Hemisphere:
North American Embassy of Anaphoria island
'''''''South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria
',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

-----Original Message-----
From: Petr Pařízek [mailto:p.parizek@chello.cz]
Sent: Sunday, March 23, 2008 04:16 AM
To: tuning@yahoogroups.com
Subject: [tuning] A weird partial 13-limit linear temperament

Hi guys,

has any of you tried this? Period = 2/1, generator = 13th root of 130. First
of all, the generator itself is close to 16/11. Then, 5 of them are close to
13/2. And finally, 8 of them are close to 20/1. Unfortunately, there's only
a poor approximation to 3/2. In contrast, the chord of 1:2:5:11:13 is
imitated so well that the intervals are just over half a cent away from JI!
Here's what it sounds like: http://download.yousendit.com/04C4C2D07E9D6FE9

Petr

I guess the other question about such temperments is how many units youhave to go out to get all your intervals. the whole premise oftemperments is that one uses less notes not more.

,',',',Kraig Grady,',',',
'''''''North/Western Hemisphere:
North American Embassy of Anaphoria island
'''''''South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria
',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

-----Original Message-----
From: kraiggrady@anaphoria.com [mailto:kraiggrady@anaphoria.com]
Sent: Sunday, March 23, 2008 04:08 PM
To: tuning@yahoogroups.com
Subject: Re: [tuning] A weird partial 13-limit linear temperament

i am lazy so what is that cents,

,',',',Kraig Grady,',',',
'''''''North/Western Hemisphere:
North American Embassy of Anaphoria island
'''''''South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria
',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

-----Original Message-----
From: Petr Pařízek [mailto:p.parizek@chello.cz]
Sent: Sunday, March 23, 2008 04:16 AM
To: tuning@yahoogroups.com
Subject: [tuning] A weird partial 13-limit linear temperament

Hi guys,

has any of you tried this? Period = 2/1, generator = 13th root of 130. First
of all, the generator itself is close to 16/11. Then, 5 of them are close to
13/2. And finally, 8 of them are close to 20/1. Unfortunately, there's only
a poor approximation to 3/2. In contrast, the chord of 1:2:5:11:13 is
imitated so well that the intervals are just over half a cent away from JI!
Here's what it sounds like: http://download.yousendit.com/04C4C2D07E9D6FE9

Petr

648.2 cents, about .4 cents lower than 16/11. I think the appeal of
these kinds of tunings is that they tend to be like scales with a
marked character, even sounding like specific modes. Personally I
think that being so close to 16/11 kind of begs the question, why not
16/11? and the scales I make along these lines, sort of no-wrong-
notes kind of things for interactive installations/triggered
performances are done with Just ratios.

-Cameron Bobro

--- In tuning@yahoogroups.com, kraiggrady@... wrote:
>
> I guess the other question about such temperments is how many
units youhave to go out to get all your intervals. the whole premise
oftemperments is that one uses less notes not more.
>
> ,',',',Kraig Grady,',',',
> '''''''North/Western Hemisphere:
> North American Embassy of Anaphoria island
> '''''''South/Eastern Hemisphere:
> Austronesian Outpost of Anaphoria
> ',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
>
>
>
> -----Original Message-----
> From: kraiggrady@... [mailto:kraiggrady@...]
> Sent: Sunday, March 23, 2008 04:08 PM
> To: tuning@yahoogroups.com
> Subject: Re: [tuning] A weird partial 13-limit linear temperament
>
> i am lazy so what is that cents,
>
>
> ,',',',Kraig Grady,',',',
> '''''''North/Western Hemisphere:
> North American Embassy of Anaphoria island
> '''''''South/Eastern Hemisphere:
> Austronesian Outpost of Anaphoria
> ',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
>
>
>
> -----Original Message-----
> From: Petr Pařízek [mailto:p.parizek@...]
> Sent: Sunday, March 23, 2008 04:16 AM
> To: tuning@yahoogroups.com
> Subject: [tuning] A weird partial 13-limit linear temperament
>
> Hi guys,
>
> has any of you tried this? Period = 2/1, generator = 13th root of
130. First
> of all, the generator itself is close to 16/11. Then, 5 of them
are close to
> 13/2. And finally, 8 of them are close to 20/1. Unfortunately,
there's only
> a poor approximation to 3/2. In contrast, the chord of 1:2:5:11:13
is
> imitated so well that the intervals are just over half a cent away
from JI!
> Here's what it sounds like: http://download.yousendit.com/
04C4C2D07E9D6FE9
>
> Petr
>

For Cameron:

If you choose 16/11 as the generator, then the 5/4 is about 3 cents sharp.
The reason why I used the 13th root of 130 was that I wanted to get as close
to JI as possible, which actually IS possible here for all the three
intervals I approximated -- 5/4, 11/8, 13/8.

Petr

how many step do you have to run out to get the harmonics up to 13

,',',',Kraig Grady,',',',
'''''''North/Western Hemisphere:
North American Embassy of Anaphoria island
'''''''South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria
',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

-----Original Message-----
From: Petr Par�zek [mailto:p.parizek@chello.cz]
Sent: Sunday, March 23, 2008 06:12 PM
To: tuning@yahoogroups.com
Subject: Re: [tuning] Re: A weird partial 13-limit linear temperament

For Cameron:

If you choose 16/11 as the generator, then the 5/4 is about 3 cents sharp.
The reason why I used the 13th root of 130 was that I wanted to get as close
to JI as possible, which actually IS possible here for all the three
intervals I approximated -- 5/4, 11/8, 13/8.

Petr

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
>
> For Cameron:
>
> If you choose 16/11 as the generator, then the 5/4 is about 3 cents
sharp.
> The reason why I used the 13th root of 130 was that I wanted to get
as close
> to JI as possible, which actually IS possible here for all the three
> intervals I approximated -- 5/4, 11/8, 13/8.
>
> Petr
>

Yes, that's clear. Did you try it using 16/11 and compare the sound?

-Cameron Bobro

Carl Lumma wrote:

> I've suggested using chordal harmonic entropy. You would rank
> all the chords of n-elements by consonance, then extract the
> most common basis sets from the top 100 of them (say), or perhaps
> even use the top 10 as basis sets directly.
> > Unfortunately, nobody knows how to calculate chordal h.e.

Yes.

> I've also suggested it could be used instead of cents error,
> but Graham has pointed out that the behavior of the h.e.
> function may make the optimization problem hard. But until
> we can see the behavior of the chordal h.e. funtion, we won't
> really know that.

If you get a smooth function, it should be easier to optimize than TOP-max. The trick is to get that function.

Graham

Carl Lumma wrote:
>> I agree. It's clear from this example that there are lots of
>> potentially interesting temperaments that are being neglected.
>> Fortunately, I think that much of the theory is still
>> applicable to these kinds of temperaments with some adaptation.
> > Absolutely. But certain problems arise. For one, how do you
> manage the explosion of basis sets? For another, the terminology
> gets impossible. Just look at the kind of trouble you can
> get into:

I don't see any problems. Maybe there's a theoretical opportunity for people to dig through all the different limits. Is that what you mean by an explosion of basis sets?

The terminology I use is that you start from prime intervals. The prime intervals should be linearly independent or, if they're numbers, relative primes. Most things still work if they're linearly dependent. You may want ratios instead of integers but you can always multiply through by the lowest common denominator.

I call the limit in question the "2.5.11.13-limit". That's based on Erv Wilson's CPS notation. Maybe calling it a prime limit is a bit of a stretch, so you can specify other kinds of limits.

What else is there?

Graham

Herman Miller wrote:

> Here's a summary of some of the best temperaments in this group, with > the generator mappings, complexity, and error for each one.

Did you compare them with my online script?

http://x31eq.com/temper/regular.html

It uses scalar complexity now, so the figures will be smaller than yours. Tell it to find more ETs if you don't change the badness.

Graham

Carl Lumma wrote:

> I don't think it likely anyone here's tried it, because
> other than a few investigations into bases lacking 2, our
> theory is entirely based on complete prime or odd limits.
> It's a huge problem I've long complained about, but this
> example does far more to convince.

I have a list of equal temperament mappings with your name on it. They may or may not have bases with 2. 87-equal in the 2.5.11.13-limit is, in fact, the best system with four primes by whatever criteria you set. And the class we're talking about does support 87-equal.

The three-prime octave-ETs that come ahead of it are:

80-equal in the 2.13.17-limit
57-equal in the 2.7.17-limit
87-equal in the 2.5.11-limit
12-equal in the 2.3.17-limit
50-equal in the 2.11.13-limit
26-equal in the 2.7.11-limit
87-equal in the 2.11.13-limit
31-equal in the 2.5.7-limit

The next few four-prime systems, which all happen to have octaves, are:

24-equal in the 2.3.11.17-limit
57-equal in the 2.7.13.17-limit
53-equal in the 2.3.5.13-limit

The first five-prime system is 37-equal in the 2.5.7.11.13-limit.

Graham

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> Carl Lumma wrote:
> >> I agree. It's clear from this example that there are lots of
> >> potentially interesting temperaments that are being neglected.
> >> Fortunately, I think that much of the theory is still
> >> applicable to these kinds of temperaments with some adaptation.
> >
> > Absolutely. But certain problems arise. For one, how do
> > you manage the explosion of basis sets? For another, the
> > terminology gets impossible. Just look at the kind of
> > trouble you can get into:
>
> I don't see any problems. Maybe there's a theoretical
> opportunity for people to dig through all the different
> limits. Is that what you mean by an explosion of basis sets?

Yes.

> The terminology I use is that you start from prime
> intervals. The prime intervals should be linearly
> independent or, if they're numbers, relative primes. Most
> things still work if they're linearly dependent. You may
> want ratios instead of integers but you can always multiply
> through by the lowest common denominator.
>
> I call the limit in question the "2.5.11.13-limit". That's
> based on Erv Wilson's CPS notation. Maybe calling it a
> prime limit is a bit of a stretch, so you can specify other
> kinds of limits.
>
> What else is there?

A way to compare different limits.

-Carl

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> Carl Lumma wrote:
>
> > I don't think it likely anyone here's tried it, because
> > other than a few investigations into bases lacking 2, our
> > theory is entirely based on complete prime or odd limits.
> > It's a huge problem I've long complained about, but this
> > example does far more to convince.
>
> I have a list of equal temperament mappings with your name
> on it.

I was quite baffled with the way you did that, but of
course appreciative that you did it. I still intend to
do my own version. One fine day.

-Carl

le 24/03/08 2:12, Petr Parízek à p.parizek@chello.cz a écrit :

For Cameron:

If you choose 16/11 as the generator, then the 5/4 is about 3 cents sharp.
The reason why I used the 13th root of 130 was that I wanted to get as close
to JI as possible, which actually IS possible here for all the three
intervals I approximated -- 5/4, 11/8, 13/8.

Petr

That's a great 2:5:11:13 system !
Apart from simple JI, 13th root of 130 could be one of the best choices
indeed.
Depending on applications, it's true you could use 16/11, or the 5th root of
13/2, or the 8th root of 20,
or even the 18th root of 845 (13.13.5) and many others generators, with good
results.

All those are actually multiple attractors, on my point of view, of one
central fractal ratio that is :
1,454 131 238 983 5
(only 0,03 cents lower from yours = 1,454 156 092 147 2)

solution of x = (x power 6) - 8

It expresses the differential coherence of the "13/11" type interval of your
system, or (x power 6) / 8,
as it appears for example in the following fractal series :

130 189 275 400 581 845 1229 1787 2600 3781 5493 7989 etc...

where
189 = 1229 less 8 times 130 (1040)
275 = 1787 less 8 times 189 (1512)
400 = 2600 less 8 times 275 (2200) and so on...

their ratios converge (slowly) towards 1,454 131 238 983 5

while the value of the coherent "13/11" converges towards 1,181 766 404 872
9
= 1 + (x / 8) = (x power 6) / 8

interesting enough, this serie, among the best possible, starts by 130,
shared by the powers of your ratio, of course.

20, 29 belong also to the same serie (29 =189 - 160) here but no more whole
numbers under 130.
All octaves combinations are of course possible to create scales, wether
rational or not.

- - - - - - - - - - -
Jacques Dudon

Graham Breed wrote:
> Herman Miller wrote:
> >> Here's a summary of some of the best temperaments in this group, with >> the generator mappings, complexity, and error for each one.
> > Did you compare them with my online script?
> > http://x31eq.com/temper/regular.html
> > It uses scalar complexity now, so the figures will be > smaller than yours. Tell it to find more ETs if you don't > change the badness.

Your complexity measure sorts differently from what I'm using, so the results aren't directly comparable.

6&12 [<6, 14, 21, 22], <0, 0, 0, 1]> (4.891266 vs. 0.706)
7&20 [<1, 2, 3, 4], <0, 2, 3, -2]> (5.071741 vs. 0.598)
10&13 [<1, 2, 5, 4], <0, 1, -5, -1]> (5.883697 vs. 0.694)

I did initially miss a couple that could be added to the list, but neither of these are better (considering both error and complexity) than the ones in the original list.

[<6, 14, 21, 22], <0, 0, 0, 1]> c = 4.891266, e = 6.995120
[<1, 2, 3, 4], <0, 2, 3, -2]> c = 5.071741, e = 4.055217

Kraig- this number is what we would call the "complexity"
of the temperament. Lower is better! Fortunately, Petr's
already answered your question (see the message below!)...
It's 1 step to 11/8, 5 steps to 13/8, and 8 steps to 5/4.
Pretty rare with this accuracy. If you want an ET, I
suggest 37. It also has a decent 7/4. If you want to
TOP tune it, it winds up being 37 equal steps to an
1199.3-cent octave.

1199.2618468907767 :2
2787.4734819623454 :5
3370.8981642335348 :7
4148.79774059512 :11
4440.510081730714 :13

-Carl

--- In tuning@yahoogroups.com, kraiggrady@... wrote:
>
> I guess the other question about such temperments is how many
> units you have to go out to get all your intervals. the whole
> premise of temperments is that one uses less notes not more.
>
> ,',',',Kraig Grady,',',',
> '''''''North/Western Hemisphere:
> North American Embassy of Anaphoria island
> '''''''South/Eastern Hemisphere:
> Austronesian Outpost of Anaphoria
> ',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
>
> //
> -----Original Message-----
> From: Petr Pařízek [mailto:p.parizek@...]
> Sent: Sunday, March 23, 2008 04:16 AM
> To: tuning@yahoogroups.com
> Subject: [tuning] A weird partial 13-limit linear temperament
>
> Hi guys,
>
> has any of you tried this? Period = 2/1, generator = 13th root
> of 130. First of all, the generator itself is close to 16/11.
> Then, 5 of them are close to 13/2. And finally, 8 of them are
> close to 20/1.

so 11/8 as the inversion, so complexity doesnot differentiate between plus or minus? or is 8 telling us that the chain is 8 long to get the desired ratios.
sorry as i said i am away from my calulator till it all arrives with the rest of my stuff. so what are the MOS's
,',',',Kraig Grady,',',',
'''''''North/Western Hemisphere:
North American Embassy of Anaphoria island
'''''''South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria
',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

-----Original Message-----
From: Carl Lumma [mailto:carl@lumma.org]
Sent: Monday, March 24, 2008 11:51 PM
To: tuning@yahoogroups.com
Subject: [tuning] Re: A weird partial 13-limit linear temperament

Kraig- this number is what we would call the "complexity"
of the temperament. Lower is better! Fortunately, Petr's
already answered your question (see the message below!)...
It's 1 step to 11/8, 5 steps to 13/8, and 8 steps to 5/4.
Pretty rare with this accuracy. If you want an ET, I
suggest 37. It also has a decent 7/4. If you want to
TOP tune it, it winds up being 37 equal steps to an
1199.3-cent octave.

1199.2618468907767 :2
2787.4734819623454 :5
3370.8981642335348 :7
4148.79774059512 :11
4440.510081730714 :13

-Carl

--- In tuning@yahoogroups.com, kraiggrady@... wrote:
>
> I guess the other question about such temperments is how many
> units you have to go out to get all your intervals. the whole
> premise of temperments is that one uses less notes not more.
>
> ,',',',Kraig Grady,',',',
> '''''''North/Western Hemisphere:
> North American Embassy of Anaphoria island
> '''''''South/Eastern Hemisphere:
> Austronesian Outpost of Anaphoria
> ',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
>
> //
> -----Original Message-----
> From: Petr Pař�­zek [mailto:p.parizek@...]
> Sent: Sunday, March 23, 2008 04:16 AM
> To: tuning@yahoogroups.com
> Subject: [tuning] A weird partial 13-limit linear temperament
>
> Hi guys,
>
> has any of you tried this? Period = 2/1, generator = 13th root
> of 130. First of all, the generator itself is close to 16/11.
> Then, 5 of them are close to 13/2. And finally, 8 of them are
> close to 20/1.

kraiggrady@anaphoria.com wrote:
> so 11/8 as the inversion, so complexity doesnot differentiate between > plus or minus? or is 8 telling us that the chain is 8 long to get the > desired ratios.

Yes, both. You need 8 steps to get any interval in the limit, which means a minimum of 9 notes. To get a full diamond you need 2*8+1=17 notes.

It looks like the complexity's 9 for the 1.5.11.13 odd-limit. So 10 notes to get the first 1.5.11.13 chord and 19 notes for a diamond.

> sorry as i said i am away from my calulator till it all arrives with the > rest of my stuff. so what are the MOS's

13 24
37
50
87
137

Graham

--- In tuning@yahoogroups.com, kraiggrady@... wrote:
>
> so 11/8 as the inversion, so complexity doesnot differentiate
> between plus or minus?

Since this is an temperament has an octave period, I just
spelled all the ratios I was talking so they'd be in one
octave. Technically one should also count the number of
octaves needed to do this, but it makes little difference
in this case.

>or is 8 telling us that the chain is 8 long to get the
>desired ratios.

Yes.

> sorry as i said i am away from my calulator till it all arrives with
the rest of my stuff. so what are the MOS's

12 25 37 50 62 75 87

or something like that

-Carl

That's a very groovy take on things, I dig the idea of converging
points and have been using Pi/4 and ln2, converging points from
alternating harmonic series, but that's all, very simple. Do you have
some articles and so on on these fractal converging points related to
tuning?

-Cameron Bobro

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:
>
> le 24/03/08 2:12, Petr Parízek à p.parizek@... a écrit :
>
> For Cameron:
>
> If you choose 16/11 as the generator, then the 5/4 is about 3 cents
sharp.
> The reason why I used the 13th root of 130 was that I wanted to get
as close
> to JI as possible, which actually IS possible here for all the three
> intervals I approximated -- 5/4, 11/8, 13/8.
>
> Petr
>
>
> That's a great 2:5:11:13 system !
> Apart from simple JI, 13th root of 130 could be one of the best
choices
> indeed.
> Depending on applications, it's true you could use 16/11, or the
5th root of
> 13/2, or the 8th root of 20,
> or even the 18th root of 845 (13.13.5) and many others generators,
with good
> results.
>
> All those are actually multiple attractors, on my point of view, of
one
> central fractal ratio that is :
> 1,454 131 238 983 5
> (only 0,03 cents lower from yours = 1,454 156 092 147 2)
>
> solution of x = (x power 6) - 8
>
> It expresses the differential coherence of the "13/11" type
interval of your
> system, or (x power 6) / 8,
> as it appears for example in the following fractal series :
>
> 130 189 275 400 581 845 1229 1787 2600 3781 5493 7989 etc...
>
> where
> 189 = 1229 less 8 times 130 (1040)
> 275 = 1787 less 8 times 189 (1512)
> 400 = 2600 less 8 times 275 (2200) and so on...
>
> their ratios converge (slowly) towards 1,454 131 238 983 5
>
> while the value of the coherent "13/11" converges towards 1,181
766 404 872
> 9
> = 1 + (x / 8) = (x power 6) / 8
>
> interesting enough, this serie, among the best possible, starts by
130,
> shared by the powers of your ratio, of course.
>
> 20, 29 belong also to the same serie (29 =189 - 160) here but no
more whole
> numbers under 130.
> All octaves combinations are of course possible to create scales,
wether
> rational or not.
>
> - - - - - - - - - - -
> Jacques Dudon
>

Oh, Petr, almost forgot- have you taken a look at the square root of
your generator? (What I did is just split the difference between half
of 16/11 and a third of 7/4, so to speak; with a very similar
resulting generator).

-Cameron Bobro

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
>
> For Cameron:
>
> If you choose 16/11 as the generator, then the 5/4 is about 3 cents
sharp.
> The reason why I used the 13th root of 130 was that I wanted to get
as close
> to JI as possible, which actually IS possible here for all the three
> intervals I approximated -- 5/4, 11/8, 13/8.
>
> Petr
>

le 25/03/08 13:39, Cameron Bobro à misterbobro@yahoo.com a écrit :

That's a very groovy take on things, I dig the idea of converging
points and have been using Pi/4 and ln2, converging points from
alternating harmonic series, but that's all, very simple. Do you have
some articles and so on on these fractal converging points related to
tuning?

-Cameron Bobro

Thanks for your interest.
Sorry, but no article in english yet, and our website is almost entirely in
french.

I use these algorithms to create what I call "fractal waveforms", that
contains many of the serie's frequencies in its spectrum - bringing perfect
adequation between timbers and intonations in such contexts.

"Differential coherence : experimenting with new areas of consonance" / in
1/1 vol.11 #2, winter 2003 explains the acoustic and musical basis of that
research.

One photosonic music CD I did some years ago ("Lumières audibles") uses
several of such fractal systems and it has some writing on it, in english.
It is distributed in the states through the Just Intonation Network, and
EMF.

One danced concert called "D'or et d'ours" ("Of gold and bear") I give with
my microtonal ensemble is entirely based on such fractals in the form of
rhythmns and sounds, and should be edited in a CD this year.
Will let it known through this list.

- - - - - - - - - - - - - - - -
Jacques Dudon
Atelier d'Exploration Harmonique
http://aeh.free.fr

Cameron wrote:

> Oh, Petr, almost forgot- have you taken a look at the square root of
> your generator? (What I did is just split the difference between half
> of 16/11 and a third of 7/4, so to speak; with a very similar
> resulting generator).

This is keemun. I have played around with it a few weeks ago and I'm still not sure what to think about its characteristic "mood". As I've learned, most people use it for the possibility of dividing the 7/4 into 3 steps, but noone seems to mention that you can also divide the 32/5 (i.e. minor sixth + two octaves) into 10 steps and get almost the same. A very good example of such a temperament is a generator of the 19th root of 512/15 which makes both 8/5 and 4/3 detuned by the same amount. The comma we are tempering out here is actually 1953125/1889568.

Petr

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
>
> Cameron wrote:
>
>
>
> > Oh, Petr, almost forgot- have you taken a look at the square root
of
> > your generator? (What I did is just split the difference between
half
> > of 16/11 and a third of 7/4, so to speak; with a very similar
> > resulting generator).
>
>
>
> This is keemun. I have played around with it a few weeks ago and
I'm still not sure what to think about its characteristic "mood". As
I've learned, most people use it for the possibility of dividing the
7/4 into 3 steps, but noone seems to mention that you can also divide
the 32/5 (i.e. minor sixth + two octaves) into 10 steps and get
almost the same. A very good example of such a temperament is a
generator of the 19th root of 512/15 which makes both 8/5 and 4/3
detuned by the same amount. The comma we are tempering out here is
actually 1953125/1889568.

Hmmmm... Graham's paper has Keemun with a 317 cent generator, which
would make the 7/4 flat. I mean a generator around 323- 324 cents,
so you can get 2:5:7:11:13 and in tasteful ways.... Lessee... yes,
"errors" from .6 cents to a max of 2 cents, and all errors in one
direction (high). My slightly detuned octave (<2.4 cents low) is just
a guess, "earballing" it using rational intervals, I'm sure Graham
and others could tweak both the generator and the octave so they're
even better as far as "errors", and all that jazz.

But my interest in this tuning isn't just the Just intervals, but the
fact that the tuning produces other intervals I keep gnawing on
because they sound great to me but can't really be completely
explained, like Margot and Dave's "metastable" third just a hair
above 14/11, also a point of "harmonic entropy", and a low minor
third I kept yammering about, etc.

Since exactly what I want is the Just intervals a tiny bit high, and
the ear-found "shadow" intervals within their fuzzy little ranges,
and a slowly moving octave because I'm using synths, it's a "tuning",
not a "temperament", AFAIC.

Whoops, sorry to ramble on. Thanks for posting the tune, by the way.

-Cameron Bobro

I wrote...
> > I have a list of equal temperament mappings with your name
> > on it.
>
> I was quite baffled with the way you did that, but of
> course appreciative that you did it. I still intend to
> do my own version. One fine day.

Turns out that day was today. Here's what I did:

In each equal temperament between 5 & 99 notes per octave
(inclusive, so 95 ETs in all), I tuned all triads, tetrads,
pentads, and hexads that can be formed from the set
{2 3 5 7 11 13 17}. That's 35 + 35 + 21 + 7 = 98 chords
in all.

For each ET, I kept only the most accurately approximated
triad, tetrad, etc. So 95 ETs * 4 chord sizes = 380 chords
in all.
I used TOP damage to assess tuning accuracy. TOP damage
tells you the maximum Tenney-weighted error of any interval
that can be formed from the factors in the chord (after
the octave has been optimally stretched/compressed).

I then threw out those chords which had more than half the
number of notes of the containing ET. So tetrads in 5-ET
were thrown out, as were hexads in 7-ET, etc. For odd ETs,
I let the larger half stay. So I kept triads in 5-ET, for
example. This left 368 chords.

I then threw out chords which were not consistent in the
given ET. There were 81 of these. The 287 remaining chords
are listed in this Excel spreadsheet:

http://lumma.org/stuff/ETsProgram.xls

They are sorted first by chord size (triads, tetrads, etc.),
and then by increasing badness. Badness here is TOP damage
times the number of notes in the ET. Of course you can use
Excel to sort however you like.

Of note: the "best" triad is 3:5:17 in 72-ET. The best
tetrad is 2:5:11:13 in 87-ET (Petr's temperament). The
best pentad is 2:5:7:11:13 in 37-ET (Petr's temperament
with 7 added). The best hexad is 2:3:5:7:11:17 in 72-ET.

You may argue that I shouldn't use weighted error here,
since it tends to keep the most consonant chords from
showing up. And you'd be right. So I'll do another
version some other time with unweighted error.

It could also be said that tuning by "patent val" and
then casting out inconsistent ETs may miss some good
mappings, but I don't think so. I think they'll show
up in the ETs where they are patent. Could be wrong
though -- I didn't look into this.

-Carl

Petr Pařízek wrote:

> has any of you tried this? Period = 2/1, generator = 13th root of 130. First > of all, the generator itself is close to 16/11. Then, 5 of them are close to > 13/2. And finally, 8 of them are close to 20/1. Unfortunately, there's only > a poor approximation to 3/2. In contrast, the chord of 1:2:5:11:13 is > imitated so well that the intervals are just over half a cent away from JI!
> Here's what it sounds like: http://download.yousendit.com/04C4C2D07E9D6FE9

Oh, yes, 2.5.11.13 is a theoretically interesting limit.

I'm sure you all know that 4:5:6 triads have some unique properties. One of them comes in two parts:

1) The first number in the ratios is 4

2) No intervals are smaller than 7:6, including with the implied octave.

No other triads fulfil both these criteria and no triads fulfil a stronger version of (1) where the first number should be 2. (1) may be important because having a power of two at the bottom makes the root of the chord agree with the fundamental of the virtual pitch. That may make the sense of rootedness stronger. (2) is important because smaller intervals will get within the critical band and so sound inherently dissonant regardless of timbre.

It's been suggested that 16:19:24 is a target tuning for a minor triad. That would make sense in the light of these criteria. However, why jump straight from 4 to 16? If 16 is good then 8 should be as well. There are two triads that fulfil the criteria if we change 4 to 8 in (1). They are 8:10:13 and 8:11:13.

So, a 2.5.11.13-limit gives us the second and third simplest chords in this family without the luxury of the simplest and most familiar of all (4:5:6). That means you can experiment with the new chords and avoid the comparison with good old-fashioned major.

I can't say if these chords really are special in sound. I did play with them a while back (in mystery) and they can make good progressions. But you can convince yourself of anything if you try hard enough. The only way to establish the theory is for people to make music including the chords and see if they make sense. It'll certainly be interesting to find out.

Graham

Carl Lumma wrote:

> In each equal temperament between 5 & 99 notes per octave
> (inclusive, so 95 ETs in all), I tuned all triads, tetrads,
> pentads, and hexads that can be formed from the set
> {2 3 5 7 11 13 17}. That's 35 + 35 + 21 + 7 = 98 chords
> in all.

These are all octave based, then?

> For each ET, I kept only the most accurately approximated
> triad, tetrad, etc. So 95 ETs * 4 chord sizes = 380 chords
> in all.
> I used TOP damage to assess tuning accuracy. TOP damage
> tells you the maximum Tenney-weighted error of any interval
> that can be formed from the factors in the chord (after
> the octave has been optimally stretched/compressed).

So it's really an incomplete prime-limit measure.

<snip>

> Of note: the "best" triad is 3:5:17 in 72-ET. The best
> tetrad is 2:5:11:13 in 87-ET (Petr's temperament). The
> best pentad is 2:5:7:11:13 in 37-ET (Petr's temperament
> with 7 added). The best hexad is 2:3:5:7:11:17 in 72-ET.

That agrees with the list I have, by a different methodology, except for the triad. But then chords in octave-based temperaments that don't include octaves are likely to be treated differently.

> You may argue that I shouldn't use weighted error here,
> since it tends to keep the most consonant chords from
> showing up. And you'd be right. So I'll do another
> version some other time with unweighted error.

Unweighted errors would show a different thing.

> It could also be said that tuning by "patent val" and
> then casting out inconsistent ETs may miss some good
> mappings, but I don't think so. I think they'll show
> up in the ETs where they are patent. Could be wrong
> though -- I didn't look into this.

The best mappings will be consistent. It's only when you look at inconsistent mappings that you need to make sure you have the best ones.

Graham

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
> Carl Lumma wrote:
>
> > In each equal temperament between 5 & 99 notes per octave
> > (inclusive, so 95 ETs in all), I tuned all triads, tetrads,
> > pentads, and hexads that can be formed from the set
> > {2 3 5 7 11 13 17}. That's 35 + 35 + 21 + 7 = 98 chords
> > in all.
>
> These are all octave based, then?

Yes. It's just a standard to compare them. If I describe
them in terms of steps to the smallest identity in the
chord they wouldn't be directly comparable.

> > For each ET, I kept only the most accurately approximated
> > triad, tetrad, etc. So 95 ETs * 4 chord sizes = 380 chords
> > in all.
> > I used TOP damage to assess tuning accuracy. TOP damage
> > tells you the maximum Tenney-weighted error of any interval
> > that can be formed from the factors in the chord (after
> > the octave has been optimally stretched/compressed).
>
> So it's really an incomplete prime-limit measure.

I don't know what that means but the answer is probably yes.

> > Of note: the "best" triad is 3:5:17 in 72-ET. The best
> > tetrad is 2:5:11:13 in 87-ET (Petr's temperament). The
> > best pentad is 2:5:7:11:13 in 37-ET (Petr's temperament
> > with 7 added). The best hexad is 2:3:5:7:11:17 in 72-ET.
>
> That agrees with the list I have, by a different
> methodology, except for the triad.

What do you get for best triad?

> But then chords in
> octave-based temperaments that don't include octaves are
> likely to be treated differently.

Howso?

> > You may argue that I shouldn't use weighted error here,
> > since it tends to keep the most consonant chords from
> > showing up. And you'd be right. So I'll do another
> > version some other time with unweighted error.
>
> Unweighted errors would show a different thing.

Different, and more sensical.

-Carl

Carl Lumma wrote:
> --- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>> Carl Lumma wrote:

>>> Of note: the "best" triad is 3:5:17 in 72-ET. The best
>>> tetrad is 2:5:11:13 in 87-ET (Petr's temperament). The
>>> best pentad is 2:5:7:11:13 in 37-ET (Petr's temperament
>>> with 7 added). The best hexad is 2:3:5:7:11:17 in 72-ET.
>> That agrees with the list I have, by a different >> methodology, except for the triad.
> > What do you get for best triad?

5:7:13 with 91 notes to the 5:1.

I don't see anything like 3:5:17 in 72-ET so I must be genuinely missing something. My search will ignore prime-limits that don't only consist of prime numbers (e.g. 7.9.11, 9/5.7.11) but it should catch 3.5.17. And even if it finds non-octave temperaments you don't have, if a good temperament happens to have octaves it should still be there, and score as well. Maybe I'm scoring complexity by the number of steps to the lowest prime instead of to the octave.

>> But then chords in >> octave-based temperaments that don't include octaves are >> likely to be treated differently.
> > Howso?

Dunno, but there's something going on here.

>>> You may argue that I shouldn't use weighted error here,
>>> since it tends to keep the most consonant chords from
>>> showing up. And you'd be right. So I'll do another
>>> version some other time with unweighted error.
>> Unweighted errors would show a different thing.
> > Different, and more sensical.

Weighted errors give you a badness for the non-consecutive prime-limit that includes the chord. Unweighted errors give you a badness for the chord itself.

Graham

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
> >>> Of note: the "best" triad is 3:5:17 in 72-ET. The best
> >>> tetrad is 2:5:11:13 in 87-ET (Petr's temperament). The
> >>> best pentad is 2:5:7:11:13 in 37-ET (Petr's temperament
> >>> with 7 added). The best hexad is 2:3:5:7:11:17 in 72-ET.
> >> That agrees with the list I have, by a different
> >> methodology, except for the triad.
> >
> > What do you get for best triad?
>
> 5:7:13 with 91 notes to the 5:1.

No wonder -- I only searched up to 99 notes to the 2:1.

I stopped at 99 because I don't think ETs beyond this
are very useful. They're useful as tunings, but I think
it's better at that point to use LTs. I suppose Gene
would argue that large ETs are useful in computer music
when switching between accurate LTs. OK, fine. But
for most purposes 99 is plenty enough.

> I don't see anything like 3:5:17 in 72-ET so I must be
> genuinely missing something. My search will ignore
> prime-limits that don't only consist of prime numbers (e.g.
> 7.9.11, 9/5.7.11)

I didn't consider these either.

> Maybe I'm scoring complexity by
> the number of steps to the lowest prime instead of to the
> octave.

That should give the same results. But if you're *naming*
things by steps to the lowest prime, you'll have to
translate... do you have 3:5:17 with 114 steps to the 3:1?

> >>> You may argue that I shouldn't use weighted error here,
> >>> since it tends to keep the most consonant chords from
> >>> showing up. And you'd be right. So I'll do another
> >>> version some other time with unweighted error.
> >> Unweighted errors would show a different thing.
> >
> > Different, and more sensical.
>
> Weighted errors give you a badness for the non-consecutive
> prime-limit that includes the chord. Unweighted errors give
> you a badness for the chord itself.

I've been thinking about this. It is apparently true only
for log weighting, which is of course what most people use.
However it strongly biases the results away from chords
containing lower identities, so it's simply not appropriate
here. And anyway, while the benefit is clear for intervals
like 9:8 and 15:8, beyond that it's not so clear. Who cares
if the weighted error of 135/whatever is below some value?
Paul placed a big emphasis on this, and I'd like to know
the reasoning. In generalized diatonic music, there are
usually only a few chords used. And he would be the first
to decry that intervals like 135/whatever have any justness
to them in the first place.

-Carl

Carl Lumma wrote:
> --- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>>>>> Of note: the "best" triad is 3:5:17 in 72-ET. The best
>>>>> tetrad is 2:5:11:13 in 87-ET (Petr's temperament). The
>>>>> best pentad is 2:5:7:11:13 in 37-ET (Petr's temperament
>>>>> with 7 added). The best hexad is 2:3:5:7:11:17 in 72-ET.
>>>> That agrees with the list I have, by a different >>>> methodology, except for the triad.
>>> What do you get for best triad?
>> 5:7:13 with 91 notes to the 5:1.
> > No wonder -- I only searched up to 99 notes to the 2:1.

Yes. 91 notes to the 5:1 gives less than 99 notes to the 2:1.

The code I have now says it's searching for 5 to 100 notes to the octave. And non-octave temperaments are specified by an octave size but the octave isn't considered when choosing the best mapping. Which sounds very confusing, but means it should be consistent with what you're doing.

> I stopped at 99 because I don't think ETs beyond this
> are very useful. They're useful as tunings, but I think
> it's better at that point to use LTs. I suppose Gene
> would argue that large ETs are useful in computer music
> when switching between accurate LTs. OK, fine. But
> for most purposes 99 is plenty enough.
> >> I don't see anything like 3:5:17 in 72-ET so I must be >> genuinely missing something. My search will ignore >> prime-limits that don't only consist of prime numbers (e.g. >> 7.9.11, 9/5.7.11)
> > I didn't consider these either.

I thought you were taking all chords.

>> Maybe I'm scoring complexity by >> the number of steps to the lowest prime instead of to the >> octave.
> > That should give the same results. But if you're *naming*
> things by steps to the lowest prime, you'll have to
> translate... do you have 3:5:17 with 114 steps to the 3:1?

Now here's a funny thing. I checked the code, and it is counting steps to the octave. So I ran it to check it's the same as the output file I was looking at, so I could play with it and see what's wrong. The result is the output is formatted completely differently to the old output file! And, sitting there at the top of the list is

3.5.17-limit <114, 167, 294] 17 cent steps

It really wasn't there before. Not in the list at all. Other results still agree with you.

>>>>> You may argue that I shouldn't use weighted error here,
>>>>> since it tends to keep the most consonant chords from
>>>>> showing up. And you'd be right. So I'll do another
>>>>> version some other time with unweighted error.
>>>> Unweighted errors would show a different thing.
>>> Different, and more sensical.
>> Weighted errors give you a badness for the non-consecutive >> prime-limit that includes the chord. Unweighted errors give >> you a badness for the chord itself.
> > I've been thinking about this. It is apparently true only
> for log weighting, which is of course what most people use.
> However it strongly biases the results away from chords
> containing lower identities, so it's simply not appropriate
> here. And anyway, while the benefit is clear for intervals
> like 9:8 and 15:8, beyond that it's not so clear. Who cares
> if the weighted error of 135/whatever is below some value?
> Paul placed a big emphasis on this, and I'd like to know
> the reasoning. In generalized diatonic music, there are
> usually only a few chords used. And he would be the first
> to decry that intervals like 135/whatever have any justness
> to them in the first place.

I thought Paul's reasoning was that it didn't matter where you drew the line.

I'm writing about the RMS errors of finite sets of intervals this month. It looks like TOP-RMS is a good bet if you don't know exactly what intervals you want but do know the prime limit. Possibly the lower primes should be given more weight. I haven't done the number crunching yet.

Graham

Graham wrote...

> >>> What do you get for best triad?
> >>
> >> 5:7:13 with 91 notes to the 5:1.
> >
> > No wonder -- I only searched up to 99 notes to the 2:1.
>
> Yes. 91 notes to the 5:1 gives less than 99 notes to the 2:1.

Whoops, got that backwards. Yup, should have seen that.
Let's see, it should have about 39 notes to the 2:1.
For 39's triad I get 7:13:17. 38 & 40 aren't 5:7:13 either.
In fact, 5:7:13 doesn't show up in my list.

> The code I have now says it's searching for 5 to 100 notes
> to the octave. And non-octave temperaments are specified by
> an octave size but the octave isn't considered when choosing
> the best mapping. Which sounds very confusing, but means it
> should be consistent with what you're doing.

That does sound like what I'm doing.

> > I stopped at 99 because I don't think ETs beyond this
> > are very useful. They're useful as tunings, but I think
> > it's better at that point to use LTs. I suppose Gene
> > would argue that large ETs are useful in computer music
> > when switching between accurate LTs. OK, fine. But
> > for most purposes 99 is plenty enough.
> >
> >> I don't see anything like 3:5:17 in 72-ET so I must be
> >> genuinely missing something. My search will ignore
> >> prime-limits that don't only consist of prime numbers (e.g.
> >> 7.9.11, 9/5.7.11)
> >
> > I didn't consider these either.
>
> I thought you were taking all chords.

I test k-combinations of {2 3 5 7 11 13 17} for each ET.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Graham wrote...
>
> > >>> What do you get for best triad?
> > >>
> > >> 5:7:13 with 91 notes to the 5:1.
> > >
> > > No wonder -- I only searched up to 99 notes to the 2:1.
> >
> > Yes. 91 notes to the 5:1 gives less than 99 notes to the 2:1.
>
> Whoops, got that backwards. Yup, should have seen that.
> Let's see, it should have about 39 notes to the 2:1.
> For 39's triad I get 7:13:17.

Possibly because 39 is inconsistent over 7:13:17.
Which val are you using?

-Carl

Carl Lumma wrote:
> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>> Graham wrote...
>>
>>>>>> What do you get for best triad?
>>>>> 5:7:13 with 91 notes to the 5:1.
>>>> No wonder -- I only searched up to 99 notes to the 2:1.
>>> Yes. 91 notes to the 5:1 gives less than 99 notes to the 2:1.
>> Whoops, got that backwards. Yup, should have seen that.
>> Let's see, it should have about 39 notes to the 2:1.
>> For 39's triad I get 7:13:17.
> > Possibly because 39 is inconsistent over 7:13:17.
> Which val are you using?

5.7.13-limit <91, 110, 145] 31 cent steps

Still at number 2.

Graham

Graham wrote...

> >> For 39's triad I get 7:13:17.
> >
> > Possibly because 39 is inconsistent over 7:13:17.
> > Which val are you using?
>
> 5.7.13-limit <91, 110, 145] 31 cent steps
>
> Still at number 2.

So we agree on number 1 now? Still, this is worrisome.
The 2nd-best triad doesn't use the patent val in its
parent ET. My approach of casting out inconsistent ETs
must not be a good one.

-Carl

Carl Lumma wrote:
> Graham wrote...
> >>>> For 39's triad I get 7:13:17.
>>> Possibly because 39 is inconsistent over 7:13:17.
>>> Which val are you using?
>> 5.7.13-limit <91, 110, 145] 31 cent steps
>>
>> Still at number 2.
> > So we agree on number 1 now? Still, this is worrisome.
> The 2nd-best triad doesn't use the patent val in its
> parent ET. My approach of casting out inconsistent ETs
> must not be a good one.

Yes, number 1 is the same.

31 cent scale steps give 38.7 steps to the octave. Obviously, this won't come out as the nearest-prime mapping for pure octaves. And if you're looking for octave-base ETs it's quite reasonable to leave it out.

As a search for non-octave ETs what I'm doing is still woefully incomplete. It should include 4, 8, or 16 as an independent prime interval when it excludes 2. Even for octave-based scales 9 should be there.

Graham

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> >>> Which val are you using?
> >>
> >> 5.7.13-limit <91, 110, 145] 31 cent steps
//
> > The 2nd-best triad doesn't use the patent val in its
> > parent ET. My approach of casting out inconsistent ETs
> > must not be a good one.
//
> if you're looking for octave-base ETs
> it's quite reasonable to leave it out.

But I'm not looking for octave-based ETs, I'm just using
them as a naming scheme. The patent val looks like
<91 109 144], so this really seems to be a case of a
very good but inconsistent equal-step tuning. Or do
you disagree about the patent val here?

> As a search for non-octave ETs what I'm doing is still
> woefully incomplete. It should include 4, 8, or 16 as an
> independent prime interval when it excludes 2. Even for
> octave-based scales 9 should be there.

Good point! 15 too.

-Carl

Carl Lumma wrote:
> --- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>>>>> Which val are you using?
>>>> 5.7.13-limit <91, 110, 145] 31 cent steps
> //
>>> The 2nd-best triad doesn't use the patent val in its
>>> parent ET. My approach of casting out inconsistent ETs
>>> must not be a good one.
> //
>> if you're looking for octave-base ETs >> it's quite reasonable to leave it out.
> > But I'm not looking for octave-based ETs, I'm just using
> them as a naming scheme. The patent val looks like
> <91 109 144], so this really seems to be a case of a
> very good but inconsistent equal-step tuning. Or do
> you disagree about the patent val here?

I disagree about using a patent val based on octaves for a system that doesn't have octaves. If you take the nearest primes approximation to 91-ED3 you should get the correct mapping. That's the same as 39.2 steps to the octave.

>> As a search for non-octave ETs what I'm doing is still >> woefully incomplete. It should include 4, 8, or 16 as an >> independent prime interval when it excludes 2. Even for >> octave-based scales 9 should be there.
> > Good point! 15 too.

Yes.

Graham

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> Carl Lumma wrote:
> > --- In tuning@yahoogroups.com, Graham Breed <gbreed@> wrote:
> >>>>> Which val are you using?
> >>>> 5.7.13-limit <91, 110, 145] 31 cent steps
> > //
> >>> The 2nd-best triad doesn't use the patent val in its
> >>> parent ET. My approach of casting out inconsistent ETs
> >>> must not be a good one.
> > //
> >> if you're looking for octave-base ETs
> >> it's quite reasonable to leave it out.
> >
> > But I'm not looking for octave-based ETs, I'm just using
> > them as a naming scheme. The patent val looks like
> > <91 109 144], so this really seems to be a case of a
> > very good but inconsistent equal-step tuning. Or do
> > you disagree about the patent val here?
>
> I disagree about using a patent val based on octaves for a
> system that doesn't have octaves. If you take the nearest
> primes approximation to 91-ED3 you should get the correct
> mapping.

That's what I thought I was doing. I'm dividing through
by log(2) instead of log(3), but I thought it shouldn't
matter, especially since both of the numbers you've given
for steps/octave (38.7 and 39.2) round to 39.

-Carl

Carl Lumma wrote:

> That's what I thought I was doing. I'm dividing through
> by log(2) instead of log(3), but I thought it shouldn't
> matter, especially since both of the numbers you've given
> for steps/octave (38.7 and 39.2) round to 39.

39.2 is correct, so it's more precisely 30.6 cent steps.

The nearest-primes mapping for 39.0 steps to the octave differs from that for 39.2 steps. So rounding off isn't enough. You need to use 91log(2)/log(5). (Yes that's 91ED5, not ED3, and I introduced that mistake. Sorry.) As long as you always take equal divisions of the smallest prime interval, rather than a shadow octave, nearest-primes will get you all the consistent ETs in any non-consecutive prime limit.

Graham

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> Carl Lumma wrote:
>
> > That's what I thought I was doing. I'm dividing through
> > by log(2) instead of log(3), but I thought it shouldn't
> > matter, especially since both of the numbers you've given
> > for steps/octave (38.7 and 39.2) round to 39.
>
> 39.2 is correct, so it's more precisely 30.6 cent steps.
>
> The nearest-primes mapping for 39.0 steps to the octave
> differs from that for 39.2 steps. So rounding off isn't
> enough. You need to use 91log(2)/log(5). (Yes that's
> 91ED5, not ED3, and I introduced that mistake. Sorry.) As
> long as you always take equal divisions of the smallest
> prime interval, rather than a shadow octave, nearest-primes
> will get you all the consistent ETs in any non-consecutive
> prime limit.

That makes sense; thanks for catching that.

-Carl

Graham wrote...

> >> As a search for non-octave ETs what I'm doing is still
> >> woefully incomplete. It should include 4, 8, or 16 as an
> >> independent prime interval when it excludes 2. Even for
> >> octave-based scales 9 should be there.
> >
> > Good point! 15 too.
>
> Yes.

Actually, I don't think there's a need to include squares
and cubes like 4 and 8, because their errors will always
be double, triple etc. that of their prime *and so will
their complexities*. The only difference it makes is that
for a given complexity cutoff in your search, it gives you
more chances to detect these chords (e.g. those cases where
4 is approximated by an odd number of steps). I'm not sure
they deserve that chance, but I'm willing to be convinced.

15, on the other hand, must be included because its error
could be less than the errors of its primes (when the
errors of 3 & 5 are opposite in sign).

Make any sense?

-Carl

Carl Lumma wrote:

> Actually, I don't think there's a need to include squares
> and cubes like 4 and 8, because their errors will always
> be double, triple etc. that of their prime *and so will
> their complexities*. The only difference it makes is that
> for a given complexity cutoff in your search, it gives you
> more chances to detect these chords (e.g. those cases where
> 4 is approximated by an odd number of steps). I'm not sure
> they deserve that chance, but I'm willing to be convinced.
> > 15, on the other hand, must be included because its error
> could be less than the errors of its primes (when the
> errors of 3 & 5 are opposite in sign).
> > Make any sense?

What happens when a square approximates to an odd number of steps? The prime has to be rounded off so that the square's nearly a step out. If you don't actually want the prime you can do better. If you're looking at chords in a weighted prime limit then you can ignore the squares.

Graham

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> Carl Lumma wrote:
>
> > Actually, I don't think there's a need to include squares
> > and cubes like 4 and 8, because their errors will always
> > be double, triple etc. that of their prime *and so will
> > their complexities*. The only difference it makes is that
> > for a given complexity cutoff in your search, it gives you
> > more chances to detect these chords (e.g. those cases where
> > 4 is approximated by an odd number of steps). I'm not sure
> > they deserve that chance, but I'm willing to be convinced.
> >
> > 15, on the other hand, must be included because its error
> > could be less than the errors of its primes (when the
> > errors of 3 & 5 are opposite in sign).
> >
> > Make any sense?
>
> What happens when a square approximates to an odd number of
> steps?

Let's say 4 is approximated by 5 steps. You get this system
when 2 is approximated by 5 steps, with twice the complexity
but half the 2\4-error. So there's only an issue where these
2x (or 3x or 4x) systems are above your complexity cutoff.
Composite odds like 15 are different. But I suppose you're
right: to avoid all problems, just include them.

-Carl

Where we left off...

> In each equal temperament between 5 & 99 notes per octave
> (inclusive, so 95 ETs in all), I tuned all triads, tetrads,
> pentads, and hexads that can be formed from the set
> {2 3 5 7 11 13 17}. That's 35 + 35 + 21 + 7 = 98 chords
> in all.
>
> For each ET, I kept only the most accurately approximated
> triad, tetrad, etc. So 95 ETs * 4 chord sizes = 380 chords
> in all.
> I used TOP damage to assess tuning accuracy. TOP damage
> tells you the maximum Tenney-weighted error of any interval
> that can be formed from the factors in the chord (after
> the octave has been optimally stretched/compressed).
>
> I then threw out those chords which had more than half the
> number of notes of the containing ET. So tetrads in 5-ET
> were thrown out, as were hexads in 7-ET, etc. For odd ETs,
> I let the larger half stay. So I kept triads in 5-ET, for
> example. This left 368 chords.
>
> I then threw out chords which were not consistent in the
> given ET.

I did everything the same way, except instead of testing
for consistency of an ET, I tested 9 different vals for
each "ET" (rounded number of notes/octave) and took the
one with lowest TOP damage. So there are in fact 368
entries in this file:

http://lumma.org/stuff/ETsProgram2.xls

> It could also be said that tuning by "patent val" and
> then casting out inconsistent ETs may miss some good
> mappings, but I don't think so.

Looks like I was wrong about that.

Next time: a version with unweighted error.

-Carl

Graham & I wrote...

> >>> Of note: the "best" triad is 3:5:17 in 72-ET. The best
> >>> tetrad is 2:5:11:13 in 87-ET (Petr's temperament). The
> >>> best pentad is 2:5:7:11:13 in 37-ET (Petr's temperament
> >>> with 7 added). The best hexad is 2:3:5:7:11:17 in 72-ET.
> >>
> >> That agrees with the list I have, by a different
> >> methodology, except for the triad.
> >
> > What do you get for best triad?
>
> 5:7:13 with 91 notes to the 5:1.

In my new file, the results are:

"ET" primes val
triad 41 (5 13 17) <96 153 169]
tetrad 41 (5 11 13 17 <96 143 153 169]
pentad 41 (5 7 11 13 17) <96 116 143 153 169]
hexad 83 (3 5 7 11 13 17) <131 192 232 286 306 338]

What do you think of this, Graham?

-Carl

Carl Lumma wrote:

> In my new file, the results are:
> > "ET" primes val
> triad 41 (5 13 17) <96 153 169]
> tetrad 41 (5 11 13 17 <96 143 153 169]
> pentad 41 (5 7 11 13 17) <96 116 143 153 169]
> hexad 83 (3 5 7 11 13 17) <131 192 232 286 306 338]
> > What do you think of this, Graham?

Dunno. None of them are in my results.

Graham

> > In my new file, the results are:
> >
> > "ET" primes val
> > triad 41 (5 13 17) <96 153 169]
> > tetrad 41 (5 11 13 17 <96 143 153 169]
> > pentad 41 (5 7 11 13 17) <96 116 143 153 169]
> > hexad 83 (3 5 7 11 13 17) <131 192 232 286 306 338]
> >
> > What do you think of this, Graham?
>
> Dunno. None of them are in my results.
>
> Graham

Are you finding the val with least TOP damage? I thought
you were. Can you check if your best results have lower
TOP damage/step size than these?

Here are the best 5- and 7-limit:

(2 3 5) <53 84 123] 0.303 16.07
(2 3 5 7) <99 157 230 278] 0.339 33.57

-Carl

Carl Lumma wrote:
>>> In my new file, the results are:
>>>
>>> "ET" primes val
>>> triad 41 (5 13 17) <96 153 169]
>>> tetrad 41 (5 11 13 17 <96 143 153 169]
>>> pentad 41 (5 7 11 13 17) <96 116 143 153 169]
>>> hexad 83 (3 5 7 11 13 17) <131 192 232 286 306 338]
>>>
>>> What do you think of this, Graham?
>> Dunno. None of them are in my results.
> > Are you finding the val with least TOP damage? I thought
> you were. Can you check if your best results have lower
> TOP damage/step size than these?

I'm not finding the right vals. The routine gets the "best val" for each number of steps to the octave. But there's no number of steps to the octave that gives 96 steps to a 3:1, and the optimization formula misses them.

> Here are the best 5- and 7-limit:
> > (2 3 5) <53 84 123] 0.303 16.07
> (2 3 5 7) <99 157 230 278] 0.339 33.57

Yes, that's right.

Here we go, fixed it:

5.13.17-limit <96, 153, 169] 29 cent steps
7.11.13-limit <198, 244, 261] 17 cent steps
5.13.17-limit <192, 306, 338] 15 cent steps
3.5.17-limit <114, 167, 294] 17 cent steps
5.7.13-limit <91, 110, 145] 31 cent steps
2.5.11-limit <87, 202, 301] 14 cent steps

Graham

I wrote...

> >>> In my new file, the results are:
> >>>
> >>> "ET" primes val
> >>> triad 41 (5 13 17) <96 153 169]
> >>> tetrad 41 (5 11 13 17 <96 143 153 169]
> >>> pentad 41 (5 7 11 13 17) <96 116 143 153 169]

This looks like 1/3 of 88CET.

> >>> hexad 83 (3 5 7 11 13 17) <131 192 232 286 306 338]

Looks suspiciously like 2 * the above.

Graham wrote...

> Here we go, fixed it:
>
> 5.13.17-limit <96, 153, 169] 29 cent steps
> 7.11.13-limit <198, 244, 261] 17 cent steps
> 5.13.17-limit <192, 306, 338] 15 cent steps
> 3.5.17-limit <114, 167, 294] 17 cent steps

Same as my results, except I have the 2nd and 3rd
entries reversed (but they're close and it could be
rounding error). The 3rd entry is 2 * the first.

> 5.7.13-limit <91, 110, 145] 31 cent steps

I have two before this:

(7 13 17) <261 344 380] '93-ET'
(7 13 17) <22 29 32] '8-ET'

> 2.5.11-limit <87, 202, 301] 14 cent steps

I then have 5 before this:

(5 7 13) <91 110 145]
(2 13 17) <80 296 327]
(2 7 17) <57 160 233]
(7 11 17) <125 154 182]
(11 13 17) <259 277 306]

-Carl

Carl Lumma wrote:

>> 5.7.13-limit <91, 110, 145] 31 cent steps
> > I have two before this:
> > (7 13 17) <261 344 380] '93-ET'
> (7 13 17) <22 29 32] '8-ET'

I also use a complexity for the basis. The sum of the logs of the primes I think. So 5 wins out over 13.

Graham

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> Carl Lumma wrote:
>
> >> 5.7.13-limit <91, 110, 145] 31 cent steps
> >
> > I have two before this:
> >
> > (7 13 17) <261 344 380] '93-ET'
> > (7 13 17) <22 29 32] '8-ET'
>
> I also use a complexity for the basis. The sum of the logs
> of the primes I think. So 5 wins out over 13.
>
>
> Graham

That's one way to partially counteract the effects of
the Tenney weighting (depending on when you apply it).

-Carl