Topic: Interval of equivalence and difference tones

2 scales

File Description Notes Period (¢)
gldspec The golden spectrum 12 1200.0
recurrence4-3 step = 1.22074408460576, quantiz 10c, f0=25Hz 68 11888.3

Thread (21 messages)

From: Hudson Lacerda (2006-04-01)
Subject: Interval of equivalence and difference tones

Hello,

I have a question on intervals (i) with a specific property:
Each element in the set i^n (n=0,1,2,3,4,...) belongs to the set i*n.


Thinking on intervals for equivalence I recalled a special property of 
the 2:1 octave:

A series of superimposed octaves, as ratios, is:
1 2 4 8 16...
The difference tones for adjacent octaves are:
1 2 4 8...

For Bohlen-Pierce, the interval of equivalence is 3:1, one have the series:
1 3 9 27 81...
The difference tones have ratios:
2 6 18 54...
i.e., one octave up the series (but not elements of the series).

I am wondering what intervals (i) have a property such as the octave, 
somewhat generalized as:

Each element in the set i^n (n=1,2,3,4,...) belongs to the set i*n.

Here is a solution (the limit of fibonacci(n+1)/fibonacci(n), n>=1): 
1.61803398874989

i^n series with i=1.61803398874989 (n=0,1,2...):

    1.00000000000000
    1.61803398874989
    2.61803398874989
    4.23606797749979
    6.85410196624969
   11.09016994374948
   17.94427190999916

i*n:

   1.000000000000000
   1.618033988749895
   2.618033988749896
   4.236067977499791
   6.854101966249686

Are there other intervals with that property?


Cheers,
Hudson

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From: Hudson Lacerda (2006-04-01)
Subject: Interval of equivalence and difference tones

(Sorry, the property was described wrong...)


Hello,

I have a question on intervals (i) with a specific property.

Thinking on intervals for equivalence I recalled a special property of
the 2:1 octave:

A series of superimposed octaves, as ratios, is:
1 2 4 8 16...
The difference tones for adjacent octaves are:
1 2 4 8...

For Bohlen-Pierce, the interval of equivalence is 3:1, one have the series:
1 3 9 27 81...
The difference tones have ratios:
2 6 18 54...
i.e., one octave up the series (but not elements of the series).

I am wondering what intervals (i) have a property such as the octave.

Here is a solution (the limit of fibonacci(n+1)/fibonacci(n), n>=1):
1.61803398874989

i^n series with i=1.61803398874989 (n=0,1,2...):

    1.00000000000000
    1.61803398874989
    2.61803398874989
    4.23606797749979
    6.85410196624969
   11.09016994374948
   17.94427190999916

difference between adjacent elements of the series:

   0.618033988749895
   1.000000000000000
   1.618033988749895
   2.618033988749896
   4.236067977499791
   6.854101966249686

Are there other intervals with that property?


Cheers,
Hudson

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From: Gene Ward Smith (2006-04-01)
Subject: Re: Interval of equivalence and difference tones

--- In tuning@yahoogroups.com, Hudson Lacerda <hfmlacerda@...> wrote:

> A series of superimposed octaves, as ratios, is:
> 1 2 4 8 16...
> The difference tones for adjacent octaves are:
> 1 2 4 8...

> I am wondering what intervals (i) have a property such as the octave.

If we have the recurrence relation a(n+1) - a(n) = a(n), then
a(n+1) = 2a(n), and so up to scaling we have only the one solution.
However, other things come close. For instance, f(n+2) = f(n+1)+f(n)
implies f(n+2)-f(n+1) = f(n); so for example, taking the fibonacci
sequence 1,1,2,3,5,8,13 ..., successive differences are 0,1,1,2,3,5,...
From: Hudson Lacerda (2006-04-02)
Subject: Re: [tuning] Re: Interval of equivalence and difference tones

Gene Ward Smith escreveu:
> If we have the recurrence relation a(n+1) - a(n) = a(n), then
> a(n+1) = 2a(n), and so up to scaling we have only the one solution.
> However, other things come close. For instance, f(n+2) = f(n+1)+f(n)
> implies f(n+2)-f(n+1) = f(n); so for example, taking the fibonacci
> sequence 1,1,2,3,5,8,13 ..., successive differences are 0,1,1,2,3,5,... 

Yes, Gene, but the Fibonacci sequence is not based on a single interval 
(for superimposing as i^m, where m is an integer; and for use as 
equivalence interval for scales), though it *approaches* a single 
interval (~1.618).

The set of intervals of octaves i=2, (i^m) is:
1 2 4 8 16 32 64 ...
The successive differences (i^n - i^(n-1)) are:
1 2 4 8 16 32 ...

For the golden proportion i=1.61803398874989, (i^m) is:
    1.00000000000000
    1.61803398874989
    2.61803398874989
    4.23606797749979
    6.85410196624969
   11.09016994374948
   17.94427190999916 ...
The successive differences are:
   0.618033988749895
   1.000000000000000
   1.618033988749895
   2.618033988749896
   4.236067977499791
   6.854101966249686 ...
Only the first difference is not in i^m.

Also, please note that 1/0.618033988749895 = 1.618033988749895, thus 
0.618.. belongs to i^m if m can be -1. This way, all difference tones in 
the form f2-f1 (for successive tones) don't bring any different intervals.

Then, I want know values for i so that [i^m] is a subset of [i^n - 
i^(n-1)] for integers m,n. How to obtain such values for i?

BTW, I've made some experiments with spectra based on the interval 
1.618. The difference tones (like 0.618) are very audible, thus the 
pitch of the sound is very ambiguous. (They sound like ``generalized'' 
Shepard sounds.)

Here are two test .ogg files:
http://geocities.yahoo.com.br/hfmlacerda/1.618.zip

Regards,
Hudson

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From: Gene Ward Smith (2006-04-02)
Subject: Re: Interval of equivalence and difference tones

--- In tuning@yahoogroups.com, Hudson Lacerda <hfmlacerda@...> wrote:

> Then, I want know values for i so that [i^m] is a subset of [i^n - 
> i^(n-1)] for integers m,n. How to obtain such values for i?

Why not ask this on tuning-math? I'm unwilling to think about it here.
From: Petr Parízek (2006-04-02)
Subject: Re: [tuning] Re: Interval of equivalence and difference tones

Hudson wrote:

> Here are two test .ogg files:
> http://geocities.yahoo.com.br/hfmlacerda/1.618.zip

And here is a chain of 11 consecutive "Phis" (i.e. ~1.618s) reduced to 2/1:

! gldspec.scl
!
The golden spectrum
 12
!
 99.27089
 198.54178
 267.63881
 366.90970
 466.18059
 565.45148
 634.54852
 733.81941
 833.09030
 932.36119
 1100.72911
 2/1

Petr
From: Kraig Grady (2006-04-02)
Subject: Re: Interval of equivalence and difference tones

this is ground pretty much covered in Erv's paper on Mt Meru.
 Likewise it is the basis of the tuning i have been using for the last 7 
or 8 years.
 It is, as i have pointed out in the past, produces better results in 
explaining many of the tunings we find in africa and indonesia.
 Which of course must be pointed out that many of these lie out side of 
this framework too.

It makes perfect sense that people would be attracted to having 
differance tones generate other pitches already in the scale.
 Some of these are anything but simple ratio relationships .
 Still they often are consonant with themselves which made me abandon 
the idea of consonance being based on simple ratios and more in line 
with "acoustical coincidences" which includes simple ratios as a sub 
category.
 
-- 
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles
From: Hudson Lacerda (2006-04-02)
Subject: Interval of equivalence and difference tones

Hi.


I have a question on intervals [i] with a specific property: a chain of 
[i] should yield combination tones (in the form f2-f1, or even 2*f1-f2 
and/or 3*f1-2*f2) belonging to the chain itself. Such intervals may be 
useful as interval of equivalence.


The set (chain) of intervals of octave i=2, (i^m) is:
1 2 4 8 16 32 64 ...
The successive differences (i^n - i^(n-1), i.e. f2-f1) are:
1 2 4 8 16 32 ...

For the golden proportion i=1.61803398874989, (i^m) is:
    1.00000000000000
    1.61803398874989
    2.61803398874989
    4.23606797749979
    6.85410196624969
   11.09016994374948
   17.94427190999916 ...
The successive differences are:
   0.618033988749895
   1.000000000000000
   1.618033988749895
   2.618033988749896
   4.236067977499791
   6.854101966249686 ...

Note that only the first difference is not in i^m.
However, please note that 1/0.618033988749895 = 1.618033988749895, thus
~0.618 belongs to i^m if [m] can be -1. This way, all difference tones 
in the form f2-f1 (for successive tones) don't bring any different (new) 
intervals.

*************
Then, I want know values for [i] so that [i^m] is a subset of [i^n -
i^(n-1)] for integers m,n. How to obtain such values for [i]?
*************

Still for ~1.618, if we let [m] have negative values, more combinations 
tones coincide with the original set:

i^m, for [m] in -6 to 6:
    0.0557280900008412
    0.0901699437494742
    0.1458980337503154
    0.2360679774997897
    0.3819660112501052
    0.6180339887498948
    1.0000000000000000
    1.6180339887498949
    2.6180339887498949
    4.2360679774997898
    6.8541019662496856
   11.0901699437494763
   17.9442719099991628

The main combination tones are (f1 is the lower frequency of a pair, f2 
is the higher frequency):

f2 - f1:
   0.0344418537486330
   0.0557280900008412
   0.0901699437494742
   0.1458980337503155
   0.2360679774997896
   0.3819660112501052
   0.6180339887498949
   1.0000000000000000
   1.6180339887498949
   2.6180339887498958
   4.2360679774997907
   6.8541019662496865

2*f1 - f2:
   0.0212862362522082
   0.0344418537486330
   0.0557280900008412
   0.0901699437494742
   0.1458980337503155
   0.2360679774997896
   0.3819660112501051
   0.6180339887498949
   1.0000000000000000
   1.6180339887498940
   2.6180339887498949
   4.2360679774997898

3*f1 - 2*f2:
   -0.0131556174964248
   -0.0212862362522082
   -0.0344418537486330
   -0.0557280900008413
   -0.0901699437494741
   -0.1458980337503157
   -0.2360679774997898
   -0.3819660112501051
   -0.6180339887498949
   -1.0000000000000018
   -1.6180339887498967
   -2.6180339887498931

BTW, I've made some experiments with spectra based on the interval
1.618. The combination tones (like 0.618) are very audible, thus the
pitch of the sound is very ambiguous. (They sound like ``generalized''
Shepard sounds.)

Here are two test .ogg files:
http://geocities.yahoo.com.br/hfmlacerda/1.618.zip

Regards,
Hudson

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From: Hudson Lacerda (2006-04-03)
Subject: Re: [tuning] Re: Interval of equivalence and difference tones

Hi Kraig,

Thanks for the response.

Kraig Grady escreveu:
> this is ground pretty much covered in Erv's paper on Mt Meru.

Which paper?

Searching on the net, I found this address, but it *is not* valid:
http://www.iii.rmit.edu.au/sonology/acmc2002/ACMC2002_Web_Proceedings/123-132_Burt.pdf
But the Google's cache (as HTML) is available, though:
http://72.14.203.104/search?q=cache:NR6gjGPQ2DQJ:www.iii.rmit.edu.au/sonology/acmc2002/ACMC2002_Web_Proceedings/123-132_Burt.pdf+123-132_Burt.pdf&hl=pt-BR&gl=br&ct=clnk&cd=2
I've found the W. Burt page, but not the article...
http://www.emf.org/subscribers/burt/

Anyway, these are interesting:
http://www.anaphoria.com/wilsonintroMERU.html
http://arxiv.org/pdf/physics/0411195

[...]
> It makes perfect sense that people would be attracted to having 
> differance tones generate other pitches already in the scale.
>  Some of these are anything but simple ratio relationships .
>  Still they often are consonant with themselves which made me abandon 
> the idea of consonance being based on simple ratios and more in line 
> with "acoustical coincidences" which includes simple ratios as a sub 
> category.

My aim is to use such intervals (the limits of recurrences) for 
equivalence, using them in synthesized spectra. The coincidence of 
combination tones ((n+1)*f1 - (n)*f2) can reinforce the equivalence 
relationship and avoid extraneous resulting tones.

{{ As a (counter-)example, BP scale in the form 3^(n/13) has several 
difference tones out the scale, even when using only odd harmonic 
partials. Other intervals should be used as partials, in special to 
reduce the harshness of the pseudo-fifths (3^(5/13)) and pseudo-octaves 
(3^(8/13)). }}

Back to the recurrences, the sequence 1,1,1,2,3,4,6,9,13,19,28 converges 
to 1.46557123188, which has the property I was looking for. Here is my 
GNU Octave test:

octave:1> x=1.46557123188
x = 1.46557123188000

	[...]

octave:4> k=x.^[-6:6]'
k =

   0.100915624219394
   0.147899035703157
   0.216756571949339
   0.317672196169879
   0.465571231874715
   0.682327803826515
   1.000000000000000
   1.465571231880000
   2.147899035714261
   3.147899035725613
   4.613470267622252
   6.761369303360897
   9.909268339122248

octave:5> diff(k)
ans =

   0.0469834114837625
   0.0688575362461827
   0.1009156242205397
   0.1478990357048356
   0.2167565719518000
   0.3176721961734854
   0.4655712318800000
   0.6823278038342606
   1.0000000000113527
   1.4655712318966385
   2.1478990357386447
   3.1478990357613510

Regards,
Hudson

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From: Gene Ward Smith (2006-04-03)
Subject: Re: Interval of equivalence and difference tones

--- In tuning@yahoogroups.com, Hudson Lacerda <hfmlacerda@...> wrote:

> Back to the recurrences, the sequence 1,1,1,2,3,4,6,9,13,19,28
converges 
> to 1.46557123188, which has the property I was looking for. Here is my 
> GNU Octave test:

Recurrence sequences L_{n+1} = L_n + L_{n-k} for fixed k and such that
the first k+1 L_i's are equal to 1 are called "Lame sequences of
higher order" (with an accent on Lame') by de Villiers; the ratios of
successive terms all converge to the corresponding positive real root of
x^(k+1) - x^k - 1 = 0.
From: Carl Lumma (2006-04-03)
Subject: Re: Interval of equivalence and difference tones

> Kraig,
> 
> It seems likely that cultures employing primarily 
> monophonic/heterophonic forms of music-- as opposed to harmonic 
> (chordal) --might choose to delight in complex melodic intervals
> that might provide more variety, intensity or nuance.
> 
> Just a thought. :-)
> 
> jl

Many of the same principles that guide harmony guide melody,
since prominant partials suggest future notes.  But difference
tones are unlikely to be a primary factor in either in
traditional musics, in my opinion.

-Carl
From: Yahya Abdal-Aziz (2006-04-04)
Subject: Re: Interval of equivalence and difference tones

Hi all,

On Sun, 02 Apr 2006, Kraig Grady wrote, in reply to Hudson
Lacerda's thread on forming scales that include their own
difference tones:
>
> this is ground pretty much covered in Erv's paper on Mt Meru.
>  Likewise it is the basis of the tuning i have been using for the last 7
> or 8 years.
>  It is, as i have pointed out in the past, produces better results in
> explaining many of the tunings we find in africa and indonesia.
>  Which of course must be pointed out that many of these lie out side of
> this framework too.
>
> It makes perfect sense that people would be attracted to having
> differance tones generate other pitches already in the scale.
>  Some of these are anything but simple ratio relationships .
>  Still they often are consonant with themselves which made me abandon
> the idea of consonance being based on simple ratios and more in line
> with "acoustical coincidences" which includes simple ratios as a sub
> category.


This is a pretty deep generalisation of the notion
of consonant or harmonic sound.  It gives us much
more to play with than the overtone series by itself,
which used as sole tonal resource, may give a rather
bland sound.

Take an arbitrary (countable) set of n pitches:
P = {p1, p2, ... pn}

Form all the difference tones, dij = pi - pj for all
i and j between 1 and n inclusive, and label the set
of such difference tones D(P).  Repeat the process
on D(P) to obtain D(D(P)), and so on, until no new
tones are obtained.

The set P union D(P) union D(D(P)) union ..., or "the
closure of P under difference", is a gamut of
pitches which contains all of its difference tones.

So by suitable choice of the original set P, one can
form a gamut with just about any desired degree
of "colour" (or should I say, inharmonicity), knowing
that including all the difference tones will not
materially affect that degree of colour.

From this gamut, of say m pitches, one may draw
2^m different subsets, each of which can serve
as either an ascending or a descending scale.
Combining both an ascending and a descending
scale, one has 2^(m+1) different scale resources
formed in this way, where m >= n.  This may be
quite a large number, but selecting from among
these those scales that conform to our normal bias
towards providing "reasonably" sized melodic steps
would often reduce this to a mere handful, or
perhaps even non.

Hudson's choice of the initial set of pitches P as
being members of a mathematical series such as
Fibonacci or Lam\ufffd is one way to go.  Another
approach would be to select the initial pitches
as random frequencies in the audible range
(taken as 20 Hz to 20,000 Hz for convenience).
Limiting the precision of the original pitches to be
0 decimal places, ie whole numbers, means that the
smallest possible difference tone would be 1 Hz.
Likewise, if the precision were 1 decimal place, the
smallest possible difference tone would be 0.1 Hz,
and so on.  Rather than measuring the pitches in Hz,
it might be more practical to do so in cents.  By
always including pitch 0 cents in P, it follows that
D(P) will include all of P, so the closure will be one
of the D(D(D(...D(P)...))) or D^k(P) for some k.

For example, take the set P = {0, 387, 492, 871}.
Then D(P) = P union {105, 379, 486}.
Then D(D(P)) = P union {105, 379, 486} union
{6, 8, 99, 113, 274, 282, 381, 385, 389, 492, 766}.
And D^3(P) includes 8-6 = 2; so if you don't want
differences that small in your gamut, another, less
random, approach may serve better.

For example, suppose you decide that you do want
387 cents in your initial pitch set P, and that you do
NOT want any interval smaller than 17 cents.  Can
we find some x such that the closure of
P = {0, 387, x}
under difference does not contain any value y < 17?

Putting x = 17 gives us successive differences:
387 - 17 = 370
370 - 17 = 353
...
47 - 17 = 30
30 - 17 = 13.
But 13 < 17.   So that doesn't work.  Generally, the
smallest member of the closure of P = {0, p1, p2, ... pn}
under difference should be a common divisor of all
the pi.  The only way we can ensure that
P = {0, p1=17, p2, ... pn} does not contain any member
(other than 0) less than 17 is to make every pi a
multiple of 17, for i between 1 and n inclusive.  We
can write P = 17Q, where Q is the reduced initial pitch
set: Q = {0, q1, q2, ... qn} and qi = pi/17 for all i.
We can choose the qi as random integers, confident
that the closure of P = 17Q under difference will
contain no member (other than 0) smaller than 17.

So here is a procedure for generating a pitch set
(gamut) that:
a) contains no step smaller than s cents; and
b) contains all its own difference pitches:

Select random positive integers q1, q2, ... qn.
Form the inital pitch class P = {0, s*q1, s*q2, ... s*qn}.
Form the difference sets D(P), D(D(P)) ... as above
until no new members are created.  The last such
difference set ie the desired set.

To keep the sums simpler, form the difference sets
of Q rather than P.  Since D(P) = D(s*Q) = s*D(Q),
you only need to multiply the final difference set
D^k(Q) by s to get the desired gamut D^k(P).

Example:
Select random positive integers 3, 6, 8, 19.
Q = {0, 3, 6, 8, 19}.
D(Q) = Q union {2, 5, 11, 13, 16}.
D^2(Q) = D(Q) union {1, 4, 7, 9, 10, 14, 17}.
D^3(Q) = D^2(Q) union {12, 15, 18}.
D^4(Q) = D^3(Q) union {}.

At this point it is clear that we have generated
D^3(Q) = {0, 1, 2, ... 18, 19}.  Provided the random
integers Q contain at least two positive integers
with no common factor (such as 6 and 19 in this
example), the procedure above will always
generate an initial segment (IS) of the natural
(counting) numbers, {1, 2, 3, ... qn}.

Choose the minimum interval, s = 59 cents say.
Then the gamut is
P = 59 D^3(Q)
   = 59 {0, 1, 2, ... 18, 19}
   = {0, 59, 118, 177, ... 1062, 1121}

Note that this gamut is NOT octave equivalent!
The next two additions of a 59-cent interval give
1180 cents, just 20 cents short of, and 1239 cents,
just 39 cents past,  the octave.  Since neither 20
cents nor 39 cents were included in the gamut,
neither is the octave.  Such a gamut, and scales
derived from it, are poorly suited to timbres with
strong harmonic partials.

To get a gamut, and thus scales, with a suitable
total range for a composition, it may be necessary
to artificially make the range from which the qi are
randomly drawn a little larger than the desired
total range.  For example, if a five-octave range is
required in practice for the pi, that corresponds to
a [5 * 1200 / 59] = 101 (cent) range for the qi.
Setting the maximum possible qi  value to about
one-sixth more, say, 115, then choosing random
integers qi in the range 1 to 115, should provide a
suitable range of values for the pi.

Note also that this gamut (because it has two
co-prime generating intervals) is an "equal division
of the maximal generator" gamut, which we might
call an EDMG gamut.  An EDMG scale would, of
course, be any scale of pitches selected from this
gamut.

You can also control the density of pitches in the
EDMG gamut to some extent by deciding how
many distinct pitches you want, on average, in
each octave, and using that to determine the
total number n of qi that you randomly select.

For example, suppose  you want about nine pitches
per octave, and a range of five octaves.  That
means you will need to select _or generate_
9 * 5 = 45 distinct pitches.  A simple plan is to
randomly choose integers from the range 1 to 115,
until you have 45 distinct integers.  Using the first
few of these, form their difference closure,
counting the new members added.  Keep adding one
member from the random list at a time, together
with its differences with all preceding members,
still keeping count.  When you have enough, stop.
You could, of course, delay drawing another distinct
random number from 1 to 115 until necessary.

Regards,
Yahya

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From: Petr Parízek (2006-04-04)
Subject: Re: [tuning] Re: Interval of equivalence and difference tones

Hi Hudson.

You wrote:

> Back to the recurrences, the sequence 1,1,1,2,3,4,6,9,13,19,28 converges 
> to 1.46557123188, which has the property I was looking for.

How did you find this value?

Petr
From: Gene Ward Smith (2006-04-04)
Subject: Re: Interval of equivalence and difference tones

--- In tuning@yahoogroups.com, Petr Parízek  wrote:
>
> Hi Hudson.
> 
> You wrote:
> 
> > Back to the recurrences, the sequence 1,1,1,2,3,4,6,9,13,19,28
converges 
> > to 1.46557123188, which has the property I was looking for.
> 
> How did you find this value?

The recurrence relationship is a(n) = a(n-1) + a(n-3). Hence, the
characteristic polynomial is x^3-x^2-1, and solving that shows that it
has a single positive real root, given above. This is a Pisot number,
meaning all the conjugate roots are less than one in absolute value.
From: Hudson Lacerda (2006-04-04)
Subject: Re: [tuning] Re: Interval of equivalence and difference tones

Hello Yahya,

Thanks for the input.

By using a simplified version of your other approach (on frequencies), I 
got the scale below (whithout interval of equivalence).

It is based on the interval 1.22074408460576, which is the limit of 
n[i]/n[i-1] from the series n[i] = [i-4]-n[i-3], with initial values 
equal to 1.

The minimal interval was set to 10 cents, the range was 25-25250Hz, and 
the minimal difference of frequencies was 25Hz (all frequencies were 
quantized to harmonics of 25Hz).

yahya(P) doesn't compute all differences to add new frequencies, but 
only the differences between successive frequencies in P instead (just 
to save computation time).

Set the base pitch as 25Hz to try with the scale.
The GNU Octave functions I used are listed below the scale.

Cheers,
Hudson

--------------------------------------------------------------------

! recurrence4-3.scl
! [ans,lim]=recorrencia(1e3,4,3);
! lim = 1.22074408460576
! P=yahya(25.* 1.22074408460576 .^[0:128]',[25,25250],25,10);
! fprintf("%.2f\n",rat2cent(P./25)(2:end))
step = 1.22074408460576, quantiz 10c, f0=25Hz
68
!
1200.00
1901.96
2400.00
2786.31
3101.96
3368.83
3600.00
3803.91
3986.31
4151.32
4301.96
4440.53
4568.83
4688.27
4800.00
5003.91
5097.51
5186.31
5351.32
5501.96
5640.53
5768.83
5829.58
5888.27
6053.27
6155.14
6203.91
6342.48
6470.78
6511.52
6551.32
6701.96
6840.53
6873.50
6905.87
7059.17
7200.00
7226.84
7253.27
7403.91
7542.48
7586.31
7751.32
7901.96
7937.65
8089.76
8288.27
8426.84
8627.79
8786.31
8980.58
9119.90
9321.82
9473.78
9666.34
9815.89
10013.16
10161.13
10359.59
10505.87
10706.21
10849.99
11051.34
11197.10
11395.03
11542.06
11741.56
11888.27

--------------------------------------------------------------------

function [ P ] = yahya (P,m,r,c)
if(size(P)(2)>0)
   P=reshape(P,[length(P),1]); # P as column
endif
if(nargin<2 || length(m)~=2 || isempty(m))
   m=[0,0]; # minimum
endif
if(nargin<3 || isempty(r))
   r=0; # resolution
endif
if(nargin<4 || isempty(c))
   c=0;
endif
x=-1;y=0;
if(r) # pre-quantization
   P=round(P./r).*r;
endif
while(x~=y)
   x=length(P);
   D=diff(P); # here only the successive differences, not all differences
   P=union(D,P);
   if(m)
     P=P(find(P>=m(1)));
     P=P(find(P<=m(2)));
   endif
   y=length(P);
endwhile
if(r) # post-quantization
   P=unique(round(P./r).*r);
endif
if(c) # quantization in cents ; check this!!!
   C=diff(1200.*(log2([P(1);P])));
   C=find(abs(C)>=c);
   P=unique([P(1);P(C)]); # TODO: check this
endif
endfunction

----------------------------------------------------------

function [ H,lim ] = recorrencia (n,x,y,seed)
H=ones(n,1);
if(nargin>3 && ~isempty(seed))
   H(1:length(seed))=seed;
endif
for(i=(max([x,y])+1):n)
   H(i)=H(i-x)+H(i-y);
endfor
lim=H(n)/H(n-1)
endfunction

-----------------------------------------------------------

		
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From: Hudson Lacerda (2006-04-05)
Subject: Re: [tuning] Re: Interval of equivalence and difference tones

Petr Par\ufffdzek escreveu:
> Hi Hudson.
> 
> You wrote:
> 
> 
>>Back to the recurrences, the sequence 1,1,1,2,3,4,6,9,13,19,28 converges 
>>to 1.46557123188, which has the property I was looking for.
> 
> 
> How did you find this value?

Gene showed us an analytical approach. I used brute force...  I 
generated a part of the sequence and taken the ratio of the last two 
generated elements:

function [ H,lim ] = recorrencia (n,x,y,seed)
H=ones(n,1);
if(nargin>3 && ~isempty(seed))
   H(1:length(seed))=seed;
endif
for(i=(max([x,y])+1):n)
   H(i)=H(i-x)+H(i-y);
endfor
lim=H(n)/H(n-1)
endfunction

In that case, I called somethink like this to obtain the limit (lim):
[H,lim]=recorrencia(1000,3,2);



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From: Jon Szanto (2006-04-06)
Subject: Re: Interval of equivalence and difference tones

--- In tuning@yahoogroups.com, Hudson Lacerda <hfmlacerda@...> wrote:
> function [ H,lim ] = recorrencia (n,x,y,seed)

Is everyone around here aware of the tuning-math list? Hmmm??

Jon
From: Hudson Lacerda (2006-04-06)
Subject: Re: [tuning] Re: Interval of equivalence and difference tones

Jon Szanto escreveu:
> --- In tuning@yahoogroups.com, Hudson Lacerda <hfmlacerda@...> wrote:
> 
>>function [ H,lim ] = recorrencia (n,x,y,seed)
> 
> 
> Is everyone around here aware of the tuning-math list? Hmmm??
> 
> Jon

I got no feedback from there...
Cheers,
Hudson

-- 
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*N\ufffdo deixe seu voto sumir! http://www.votoseguro.org/
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From: Jon Szanto (2006-04-06)
Subject: Re: Interval of equivalence and difference tones

Hi Hudson,

--- In tuning@yahoogroups.com, Hudson Lacerda <hfmlacerda@...> wrote:
> I got no feedback from there...

I'm sorry about that, and I *don't* mean to make you feel bad. I'm
willing to bet that virtually every member of tuning-math is a member
here, and they could certainly have taken up the dialog there, where
entire posts of calculations are not only common, but welcome! :)

Cheers,
Jon
From: Kraig Grady (2006-04-07)
Subject: Re: Re: Interval of equivalence and difference tones

this one is the first diagonal off of mt meru after the fibonacci.
 except it is is a far easier form to actually understand
 

> Subject: Re: Re: Interval of equivalence and difference tones
> Back to the recurrences, the sequence 1,1,1,2,3,4,6,9,13,19,28 converges 
> >>to 1.46557123188, which has the property I was looking for.
>
>   

-- 
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles
From: Kraig Grady (2006-04-07)
Subject: Re: Interval of equivalence and difference tones

somehow i hadn't mailed this earlier


Hello J.Smith
	The scales that are formed have exactly the property of making scales where quite often intervals will differ more than repeating.
at least when you are looking at recurrent sequences, especially in the area where the pattern is still forming.
 In cultures who use a limited number of pitches, they never get a chance to converge.
 
I am not sure how much music in the world today exist in which does not revel in the delight of multiple tones .
 most of these have developed some idea of what  a acquired sounding together  sounds like.

the latter most common as vocal traditions where the intonation is constantly being adjusted to capture the desired nuance in the intent of the words and music.
certain patterns are built up culturally that are as a consistent as spelling in English; applied in only particular musical situations





Message: 5         
   Date: Sun, 02 Apr 2006 22:20:56 -0000
   From: "J.Smith" 

Kraig,

It seems likely that cultures employing primarily 
monophonic/heterophonic forms of music-- as opposed to harmonic 
(chordal) --might choose to delight in complex melodic intervals that 
might provide more variety, intensity or nuance.

Just a thought.  :-) 

-- 
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles