Prime factorization and maqams

I think I begin to understand basic math operations in music theory Paul.

I like the way you decribe temperament. So, if I understand you correctly, it is only with the concern of equating 5:4 to 81:64 that many meantone temperaments are derived? Is that why it is the syntonic comma (5:4 / 81:64) that is tempered here?

I keep ending up with a lot of JI intervals in maqam music. For example, this it the latest Huzzam scale I cooked up:

1/1
13/12
39/32
4/3 (33/25 when descending)
3/2
13/8
17/9
2/1

C Dd Eb F G Ab B C

Alterations that are part of this maqam allow for this interlude:

1/1
13/12
39/32
351/256
13/9
13/8
117/64
243/128

C Db Eb F Gb Ab Bb Cb (play descending)

As for the Ptolemy commas that come into play in the Rast 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 only assumed that subtracting two (5-limit) Ptolemy commas from 9/8 is a more `just` way of reaching 27/22 (which is an 11-limit ratio). Am I stretching the definition of temperament here? Is not temperament a slight adjustment of basic JI intervals to represent more complex JI intervals with the same ratio? Ah, but now I see that 88209/80000 is as complex in its prime content as 27/22. So I guess it is not permissable to temper one of these.

But I don't understand how you obtain the generators here:

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?

Almost... but the generators? Is prime 2, 2:1? and prime 3, 3:1 and prime 5, 5:1? If so, I begin to understand. The math is most entertaining... temperament seems to be a lot of fun. But how do you correlate 2^5 with 5*g1 and 5^3 with 3*g3? In the lattice diagram?

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?

Ah, almost... almost... But please elucidate the correlations for me.

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.

Spledid! So, this is what is called tempering out 4000:3993. Are there other equations such as this for tempering out the same interval?

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.

Marvelous. So, is there a scala file I can obtain or can you show me how to prepare one for experimenting on this?

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

Ah! So, 72-tone equal temperament tempers out 4000:3993 along with some other commas?

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

Don't say that... say you borrowed. I'm sure there will be no penalty for that. ;)

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.

Yes, I follow you.

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.

Ah, I get it! So, the numerator of 4000:3993 is prime two times five * prime five times three. Since prime 2 of 72tET is the 72nd step and prime 3 the 167th, you say 4000 is 360 (five octaves) + 501 (three octave-fifths) = 861 steps. The same analogy applies to the denumerator, and hence, the same number of steps, with the difference being 0, resulting in 4000:3993 to vanish.

Cordially,
Ozan

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

> I only assumed that subtracting two (5-limit) Ptolemy commas from
9/8 is a more `just` way of reaching 27/22 (which is an 11-limit
ratio). Am I stretching the definition of temperament here?

No, you can temper anything you like. However, 27/22 is an interval of
12/11 above 9/8, so I don't know what you mean by subtraction. The
Ptolemy comma is a 3-limit comma, 3^12/2^19, so I also don't know what
you are subtracting, or adding.

Is not temperament a slight adjustment of basic JI intervals to
represent more complex JI intervals with the same ratio? Ah, but now I
see that 88209/80000 is as complex in its prime content as 27/22. So I
guess it is not permissable to temper one of these.

You can temper either one or both, but you need to decide what they
will be made equivalent to. For instance, a logical way to temper
27/22 would be to equate it to 11/9, which would mean tempering out
243/242. A logical way to temper 88209/80000 would be to equate it to
11/10, which requires tempering out 8019/8000. You can certainly do
both together, tempering out 4000/3993 and 243/242 both; one good
means of doing that is using 72-equal. 72-et tempers out 441/440 as
well, and you can get a linear temperament by adding this to 243/242
and 4000/3993, or you can get a planar temperament by simply tempering
out those two.

> Almost... but the generators? Is prime 2, 2:1? and prime 3, 3:1 and
prime 5, 5:1? If so, I begin to understand. The math is most
entertaining... temperament seems to be a lot of fun. But how do you
correlate 2^5 with 5*g1 and 5^3 with 3*g3? In the lattice diagram?

The generators have been given by Paul in logatrithmic form, in cents.
1200 cents is exactly 2, but 1902.0441876 cents is 3.000155 and
2785.912380 cents is 4.998841. Hence instead of 5^3 we would tune to
(4.998841)^3, and so forth.

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

> I think I begin to understand basic math operations in music
>theory Paul.
>
> I like the way you decribe temperament. So, if I understand you
>correctly, it is only with the concern of equating 5:4 to 81:64
>that many meantone temperaments are derived?

Or the concern with equating 5:3 to 27:16, or the concern with
equating 10:9 to 9:8.

>Is that why it is the syntonic comma (5:4 / 81:64) that is tempered
>here?

>Yes, and of course the syntonic comma is also equal to (5:3 /
>27:16) and to (10:9 / 9:8).
> As for the Ptolemy commas that come into play in the Rast
>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 only assumed that subtracting two (5-limit) Ptolemy commas from
>9/8 is a more `just` way of reaching 27/22 (which is an 11-limit
>ratio).

More 'just'? I'm afraid I don't understand. Can you explain what you
mean? I was wondering where these Ptolemy commas come in, musically
speaking, since you weren't proposing they be tempered out, so I
thought they must somehow be relevant to some distinctions present
within the musical materials themselves.

>Am I stretching the definition of temperament here?

I can't tell yet. I'll wait for further clarification from you.

>Is not
>temperament a slight adjustment of basic JI intervals

Yes.

>to represent
>more complex JI intervals with the same ratio?

To represent two or more different complex ratios with a single
interval, yes.

>Ah, but now I see that 88209/80000 is as complex in its prime
>content as 27/22.

?

>So I guess it is not permissable to temper one of
>these.

Not permissable? Why not?

> But I don't understand how you obtain the generators here:
>
> 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?
>
> Almost... but the generators?

It takes a little algebra to find the generators, which is why I
copied them from Gene's post :) But as you can see, I verified that
these generators do satisfy the condition that 4000:3993 be tempered
out . . . and it seems, from the below, that you may now understand
this verification -- but let me know if I need to clarify further.

>Is prime 2, 2:1? and prime 3, 3:1 and prime 5, 5:1?

Yes.

>If so, I begin >to understand. The math is most entertaining...
>temperament seems >to be a lot of fun. But how do you correlate 2^5
>with 5*g1 and 5^3 >with 3*g3?

The generators are given in cents, which is a logarithm-of-ratio
measure of interval size. A basic property of logarithms is that

log(a^b) = b*log(a)

So if we say that g1=1200 cents represents prime 2 = 2:1, then
2*g1=2400 cents represents 2^2 = 4:1, 3*g1=3600 cents represents 2^3
= 8:1, 4*g1=4800 cents represents 2^4 = 16:1, and 5*g1=6000 cents
represents 2^5 = 32:1, etc. And if g3=2786 cents represents prime 5
= 5:1, then 2*g1=5572 cents represents 5^2 = 25:1, and 3*g1=8358
cents represents 5^3 = 125:1.

Making sense?

> 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.
>
> Spledid! So, this is what is called tempering out 4000:3993. Are
>there other equations such as this for tempering out the same
>interval?

I don't really understand the question, but maybe the below answered
it . . .

> 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.
>
> Marvelous. So, is there a scala file I can obtain or can you show
>me how to prepare one for experimenting on this?

You'll have to wait until after the weekend, when I return to the
office, where I have my software . . . then I'll be able to propose
some reasonable scales that are constructed from these generators.
Unless Gene or someone beats me too it, which I hope they will!

BTW, how do you feel about tempered octaves? If you approve of them,
I might use the TOP temperament of 4000:3993 alone, whose
approximations to the primes can be calculated with the formulas in
the paper I sent you.

> Ah! So, 72-tone equal temperament tempers out 4000:3993 along with
>some other commas?

Yup! For example, 225:224, 385:384, and 2401:2400.

> 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.
>
>
> Yes, I follow you.

This is no different than the first case I mentioned above, except
that here there's only one generator: 1/72 octave.

> 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.
>
>
>
> Ah, I get it! So, the numerator of 4000:3993 is prime two times
>five * prime five times three. Since prime 2 of 72tET is the 72nd
>step and prime 3 the 167th, you say 4000 is 360 (five octaves) +
>501 (three octave-fifths) = 861 steps. The same analogy applies to
>the denumerator, and hence, the same number of steps, with the
>difference being 0, resulting in 4000:3993 to vanish.

Hopefully the first case above is now equally clear to you. Let me
know if it isn't. Also, I'd encourage you to write a similar
paragraph explaining how 81:80 vanishes in meantone temperament -- I
think you'd learn something about both the general idea of
temperament and about the specific example in which Western common
practice music arose.

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

> I only assumed that subtracting two (5-limit) Ptolemy commas

Hi Ozan, didn't you define the Ptolemy comma as 100:99?
That's 2^2 * 3^(-2) * 5^2 * 11^(-2), hence an 11-limit comma, and
not a 5-limit one. Or did our lines get crossed somehow?

>
> I only assumed that subtracting two (5-limit) Ptolemy commas from
>9/8 is a more `just` way of reaching 27/22 (which is an 11-limit
>ratio).

More 'just'? I'm afraid I don't understand. Can you explain what you
mean? I was wondering where these Ptolemy commas come in, musically
speaking, since you weren't proposing they be tempered out, so I
thought they must somehow be relevant to some distinctions present
within the musical materials themselves.

So sorry for that error Paul, I meant to say that I somehow hear 9801:9000 (not 88209/80000) to be a better substitute for 27:22, as in:

1/1 12/11 9801/8000 4/3

Don't ask me why, beats me...

It takes a little algebra to find the generators, which is why I
copied them from Gene's post :) But as you can see, I verified that
these generators do satisfy the condition that 4000:3993 be tempered
out . . . and it seems, from the below, that you may now understand
this verification -- but let me know if I need to clarify further.

But that is the crux of the issue! I want to know how to find the generators, so that I can manage to temper any interval I choose after that.

The generators are given in cents, which is a logarithm-of-ratio
measure of interval size. A basic property of logarithms is that

log(a^b) = b*log(a)

So if we say that g1=1200 cents represents prime 2 = 2:1, then
2*g1=2400 cents represents 2^2 = 4:1, 3*g1=3600 cents represents 2^3
= 8:1, 4*g1=4800 cents represents 2^4 = 16:1, and 5*g1=6000 cents
represents 2^5 = 32:1, etc. And if g3=2786 cents represents prime 5
= 5:1, then 2*g1=5572 cents represents 5^2 = 25:1, and 3*g1=8358
cents represents 5^3 = 125:1.

Making sense?

I'm sure it does to math-people like yourself, but give me some more time to understand the relationships.

> 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.
>
> Marvelous. So, is there a scala file I can obtain or can you show
>me how to prepare one for experimenting on this?

You'll have to wait until after the weekend, when I return to the
office, where I have my software . . . then I'll be able to propose
some reasonable scales that are constructed from these generators.
Unless Gene or someone beats me too it, which I hope they will!

Cool.

BTW, how do you feel about tempered octaves? If you approve of them,
I might use the TOP temperament of 4000:3993 alone, whose
approximations to the primes can be calculated with the formulas in
the paper I sent you.

That would be swell Paul. You mean widening the octaves of course, don't you? Otherwise I would hear them to be unduly dissonant. My Bechstein grand has the octaves stretched by a few cents you know (I instructed the tuners myself). That creates a brilliant effect in fff passages.

Hopefully the first case above is now equally clear to you. Let me
know if it isn't. Also, I'd encourage you to write a similar
paragraph explaining how 81:80 vanishes in meantone temperament -- I
think you'd learn something about both the general idea of
temperament and about the specific example in which Western common
practice music arose.

Urm... Let's see... For 81:80 to vanish, I need the numerator of this interval to be equal to the denumerator. Since the syntonic comma is a 5-limit interval, the system will have the primes 2,3 and 5, making this a three dimensional system of tuning. Right? Tempering out the syntonic comma will result in the upper-most dimension (prime) to collapse, creating a temperament of 2 dimensions. So, the first dimension is the octave with 2:1 which will remain untouched for my purposes, and the second dimension is the tempered perfect fifth with the ratio 3:2 and the third dimension is the tempered pure third with the ratio 5:4.

Prime factorizing 81:80 gives me

3*3*3*3 (3^4) for the numerator

and

2*2*2*2*5 (2^4 * 5^1) for the denumerator.

So, will you help me figure out the rest?

Cordially,
Ozan

No, you can temper anything you like. However, 27/22 is an interval of
12/11 above 9/8, so I don't know what you mean by subtraction. The
Ptolemy comma is a 3-limit comma, 3^12/2^19, so I also don't know what
you are subtracting, or adding.

Ah, yes, that was my mistake there. I meant to say 9801:9000, which is two Ptolemy commas less than 5:4. Now I see that both 9801:9000 and 27:22 are 11-limit intervals. Is it ok to temper out 4000:3993 to equate these two intervals?

Gene, I know the generator values in cents, my question was, how he derived them in the first place.

Cordially,
Ozan

hi Ozan,

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

> Prime factorizing 81:80 gives me
>
> 3*3*3*3 (3^4) for the numerator
>
> and
>
> 2*2*2*2*5 (2^4 * 5^1) for the denumerator.

it looks like Paul made a typo and wrote "denumerator",
and then you copied the error. the correct term for the
bottom number in a fraction or ratio is "denominator".

we write the monzo of 81:80 as: [-4 4, -1>

in a monzo, positive exponents denote factors of the
numerator, and negative exponents denote factors of
the denominator.

-monz

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

> It takes a little algebra to find the generators, which is why I
> copied them from Gene's post :) But as you can see, I verified that
> these generators do satisfy the condition that 4000:3993 be tempered
> out . . . and it seems, from the below, that you may now understand
> this verification -- but let me know if I need to clarify further.

> But that is the crux of the issue! I want to know how to find the
generators, so that I can manage to temper any interval I choose after
that.

You might go over to tuning-math, give an example of what you want to
find generators for, and we could take it from there. In general,
however, finding generators depends on how you want to optimize--you
define what you mean by "best", and then calculate the tuning which is
"best" in that sense.

Ay ay ay... that was a major bummer. Thanks for the correction Monz!

----- Original Message -----
From: monz
To: tuning@yahoogroups.com
Sent: 06 Mart 2005 Pazar 19:58
Subject: [tuning] Re: Prime factorization and maqams

hi Ozan,

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

> Prime factorizing 81:80 gives me
>
> 3*3*3*3 (3^4) for the numerator
>
> and
>
> 2*2*2*2*5 (2^4 * 5^1) for the denumerator.

it looks like Paul made a typo and wrote "denumerator",
and then you copied the error. the correct term for the
bottom number in a fraction or ratio is "denominator".

we write the monzo of 81:80 as: [-4 4, -1>

in a monzo, positive exponents denote factors of the
numerator, and negative exponents denote factors of
the denominator.

-monz

Splendid Gene, now suppose I want to temper out 4000:3993, what is the procedure to follow?

Cordially,
Ozan
----- Original Message -----
From: Gene Ward Smith
To: tuning@yahoogroups.com
Sent: 06 Mart 2005 Pazar 20:02
Subject: [tuning] Re: Prime factorization and maqams

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

> It takes a little algebra to find the generators, which is why I
> copied them from Gene's post :) But as you can see, I verified that
> these generators do satisfy the condition that 4000:3993 be tempered
> out . . . and it seems, from the below, that you may now understand
> this verification -- but let me know if I need to clarify further.

> But that is the crux of the issue! I want to know how to find the
generators, so that I can manage to temper any interval I choose after
that.

You might go over to tuning-math, give an example of what you want to
find generators for, and we could take it from there. In general,
however, finding generators depends on how you want to optimize--you
define what you mean by "best", and then calculate the tuning which is
"best" in that sense.

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

> Splendid Gene, now suppose I want to temper out 4000:3993, what is
the procedure to follow?

An easy way to do it is to find an equal temperament which tempers it
out, such as 72, 94, 130, 224, 311 or 441. If you want an optimized
tuning, you need to decide what you mean by "optimal". If you factor
4000/3993 into primes, you get 2^5 3^(-1) 5^3 11^(-3), or
|5 -1 3 0 -3>. If g2, g3, g5, g7 and g11 are the generators in
logaritmic form, say cents, then you can set g7=cents(7)=3368.826
cents right away, since it doesn't appear in the factorization. Beyond
that, 5g2-g3+3g5-3g11 = 0, which is a single linear condition, and not
enough by itself to determine g2, g3, g5 and g11. If you have a list
of consonances, you can, for instance, do a least-squares
optimization. You might also decide octaves are to be precise, and so
set g2=1200 cents, which means you would optimize the rest. This is
just an example, as there are various ways we might choose to optimize.

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> we write the monzo of 81:80 as: [-4 4, -1>
>
> in a monzo, positive exponents denote factors of the
> numerator, and negative exponents denote factors of
> the denominator.
>

This is essential the same as in Dr. John H. Chalmers Jr.'s table in
Xenharmonikon Nr. 1. What makes it a monzo and not a chalmers?

Gabor

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

> This is essential the same as in Dr. John H. Chalmers Jr.'s table in
> Xenharmonikon Nr. 1. What makes it a monzo and not a chalmers?

It's my fault in the main. I wanted a name for my own private purposes
in my Maple programs, and I started using it in public. The reason for
the name is that while lots of people used this sort of notation, Joe
was the one who was enthusiastic about it. It gives us a name specific
to this particular use, and assoicated to the |...> notation.

hi Gabor (and Pete, Gene, and Carl),

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@c...>
wrote:

> --- In tuning@yahoogroups.com, "alternativetuning"
> <alternativetuning@y...> wrote:
>
> > This is essential the same as in Dr. John H. Chalmers Jr.'s
> > table in Xenharmonikon Nr. 1. What makes it a monzo and
> > not a chalmers?
>
> It's my fault in the main. I wanted a name for my own
> private purposes in my Maple programs, and I started using
> it in public. The reason for the name is that while lots
> of people used this sort of notation, Joe was the one who
> was enthusiastic about it. It gives us a name specific to
> this particular use, and assoicated to the |...> notation.

i advocated this notation instead of ratios, in the
first draft of my book which appeared in 1995, and in
my paper _JustMusic Prime-factor Notation_ in 1997.

i have since found out that Fokker used a similar concept
in his last papers (late 1960's) ... so AFAIK he was
the first one to use it. but also AFAIK neither Fokker
nor Chalmers ever expressly proposed to use it as a
musical notation.

-monz

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

> So sorry for that error Paul, I meant to say that I somehow hear
9801:9000

I think you mean 9801:8000?

>(not 88209/80000) to be a better substitute for 27:22, as in:
>
> 1/1 12/11 9801/8000 4/3
>
> Don't ask me why, beats me...

Hi Ozan,

I'm impressed that you're sensitive to a 3 cent change in the
intonation of one note. What means are you using to render and listen
to these intervals?

I have a bunch of questions/reactions, but let me start off with this:

(1) How do you know it's 9801/8000 and not 49/40 that your ear is
guiding you towards?

>> It takes a little algebra to find the generators, which is why I
>> copied them from Gene's post :) But as you can see, I verified
>>that
>> these generators do satisfy the condition that 4000:3993 be
tempered
>> out . . . and it seems, from the below, that you may now
understand
>> this verification -- but let me know if I need to clarify further.
>
>
> But that is the crux of the issue! I want to know how to find the
>generators, so that I can manage to temper any interval I choose
>after that.

Kalle has just posted basically the same question here. I hope you
and Kalle will both post your questions on tuning-math, since that's
where this has been discussed in the past, particularly by Gene and
Graham.

> The generators are given in cents, which is a logarithm-of-ratio
> measure of interval size. A basic property of logarithms is that
>
> log(a^b) = b*log(a)
>
> So if we say that g1=1200 cents represents prime 2 = 2:1, then
> 2*g1=2400 cents represents 2^2 = 4:1, 3*g1=3600 cents represents
2^3
> = 8:1, 4*g1=4800 cents represents 2^4 = 16:1, and 5*g1=6000 cents
> represents 2^5 = 32:1, etc. And if g3=2786 cents represents prime 5
> = 5:1, then 2*g1=5572 cents represents 5^2 = 25:1, and 3*g1=8358
> cents represents 5^3 = 125:1.
>
> Making sense?
>
>
> I'm sure it does to math-people like yourself, but give me some
>more time to understand the relationships.

OK . . . let's take a step back. Do you normally consider interval
sizes in cents? Or in some other system of units? Would you agree
than an octave is a 2:1 ratio, and that a triple-octave is an 8:1
ratio? Or does this mystify you?

> BTW, how do you feel about tempered octaves? If you approve of
them,
> I might use the TOP temperament of 4000:3993 alone, whose
> approximations to the primes can be calculated with the formulas in
> the paper I sent you.
>
>
> That would be swell Paul. You mean widening the octaves of course,
>don't you? Otherwise I would hear them to be unduly dissonant. My
>Bechstein grand has the octaves stretched by a few cents you know (I
>instructed the tuners myself). That creates a brilliant effect in
>fff passages.

I wasn't thinking of a piano, because the scale in question will have
too many notes per octave to be expressible on a piano. A piano has
stretched overtones, which is why stretched octaves sound great on
it. However, many other instruments, such as bowed strings,
brass/wind instruments, and the human voice have harmonic overtones.
For such instruments, a stretched octave won't sound much less
discordant than a similarly compressed octave, based on my experience
and knowledge. But perhaps you have found otherwise? Perhaps you're
only speaking of *melodic* octaves, not harmonic (simultaneously-
sounding) ones?

>> Hopefully the first case above is now equally clear to you. Let me
>> know if it isn't. Also, I'd encourage you to write a similar
>> paragraph explaining how 81:80 vanishes in meantone temperament --
I
>> think you'd learn something about both the general idea of
>> temperament and about the specific example in which Western common
>> practice music arose.
>
>
>> Urm... Let's see... For 81:80 to vanish, I need the numerator of
>this interval to be equal to the denumerator. Since the syntonic
>comma is a 5-limit interval, the system will have the primes 2,3 and
>5, making this a three dimensional system of tuning. Right?

Most likely, though for your example of 4000:3993, which is an 11-
limit interval, I was thinking, instead of letting

>Tempering out the syntonic comma will result in the upper-most
>dimension (prime) to collapse,

What do you mean uppermost? The largest? How do you know this? What
is it about the syntonic comma that allows you to reach this
conclusion?

>creating a temperament of 2 dimensions. So, the first dimension is
>the octave with 2:1 which will remain untouched for my purposes, and
>the second dimension is the tempered perfect fifth with the ratio
>3:2 and the third dimension is the tempered pure third with the
>ratio 5:4.

I thought you said 2 dimensions?

> Prime factorizing 81:80 gives me
>
> 3*3*3*3 (3^4) for the numerator
>
> and
>
> 2*2*2*2*5 (2^4 * 5^1) for the denumerator.
>
> So, will you help me figure out the rest?

You seem to have left out some of your reasoning above. Perhaps it
was valid, perhaps it wasn't . . . I'm not sure unless you fill it in
for me.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@c...>
wrote:
>
> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...>
wrote:
>
> > It takes a little algebra to find the generators, which is why I
> > copied them from Gene's post :) But as you can see, I verified
that
> > these generators do satisfy the condition that 4000:3993 be
tempered
> > out . . . and it seems, from the below, that you may now
understand
> > this verification -- but let me know if I need to clarify further.
>
> > But that is the crux of the issue! I want to know how to find the
> generators, so that I can manage to temper any interval I choose
after
> that.
>
> You might go over to tuning-math, give an example of what you want
to
> find generators for, and we could take it from there. In general,
> however, finding generators depends on how you want to optimize--you
> define what you mean by "best", and then calculate the tuning which
is
> "best" in that sense.

Gene, I might express the situation a bit differently.

You can express the generators, for example, in terms of ratios which
they represent in the temperament, in which case the question of
optimization or what is "best" is moot. The set of generators will
not be unique, but different optimizations don't change this; they
simply tell you the precise tuning of a given set of generators that
will be "best" under various criteria.

Isn't it true that when *you* determine the generators of a
temperament, your software first determines them abstractly of any
specific optimization criterion?

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

> You can express the generators, for example, in terms of ratios which
> they represent in the temperament, in which case the question of
> optimization or what is "best" is moot.

I thought that by "finding the generators" he simply meant finding the
tuning.

> Isn't it true that when *you* determine the generators of a
> temperament, your software first determines them abstractly of any
> specific optimization criterion?

"Finding the generators" in what sense? You can use Hermite normal
form, and "find" the generators of 225/224-planar in the sense of
saying that
some tuning of 2, 3, and 5 will work; but you also could, for
instance, find TOP tunings for 2, 3, 5, and 7. Most of the time people
are talking about linear temperaments, in which case you find the
period and, perhaps, the smallest generator > 1 according to some
optimization.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@c...>
wrote:
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
>
> > You can express the generators, for example, in terms of ratios
which
> > they represent in the temperament, in which case the question of
> > optimization or what is "best" is moot.
>
> I thought that by "finding the generators" he simply meant finding
the
> tuning.
>
> > Isn't it true that when *you* determine the generators of a
> > temperament, your software first determines them abstractly of
any
> > specific optimization criterion?
>
> "Finding the generators" in what sense? You can use Hermite normal
> form, and "find" the generators of 225/224-planar in the sense of
> saying that
> some tuning of 2, 3, and 5 will work; but you also could, for
> instance, find TOP tunings for 2, 3, 5, and 7.

I could have easily told Ozan how the primes would be tuned by TOP
for the temperament in question. But finding a set of generators
(even abstractly) brings you a step closer to actually constructing a
reasonable scale in that temperament. And for that I relied on your
post. Now I'm asking you a question about your methods, but as far as
I can tell, you're not answering.

> Most of the time people
> are talking about linear temperaments,

I never use that term anymore, and George Secor agrees with my new
usage. It partly came down to, believe it or not Pete, respect for
Erv Wilson's original usage of and meaning for the term, which we
believe would not encompass a lot of the systems where the octave is
more than one period that we used to call "linear temperaments."

> in which case you find the
> period and, perhaps, the smallest generator > 1 according to some
> optimization.

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

> I could have easily told Ozan how the primes would be tuned by TOP
> for the temperament in question. But finding a set of generators
> (even abstractly) brings you a step closer to actually constructing a
> reasonable scale in that temperament. And for that I relied on your
> post. Now I'm asking you a question about your methods, but as far as
> I can tell, you're not answering.

OK, I replied over on tuning-math, but the method I used is not going
to be the best thing for Ozan, or anyone who doesn't have a package
giving them reduction to Hermite normal form.

> > Most of the time people
> > are talking about linear temperaments,
>
> I never use that term anymore, and George Secor agrees with my new
> usage.

Geez, and here you reamed me out when I first came here for not using
it! What is the new usage?

>> Most of the time people
>> are talking about linear temperaments,
>
>I never use that term anymore, and George Secor agrees with my new
>usage.

Oh my.

>It partly came down to, believe it or not Pete, respect for
>Erv Wilson's original usage of and meaning for the term, which we
>believe would not encompass a lot of the systems where the octave is
>more than one period that we used to call "linear temperaments."

What purpose can such anal-retentive re-engineering of terminology
serve?

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> Most of the time people
> >> are talking about linear temperaments,
> >
> >I never use that term anymore, and George Secor agrees with my new
> >usage.
>
> Oh my.
>
> >It partly came down to, believe it or not Pete, respect for
> >Erv Wilson's original usage of and meaning for the term, which we
> >believe would not encompass a lot of the systems where the octave
is
> >more than one period that we used to call "linear temperaments."
>
> What purpose can such anal-retentive re-engineering of terminology
> serve?
>
> -Carl

It's not re-engineering when it conforms with an older, existing,
published usage. And it's just plain logical, regardless of previous
usage. I know George Secor agrees, and hope he will chime in.

>> What purpose can such anal-retentive re-engineering of terminology
>> serve?
>
>It's not re-engineering when it conforms with an older, existing,
>published usage. And it's just plain logical, regardless of previous
>usage. I know George Secor agrees, and hope he will chime in.

What is your new usage?

I seriously doubt Erv would want this distinction re. octave/nonoctave
periods.

-Carl

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

> It's not re-engineering when it conforms with an older, existing,
> published usage.

The older, existing, published usage being?

My personal preference would be to say that equal temperaments are
rank one, The Artistic Temperaments Formerly Known as "Linear" rank
two, and so forth. This corresponds to an older, existing, firmly
established and often published usage in mathematics, if not in music.

See

http://en.wikipedia.org/wiki/Rank_of_an_abelian_group

http://mathworld.wolfram.com/GroupRank.html

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@c...>
wrote:
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
>
> > It's not re-engineering when it conforms with an older, existing,
> > published usage.
>
> The older, existing, published usage being?

See Erv Wilson's articles in the first few issues of Xenharmonikon.

> My personal preference would be to say that equal temperaments are
> rank one, The Artistic Temperaments Formerly Known as "Linear" rank
> two, and so forth.

I called them "one-dimensional" and "two-dimensional" temperaments in
my paper, respectively, but I suppose "rank one" and "rank two" are
nice and shorter, and probably more mathematically correct as well.

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
>
> --- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> > >> Most of the time people
> > >> are talking about linear temperaments,
> > >
> > >I never use that term anymore, and George Secor agrees with my
new
> > >usage.
> >
> > Oh my.

Oh my, indeed! I don't know exactly what it is that I'm agreeing
with. All I know is that I agreed with the way Paul *was*
using "linear" late last year to classify temperaments.

For the record, here's the text of our conversation:

George Secor, 8 Dec 2004, to Paul Erlich & Dave Keenan
Begin text
------------------

Paul, Dave,

While I'm leaving it to the two of you to hash out a definition
of "linear" vs. non-linear, I thought that I should throw in a few
observations, opinions, and/or comments that might help either to
clarify or further muddle your discussion (and likewise, I'll leave
it to you to figure out which arguments I'm supporting or refuting).

I've selected a few things to respond to:

--- wally paulrus <wallyesterpaulrus@yahoo.com> wrote:

> >How would you construct a homogeneous multiple-chain keyboard?
>
> Using Dave Keenan's keyboard-design spreadsheet, it presents no
> special difficulties. Don't use that? Then for N-chain, do something
> like this:
>
> First, scale all the intervals by multiplying them by N so that the
> period becomes an octave. Then, design a keyboard for the resulting
> tuning, with the "aspect ratio" deliberately chosen to be N times
too
> wide. Finally, scale the horizontal axis by 1/N, and apply the
> original temperament to the keyboard.
>
> >For
> >multiple chains of whatever interval, I imagine that the tones in a
> >single chain would have to be arranged in a non-slanting row,
> >typewriter-style.
>
> I don't think so -- rather, its the parallel notes in different
> chains that would form a non-slanting row.

Yes, I see. That's something I hadn't considered, and I can't say
that this is something that would have occurred to Erv. If you
haven't found anything like that in any of his many drawings, then I
think it's safe to say that he didn't think of it -- or if he did,
then he didn't think it was worth-while, for the simple reason that
keys or notes at the same point along the y-axis of the keyboard
would have to be 1/N-octave, where N is a small integer. That's not
very useful if your goal is to have a regular temperament (other than
an EDO) that accurately approximates small-number ratios. You're
stuck with a proliferation of 600, 400, 300, or 240-cent intervals
that don't come within a few cents of any consonance.

On the other hand, if you're mapping JI to certain divisions of the
octave that don't lend themselves well to a single chain of fifths
(or other generator), then this might offer a few possibilities.

Assuming octave equivalence, would this be linear? I would have to
say definitely not (because there are *separate* chains, each capable
of having either a finite or infinite number of tones. Each of the
chains is, by itself, linear, and since you're adding another
dimension by way of the interval separating adjacent chains, the
result (even if you try to justify it as "multiple lines") is no
longer linear. But it's not exactly the same as what we would
call "planar" either, because one of your dimensions wraps back on
itself after N tones. I would be inclined to call it "cylindrical".
But having one of the dimensions wrap back on itself doesn't decrease
the number of dimensions -- it merely limits the number of tones
occurring along that particular dimension.

Interestingly, removing octave equivalence would not increase the
number of dimensions -- it would only make both dimensions capable of
having an infinite number of tones (limited only by the range of
audibility).

In this discussion it must be observed that the chain wraps back on
itself -- not onto an *adjacent* chain (where the number of
dimensions *would* be decreased by one).

> Well, the obvious approach seems quite different to me. It seems to
> me you're viewing multiple-chain tunings too differently from the
> usual case, where the single chain is in fact multiple chains at
> octave intervals apart from one another.

Hmmm, when you come down to what you might actually end up with for a
keyboard, it may not necessarily be all that different. E.g., if you
have your parallel chains 1/5-octave apart and use a generator on the
order of a minor 2nd, you would end up with something very close to
the diagram I sent you.

> >BTW, I've attached a keyboard diagram for a 25-EDO mapping, which
could
> >be used for 5 chains of fifths with 5 fifths in each chain.
>
> I'll look at this in a bit . . .

The peculiar thing about it is that, although like intervals all
subtend like numbers of degrees, the tones are not necessarily in
order of pitch.

--- wally paulrus <wallyesterpaulrus@yahoo.com> wrote:
>
> Why do you say 25-EDO and not 25-tET?

I should have said 25-division, or 25-DO. The pitches are JI, mapped
modulo 25.

--George

------------------
End of text

Paul Erlich, 9 Dec 2004, to George Secor, cc: Dave Keenan
Begin text
------------------

Hi George! (and Dave!!)

>> I don't think so -- rather, its the parallel notes in different
>> chains that would form a non-slanting row.

>Yes, I see. That's something I hadn't considered, and I can't say
that
>this is something that would have occurred to Erv. If you haven't
>found anything like that in any of his many drawings, then I think
it's
>safe to say that he didn't think of it -- or if he did, then he
didn't
>think it was worth-while, for the simple reason that keys or notes at
>the same point along the y-axis of the keyboard would have to be
>1/N-octave, where N is a small integer. That's not very useful if
your
>goal is to have a regular temperament (other than an EDO) that
>accurately approximates small-number ratios.

George, I must respectfully disagree with your assessment of utility
here. Have you received my paper yet? I believe I've explored this
question in ever-increasing depth over the past decade or so and my
paper is a thumbnail sketch of part of this issue.

>You're stuck with a
>proliferation of 600, 400, 300, or 240-cent intervals that don't come
>within a few cents of any consonance.

Other regular temperaments also have a proliferation of intervals
that don't come within a few cents of any consonance. I believe my
paper compares 2D temperaments on a truly level basis and some of the
best do contain fraction-of-octave intervals. For example, the srutal
system arises from tempering out the diaschisma or 2048:2025. This
necessarily introduces half-octave intervals, which arise as
representations of both 45:32 and 64:45. But the consonances in the
system are about twice as accurate as in meantone, yet the complexity
of the system is not unmanageable. A much more complex system is
Ennealimmal, which divides the octave into 9 equal parts, but it is a
far more accurate 7-limit 2D temperament than any other with similar
or lower complexity, and is far simpler than other with similar
accuracy (of the consonances).

I'm completely convinced that you and Erv have genuinely missed some
important possibilities due to your prejudice here, and I'm certain
that, given enough time, I can convince both of you of that.

>Assuming octave equivalence, would this be linear? I would have to
say
>definitely not (because there are *separate* chains, each capable of
>having either a finite or infinite number of tones. Each of the
chains
>is, by itself, linear, and since you're adding another dimension by
way
>of the interval separating adjacent chains, the result (even if you
try
>to justify it as "multiple lines") is no longer linear. But it's not
>exactly the same as what we would call "planar" either, because one
of
>your dimensions wraps back on itself after N tones. I would be
>inclined to call it "cylindrical". But having one of the dimensions
>wrap back on itself doesn't decrease the number of dimensions -- it
>merely limits the number of tones occurring along that particular
>dimension.

Exactly! We seem to be in total agreement on this question, George.

>Interestingly, removing octave equivalence would not increase the
>number of dimensions -- it would only make both dimensions capable of
>having an infinite number of tones (limited only by the range of
>audibility).

Right -- and it would also increase the field of possibilities for
what pair of intervals you can usefully take to be your
two "generators".

>> Well, the obvious approach seems quite different to me. It seems to
>> me you're viewing multiple-chain tunings too differently from the
>> usual case, where the single chain is in fact multiple chains at
>> octave intervals apart from one another.

>Hmmm, when you come down to what you might actually end up with for a
>keyboard, it may not necessarily be all that different.

Right, that's what I was trying to say before -- just treat the
period as if it were an octave when you design the keyboard,
then "squeeze" the design so that the number of periods making up an
octave can be reached with one hand.

>The peculiar thing about it is that, although like intervals all
>subtend like numbers of degrees, the tones are not necessarily in
order
>of pitch.

This sort of thing has come up on the tuning-math list over and over
in the past.

Best,
Paul

------------------
End of text

George Secor, 9 Dec 2004, to Paul Erlich, cc: Dave Keenan
Begin text
------------------

--- wally paulrus <wallyesterpaulrus@yahoo.com> wrote:

> Hi George! (and Dave!!)
>
> >> I don't think so -- rather, its the parallel notes in different
> >> chains that would form a non-slanting row.
>
> >Yes, I see. That's something I hadn't considered, and I can't say
that
> >this is something that would have occurred to Erv. If you haven't
> >found anything like that in any of his many drawings, then I think
it's
> >safe to say that he didn't think of it -- or if he did, then he
didn't
> >think it was worth-while, ...
>
> George, I must respectfully disagree with your assessment of utility
> here. Have you received my paper yet?

Yes, you should remember that I did offer a couple of very minor
criticisms.

> I believe I've explored this
> question in ever-increasing depth over the past decade or so and my
> paper is a thumbnail sketch of part of this issue.

Reading the paper and studying in depth some of the examples given in
your tables and figures amounts to two different things. I've done
the former but have not yet taken the time to do the latter. What
has taken you a decade cannot be fully absorbed by another overnight.

Anyway, you've put your finger on the *reason* why I've overlooked
multiple-chain temperaments.

> ...
> I'm completely convinced that you and Erv have genuinely missed some
> important possibilities due to your prejudice here, and I'm certain
> that, given enough time, I can convince both of you of that.

You're probably right about that on all counts. Perhaps our
discussion (and your paper) should be printed out and mailed to Erv
so he can respond.

> ...
> >> Well, the obvious approach seems quite different to me. It seems
to
> >> me you're viewing multiple-chain tunings too differently from the
> >> usual case, where the single chain is in fact multiple chains at
> >> octave intervals apart from one another.
>
> >Hmmm, when you come down to what you might actually end up with
for a
> >keyboard, it may not necessarily be all that different.
>
> Right, that's what I was trying to say before -- just treat the
> period as if it were an octave when you design the keyboard, then
> "squeeze" the design so that the number of periods making up an
> octave can be reached with one hand.

The essential difference is that I was thinking of chains going along
the x-axis and having the interval separating them breaking out into
the y-dimension, whereas you're doing the opposite. I would say that
yours is the more proper way of looking at it. Kudos!

Best,

--George

------------------
End of text

> > >It partly came down to, believe it or not Pete, respect for
> > >Erv Wilson's original usage of and meaning for the term, which
we
> > >believe would not encompass a lot of the systems where the
octave is
> > >more than one period that we used to call "linear temperaments."
> >
> > What purpose can such anal-retentive re-engineering of terminology
> > serve?
> >
> > -Carl
>
> It's not re-engineering when it conforms with an older, existing,
> published usage. And it's just plain logical, regardless of
previous
> usage. I know George Secor agrees, and hope he will chime in.

As I said I my previous message, I did write Erv, but his response
was so brief that it did not even begin to address these issues. In
the above letters I expressed my own opinion as to what I thought
should be covered by the term "linear" temperament, and you can take
that as my opinion, for what it's worth -- no more, no less. While I
attempted to speculate on what Erv might have thought about all this,
again it's mere speculation -- no more, no less. Take it for what
it's worth.

Best,

--George

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> What purpose can such anal-retentive re-engineering of
terminology
> >> serve?
> >
> >It's not re-engineering when it conforms with an older, existing,
> >published usage. And it's just plain logical, regardless of
previous
> >usage. I know George Secor agrees, and hope he will chime in.
>
> What is your new usage?

You have my new paper, Carl.

> I seriously doubt Erv would want this distinction re.
octave/nonoctave
> periods.

Well, George sent him a long letter, in which the specific question
of whether 'linear temperament' should apply to systems with more
than one period per octave was asked in explicit detail. Included
with the letter was my new paper and other supporting material. Erv's
response, unless he's sent something else since I last heard from
George, was mostly about unrelated matters, and the closest he came
to answering the question was in this passage:

' Reference; _The Penguin Desk Encyclopedia of Science and
Mathematics_*. The term "linear" is also found in said encyclopedia,
as is _Rene Descartes_ [1596-1650] who gave us the method of
_Cartesian
coordinates_. *U.S. $25.00. The term "Horagram" comes to us from
John
Harrison the horologist via Charles Lucy. ref: "Is This the Lost
Music
of the Spheres?" by Charles E. H. Lucy also "An account of the
Discovery of Musick" (1775) by John Harrison, both in _Pitch_ Vol. 1
No. 2, Editor Johny Reinhard '

I was very disappointed Erv didn't include a word about my paper (in
which I gave him credit wherever I knew it was due, and didn't
reference any of my other papers), which I'm sure he could easily
expand into a long book with all the theory, history, and musicality
in that head of his. Perhaps he'll write George again at some
point . . .

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...> wrote:

> Assuming octave equivalence, would this be linear? I would have to
> say definitely not (because there are *separate* chains, each capable
> of having either a finite or infinite number of tones.

What we were calling "linear" temperaments *always* have separate
chains, which is why I objected to the term, and *never* have a finite
number save that you cut them off at some point. However, I was told
it was firmly established, and now I wonder if that was so.

Each of the
> chains is, by itself, linear, and since you're adding another
> dimension by way of the interval separating adjacent chains, the
> result (even if you try to justify it as "multiple lines") is no
> longer linear. But it's not exactly the same as what we would
> call "planar" either, because one of your dimensions wraps back on
> itself after N tones.

Nothing wraps back on itself. All that happens is that we can express
an octave in terms of the temperament, which is always true if it
isn't a non-octave temperament. There isn't any "torsion" in pitch, so
there is not and cannot be any in temperaments. Of course you can
identify 2 with 1 if you like, but this is hardly a useful thing to do
when discussion pitch.

>> I seriously doubt Erv would want this distinction re.
>> octave/nonoctave periods.
>
>Well, George sent him a long letter, in which the specific question
>of whether 'linear temperament' should apply to systems with more
>than one period per octave was asked in explicit detail. Included
>with the letter was my new paper and other supporting material. Erv's
>response, unless he's sent something else since I last heard from
>George, was mostly about unrelated matters,

Right, and that's pure Erv, it's not on accident. In his 2001 talk
at Microfest, he spent more than 50% of his talk warning and making
disclaimers that his language should not be taken seriously. 'These
words are words that I've used, and you may like other words better,'
'I looked at it this way, and I hope that you will find your own way,'
etc. etc. The picture of Erv painted on these lists (as I see it)
is not a remotely accurate one. Erv's thinking is delicate,
sensitive, and innocent.

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

Erv's thinking is delicate,
> sensitive, and innocent.
>
> -Carl

That's at least close! Well said, Carl!

>> Erv's thinking is delicate,
>> sensitive, and innocent.
>>
>> -Carl
>
>That's at least close! Well said, Carl!

Compare/contrast to your thinking in your last several
posts.

-Carl

As Rich has pointed out, I think the application of "rank" instead of dimension or linear etc... has a certain charm to it.

And I find his latest metholodogy for temperament most clarifying. Thank you Rich!

Cordially,
Ozan
----- Original Message -----
From: George D. Secor
To: tuning@yahoogroups.com
Sent: 08 Mart 2005 Salı 23:20
Subject: [tuning] Re: Prime factorization and maqams

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
>
> --- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> > >> Most of the time people
> > >> are talking about linear temperaments,
> > >
> > >I never use that term anymore, and George Secor agrees with my
new
> > >usage.
> >
> > Oh my.

Oh my, indeed! I don't know exactly what it is that I'm agreeing
with. All I know is that I agreed with the way Paul *was*
using "linear" late last year to classify temperaments.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@c...>
wrote:
>
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
>
> > Assuming octave equivalence, would this be linear? I would have
to
> > say definitely not (because there are *separate* chains, each
capable
> > of having either a finite or infinite number of tones.
>
> What we were calling "linear" temperaments *always* have separate
> chains, which is why I objected to the term,

Sadly, I'm finding myself being sucked into another debate when I
have precious little time to spare. This is one battle I would just
as soon forego. (Sigh!)

That phrase, "assuming octave equivalence", must be kept constantly
in mind. What I consider to be a "linear" temperament is one in
which the tones all occur in a *single* chain generated by an
interval of constant size, e.g., a Pythagorean or 1/4-comma meantone
sequence of fifths, or the Miracle temperament. Given the principle
of octave equivalence, so-called "separate" chains an octave apart
are not really distinct or separate from one another.

> and *never* have a finite
> number save that you cut them off at some point. However, I was told
> it was firmly established, and now I wonder if that was so.

Should one happen to use a generator (such as the fifth of 12-ET)
that results in a return to the starting tone in the sequence after a
finite number of places, does this cease to make the sequence
linear? I would think not, only that it's a special case of a linear
sequence that, due to the "wormhole" principle of octave equivalence,
makes it seem to "wrap around" back to its starting point.

Now if you have two infinite chains 1/2-octave apart, you then have
two dimensions, one infinite and the other finite (as a sort of wire-
frame cylinder). And there's no way I'm going to accept 1/2-octave
equivalence. (For that matter, I won't even accept 1:3 equivalence,
because I perceive those pitches as having completely different
harmonic identities, as different as green is from violet.)

> Each of the
> > chains is, by itself, linear, and since you're adding another
> > dimension by way of the interval separating adjacent chains, the
> > result (even if you try to justify it as "multiple lines") is no
> > longer linear. But it's not exactly the same as what we would
> > call "planar" either, because one of your dimensions wraps back
on
> > itself after N tones.
>
> Nothing wraps back on itself. All that happens is that we can
express
> an octave in terms of the temperament,

Hardly a useful thing to do when discussing temperament.

> which is always true if it
> isn't a non-octave temperament. There isn't any "torsion" in pitch,
so
> there is not and cannot be any in temperaments. Of course you can
> identify 2 with 1 if you like, but this is hardly a useful thing to
do
> when discussion pitch.

Huh? It's done all of the time in music theory books. A "G dominant
7th chord," which is normally considered to be composed of 4 separate
pitches, maintains its identity regardless of the order, spacing
(octave displacement), or doubling (at another octave) of its tones.
We perceive pitch in such a way that multiplication or division by 2
does not change its identity, and I would say that a classification
of temperaments that doesn't take that into account would be less
useful than one that does.

--George

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...> wrote:

> Should one happen to use a generator (such as the fifth of 12-ET)
> that results in a return to the starting tone in the sequence after a
> finite number of places, does this cease to make the sequence
> linear?

But it doesn't return to its starting point without octave
equivalence, which simply confuses the issue. Otherwise, you get a
cyclic group of order 12, but the elements do not represent pitches,
but pitch classes.

> Now if you have two infinite chains 1/2-octave apart, you then have
> two dimensions, one infinite and the other finite (as a sort of wire-
> frame cylinder).

If you want to think of it in a confusing way. But why would you wish
to do that?

> Huh? It's done all of the time in music theory books. A "G dominant
> 7th chord," which is normally considered to be composed of 4 separate
> pitches, maintains its identity regardless of the order, spacing
> (octave displacement), or doubling (at another octave) of its tones.

Yes, but that is in terms of pitch *classes*. When discussing
temperament, it is not helpful to introduce those. It makes far more
sense to stick to pitches, in which case, since there isn't any
torsion in the positive reals under multiplication (quite unlike the
case with complex numbers!) there can't be any in a subgroup.

> We perceive pitch in such a way that multiplication or division by 2
> does not change its identity, and I would say that a classification
> of temperaments that doesn't take that into account would be less
> useful than one that does.

You can take it into account by making one of the generators the nth
part of an octave, but introducing octave equivalence into the theory
of temperaments simply confuses the issue.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@c...>
wrote:

> You can take it into account by making one of the generators the nth
> part of an octave, but introducing octave equivalence into the theory
> of temperaments simply confuses the issue.

It also makes you blind to the possibility of not doing this. It will
be done for you automatically if you put in a reduction to Hermite
normal form step, which is what I do. However, there are other
possibilities, such as generators dual to some set of equal
temperaments. You can do meantone, for instance, by using tempered
values of the limma and apotome as generators, which correspond to the
5 and 7 meantone equal temperaments; each sends one of this pair to a
single step, and tempers out the other. An array of notes separated
by these two different sizes of semitone is one possible way to get a
keyboard layout for meantone.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@c...>
wrote:

You can do meantone, for instance, by using tempered
> values of the limma and apotome as generators, which correspond to the
> 5 and 7 meantone equal temperaments; each sends one of this pair to a
> single step, and tempers out the other. An array of notes separated
> by these two different sizes of semitone is one possible way to get a
> keyboard layout for meantone.

In case it wasn't clear, this works just as well for rank 2
temperaments where the period is not an octave; no wrapping is in
sight. For instance, tempering out 2048/2025, we can use 10 and 12,
corresponding to generators 25/24 and 81/80. This is related to what
I've termed Bosanquet lattices, and you can see, under "10&12
diaschismic", pictures of a symmetrical lattice with octaves along the
horizonal axis corresponding to this, as well as the 5&7 meantone I
discussed previously.

http://66.98.148.43/~xenharmo/bosanquet.html

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@c...>
wrote:
>
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
>
> > Should one happen to use a generator (such as the fifth of 12-ET)
> > that results in a return to the starting tone in the sequence
after a
> > finite number of places, does this cease to make the sequence
> > linear?
>
> But it doesn't return to its starting point without octave
> equivalence, which simply confuses the issue. Otherwise, you get a
> cyclic group of order 12, but the elements do not represent pitches,
> but pitch classes.

Okay, I stand corrected -- it's pitch classes.

> > Now if you have two infinite chains 1/2-octave apart, you then
have
> > two dimensions, one infinite and the other finite (as a sort of
wire-
> > frame cylinder).
>
> If you want to think of it in a confusing way. But why would you
wish
> to do that?

Because it's simpler to disregard the octaves and think of the tones
as being in two infinite chains than in an (infinite) 2-dimensional
lattice. (At least I think it's simpler from a musician's point of
view.)

> > Huh? It's done all of the time in music theory books. A "G
dominant
> > 7th chord," which is normally considered to be composed of 4
separate
> > pitches, maintains its identity regardless of the order, spacing
> > (octave displacement), or doubling (at another octave) of its
tones.
>
> Yes, but that is in terms of pitch *classes*. When discussing
> temperament, it is not helpful to introduce those. It makes far more
> sense to stick to pitches, in which case, since there isn't any
> torsion in the positive reals under multiplication (quite unlike the
> case with complex numbers!) there can't be any in a subgroup.

Well, I guess what you're saying makes more sense from a
mathematician's point of view.

> > We perceive pitch in such a way that multiplication or division
by 2
> > does not change its identity, and I would say that a
classification
> > of temperaments that doesn't take that into account would be less
> > useful than one that does.
>
> You can take it into account by making one of the generators the nth
> part of an octave, but introducing octave equivalence into the
theory
> of temperaments simply confuses the issue.

If n=1 the temperament is much simpler from a musical point of view
(in which octave equivalence is assumed). Historical tunings or
temperaments (e.g., Pythagorean, meantone, equal, etc.) with (pure)
octave and (pure or tempered) fifth generators have traditionally
been regarded as "linear" in that they consist of a single sequence
of fifths, with octave-equivalence assumed.

But change the octave generator to 1/n octaves and you have something
that's musically more complicated, with n chains of fifths. It would
be very confusing to call something like that "linear." Better not
to use the term at all. I think I saw that you're in the process of
adopting different terminology, yes? If so, is there a
terminological distinction between temperaments with octave and non-
octave generators? (I would include octaves tempered by no more than
a few cents in the octave-generator category.) Such a distinction
would be very useful from the ordinary musician's point of view.

--George

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...> wrote:
> > > Now if you have two infinite chains 1/2-octave apart, you then
> have
> > > two dimensions, one infinite and the other finite (as a sort of
> wire-
> > > frame cylinder).
> >
> > If you want to think of it in a confusing way. But why would you
> wish
> > to do that?
>
> Because it's simpler to disregard the octaves and think of the
tones
> as being in two infinite chains than in an (infinite) 2-dimensional
> lattice. (At least I think it's simpler from a musician's point of
> view.)

That's what I was saying--you should think of it as a system with two
generators, where one of the generators may, or may not, be the nth
part of an octave. Division of the octave into parts is very much a
side issue.

> Well, I guess what you're saying makes more sense from a
> mathematician's point of view.

Pretty much, but I also don't see how insisting on dragging in octave
equivalence helps us understand regular temperaments in general.

> If n=1 the temperament is much simpler from a musical point of view
> (in which octave equivalence is assumed).

I don't see why. I do see we are more familiar with such systems, but
how does that make them much simpler?

> But change the octave generator to 1/n octaves and you have
something
> that's musically more complicated, with n chains of fifths. It
would
> be very confusing to call something like that "linear." Better not
> to use the term at all. I think I saw that you're in the process
of
> adopting different terminology, yes?

When I first showed up on these lists I wanted to call them rank two
temperaments, and that seems to be suddenly gaining ground because of
the objection to "linear" when the period is not an octave.

If so, is there a
> terminological distinction between temperaments with octave and non-
> octave generators? (I would include octaves tempered by no more
than
> a few cents in the octave-generator category.) Such a distinction
> would be very useful from the ordinary musician's point of view.

Why? How does it impact musical practice so powerfully? If you set up
a Bosanquet keyboard arragement, it all looks pretty much the same.