Sonance degree (DEFINITION)

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Sonance degree
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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)

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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

</tuning/topicId_18294.html#18625>
</tuning/topicId_20642.html#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

----- Original Message -----
From: Pierre Lamothe <plamothe@aei.ca>
To: <tuning-math@yahoogroups.com>
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
http://www.monz.org
"All roads lead to n^0"

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--- 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!

--- 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

Hi Monz,

I used abstract language to establish a generalized property, but the thing
is much more simple than it seems.

-----

You know surely implicitely what is a morphism since you could probably write

log(15) = log(3) + log(5)
log(5/3) = log(5) - log(3)
log(243) = 5 log(3)

The D function which gives the degree of any interval in a system is also,
like the log function, a morphism, having that property

D(ab) = D(a) + D(b)

and the following is simply included in it

D(a/b) = D(a) - D(b)
D(a^b) = b D(a)

-----

Taking, for instance, all the intervals in the prime 5-limit, there exist
many functions D depending of the system used, the most simple
corresponding to the chinese pentatonic, the japanese pentatonic and the
Zarlino heptatonic.

For a such function D in 5-limit applied to any interval

X = (2^a)(3^b)(5^c)

the morphism property implies a decomposition of the degree function like this

D(X) = D(2^a) + D(3^b) + D(5^c)

= a D(2) + b D(3) + c D(5)

Thus, it is sufficient to know the degree of the primes D(2), D(3) and D(5)
to easily calculate the degree of any interval using the powers a, b and c.

-----

You know also the classes of interval generated by the tonic rotation in
the Zarlino scale. Since the function D has to give for any interval its
degree or interval rank, all that would have to be respected

D(1) = 0
D(16/15) = D(10/9) = D(9/8) = 1
D(32/27) = D(6/5) = D(5/4)= 2
D(4/3) = D(27/20) = D(45/32) = 3
D(64/45) = D(40/27) = D(3/2) = 4
D(8/5) = D(5/3) = D(27/16) = 5
D(16/9) = D(9/5) = D(15/8) = 6
D(2) = 7

You could verify that this D respect here the morphism property with the
first values found for D(2), D(3) and D(5) like this

D(2) = 7 (simply given)

D(3/2) = D(3) - D(2)
4 = D(3) - 7 --> D(3) = 11

D(5/4) = D(5) - 2 D(2)
2 = D(5) - 14 --> D(5) = 16

e.g. D(15/8) = -3 D(2) + 1 D(3) + 1 D(5)
= -21 + 11 + 16 = 6

-----

It is sufficient to know the unison vectors of a such system to calculate
its function D. The unison vectors in our Zarlino example are (the srutis
between the steps)

(10/9)/(16/15) = 25/24
(9/8)/(10/9) = 81/80

The class of a such sruti is forcely 0 (unison vector) since if a and b are
steps, their common class is 1 and

D(a/b) = D(a) - D(b) = 1 - 1 = 0

Now, let us see how the following equations permit to calculate D.

D(25/24) = 0
D(81/80) = 0

is decomposed like this

-3 D(2) - 1 D(3) + 2 D(5) = 0 (*)
-4 D(2) + 4 D(3) - 1 D(5) = 0 (**)

Doubling (**) and adding to (*) gives

-11 D(2) + 7 D(3) = 0

Quadrupling (*) and adding to (**) gives

-16 D(2) + 7 D(5) = 0

We have thus

D(2) D(3) D(5)
---- = ---- = ----
7 11 16

The minimal integer solution in this periodic system is obviously

D(2) = 7
D(3) = 11
D(5) = 16

-----

Now, for the sonance degree. We had with 15/8 in Zarlino

D(15/8) = -3 D(2) + 1 D(3) + 1 D(5)
= -21 + 11 + 16 = 6

Taking the absolute value for the powers and dividing by D(2) we have for
the sonance degree

SD(15/8) = ( 3 D(2) + 1 D(3) + 1 D(5) ) / D(2)
= ( 21 + 11 + 16 ) / 7 = 48/7

Hoping that seems more simple.

Pierre

> ----- Original Message -----
> From: Pierre Lamothe <plamothe@aei.ca>
> To: <tuning-math@yahoogroups.com>
> Sent: Thursday, June 21, 2001 11:10 PM
> Subject: [tuning-math] Re: Sonance degree (DEFINITION)
>
>
>
>
> Hi Monz,
>
> I used abstract language to establish a generalized property, but the
thing
> is much more simple than it seems.
> <etc.>

Thanks very much, Pierre. I hope that you were not offended by
my asking others for help instead of approaching you directly.

In the past it seemed to me that you were struggling a bit
trying to express yourself in English, and so I thought I was
simply avoiding putting you in that position.

But this post is indeed very clear! Thanks so much. I have
only glanced quickly at it, but will study it more fully.

-monz
http://www.monz.org
"All roads lead to n^0"

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