Kleismic Temperament

Dear Paul, you are right of course, but remember that I'm still a novice who is trying to find his way in the tuning jungle by trial and error. So, would it be a kleismic temperament if I were to widen the third by a small amount so as to make 4000:3993 dissapear?

386.313714 cents + 3.032314 cents = 389.346028 cents

Subtract two Ptolemy commas (2*100:99=10000:9801) from a pure third tempered thus (389.346028 cents - 34.798967 cents = 354.547061 cents)

Add a major whole tone to the undecimal neutral second (9:8 * 12:11= 27:22 Zalzal Wosta = 354.547060 cents)

----- Original Message -----
From: wallyesterpaulrus
To: tuning@yahoogroups.com
Sent: 24 Şubat 2005 Perşembe 19:40
Subject: [tuning] Re: Kleismic Yarman Temperament

I'd condemn neither of you. Part of the problem I'm trying to get at
is in the title of this thread. Whenever I've seen it used, "Kleismic
Temperament" means a temperament in which the Kleisma vanishes. So
shouldn't "Kleismic Yarman Temperament" refer to a temperament in
which the "Yarman Kleisma" vanishes? I think it should.

Dear Ozan,

> you are right of course, but remember that I'm still a
>novice who is trying to find his way in the tuning jungle by trial
>and error.

Of course -- please excuse my poor manners and excessive zeal for
logic. I'm smiling in friendship even as I write seemingly harsh
critiques, Ozan. Too bad we're not in the same room!

>So, would it be a kleismic temperament if I were to widen the third
>by a small amount so as to make 4000:3993 dissapear?

You'd probably narrow, rather than widen, the ~5:4, as I'll explain
below. But first a terminological point. "Kleismic" seems to be a
word invented by Graham Breed. Perhaps referred to as "kleismatic" by
others earlier. Either way, if it refers to temperament, it's a
temperament where 15652:15552 vanishes. I like "Hanson" for this, as
you can see (along with much info on it) in the paper I mailed you,
because the late Larry Hanson seems to have been the first to write
about this as a distinct class of tuning systems.

> 386.313714 cents + 3.032314 cents = 389.346028 cents

This is just 5:4 * 4000:3993, or 5000:3993.

> Subtract two Ptolemy commas (2*100:99=10000:9801) from a pure third
>tempered thus (389.346028 cents - 34.798967 cents = 354.547061 cents)

This is just 5000:3993 / 10000:9801, or 27:22.

> Add a major whole tone to the undecimal neutral second (9:8 *
>12:11= 27:22 Zalzal Wosta = 354.547060 cents)

And that's another way of getting 27:22.

So, what do these various JI relationships tell you? And what do they
have to do with temperament?

Anyhow, if you're going to make 4000:3993 disappear -- that is,
temper it out -- you'll probably want to temper prime 5 narrow, since
5 is a factor of the larger number in the ratio (4000). What you want
is for the tuning of the primes to be adjusted so that '4000'
= '3993'. Then you'll have a bona-fide temperament based on the
vanishing of 4000:3999. If you keep the octaves (prime 2) pure, this
means that the 5:4 will be tempered narrow. But if you allow the
octaves to be tempered too, you'd probably narrow them as well (since
2 is a factor of 4000), so 5:4 might end up either wide or narrow
depending on how you're doing the tempering.

Please do excuse my manner again . . . I hope you'll get a chance to
reread the paper I sent you. It sets forth the basic concepts of
temperament with the examples most basic and central to Western
practice -- 12-tone equal temperament (understood as making
531441:524288 vanish), and meantone temperament (understood as making
81:80). That might make these more exotic ideas of ours a little less
elusive.

And I apologize, yet again, for I have not had a chance to compliment
you for your incredible work in music and theory, and for your
obvious intelligence in assimilating so many new ideas so quickly, as
demonstrated on and off this list. You may be a novice now, but won't
be for long, I'm sure!

-Paul

Dear Paul, your appreciation is very much appreciated! I thank you for your kind and encouraging words. But there is no need to apologize, you are just being frank and I like the way you grasp the obvious facts to which I'm quite oblivious as yet.

Your paper is still a bit too complicated for the likes of me, as I am not very knowledged in mathematical operations such as prime factorization. So, bear with me with patience as I tackle the intricate phenomenon of temperament.

As to your question, I deem the JI ratio of Zalzal Wusta a significant element in the construction of many maqams, as in the tetrachord:

1/1
9/8
27/22
4/3

Play it like this, and you have a Usshaq genre:

4/3 27/22 9/8 1/1 9/8

If I were to temper the 4000:3993 kleisma that results from comparing 27:22 to a whole tone minus two Ptolemy commas, I figure that would somewhat justify the inclination of Zalzal Wusta to the whole tone ratios as demonstrated above.

Can you now give an example of a `bona fide` temperament where 4000:3993 vanishes?

Cordially,
Ozan

----- Original Message -----
From: wallyesterpaulrus
To: tuning@yahoogroups.com
Sent: 01 Mart 2005 Salı 4:21
Subject: [tuning] Re: Kleismic Temperament

Dear Ozan,

Of course -- please excuse my poor manners and excessive zeal for
logic. I'm smiling in friendship even as I write seemingly harsh
critiques, Ozan. Too bad we're not in the same room!

Please do excuse my manner again . . . I hope you'll get a chance to
reread the paper I sent you. It sets forth the basic concepts of
temperament with the examples most basic and central to Western
practice -- 12-tone equal temperament (understood as making
531441:524288 vanish), and meantone temperament (understood as making
81:80). That might make these more exotic ideas of ours a little less
elusive.

And I apologize, yet again, for I have not had a chance to compliment
you for your incredible work in music and theory, and for your
obvious intelligence in assimilating so many new ideas so quickly, as
demonstrated on and off this list. You may be a novice now, but won't
be for long, I'm sure!

-Paul

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:

> Dear Paul, your appreciation is very much appreciated! I thank you
>for your kind and encouraging words. But there is no need to
>apologize, you are just being frank and I like the way you grasp the
>obvious facts to which I'm quite oblivious as yet.
>
> Your paper is still a bit too complicated for the likes of me, as I
>am not very knowledged in mathematical operations such as prime
>factorization.

That is probably my fault -- I could probably have done a better job
explaining it. This is not a higher math operation. See this webpage
and try the example problems -- it will probably help:

http://amby.com/educate/math/2-1_fact.html

> So, bear with me with patience as I tackle the intricate phenomenon
>of temperament.

Temperament is the slight adjustment of the basic intervals of JI so
that different (possibly complex) JI intervals get represented by the
same tempered intervals. That way you end up with tuning systems with
fewer notes, and fewer interval sizes, than would be required in JI.

Kyle Gann has a mostly reasonable page on Western historical tunings
that may help make this less abstract:

http://home.earthlink.net/~kgann/histune.html

> As to your question, I deem the JI ratio of Zalzal Wusta a
>significant element in the construction of many maqams, as in the
>tetrachord:
>
> 1/1
> 9/8
> 27/22
> 4/3
>
> Play it like this, and you have a Usshaq genre:
>
> 4/3 27/22 9/8 1/1 9/8
>
> If I were to temper the 4000:3993 kleisma that results from
>comparing 27:22 to a whole tone minus two Ptolemy commas, I figure
>that would somewhat justify the inclination of Zalzal Wusta to the
>whole tone ratios as demonstrated above.

I'm completely lost. Where do the two Ptolemy commas come in,
musically speaking? And what inclination to the whole tone ratios do
you speak of? Can you write a much longer paragraph elaborating on
the above?

> Can you now give an example of a `bona fide` temperament where
>4000:3993 vanishes?

If you start with 11-limit Just Intonation, which is 5-dimensional
{2,3,5,7,11}, and temper out 4000:3993, you end up with a 4-
dimensional temperament which (in one optimized form) can be
constructed from these four generators, in cents: g1=1200., g2=-
165.9852708, g3=2785.912380, g4=3368.915093.

Prime 2 is being approximated, trivially, by the g1, 1200 cents.

Prime 3 is approximated by 2*g1 + 3*g2: 2*1200 - 3*165.9852708 =
1902.0441876 cents.

Prime 5 is approximated by g3, 2785.912380 cents.

Prime 7 is approximated by the g4, 3368.915093 cents. In fact you can
leave this generator out altogether, and have a 3-dimensional system,
if you're only interested in {2,3,5,11}-JI and its temperaments.

Prime 11 is approximated by g1 - g2 + g3: 1200 + 165.9852708 +
2785.912380 = 4151.8976508 cents.

Other optimizations will lead to slightly different values for these
four generators. But the key point to notice is that no matter how
you choose the generators, the representation of 4000:3993 will
vanish. Why? 4000 is 2*2*2*2*2*5*5*5, or 2^5 * 5^3. So in terms of
the generators above, it will be

5*g1 + 3*g3

Do you follow?

Meanwhile, 3993 is 3*11*11*11, or 3^1 * 11^3. So in terms of the
generators above, it will be

2*g1 + 3*g2 + 3*(g1 - g2 + g3)
= 2*g1 + 3*g2 + 3*g1 - 3*g2 + 3*g3
= 5*g1 + 3*g3.

Still following?

Thus 4000 amounts to the same thing as 3993 in any tuning system that
approximates the primes in this way. So 4000:3993 would be
represented by 5*g1 + 3*g3 - (5*g1 + 3*g3) = 0; i.e., it vanishes.

The only other ratios that vanish in such a system are powers of
4000:3993 -- 16000000:15944049, 64000000000:63664587657,
256000000000000:254212698514401, etc. No other ratios vanish in this
system.

If you don't mind that other ratios vanish as well as 4000:3993 and
its powers, here are some other temperaments of 11-limit JI:

3-dimensional:
() 540:539 /\ 4000:3993 (generators 600, 433.9278373, 2785.854450
cents)

2-dimensional:
() Wizard (generators 600, ~217 cents)
() Octoid (generators 150, ~16 cents)
() 385:384 /\ 2401:2400 /\ 4000:3993 (generators 1200, ~583 cents)
() 225:224 /\ 385:384 /\ 4000:3993 (generators 600, ~383.2 cents)
() 243:242 /\ 385:384 /\ 4000:3993 (generators 1200, ~516.68 cents)
() 225:224 /\ 243:242 /\ 4000:3993 (generators 1200, ~516.69 cents)

1-dimensional:
() 72-tone equal temperament
() 311-tone equal temperament

I stole most of this information from Gene's posts on tuning-math;
any errors are of course my fault.

Let's verify that 4000:3993 vanishes in 72-tone equal temperament. In
72-tone equal temperament, prime 2 is approximated by 72 steps, prime
3 is approximated by 114 steps, prime 5 is appproximated by 167
steps, (prime 7 is approximated by 202 steps,) and prime 11 is
approximated by 249 steps. So 4000, or 2^5 * 5^3, will be represented
by 5*72 + 3*167 = 861 steps, while 3993, or 3^1 * 11^3, will be
represented by 114 + 3*249 = 861 steps. Hence 4000:3993 will be
represented by 0 steps, hence vanish, in 72-equal.

Please come back to me with questions about any or all of this -- I
hope to help you understand all this, and will spend as much time
clarifying as need be.

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

> > Can you now give an example of a `bona fide` temperament where
> >4000:3993 vanishes?
>
> If you start with 11-limit Just Intonation, which is 5-dimensional
> {2,3,5,7,11}, and temper out 4000:3993, you end up with a 4-
> dimensional temperament which (in one optimized form) can be
> constructed from these four generators, in cents: g1=1200., g2=-
> 165.9852708, g3=2785.912380, g4=3368.915093.

As Paul mentions below, another interesting temperament which tempers
out 4000/3993 is 311-equal. This can be used as a surrogate for just
intonation up through the 41 limit, so long as you do not demand ultra
precision. Of course, this will also temper out a huge list of other
commas. Combining it with another equal temperament which tempers out
4000/3993, for instance 224-equal, will lead to linear temperaments,
and so forth.