Tuning-Math messages 277 - 301

This is an Opt In Archive . We would like to hear from you if you want your posts included. For the contact address see About this archive. All posts are copyright (c).

Contents Hide Contents S 1

Previous Next

0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950

250 - 275 -



top of page bottom of page down


Message: 277

Date: Thu, 21 Jun 2001 19:18 +0

Subject: Re: recap of decimal notation

From: graham@m...

In-Reply-To: <9gsrt9+o9t0@e...>
Joseph Pehrson wrote:

> And *yes*, I have been on that page of Graham's SEVERAL times, and I 
> *never* get it.  It's all very sophisticated, but I get lost in the 
> presentation.
> 
> Would you mind recapping that in another way, so I can understand 
> *that* notation.

What's sophisticated about it?  You understand staff notation don't you?  
There are 7 notes (A, B, C, D, E, F and G) and 2 things you can do to 
them.  I decimal notation, there are 10 notes (0, 1, 2, 3, 4, 5, 6, 7, 
8 and 9) and still 2 things you can do to them (^ and v or > and < 
depending on which page you look at).

The 10 notes are this scale in 72-equal:

0   1   3   4   5   6   7   8   9   0
  7   7   7   7   7   7   7   7   9

The ^,v,>,< or whatever tell you to move by 2 steps of 72-equal.


                 Graham


top of page bottom of page up down


Message: 278

Date: Thu, 21 Jun 2001 19:18 +0

Subject: Re: 7/72 generator in blackjack

From: graham@m...

In-Reply-To: <9gpf0f+i66g@e...>
> By what figure-of-demerit and at what odd-limits can we claim that the 
> MIRACLE generator is the best?

11-limit by most figures of demerit, although that is ignoring some 
obviously too complex scales.

7-limit it's second by the default FOD, whatever that was.  9-limit it 
doesn't score so well.

> Does cardinality_of_smallest_MOS_containing_a_complete_otonality 
> divided by exp(-(minimax_error/17c)^2) do it at 7 and 11 limits? 
> What's the best 9-limit generator by this FoD?

13/41, consistent with 19 and 22 note scales.  Schismic is second.

Miracle does really badly in the 7-limit with this measure.  See

<5 12 19 22 26 27 29 31 41 46 50 53 58 60 68 70 72 77 80 84 *>

etc.

> I'm sure some folks would be interested in the 13-limit result too.

Right.  I've added, but not checked, that.

<Automatically generated temperaments *>

It should be easily hackable now.  Although there's a lot of code in 
writetemper.py, you should be able to work out how to supply different 
limits, or figures of demerit.

Obviously, you don't need the bit that uploads to my website.

           Graham


top of page bottom of page up down


Message: 279

Date: Thu, 21 Jun 2001 18:24:16

Subject: Re: Sonance degree (DEFINITION)

From: Paul Erlich

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:

> Paul (or Dave or...),
> 
> Please explain in simplified terms what Pierre wrote here.

I wish I could!


top of page bottom of page up down


Message: 280

Date: Fri, 22 Jun 2001 00:40:43

Subject: Re: 7/72 generator in blackjack

From: Dave Keenan

31-EDO is "MIRACLE" in exactly the same way that 12-tET is meantone. 
i.e. Not really. At best borderline, with large deviations from JI.

We have the following approximate analogy

Meantone MIRACLE
----------------
12-EDO   31-EDO
31-EDO   72-EDO
19-EDO   41-EDO


top of page bottom of page up down


Message: 281

Date: Fri, 22 Jun 2001 01:26:27

Subject: Re: 7/72 generator in blackjack

From: Dave Keenan

> <Automatically generated temperaments *>

Your page says "11 limit" where it should be "13 limit".


top of page bottom of page up down


Message: 282

Date: Fri, 22 Jun 2001 01:42:55

Subject: Re: Sonance degree (DEFINITION)

From: Dave Keenan

--- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:
> Hi Joe,
> 
> <<Paul (or Dave or...), Please explain in simplified terms what 
Pierre
> wrote here.>>
> 
> Pierre's posts are really the only ones where I consistently find
> myself in need of a 'math to English' translator! Robert Walker has
> been very helpful in the past in this regard, you might want to ask
> him.
> 
> --Dan Stearns

My apologies Dan, Monz, and Pierre. Here's where it becomes obvious 
that I'm not a "real" mathematician. It's such a shame because the 
math should be the universal language that overcomes the 
French/English barrier. But I don't know all the math that Pierre 
knows. Maybe I could get a handle on it if I spent a long time, but I 
just can't justify the time. Reminds me of one time when I 
_did_ spend the time to "translate" some stuff of yours Dan. :-) 
You've come a long way since then.

Dan, you asked once before if I could explain some stuff Pierre wrote, 
generalising the golden mean to two dimensions. I didn't respond 
because I had already explained as much of it as I understood, but I 
didn't understand how to relate it to scales.

Regards,
-- Dave Keenan


top of page bottom of page up down


Message: 293

Date: Fri, 22 Jun 2001 19:00:31

Subject: Hypothesis revisited

From: Paul Erlich

Progress seems to have halted on the paper that was to introduce 
MIRACLE . . .

I suggest the title

_The Relationship Between Just Intonation and Well-Formed Scales_

and some sort of "proof" of the hypothesis (I know, it doesn't always 
work).

If we can do the following math problem, we'll be fine:

Given a k-by-k matrix, containing k-1 commatic unison vectors and 1 
chromatic unison vector, delimiting a periodicity block, find:

(a) the generator of the resulting WF (MOS) scale;

(b) the integer N such that the interval of repetition is 1/N octaves.

If we can derive a general formula of this nature, the status of the 
pathological cases (e.g., Monz' shruti block) should become clear 
(hopefully). Then we can give a few examples, including the diatonic 
and MIRACLE scales.

So, who's going to be our hero?


top of page bottom of page up down


Message: 294

Date: Fri, 22 Jun 2001 19:04:50

Subject: Re: 7/72 generator in blackjack

From: Paul Erlich

--- In tuning-math@y..., jpehrson@r... wrote:

> But isn't it a bit strange that a generator that is 7 units of  72-
> tET should create FIVES... and that 5+2, the small interval = 7.
> 
> I'm not entirely understanding why that is...
> 
> ??

It's very simple.

72 divided by 7 is 10, with a remainder of 2.

So after 10 generators in the cycle, you're 2 units short of where 
you started.

Another generator later, you're 7-2 = 5 units beyond where you 
started, and 2 units short of the second note in the chain.

Similarly, you'll divide each of the 7-unit steps of the first cycle 
into a large (5-unit) step and a small (2-unit) step as you go around 
the second cycle.

When you've completed the second cycle, you've completed the 
blackjack scale.

When you've completed the third cycle, you've completed the canasta 
scale.

When you've completed the fourth cycle, you've completed the MIRACLE-
41 scale.


top of page bottom of page up down


Message: 295

Date: Fri, 22 Jun 2001 21:39 +0

Subject: Re: Hypothesis revisited

From: graham@m...

Paul wrote:

> Given a k-by-k matrix, containing k-1 commatic unison vectors and 1 
> chromatic unison vector, delimiting a periodicity block, find:
> 
> (a) the generator of the resulting WF (MOS) scale;

That's the bit I'm not sure about

> (b) the integer N such that the interval of repetition is 1/N octaves.

Easy.  It'll usually be the determinant of the matrix.  You can always get 
it by solving the matrix equation.  Say you have


Where H is the logs of the primes, H' is the approximation, a1...ak are 
the unison vectors, where ak is chromatic, and a0 is the octave (1 0 0 
... 0).  You solve it to get


     (a0)-1 (a0)
     (a1)   (a1)
H' = (a2)   ( 0)H
     (..)   (..)
     (ak)   ( 0)

From which you know the first column of

   (a0) (a0)-1 (a0)
   (a1) (a1)   (a1)
det(a2) (a2)   ( 0)
   (..) (..)   (..)
   (ak) (ak)   ( 0)

will be a vector of integers specifying the number of steps to each prime 
interval.  You then reduce them by any common factor, and the one on top 
will be the number of steps to an octave.  Or say that it's pathological 
if there is a common factor.

> If we can derive a general formula of this nature, the status of the 
> pathological cases (e.g., Monz' shruti block) should become clear 
> (hopefully). Then we can give a few examples, including the diatonic 
> and MIRACLE scales.

If you supplied two different chromatic unison vectors, that would give 
two equal temperaments that could be plugged into my Python script to 
yield everything else we need to know.

Ideally, we could do without chromatic unison vectors altogether, but I 
don't see how to do that bit.  You could do a brute force search over all 
consistent ETs, like my program does, but that's not the elegant way of 
solving this problem.

So are we aiming for musicians or mathematicians?


                  Graham


top of page bottom of page up down


Message: 296

Date: Fri, 22 Jun 2001 20:57:33

Subject: Re: Hypothesis revisited

From: Paul Erlich

--- In tuning-math@y..., graham@m... wrote:

> > (b) the integer N such that the interval of repetition is 1/N 
octaves.
> 
> Easy.  It'll usually be the determinant of the matrix.

Huh? The determinant of the matrix is usually the number of notes, 
not the number of repetitions per octave (which is usually just 1).


  You can always get 
> it by solving the matrix equation.  Say you have
> 
> 
> Where H is the logs of the primes,

Looks like you left something out here, yes?

Let's leave out the octave, octave-equivalence will be assumed (yes, 
in a more general case it won't be, but let's not bite off more than 
we can chew).

It's fine if the paper is a bit mathematical if that helps it obtain 
a more powerful result. Music theory can get very mathematical these 
days.


top of page bottom of page up down


Message: 297

Date: Fri, 22 Jun 2001 21:04:34

Subject: Re: 7/72 generator in blackjack

From: jpehrson@r...

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

Yahoo groups: /tuning-math/message/294 *

> --- In tuning-math@y..., jpehrson@r... wrote:
> 
> > But isn't it a bit strange that a generator that is 7 units of  
72-
> > tET should create FIVES... and that 5+2, the small interval = 7.
> > 
> > I'm not entirely understanding why that is...
> > 
> > ??
> 
> It's very simple.
> 
> 72 divided by 7 is 10, with a remainder of 2.
> 
> So after 10 generators in the cycle, you're 2 units short of where 
> you started.
> 
> Another generator later, you're 7-2 = 5 units beyond where you 
> started, and 2 units short of the second note in the chain.
> 
> Similarly, you'll divide each of the 7-unit steps of the first 
cycle  into a large (5-unit) step and a small (2-unit) step as you go 
around  the second cycle.
> 
> When you've completed the second cycle, you've completed the 
> blackjack scale.

Of course... thanks Paul...  that is an easy concept.  I wonder if 
I'd been able to figure it out if I'd continued to puzzle over it...

Well, anyway, you saved me some time... thanks!

_________ ______ ______
Joseph Pehrson


top of page bottom of page up

Previous Next

0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950

250 - 275 -

top of page