Tuning-Math messages 25 - 49

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Message: 25

Date: Thu, 24 May 2001 18:13:47

Subject: Re: Hypothesis

From: Paul Erlich

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

> 
> > > Usually the chromatic vectors are an octave and a twelfth.
> > 
> > Lost me there.
> 
> The top of the equation will look like
> 
> (1 0 0 ...)     (1 0 0 ...)
> (0 1 0 ...)H' = (0 1 0 ...)H'
> 
> So the prime axes 2 and 3 aren't being tempered out.

Still confused. Since when are octaves and twelfths "chromatic"?


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Message: 26

Date: Thu, 24 May 2001 19:29:20

Subject: Re: Fwd: optimizing octaves in MIRACLE scale..

From: Paul Erlich

Oh, Monz . . . you're not expecting the result to be a stretched or 
squashed 72-tET, are you? 'Cause if you are, then it's a one-
parameter optimization -- much easier.


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Message: 27

Date: Fri, 25 May 2001 16:42:23

Subject: Re: Fwd: optimizing octaves in MIRACLE scale..

From: monz

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

> I wrote,
> 
> > [monz]
> > But for sure, the 12-integer-limit is in _Harmonielehre_.
> 
> Really? So ratios such as 16:9 would have fallen outside it?


Paul, I started a response to this but it is getting
long and interesting.  I'll post it tonight.


-monz


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Message: 28

Date: Fri, 25 May 2001 18:25 +0

Subject: Re: Hypothesis

From: graham@m...

In-Reply-To: <9ejj0r+mrao@e...>
Paul wrote:

> > > > Usually the chromatic vectors are an octave and a twelfth.
> > > 
> > > Lost me there.
> > 
> > The top of the equation will look like
> > 
> > (1 0 0 ...)     (1 0 0 ...)
> > (0 1 0 ...)H' = (0 1 0 ...)H'
> > 
> > So the prime axes 2 and 3 aren't being tempered out.
> 
> Still confused. Since when are octaves and twelfths "chromatic"?

They're intervals used to specify the temperament that aren't being 
tempered out.  I thought this was the definition of "chromatic unison 
vector".

                   Graham


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Message: 29

Date: Fri, 25 May 2001 18:53:52

Subject: Re: Hypothesis

From: Paul Erlich

--- In tuning-math@y..., graham@m... wrote:
> In-Reply-To: <9ejj0r+mrao@e...>
> Paul wrote:
> 
> > > > > Usually the chromatic vectors are an octave and a twelfth.
> > > > 
> > > > Lost me there.
> > > 
> > > The top of the equation will look like
> > > 
> > > (1 0 0 ...)     (1 0 0 ...)
> > > (0 1 0 ...)H' = (0 1 0 ...)H'
> > > 
> > > So the prime axes 2 and 3 aren't being tempered out.
> > 
> > Still confused. Since when are octaves and twelfths "chromatic"?
> 
> They're intervals used to specify the temperament that aren't being 
> tempered out.  I thought this was the definition of "chromatic 
unison 
> vector".

A unison vector defines an equivalence relation in the lattice. Hence 
the name "unison". A periodicity block results from N independent 
unison vectors in an N-dimensional lattice. A twelfth or fifth as a 
unison vector would be really crazy, as far as I can tell.

In the diatonic scale, the commatic unison vector is 81:80, and the 
chromatic unison vector is 25:24 (or 135:128).

In the decatonic scale, the commatic unison vectors are any two of 
{225:224, 64:63, 50:49}; and the chromatic unison vector is any one 
of {49:49, 28:27, 25:24}.


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Message: 30

Date: Fri, 25 May 2001 18:59:39

Subject: Re: Hypothesis

From: Paul Erlich

I wrote,

> and the chromatic unison vector is any one 
> of {49:49

Oops -- should be 49:48, of course!


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Message: 31

Date: Fri, 25 May 2001 19:59:12

Subject: Re: Fwd: optimizing octaves in MIRACLE scale..

From: Paul Erlich

I wrote:

> Oh, Monz . . . you're not expecting the result to be a stretched or 
> squashed 72-tET, are you? 'Cause if you are, then it's a one-
> parameter optimization -- much easier.

And if it is, the answer is 71.959552-tET, or 72-tET with the octave 
stretched to 1200.6745¢.


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Message: 32

Date: Fri, 25 May 2001 22:36 +0

Subject: Re: More temperaments (from Tuning List)

From: graham@m...

Paul wrote:

> > Yes, the 11- and 15-limits are riddled with them.  There's even one 
> that 
> > divides the octave into 29 equal parts!  4.7 cents for the 15-
> limit, only 
> > needs 1 generator, needing 30 for a complete chord?  Mighty 
> strange, but 
> > it does mean you can use it with two manuals tuned to 29-equal a 
> generator 
> > apart.  As I've shown before that 29-equal works quite well mapped 
> to 
> > Halberstadt, this is far from arbitrary.
> 
> My proposal of two 12-equal keyboards 15 cents apart must look pretty 
> good in the 5-limit, yes?

Doesn't make the list, but only extremely accurate temperaments do in the 
5-limit.

I think the 29+ one should be more complex than that.  I'm calculating the 
number of steps for a chord as the number of generators to the most 
complex interval times the number of equivalence intervals plus one.  I 
think the addition should be done before the multiplication, but I think 
that makes the octave-splitting ones look worse than they really are.  
This particular one looks much better, though.  I'll try and sort it out 
in the morning.

I've changed the scoring which alters the 11 and 15-limit ordering.  This 
was to balance the 9-limit for simplicity rather than accuracy.

The program and output files should be at 
<Automatically generated temperaments *> by now.


            Graham


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Message: 33

Date: Fri, 25 May 2001 21:46:10

Subject: Re: More temperaments (from Tuning List)

From: Paul Erlich

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

> The program and output files should be at 
> <Automatically generated temperaments *> by now.
> 
Excuse me if I'm misunderstanding, but isn't your top 11-limit 
generator, "11/113", just the MIRACLE generator?


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Message: 34

Date: Fri, 25 May 2001 21:52:18

Subject: Re: More temperaments (from Tuning List)

From: Paul Erlich

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

> The program and output files should be at 
> <Automatically generated temperaments *> by now.

On that page it says,

"I've written a Python script to systematically find equal 
temperaments consistent within a given prime limit."

You mean odd limit, right?


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Message: 35

Date: Sat, 26 May 2001 09:47 +0

Subject: Re: More temperaments (from Tuning List)

From: graham@m...

Paul Erlich wrote:

> Excuse me if I'm misunderstanding, but isn't your top 11-limit 
> generator, "11/113", just the MIRACLE generator?

Yes, with the new scoring system.  It's also bottom of the 9-limit and 
around the middle of the 7-limit.

The scoring could still do with some tweaking, to ensure the best 
temperaments make the list.  The real challenge will be removing the 
octave invariance.


                   Graham


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Message: 36

Date: Sat, 26 May 2001 21:53 +0

Subject: Temperament program issues

From: graham@m...

I've half-updated the stuff at

<Automatically generated temperaments *>

mostly to clean up the code.  Also temperaments that too complex or 
inaccurate get left out, which stops the 5-limit being swamped by 
impractical temperaments.


On the question of the number of notes needed for a complete chord.  
Properly, for an equal number of generators to each equivalence interval, 
the formula should be (d+1)*n where d is the generator and n is the 
number of divisions of an octave.  For the strangest example (now top of 
the 15-limit list!) which needs 29 equivalence intervals and 1 generator, 
this gives 58 notes.

However, you get 29 different complete chords within those notes.  So the 
temperament is much more efficient than one with an equivalence interval 
of an octave that requires 58 notes to get a complete chord.

For that simple case, you get a new complete chord with each new note.  So 
the number of notes to a complete chord tells you how many you'll get with 
however many notes are in the MOS.  The equivalent measure where the 
equivalence interval is a division of the octave is d*n+1, which is what 
my program uses.

In some cases, like that 29 step octave, it gives somewhat eccentric 
results.  But I think a human looking at the data can use their 
intelligence to work this out, instead of expecting the dumb computer to 
produce a magic number that explains it all.

If you're curious, you could try looking through the 7- and 9-limit lists. 
 There are some interesting scales in there I haven't seen before, along 
with some old friends.

The magic connection between Miracle and diaschismic I had in mind is that 
they both have a generator of a semitone, approximating 16:15.  This is 
the same but different to the connection between schismic and diaschismic 
(single and double-positive).

For the following discussion I'll use "formal octave" for the interval we 
choose to stand in for an octave, and "equivalence interval" for the MOS 
parameter that will be a fraction of the formal octave.

To get the program to work for inharmonic timbres, you need to express all 
intervals in terms of the formal octave.  This shouldn't be difficult to 
do.  But how do you get the formal octave?  That is our next challenge.

One approach would be to extend Dave Keenan's brute force to try all 
generators and equivalence intervals.  This is probably too complex for a 
spreadsheet, but should be easy enough as a program.  The problem is that 
it'd need quadratic time to check all combinations.  For checking an 
octave to 0.1 cents precision, that's 144 million combinations.  Okay, 
less than that, because the generator doesn't have to be greater than the 
equivalence interval.  But we still have a lot of combinations.  It 
wouldn't be efficient, but it would work.  I don't think I'll try this, 
but somebody should.  I'll have a go at the more complex method below.


The first step is to get hold of all the ranges of consistent ETs for the 
intervals we're looking at.  Paul Erlich has already done this for the 
traditional limits.  The best way is probably brute force.  It'd take a 
while but nowhere near as long as brute forcing the linear temperaments.

I think the linear temperaments can still be expressed in step sizes for 
any pair of temperaments.  The challenge is getting at the formal octave. 
 Optimising the tuning and assessing the complexity will be harder 
otherwise.  Because schismic and Miracle temperaments can work with 
different equivalence intervals, there's obviously no unique way of 
getting an equivalence interval when you know the temperament.

The first step would be to express each temperament using an arbitrary 
formal octave, probably the first one in the list of prime intervals.  
Then we can use that specification to weed out duplicates for when we come 
to the hard work.  This would mean taking each consonant interval for a 
formal octave, and seeing which ones work best.

So, what makes a good formal octave?  It's tempting to say it should be a 
prime, rather than derived, interval.  But schismic works with a fourth 
and Miracle with a fifth, so that obviously won't work.  Perhaps we only 
need to check combinations of the first few prime intervals, assuming them 
to be the most important.  But can we be sure this is always sufficient?

A post on the practical microtonality list (I don't have the reference, 
sorry) did suggest deliberately choosing poor consonances for formal 
octaves.  So we will probably have to have a lot of freedom in how the 
choice is made.

Other factors are that the scale should have a "nice" number and pattern 
of notes to its formal octave.  I'm not sure how to express this for an 
open ended temperament.


There we go, ideas are welcome.


                  Graham


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Message: 37

Date: Sat, 26 May 2001 21:44:40

Subject: Re: More temperaments (from Tuning List)

From: Dave Keenan

--- In tuning-math@y..., graham@m... wrote:
> The scoring could still do with some tweaking, to ensure the best 
> temperaments make the list.  The real challenge will be removing the 
> octave invariance.

What do you mean by this? Do you mean you want to find MA errors 
at an N+1 integer limit instead of N odd limit when the octave is 
tempered too?


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Message: 38

Date: Sat, 26 May 2001 22:49:34

Subject: Re: Temperament program issues

From: Dave Keenan

--- In tuning-math@y..., graham@m... wrote:
> On the question of the number of notes needed for a complete chord.
...

Ah yes. I see your point. So how about simply using n*d, the number of 
generators in a complete chord. Its a more useful number. If you want 
to know how many complete otonalities you get for N notes, it's simply 
N - n*d.

> If you're curious, you could try looking through the 7- and 9-limit 
lists. 
>  There are some interesting scales in there I haven't seen before, 
along 
> with some old friends.

It would be a lot easier to recognise any old friends if you had your 
program express the basis as "N chains of G cent generators".

> For the following discussion I'll use "formal octave" for the 
interval we 
> choose to stand in for an octave, and "equivalence interval" for the 
MOS 
> parameter that will be a fraction of the formal octave.

This is too confusing. How about we use the term "period" instead of 
"equivalence interval". If you thing that might be confused with a 
time period you could call it the "log period", but notice that we 
already use it in this way, without the "log" qualifier, in the term 
"periodicity block". 

> To get the program to work for inharmonic timbres, you need to 
express all 
> intervals in terms of the formal octave.  This shouldn't be 
difficult to 
> do.  But how do you get the formal octave?  That is our next 
challenge.

I'm personally not very interested in that completely general case 
just yet. Most inharmonic timbres are short lived and so coinciding 
partials isn't all that important.

I'd be more interested in the cases where the period is a 
whole-number fraction of another highly consonant (harmonic) interval, 
such as the BP 1:3, or 1:4, 2:3, 1:5, 1:6, 2:5, 3:4 etc.

And even more interested in the case where we stick to fractions of a 
real octave but allow the octave to be tempered and use N+1 integer 
limits in place of N odd-limits.

-- Dave Keenan


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Message: 39

Date: Sun, 27 May 2001 06:40:57

Subject: Re: Temperament program issues

From: Dave Keenan

--- In tuning-math@y..., graham@m... wrote:
> Okay, 
> less than that, because the generator doesn't have to be greater 
than the 
> equivalence interval.

The generator doesn't have to be greater than _half_ the period 
(equivalence interval). But yes, it's still O(n^2).


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Message: 40

Date: Sun, 27 May 2001 09:00:05

Subject: Re: Fwd: optimizing octaves in MIRACLE scale..

From: monz

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

Yahoo groups: /tuning-math/message/31 *

> I wrote:
> 
> > Oh, Monz . . . you're not expecting the result to be a
> > stretched or squashed 72-tET, are you? 'Cause if you are,
> > then it's a one-parameter optimization -- much easier.
> 
> And if it is, the answer is 71.959552-tET, or 72-tET with
> the octave stretched to 1200.6745¢.


With a step-size of 16.67603472 cents.

Thanks, Paul.  Uh... I don't think "expecting" is the way
I'd say it, but yes, I *was* *guessing* that it would be
a stretched 72-EDO.

But I'm unclear on why my expectation would have any effect
on the type of optimization.  ...?


Also, on the asking of what are probably elementary questions
like this to the rest of you on this list: is it OK for me to
ask questions like this here?  Or will it be perceived as a
nuisance to those of you who are ready to discuss nitty-gritty
tuning math?  I know that Paul is generous with his help, and so
I can keep this stuff relegated to private email if others prefer.

My hope is that the Tuning-math List can be a place for people
of all mathematical levels to be able discuss aspects of this
subject, but I would perfectly understand if most subscribers
want to keep discussion on a high level.


-monz
Yahoo! GeoCities *
"All roads lead to n^0"


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Message: 41

Date: Sun, 27 May 2001 13:27 +0

Subject: Re: Fwd: optimizing octaves in MIRACLE scale..

From: graham@m...

monz wrote:


> My hope is that the Tuning-math List can be a place for people
> of all mathematical levels to be able discuss aspects of this
> subject, but I would perfectly understand if most subscribers
> want to keep discussion on a high level.


I agree (with the first bit)

            Graham


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Message: 42

Date: Sun, 27 May 2001 13:28 +0

Subject: Re: More temperaments

From: graham@m...

Dave Keenan wrote:

> --- In tuning-math@y..., graham@m... wrote:
> > The scoring could still do with some tweaking, to ensure the best 
> > temperaments make the list.  The real challenge will be removing the 
> > octave invariance.
> 
> What do you mean by this? Do you mean you want to find MA errors 
> at an N+1 integer limit instead of N odd limit when the octave is 
> tempered too?

I mean taking an arbitrary set of intervals and finding some generators 
and periods that fit them.


              Graham


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Message: 43

Date: Sun, 27 May 2001 13:28 +0

Subject: Re: Temperament program issues

From: graham@m...

Dave Keenan wrote:

> --- In tuning-math@y..., graham@m... wrote:
> > On the question of the number of notes needed for a complete chord.
> ...
> 
> Ah yes. I see your point. So how about simply using n*d, the number of 
> generators in a complete chord. Its a more useful number. If you want 
> to know how many complete otonalities you get for N notes, it's simply 
> N - n*d.

Equally useful, they're only 1 note apart.  Originally I was using n*d for 
the scoring anyway.

> > If you're curious, you could try looking through the 7- and 9-limit 
> lists. 
> >  There are some interesting scales in there I haven't seen before, 
> along 
> > with some old friends.
> 
> It would be a lot easier to recognise any old friends if you had your 
> program express the basis as "N chains of G cent generators".

Uh, yeah, sometime.  You'll have to convert from octaves for now, or hack 
the code yourself (Python's really easy to learn, you can download the 
interpreter)

> > For the following discussion I'll use "formal octave" for the 
> interval we 
> > choose to stand in for an octave, and "equivalence interval" for the 
> MOS 
> > parameter that will be a fraction of the formal octave.
> 
> This is too confusing. How about we use the term "period" instead of 
> "equivalence interval". If you thing that might be confused with a 
> time period you could call it the "log period", but notice that we 
> already use it in this way, without the "log" qualifier, in the term 
> "periodicity block". 

Okay.

> > To get the program to work for inharmonic timbres, you need to 
> express all 
> > intervals in terms of the formal octave.  This shouldn't be 
> difficult to 
> > do.  But how do you get the formal octave?  That is our next 
> challenge.
> 
> I'm personally not very interested in that completely general case 
> just yet. Most inharmonic timbres are short lived and so coinciding 
> partials isn't all that important.

How about bells?  My wok makes some good sound if I thwack it right as 
well.

I don't think the harmonic series has many more surprises for us.  It was 
worth a look to make sure we hadn't overlooked any Miracles, but I think 
the program already shows that.  The 15-limit temperaments are too complex 
for me right now, and there's no point in finding them again.

You could try it with different sets of intervals, but that's really easy, 
no more programming.

One thing is it might be good for the program to recognize scales that 
approximate a subset of the intervals you gave it.

Thinking about it, for the non-octave case, we don't even need a 
brute-force search for ETs.  So long as we only take consistent ones, we 
can choose an arbitrary formal octave and do the same search as currently. 
 It may be best to choose a large interval for that formal octave, as 
there may be more than one consistent ET that divides a small interval 
into the same number of steps.

> I'd be more interested in the cases where the period is a 
> whole-number fraction of another highly consonant (harmonic) interval, 
> such as the BP 1:3, or 1:4, 2:3, 1:5, 1:6, 2:5, 3:4 etc.

So long as you choose the interval first, that should be easy.  You need 
to specify each prime direction in terms of that interval.  Something 
like:

newprimes = []
for interval in primes:
  newprimes.append(newprimes/pitchOfFormalOctave)

where primes is a list of the intervals in terms of octaves (current 
standard) and pitchOfFormalOctave is the pitch of this interval you chose 
in octaves (log2(3) for 3:1).


> And even more interested in the case where we stick to fractions of a 
> real octave but allow the octave to be tempered and use N+1 integer 
> limits in place of N odd-limits.

I don't really care what happens after the linear temperament's found.  
The program can be hacked for that easily enough.  If you have an 
algorithm for the optimization, it can be added as a method like 
optimizeMinimax.


            Graham


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Message: 44

Date: Sun, 27 May 2001 17:55:20

Subject: Re: Fwd: optimizing octaves in MIRACLE scale..

From: monz

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

Yahoo groups: /tuning-math/message/24 *

> I wrote,
> 
> > > hence as 33/32 and 13/12 -- differed by virtually an entire
> > > semitone (i.e., Schoenberg assumed a "unison vector" of 
143:128).
> 
> Oops! That should be 104:99, not 143:128!
> 
> > But for sure, the 12-integer-limit is in _Harmonielehre_.
> 
> Really? So ratios such as 16:9 would have fallen outside it?


(early response:)

Oops... Schoenberg doesn't actually claim that the 12th
harmonic is any kind of limit... it's simply where his
musical illustration and its accompanying explanation end.
I suppose he implies that it continues beyond into
inaudibility.

The musical illustration uses the 1st thru 12th harmonics
on F, C, and G.  So using a 12-integer-limit here would
relate it to Schoenberg's illustration, but not necessarily
to his actual theory.

In the later article, "Problems of Harmony", which BTW
was written in 1927 then revised in 1934 for presentation
in America, Schoenberg definitely explains harmony as
being based on a 13-integer-limit as harmonics 1 thru 13
on F, C, and G.

I would label this system as (1...13)/(3^(-1...1)).

Is there a better notation for that?

Still later, in _Structural Functions of Harmony_ [1949],
his "Chart of the Regions" (2 versions, in major and minor)
uses terms such as "mediant" which imply more extended
5-limit derivations for some notes than the ratios implied
by the overtone model.


====

That was my first response to this.

I was going to concede to Paul that I had been in error,
and to some extent I *was*, but guess what?... The scale
of approximated ratios implied by Schoenberg's diagram
provides only one 16:9!, between d-27 and c-48.

There are 3 other varieties of "minor 7th":

  11:6 (really a "neutral 7th") between d-54 and c-99,

  9:5 between b-45 and a-80, and

  7:4 between g-36 and f-63.


I was getting concerned that this thread was veering
off-topic, but this gives me the opportunity to remedy
that situation.   :)

(My quotes of Schoenberg are from the English translation
of _Harmonielehre_ by Roy Carter, and the page numbers
refer to that edition.)


Schoenberg [p 23] posits the existences of two "forces", one
pulling downward and one pulling upward around the tonic,
which he illustrates as: F <- C -> G and likens to resistance
against gravity.  In mathematical terms, he is referring to
the harmonic relationships of 3^-1 and 3^1, respectively.

> [Schoenberg, p 24:]
>
> ...thus it is explained how the scale that finally emerged
> is put together from the most important components of a
> fundamental tone and its nearest relatives.  These nearest
> relatives are just what gives the fundamental tone stability;
> for it represents the point of balance between their opposing
> tendencies.  This scale appears as the residue of the properties
> of the three factors, as a vertical projection, as addition:


Schoenberg then presents a diagram of the overtones and the
resulting scale, which I have adaptated, adding the partial-numbers
which relate all the overtones together as a single set:

             b-45
             g-36
       e-30
             d-27
       c-24
 a-20
       g-18  g-18
 f-16
 c-12  c-12
 f-8


 f   c   g   a   d   e   b
 8  12  18  20  27  30  45


> [Schoenberg:]
>
> Adding up the overtones (omitting repetitions) we get the seven
> tones of our scale.  Here they are not yet arranged consecutively.
> But even the scalar order can be obtained if we assume that the
> further overtones are also in effect.  And that assumption is
> in fact not optional; we must assume the presence of the other
> overtones.  The ear could also have defined the relative pitch 
> of the tones discovered by comparing them with taut strings,
> which of course become longer or shorter as the tone is lowered
> or raised.  But the more distant overtones were also a
> dependable guide.  Adding these we get the following:



Schoenberg then extends the diagram to include the
following overtones:

 fundamental  partials

     F         2...12, 16
     C         2...11
     G         2...12

(Note, therefore, that he is not systematic in his employment
of the various partials.)


Again, I adapt the diagram by adding partial-numbers:

               d-108
               c-99
               b-90
               a-81
               g-72
        f-66
 f-64
              (f-63)
        e-60
        d-54   d-54
 c-48   c-48
               b-45
 b-44
       (bb-42)
 a-40
 g-36   g-36   g-36
 f-32
        e-30
(eb-28)
               d-27
 c-24   c-24
 a-20
        g-18   g-18
 f-16
 c-12   c-12
 f-8


       (eb)            (bb)
 c   d   e   f   g   a   b   c   d   e   f   g   a   b   c   d
                       [44]            [64]
       (28)            (42)            [66]
24  27  30  32  36  40  45  48  54  60  63  72  81  90  99 108


(Note also that Schoenberg was unsystematic in his naming
of the nearly-1/4-tone 11th partials, calling 11th/F by the
higher of its nearest 12-EDO relatives, "b", while calling
11th/C and 11th/G by the lower, "f" and "c" respectively.
This, ironically, is the reverse of the actual proximity
of these overtones to 12-EDO: ~10.49362941, ~5.513179424,
and ~0.532729432 Semitones, respectively).


The partial-numbers are also given for the resulting scale
at the bottom of the diagram, showing that 7th/F (= eb-28)
is weaker than 5th/C (= e-30), and 7th/C (= bb-42) is weaker
than 5th/G (= b-45).

Also note that 11th/F (= b-44), 16th/F (= f-64) and 11th/C
(= f-66) are all weaker still, thus I have included them in
square brackets.  These overtones are not even mentioned by
Schoenberg.


Schoenberg does take note of the ambiguity present in this
collection of ratios, in his later article _Problems of Harmony_.
I won't go into that here because this is focusing on his
1911 theory.


Here is an interval matrix of Schoenberg's scale
(broken in half to fit the screen), with implied
proportions given along the left and the bottom,
and Semitone values of the intervals in the body.

Because Schoenberg's implied proportions form an
"octave"-specific pitch-set in his presentation
(not necessarily in his theory), this matrix has
no "bottom" half.


Interval Matrix of Schoenberg's implied JI scale:

108 26.04 24.00 23.37 22.18 21.06 19.02 17.20 16.35 15.55 15.16 14.04
 99 24.53 22.49 21.86 20.67 19.55 17.51 15.69 14.84 14.04 13.65 12.53
 90 22.88 20.84 20.21 19.02 17.90 15.86 14.04 13.19 12.39 12.00 10.88
 81 21.06 19.02 18.39 17.20 16.08 14.04 12.22 11.37 10.57 10.18  9.06
 72 19.02 16.98 16.35 15.16 14.04 12.00 10.18  9.33  8.53  8.14  7.02
 66 17.51 15.47 14.84 13.65 12.53 10.49  8.67  7.82  7.02  6.63  5.51
 64 16.98 14.94 14.31 13.12 12.00  9.96  8.14  7.29  6.49  6.10  4.98
 63 16.71 14.67 14.04 12.84 11.73  9.69  7.86  7.02  6.21  5.83  4.71
 60 15.86 13.82 13.19 12.00 10.88  8.84  7.02  6.17  5.37  4.98  3.86
 54 14.04 12.00 11.37 10.18  9.06  7.02  5.20  4.35  3.55  3.16  2.04
 48 12.00  9.96  9.33  8.14  7.02  4.98  3.16  2.31  1.51  1.12  0.00
 45 10.88  8.84  8.21  7.02  5.90  3.86  2.04  1.19  0.39  0.00 
 44 10.49  8.45  7.82  6.63  5.51  3.47  1.65  0.81  0.00  
 42  9.69  7.65  7.02  5.83  4.71  2.67  0.84  0.00   
 40  8.84  6.80  6.17  4.98  3.86  1.82  0.00    
 36  7.02  4.98  4.35  3.16  2.04  0.00      
 32  4.98  2.94  2.31  1.12  0.00      
 30  3.86  1.82  1.19  0.00       
 28  2.67  0.63  0.00        
 27  2.04  0.00         
 24  0.00          
      24    27    28    30    32    36    40    42    44    45    48


---

108 12.00 10.18  9.33  9.06  8.53  7.02  4.98  3.16  1.51  0.00
 99 10.49  8.67  7.82  7.55  7.02  5.51  3.47  1.65  0.00 
 90  8.84  7.02  6.17  5.90  5.37  3.86  1.82  0.00  
 81  7.02  5.20  4.35  4.08  3.55  2.04  0.00   
 72  4.98  3.16  2.31  2.04  1.51  0.00    
 66  3.47  1.65  0.81  0.53  0.00     
 64  2.94  1.12  0.27  0.00     	
 63  2.67  0.84  0.00        
 60  1.82  0.00        
 54  0.00          
      54    60    63    64    66    72    81    90    99    108



-monz
Yahoo! GeoCities *
"All roads lead to n^0"


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Message: 45

Date: Mon, 28 May 2001 03:56:34

Subject: Re: Temperament program issues

From: Dave Keenan

--- In tuning-math@y..., graham@m... wrote:
> Dave Keenan wrote:
> > It would be a lot easier to recognise any old friends if you had 
your 
> > program express the basis as "N chains of G cent generators".
> 
> Uh, yeah, sometime.  You'll have to convert from octaves for now, or 
hack 
> the code yourself (Python's really easy to learn, you can download 
the 
> interpreter)

I'm getting too old for that. I've had to learn so many new 
programming languages (and GUI libraries) in my life that I'm heartily 
sick of it.

I had a hard time recognising Paultone among your top-10 7-limit 
generators.

I think there's a bug in your MA optimiser. In the case where one or 
more intervals is purely a multiple of the period (zero generators) 
you need to either give a _range_ for the optimum generator, or 
preferably eliminate the zero-generator intervals from the 
optimisation.

If you do that you should get 109.36 cents (not 111.04) for Paultone.

> I don't really care what happens after the linear temperament's 
found.  
> The program can be hacked for that easily enough.  If you have an 
> algorithm for the optimization, it can be added as a method like 
> optimizeMinimax.

Not by me. How about giving us the top-tens ranked according to RMS 
error times (n*d)^2, generators in cents.

-- Dave Keenan


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Message: 46

Date: Mon, 28 May 2001 04:20:07

Subject: Re: Fwd: optimizing octaves in MIRACLE scale..

From: Dave Keenan

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> There are 3 other varieties of "minor 7th":
> 
>   11:6 (really a "neutral 7th") between d-54 and c-99,
> 
>   9:5 between b-45 and a-80, and
> 
>   7:4 between g-36 and f-63.

    7:4 (really a subminor 7th) between g-36 and f-63
:-)


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Message: 47

Date: Mon, 28 May 2001 05:01:00

Subject: Re: Temperament program issues

From: Dave Keenan

Doesn't the fact that meantone, (the single most popular 5-limit 
temperament of all time), doesn't even make the top-ten, mean that 
there is something very wrong with our figure of demerit?


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Message: 48

Date: Mon, 28 May 2001 08:56 +0

Subject: Re: Temperament program issues

From: graham@m...

D.KEENAN@U... (Dave Keenan) wrote:

> Doesn't the fact that meantone, (the single most popular 5-limit 
> temperament of all time), doesn't even make the top-ten, mean that 
> there is something very wrong with our figure of demerit?
> 

You're out of date.  It's this one:


66/157

basis:
(1.0, 0.419517976278)

mapping:
([1, 0], ([2, 4], [-1, -4]))
                   ^^^^^^

primeApprox:
([88, 69], [(139, 109), (204, 160)])

highest interval width: 4
notes required: 5
highest error: 0.004480  (5.377 cents)


The list I point to is the easiest way of showing that it's meantone.  A 
third is four fifths.

The figure of demerit isn't perfect but, now the really useless 
temperaments are excluded, it's good enough to bring the interesting ones 
into the 10.  You can change that to 20 if you don't trust it.


                  Graham


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Message: 49

Date: Mon, 28 May 2001 09:08:59

Subject: Re: Temperament program issues

From: Paul Erlich

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

> The magic connection between Miracle and diaschismic I had in mind 
is that 
> they both have a generator of a semitone, approximating 16:15.

Well, I usually think of diaschismic as having a ~3:2 generator and a 
half-octave "by-generator". But yes, replacing the ~3:2 with an 
approximate 16:15 is equivalent.

> This is 
> the same but different to the connection between schismic and 
diaschismic 
> (single and double-positive).

The same but different . . . pretty cryptic!
> 
> For the following discussion I'll use "formal octave" for the 
interval we 
> choose to stand in for an octave, and "equivalence interval" for 
the MOS 
> parameter that will be a fraction of the formal octave.
> 
> To get the program to work for inharmonic timbres, you need to 
express all 
> intervals in terms of the formal octave.  This shouldn't be 
difficult to 
> do.  But how do you get the formal octave?  That is our next 
challenge.

There's only one interval that sounds like a repetition of the same 
pitch to humans and other mammals. It's pretty close to 2:1.

> 
> I think the linear temperaments can still be expressed in step 
sizes for 
> any pair of temperaments.  The challenge is getting at the formal 
octave. 

Still?

>  Optimising the tuning and assessing the complexity will be harder 
> otherwise.  Because schismic and Miracle temperaments can work with 
> different equivalence intervals,

Really?

> there's obviously no unique way of 
> getting an equivalence interval when you know the temperament.
> 
> The first step would be to express each temperament using an 
arbitrary 
> formal octave, probably the first one in the list of prime 
intervals.  
> Then we can use that specification to weed out duplicates for when 
we come 
> to the hard work.  This would mean taking each consonant interval 
for a 
> formal octave, and seeing which ones work best.

You've lost me.


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