Tuning-Math messages 251 - 275

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: 251

Date: Tue, 19 Jun 2001 20:09:50

Subject: Re: Stretched tuning experiments (was: First melodic spring results)

From: Paul Erlich

--- In tuning-math@y..., "John A. deLaubenfels" <jdl@a...> wrote:
> [Paul wrote:]
> >John, what did you think of Herman's "1/7-comma meantone with 1/7-
> >comma-stretched octaves" version, as compared with, say, 55-tET?
> 
> Pulling them up right now!  Stretched 1/7 comma meantone:

Oops -- Herman should really clarify -- this is not stretched 1/7-
comma meantone, but rather a tuning where the fifths are 1/7-comma 
narrow and the octaves are 1/7-comma wide.


top of page bottom of page up down


Message: 255

Date: Wed, 20 Jun 2001 03:10:27

Subject: 7/72 generator in blackjack

From: jpehrson@r...

Well, this isn't very advanced... but, if not math, at least it's 
arithmetic...

I still don't understand how 7 of the 72-tET scale is a generator of 
blackjack.  It's a great concept  (spooky!) since we have been 
finding 7's to be very peculiar in some other instances... 

Would someone please go over that again, gently??

Thanks!

Joseph


top of page bottom of page up down


Message: 256

Date: Tue, 19 Jun 2001 21:50:44

Subject: Re: 7/72 generator in blackjack

From: monz

> ----- Original Message ----- 
> From: <jpehrson@r...>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Tuesday, June 19, 2001 8:10 PM
>Subject: [tuning-math] 7/72 generator in blackjack
>
>
> Well, this isn't very advanced... but, if not math, at least it's 
> arithmetic...

That's OK, Joe... this list is for math dummies like me, too,
as well as guys like Paul, Dave, and Graham who understand the
more esoteric stuff.

> 
> I still don't understand how 7 of the 72-tET scale is a generator of 
> blackjack.  It's a great concept  (spooky!) since we have been 
> finding 7's to be very peculiar in some other instances... 
> 
> Would someone please go over that again, gently??


Take another look at the explanation and especially the diagram
below the graph at
Definitions of tuning terms: MIRACLE scale, (c) 1998 by Joe Monzo *


Dave Keenan found the MIRACLE generator (~116.7 cents) by use
of the "brute force" approach: he had his computer perform
thousands (millions?.. billions?) of calculations and analyze
the resulting scales.

The ~116.7-cent generator came out on top as implying the
largest number of 11-limit consonances.  2^(7/72) happens
to be extremely close to the calculated MIRACLE generator
(which, I should emphasize, is only *one* possible MIRACLE
generator... there can be many, depending on the error method
selected).

The diagram on my Dictionary page shows how you cycle thru
intervals of 2^(7/72) on either side of 1/1, which in this
case really should be called 2^(0/72).  Upon reaching the
10th note on either side, you've got Blackjack.  Extending
to the 15th note on either side gives you Canasta.

This process is exactly analagous to constructing a meantone
cycle, except that instead of a "cycle of 5ths", you're
cycling thru the generator interval, whatever it may be.


An interesting digression: some scales can be thought of as
being constructed by more than one generator simultaneously.
Naturally, our familiar old 12-EDO is one such scale.  It
can be thought of as a "cycle of 5ths" where each "5th" is
7 Semitones; here's an example centered on "D" (flats and
sharps are of course equivalent to their enharmonic cousins):

 Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C# - G#
-6   -5   -4   -3  -2  -1   0   1   2   3   4    5    6

Or it can be thought of as a cycle of Semitones:

Ab - A - Bb - B - C - C# - D - Eb - E - F - F# - G - G#
-6  -5  -4   -3  -2   -1   0    1   2   3   4    5   6


In either case, the generator creates a scale of 12 distinct
pitches before producing a pitch which is an exact replica
of one already existing.



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


 


_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at Yahoo! Mail Setup *


top of page bottom of page up down


Message: 257

Date: Wed, 20 Jun 2001 06:10:54

Subject: Re: CS

From: carl@l...

>>> The Consonant-CS thing makes sense, but for the more general
>>> case I think you can leave out the computational complexity of
>>> Harmonic Entropy and just say that a scale is "CS to a tolerance
>>> of X cents".
> > 
> > Well, I suppose it would be the same difference as rating an ET
> > by max cents error, or something like Erlich's accuracy.
> 
> Not really. In the case of a CS tolerance, more cents is better.

Yes, but a scale with many (bad errors, near collisions) may have
the same (max error, min tolerance) as a scale with only one (error,
near collision).

> The probability of the average listener perceiving as the same, two 
> pitches differing by some amount. It's from that dude's experiment 
> that Paul quotes for HE. 1% change in frequency ~= 17 cents.

My goodness!  I don't think we want Goldstein's 1% bit involved here.

> i.e. ... of the scale's smallest step. Yes, that would be useful to
> know, as well as the absolute tolerance. But as a figure of merit
> in comparing improper MOS, the absolute tolerance seems more
> important to me. Can you explain further. Maybe with an example.

Well, I'd argue that a near-collision of 10 cents would not be as
important in an otherwise-even pentatonic as it would in a decatonic.


I'll note here that non-CS may be desirable.  As I once said,
'ambiguous intervals [collisions] are the common tones of melodic
modulation'.

-Carl


top of page bottom of page up down


Message: 258

Date: Wed, 20 Jun 2001 06:14:39

Subject: Re: 7/72 generator in blackjack

From: Dave Keenan

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

As a complement to Monz's explanation, see what I just added to
http://uq.net.au/~zzdkeena/Music/MiraclePitchChart.gif - Ok *

It shows the octave as a circle divided into 72 parts. Start at D> and 
follow the (new) straight line segments clockwise. You'll see each one 
jumps 7/72 of an octave. When you've done that 20 times and wound up 
at D<, you've generated Blackjack.

> Dave Keenan found the MIRACLE generator (~116.7 cents) by use
> of the "brute force" approach: he had his computer perform
> thousands (millions?.. billions?) of calculations and analyze
> the resulting scales.
> 
> The ~116.7-cent generator came out on top as implying the
> largest number of 11-limit consonances. ...

Er, no. That all came _after_ the discovery, and merely confirmed its 
"miraculous" nature (as a 7-limit or 11-limit generator, but not necc. 
9-limit). I was afraid there might have been some holes in my search 
strategy, but since then Graham Breed has performed a search using a 
completely different method to mine, and (I think?) further confirmed 
it.

Strictly speaking, the MIRACLE generator was discovered by Paul 
Erlich, who extracted it from a scale that I posted, the 31-noter that 
we now call Canasta. Paul then recognised that there was a more 
manageable (although improper) MOS with 21 notes using the same 
generator. So historically it went: Canasta - MIRACLE generator - 
Blackjack - Decimal scale (although, apart from "Blackjack" we didn't 
call them that immediately). But logically it goes: MIRACLE generator 
- Decimal scale - Blackjack - Canasta.

So Graham,

By what figure-of-demerit and at what odd-limits can we claim that the 
MIRACLE generator is the best?

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?

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

Regards,
-- Dave Keenan


top of page bottom of page up down


Message: 259

Date: Wed, 20 Jun 2001 02:43:52

Subject: Decatonic (in 72-tET)

From: Pierre Lamothe

I wish I would have prepared lattice drawings and full comments to publish
with these decatonic structures ib1183 and ib1215

<decatonic *>

but I have to quit for few weeks or months.

Since these structures are closed related to temperament 72, I think
someone could be interested, even without any comment. (I used a scale with
36 divisions but it is easy to detect points on divisions and points
in-between). 

The ib1183 structure is the minimal one corresponding to the Paul Erlich's
decatonic scales, as explained in "For all strict-JI fans . . ." at

Yahoo groups: /tuning/message/20746 *

and the ib1215 is the minimal decatonic in non-degenerated 11-limit.

-----

Since Monzo has opened a door about complexity notion, I would add soon a
new definition about complexity in a next message, before to quit.


Pierre


top of page bottom of page up down


Message: 262

Date: Wed, 20 Jun 2001 17:09:16

Subject: Challenging problem about sonance

From: Pierre Lamothe

Since my abstract thoughts seems not to be well appreciated, here is a
"concrete" challenging problem having goal only to introduce my definition
of _Sonance degree_ in my next post. If I have no feedback, I could so have
illusion that it's not only by lack of interest :-)

Could you calculate an approximative "sonance" value for any interval
represented by a coordinate vector

   (x0 x1 .. xN)*

in a primal basis

   <p0 p1 .. pN>

where only p0 is known, say as the prime 2? 

As such that question would have no sense, but with a given coherent set of
unison vectors defining a system, this problem has a deep sense : the first
coordinate (the power of 2) retains IN A COHERENT SYSTEM sufficient
information to obtain this well-approximated value for the "sonance".

More concretely. Let U = {(-4 4 -1)*,(-3 -1 2)*} a such set of unison
vectors defined by its coordinates. Without using the fact that these
vectors would represent 81/80 and 25/24 in <2 3 5>, it is possible to
easily calculate a numerical value which is the _sonance degree_ in this
system for any interval represented by (x y z)*, say (-1 1 0)* and (-3 1
1)* which would be 3/2 and 10/9 in <2 3 5>.

My definition of _Sonance degree_ simply generalises that.


Pierre


top of page bottom of page up down


Message: 263

Date: Thu, 21 Jun 2001 01:54:56

Subject: Re: Stretched tuning experiments (was: First melodic spring results)

From: Dave Keenan

--- In tuning-math@y..., Herman Miller <hmiller@I...> wrote:
> On Tue, 19 Jun 2001 14:48:00 -0600, "John A. deLaubenfels"
> >Oh, I meant that, despite fewer words.  Although, I see that Herman
> >is saying "1/7-comma meantone with 1/7-comma tempered octaves"; 
perhaps
> >if the word "tempered" were changed to "stretched", it'd be 
clearer...
> 
> Done.

No. I think maybe John still doesn't get it. It's definitely "with 
octaves tempered 1/7-comma wide" and _not_ "with a stretch of 1/7 
comma per octave". These are very different things. 

A "stretch" is in fact applied to _all_ intervals, not merely the 
octave. However it is usually specified as so many cents per octave. 
Its purpose is usually to compensate for a stretched inharmonic timbre 
or a property of human pitch perception.

A meantone with a tempered octave means that for a given note we not 
only need to know where it is on the chain of fifths, but how many 
octaves one has to reduce its stacked fifths by, to bring it back to 
the "home" octave. Herman's scale gives the optimum distributiion of 
the syntonic comma for the intervals 1:2, 2:3 and 4:5. It essentially 
doesn't give a damn about other 5-limit intervals like 3:4, 5:6, 3:5, 
5:8.

Dan Stearns, are you reading? This is an excellent example of a tuning 
that optimises only the rooted intervals (in this case including the 
octave), giving them all equal weight.

Here are the three scales to the nearest cent
1. Ordinary 1/7 comma meantone
2. 1/7 comma meantone with a stretch of 1/7 comma per octave
3. 1/7 comma meantone with octaves tempered 1/7 comma wide.

Name	Octaves	Fifths	1/7     with    with 
                        comma   stretch tempered octaves
-----------------------------------------------
C	0	0	0	0	0
C#	-4	7	92	92	80
D	-1	2	198	198	195
Eb	2	-3	303	304	309
E	-2	4	396	397	389
F	1	-1	501	502	504
F#	-3	6	593	595	584
G	0	1	699	701	699
G#	-4	8	791	793	779
A	-1	3	897	899	894
Bb	2	-2	1002	1005	1008
B	-2	5	1094	1097	1088
C	1	0	1200	1203	1203

Look at E. Notice how stretch makes the major thirds (4:5 = 386c) 
slightly worse, but wide-tempered-octaves makes them significantly 
better.

Regards,
-- Dave Keenan


top of page bottom of page up down


Message: 264

Date: Thu, 21 Jun 2001 02:04:13

Subject: Re: 7/72 generator in blackjack

From: jpehrson@r...

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

Yahoo groups: /tuning-math/message/256 *

> 
> Take another look at the explanation and especially the diagram
> below the graph at
> Definitions of tuning terms: MIRACLE scale, (c) 1998 by Joe Monzo *
> 
> 
> Dave Keenan found the MIRACLE generator (~116.7 cents) by use
> of the "brute force" approach: he had his computer perform
> thousands (millions?.. billions?) of calculations and analyze
> the resulting scales.
> 
> The ~116.7-cent generator came out on top as implying the
> largest number of 11-limit consonances.  2^(7/72) happens
> to be extremely close to the calculated MIRACLE generator
> (which, I should emphasize, is only *one* possible MIRACLE
> generator... there can be many, depending on the error method
> selected).
> 

Hello Monz!

Thank you so much for this interesting response.  Well, I read ahead, 
and I guess Dave Keenan credits Paul Erlich with the discovery of the 
MIRACLE generator.  I guess that's how I remember it, too, looking 
back at the chain (literally!) of events...

Well, from what you are saying, then, 31-EDO at with a 116.1 
generator is also a MIRACLE scale, yes... of a sort??  Of course, I 
had always heard of the "special" properties of 31-EDO...

Then, it is not too coincidental that a 31-tone NON-EDO using 116.7 
would be "miraculous" as well... I mean it's not so miraculous that 
both scales would have 31 notes, correct??

I found your "MIRACLE GENERATOR" page fascinating.  I guess I really 
hadn't read that one as carefully as the page that pertained 
PARTICULARLY to Blackjack...

I remember when Paul Erlich did the calculus to figure out the "RMS" 
or root mean square method to find the errors.

Although I can't specifically follow this in the entire, it seems 
rather related to a discussion that I had with Graham Breed and John 
deLaubenfels about finding "errors" by squaring things and then 
taking the square root...

Is that correct?  It looks as though Paul, in his calculations, is 
trying to find the least errors for all the various intervals he is 
considering, in squaring them and so forth, and then puts it all 
nicely back in a pie (not a pie chart!) to determine the MIRACLE 
generator at 116.7 cents.

Am I getting that at all...??

ANYWAY, I also very much appreciated the 72-EDO chart at the bottom 
of the MIRACLE generator page... That REALLY did a good job of 
breaking down how the various scales are related to the generator...

Well, the chart with the colors for the various scales did too...

It seems I'm gradually coming to a greater appreciation of this 
process, finally... although I have to admit I was rather "left in 
the dust" when it happened.

(In fact, I became so confused, that Paul thought I had forgotten the 
entire discussion!)

Well, anyway, thanks for the help!

_________ _______ _________
Joseph Pehrson


top of page bottom of page up down


Message: 265

Date: Thu, 21 Jun 2001 02:09:00

Subject: Re: 7/72 generator in blackjack

From: jpehrson@r...

--- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

Yahoo groups: /tuning-math/message/258 *


> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> Joseph,
> 
> As a complement to Monz's explanation, see what I just added to
> http://uq.net.au/~zzdkeena/Music/MiraclePitchChart.gif - Ok *
> 
> It shows the octave as a circle divided into 72 parts. Start at D> 
and  follow the (new) straight line segments clockwise. You'll see 
each one  jumps 7/72 of an octave. When you've done that 20 times and 
wound up  at D<, you've generated Blackjack.
> 

Hello Dave!

Thanks for this "enhancement" of your "Miracle Wheel."  I understand 
it better than ever, now.  I did have one question, though...

It doesn't really look as though one goes around 20 times with those 
lines to get blackjack...

I see a going around 10 times and then I get to D< but then there is 
not a connecting line going from that to the D> which begins a 
second "chain" of 10.  And besides, between those two notes there are 
only 4 degrees of 72, not 7.

What am I doing wrong??

Thanks!

________ _______ _______
Joseph Pehrson


top of page bottom of page up down


Message: 266

Date: Thu, 21 Jun 2001 02:29:28

Subject: Re: 7/72 generator in blackjack

From: Paul Erlich

--- In tuning-math@y..., jpehrson@r... wrote:
> 
> Is that correct?  It looks as though Paul, in his calculations, is 
> trying to find the least errors for all the various intervals he is 
> considering, in squaring them and so forth, and then puts it all 
> nicely back in a pie (not a pie chart!) to determine the MIRACLE 
> generator at 116.7 cents.
> 
> Am I getting that at all...??

Yup . . . it's identical in form to the derivation Woolhouse did (in 
1835?) of the optimal meantone temperament, which turned out to have a 
fifth (the generator) tempered by 7/26 of a syntonic comma. Monz has a 
webpage on that . . .


top of page bottom of page up down


Message: 267

Date: Thu, 21 Jun 2001 02:32:42

Subject: Re: 7/72 generator in blackjack

From: Paul Erlich

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

> It doesn't really look as though one goes around 20 times with those 
> lines to get blackjack...
> 
> I see a going around 10 times and then I get to D< but then there is 
> not a connecting line going from that to the D> which begins a 
> second "chain" of 10.  And besides, between those two notes there are 
> only 4 degrees of 72, not 7.
> 
> What am I doing wrong??
> 
There is only one chain. It _starts_ at D>. It goes around the circle 
once, producing 9 more notes. Then D pops up. Then 9 more notes. Then 
D< is last.


top of page bottom of page up down


Message: 269

Date: Thu, 21 Jun 2001 02:46:11

Subject: Re: 7/72 generator in blackjack

From: jpehrson@r...

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

Yahoo groups: /tuning-math/message/267 *

> --- In tuning-math@y..., jpehrson@r... wrote:
> 
> > It doesn't really look as though one goes around 20 times with 
those 
> > lines to get blackjack...
> > 
> > I see a going around 10 times and then I get to D< but then there 
is 
> > not a connecting line going from that to the D> which begins a 
> > second "chain" of 10.  And besides, between those two notes there 
are 
> > only 4 degrees of 72, not 7.
> > 
> > What am I doing wrong??
> > 
> There is only one chain. It _starts_ at D>. It goes around the 
circle 
> once, producing 9 more notes. Then D pops up. Then 9 more notes. 
Then 
> D< is last.

Thanks, Paul... I just started on the wrong note... :)

________ _______ ______
Joseph Pehrson


top of page bottom of page up down


Message: 271

Date: Wed, 20 Jun 2001 23:20:51

Subject: Re: 7/72 generator in blackjack

From: monz

> ----- Original Message ----- 
> From: <jpehrson@r...>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Wednesday, June 20, 2001 7:04 PM
> Subject: [tuning-math] Re: 7/72 generator in blackjack
>
> 
> Well, from what you are saying, then, 31-EDO at with a 116.1 
> generator is also a MIRACLE scale, yes... of a sort??  Of course, I 
> had always heard of the "special" properties of 31-EDO...
> 
> Then, it is not too coincidental that a 31-tone NON-EDO using 116.7 
> would be "miraculous" as well... I mean it's not so miraculous that 
> both scales would have 31 notes, correct??


Yes, Joe, you're on the right track.  If I understand him correctly,
Graham considers 31-EDO to be a MIRACLE temperament "of a sort".
But that ~0.7-cent error accumulates further down the chain to
produce intervals that are rather farther away from JI than the
72-EDO-based MIRACLE scales.

The MIRACLE *generator* is the magical element in these tunings.
You can generate a number of different-sized scales from it
and they will all have essentially the same harmonic properties,
the differences being simply a matter of the gaps in the smaller
scales.

So the 10-tone version *is* a very useable scale for a composer
who wants many JI implications with a tiny pitch-set.  You've
become convinced that Blackjack (21-tone MIRACLE) will do the
trick for you, and I've decided to focus on Canasta (31-tone
MIRACLE).  The 41-tone MIRACLE scale is yet another scale
offering even *more* JI implications, and Graham has recommended
one a bit larger than that (was it 46?).  And of course, the
full 72-EDO set closes the possibilities of the 2^(7/72) generator.


> ANYWAY, I also very much appreciated the 72-EDO chart at the bottom 
> of the MIRACLE generator page... That REALLY did a good job of 
> breaking down how the various scales are related to the generator...


Yes, quite a bit of the MIRACLE discussion was elaborated via
Graham's decimal notation, which is (I think) a lot harder to
grasp for a "regular" performer/composer microtonalist who's
used to using deviations from 12-EDO, altho from a theoretical
perspective decimal *is* more elegant to describe these tunings.

So even for me, 72-EDO notation makes it easier to understand.
(I'm just sorry that I'm so devoted to my own 72-EDO notation that
now it goes up against the version everyone else has decided to use.)


> It seems I'm gradually coming to a greater appreciation of this 
> process, finally... although I have to admit I was rather "left in 
> the dust" when it happened.
> 
> (In fact, I became so confused, that Paul thought I had forgotten the 
> entire discussion!)


You're not alone, Joe.  I too was quite mystified during the first
week or so of the MIRACLE discussion, and it was only when I realized
the theoretical *AND* practical importance of it (i.e, how easily it
can be mapped to a Ztar) that I devoted some serious study to it and
began to understand.

The MIRACLE tunings, especially Canasta for me, solve a lot of the
riddles and problems I have been facing in dealing with large
extended JI systems - specifically, how to map so many damn pitches
to a playable instrument.  MIRACLE solves the problem by distributing
the small errors so well that a very small pitch-set can represent
a huge number of JI structures.

I've begun making a lattice of the implications of the Canasta
scale (to go on my Canasta page), but it's got so many ratios
on it that I fear it will be another "spaghetti lattice"...
and I'm only up to 7-limit, haven't even plotted 11 yet!


If you think of using the 2^(7/72) generator to create extended
(i.e., >12) scales in the same way that you can use a tempered
(i.e., narrowed) meantone "5th" or the wider-than-12-EDO Pythagorean
"5th" to create >12 "extended" cycles, I think that will help
make the whole process clearer.

The meantone and Pythagorean "5ths" don't close the cycle at the
13th note, because the 1st and 13th are separated by a small
interval which is exactly or approximately one of the commas.

Similarly, if you call your origin 0 and create an 11-tone
MIRACLE scale, the notes at either end (-5 and +5 generators
away from the origin) will be separated by 2^(2/72), or
33&1/3 cents.

This is a bit bigger than a "comma", but the process is the same.
By continuing to extend the cycle beyond these 11 notes, you
get pairs of pitches all separated by 2^(2/72) - this is exactly
why the Blackjack scale has L=5 s=2 (in terms of 72-EDO degrees).

By the time you reach a cycle bounded by -15 and +15 generators,
you've filled out the "octave" pitch-space pretty evenly, hence
the Canasta scale.

Extending it to one more note on either side would give a pitch
separated from the one on the other end by only 2^(1/72), or
16&2/3 cents.  So now all MIRACLE scales above cardinality 31
will have pairs of pitches separated by *that* interval.

Then finally the 72nd generator closes the cycle... in other
words, at that point the separation of pitches on either end
finally reduces to 2^(0/72), which is a unison.



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


 


_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at Yahoo! Mail Setup *


top of page bottom of page up down


Message: 272

Date: Thu, 21 Jun 2001 03:46:51

Subject: Sonance degree (DEFINITION)

From: Pierre Lamothe

--------------
                               Sonance degree
                               --------------


Complexity (n*d) and _Sonance_ (log n + log d) are microtonal concepts on
which it seems there exist now (with varied terminology) a large agreement.

I want to introduce here the _Sonance degree_ concept, a macrotonal one.

While the _Sonance_ corresponds to a universal rational function applicable
to any isolated irreducible ratio n/d, the _Sonance degree_ is defined only
inside a discrete Z-module (lattice) with a coherent set of unison vectors.
(A coherent set implies a CS structure, in other words a periodicity
block). For an isolated ratio n/d, the _Sonance degree_ has no sense like
the usual degree (tone rank in a scale).

Dividing the usual degree of an interval by the degree number of the
octave, we obtain the _Width degree_ of this interval which is the
counterpart of the _Sonance degree_ at macrotonal level. To help the
understanding of that concept the microtonal dyad (Width, Sonance) is
compared here with the macrotonal dyad (Width_degree, Sonance_degree) at
definition level.  

The microtonal definitions of _Width_ and _Sonance_ in the context of a
lattice might be written

     _Width_ (X) = log(B) X / log(2)
   _Sonance_ (X) = log(B) |X| / log(2)

where 

   X = (X0 X1 .. XN)*

is the coordinate vector (* indicate a column vector) of any interval in
the basis

   B = <B0 B1 .. BN>

where B0 = 2 (octaviant system) and the other independant components are
normally simple primes (primal basis). The log operator applied to B gives 

  log(B) = log <2 B1 .. BN>
         = [log(2) log(B1) .. log(BN)]

and the absolute operator || applied to X gives

  |X| = (|X0| |X1| .. |XN|)*

so these definitions might also be written 
  
     _Width_ (X) = Sum(log(Bi)* Xi ) / log(2)
   _Sonance_ (X) = Sum(log(Bi)*|Xi|) / log(2)

-----

Now, the main property of a coherent musical system G is the existence of
an epimorphism D applying G on the relative integers Z. That implies

   D(xy) = D(x) + D(y)
  
and considering the octave modularity

  D(xy mod <2>) = D(x) + D(y) mod [d]

where d is the number of degree in the octave.

The quotient G/D defines the congruence classes of interval in the system.

-----

I already shown how to explicit a such epimorphism in the form of the
degree function D(X) using the unison vectors to calculate it. See

<Yahoo groups: /tuning/message/18625 *>
<Yahoo groups: /tuning/message/20746 *>

I had calculated the D(X) function for 3 systems using two distinct methods. 

   D(X) = 5x + 8y + 14z             (Slendro)
   D(X) = 7x + 11y + 16z            (Zarlino)
   D(X) = 10x + 16y + 23z + 28t     (Erlich decatonic)

Using the formalism inside the precedent definitions we have

   D(X) = [5 8 14] X
   D(X) = [7 11 16] X
   D(X) = [10 16 23 28] X

So D might be represented by a matrix operator [D0 D1 .. DN] applied to X.

We have to understand here that the multiple ratios

   D0:D1:..:DN

of the components in a such epimorphism is a rational approximation of this
irrational one

   log(B0):log(B1):..:log(BN)

depending of the unison vectors used. Here we have

        5:8:14 ~ log(2):log(3):log(7)
       7:11:16 ~ log(2):log(3):log(5)
   10:16:23:28 ~ log(2):log(3):log(5):log(7)

-----

Recalling

     _Width_ (X) = log(B) X / log(2)
   _Sonance_ (X) = log(B) |X| / log(2)

   ----------------------------------
    BY DEFINITION 
 
      _Width degree_ (X) = D X / d
    _Sonance degree_ (X) = D |X| / d
   ----------------------------------

where the matrix 

   D = [d D1 .. DN] 

is the degree operator expliciting the system epimorphism in which the
first component D0 = d is the degree of the octave and where

   X = (X0 X1 X2 .. XN)*

is the coordinate vector (* indicate a column vector) of any interval in a
basis

   <2 B1 B2 .. BN>

where only the first component B0 = 2 (octaviant system) has to be known.

The absolute operator || applied to X gives

  |X| = (|X0| |X1| .. |XN|)*

so these definitions might also be written 

     _Width degree_ (X) = Sum(Di* Xi ) / d
   _Sonance degree_ (X) = Sum(Di*|Xi|) / d

-----

In the challenging problem we had two unison vectors

   U = {(-4 4 -1)*,(-3 -1 2)*}

Using the determinant method to calcultate D

              (x -4 -3)
   D(X) = det (y  4 -1) = 7x + 11y + 16z
              (z -1  2)

then D = [7 11 16].

Then SD(X) the _Sonance degree_ of the two intervals

   (-1 1 0)*
   (-3 1 1)*

is simply

   [7 11 16](1 1 0)* / 7 = 18/7 
   [7 11 16](3 1 1)* / 7 = 48/7

-----

Comparing values rounded to three decimals for the sonance in <2 3 5> with
the sonance_degree using U we have

  sonance_degree (-1 1 0) = 2,571
            sonance (3/2) = 2,585  

   sonance_degree (3 1 1) = 6,857
           sonance (15/8) = 6,907


Pierre


top of page bottom of page up down


Message: 273

Date: Thu, 21 Jun 2001 00:59:10

Subject: Re: Sonance degree (DEFINITION)

From: monz

----- Original Message ----- 
From: Pierre Lamothe <plamothe@a...>
To: <tuning-math@xxxxxxxxxxx.xxx>
Sent: Thursday, June 21, 2001 12:46 AM
Subject: [tuning-math] Sonance degree (DEFINITION)


> 
>                                --------------
>                                Sonance degree
>                                --------------


Paul (or Dave or...),

Please explain in simplified terms what Pierre wrote here.

It seems from what I gleaned from it that he's talking
about something which relates to my "finity" concept.
I'm very interested, but having a hard time following
all the math.


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


 


_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at Yahoo! Mail Setup *


top of page bottom of page up down


Message: 274

Date: Thu, 21 Jun 2001 13:13:13

Subject: recap of decimal notation (posted to "biggie" also)

From: jpehrson@r...

Monz... this addition to the MIRACLE page is just terrific.  It 
relates the whole "generator" process to our "traditional" 
Pythagorean comma in 12-tET, etc...  It *really* put things together 
for me...

Now, could I please ask you to elaborate a bit on Graham Breed's 
decimal notation, so I can understand it??

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.

I would really appreciate it!

Thanks!

Joe

________ ________ ________
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