A catalog of chords

I think a catalog of the important JI chords up to some limit would be
a useful thing to have. A start might be made with the list of chords
buried inside of Scala somewhere, which could be weeded out. Or
perhaps someone else has such a catalog of chords already?

>I think a catalog of the important JI chords up to some limit would be
>a useful thing to have. A start might be made with the list of chords
>buried inside of Scala somewhere, which could be weeded out. Or
>perhaps someone else has such a catalog of chords already?

I like product-limited chords, ie for a:b:c, a*b*c is <= some bound.
Paul believes this will agree with harmonic entropy for JI chords,
but it'd be nice to have a working formulation of harmonic entropy
to include irrational chords.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> I like product-limited chords, ie for a:b:c, a*b*c is <= some bound.
> Paul believes this will agree with harmonic entropy for JI chords,
> but it'd be nice to have a working formulation of harmonic entropy
> to include irrational chords.

That sounds like a good place to start. I'd add that a, b, and c are
odd numbers and that gcd(a,b,c)=1. We might also filter with respect
to the rms of the octave-reduced 1, b/a, c/a compared to 3-equal, or
at least calculate this for purposes of comparison. Of course, I
assume that in every case we also will add 1/a:1/b:1/c to the catalog,
which would not need an independent listing, but which might need
weeding to keep the same chord from appearing twice.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> I like product-limited chords, ie for a:b:c, a*b*c is <= some bound.
> Paul believes this will agree with harmonic entropy for JI chords,
> but it'd be nice to have a working formulation of harmonic entropy
> to include irrational chords.

I'm looking at what you get in this way, and I don't like it. More is
needed than just bounding the product; as it is, 3:7:9 scores a lot
lower than a great many chords I would regard as of less significance.
One feature of 3:7:9 is that the top number is bounded at the rather
low bound of 9, and that should be worth something. I tried
a^2+b^2+c^2 instead, and the results seem much better

1:3:5 35
1:3:7 59
1:5:7 75
3:5:7 83
1:3:9 91
1:5:9 107
3:5:9 115
1:7:9 131
1:3:11 131
3:7:9 139
1:5:11 147
5:7:9 155
3:5:11 155

This is much better, but it is not taking into consideration how
evenly spaced the chord elements are within the octave.

>> I like product-limited chords, ie for a:b:c, a*b*c is <= some bound.
>> Paul believes this will agree with harmonic entropy for JI chords,
>> but it'd be nice to have a working formulation of harmonic entropy
>> to include irrational chords.
>
>That sounds like a good place to start. I'd add that a, b, and c are
>odd numbers

Why?

>and that gcd(a,b,c)=1.

Yes.

>We might also filter with respect to the rms of the octave-reduced
>1, b/a, c/a compared to 3-equal, or at least calculate this for
>purposes of comparison.

I don't understand this. You've divided everything by a, which
changes nothing and then mentioned rms deviation from an ET that
may be bad.

>Of course, I assume that in every case we also will add 1/a:1/b:1/
>to the catalog,

Why assume that? There's little empirical support for such an
idea.

-Carl

>> I like product-limited chords, ie for a:b:c, a*b*c is <= some bound.
>> Paul believes this will agree with harmonic entropy for JI chords,
>> but it'd be nice to have a working formulation of harmonic entropy
>> to include irrational chords.
>
>I'm looking at what you get in this way, and I don't like it. More is
>needed than just bounding the product; as it is, 3:7:9 scores a lot
>lower than a great many chords I would regard as of less significance.
>One feature of 3:7:9 is that the top number is bounded at the rather
>low bound of 9, and that should be worth something. I tried
>a^2+b^2+c^2 instead, and the results seem much better
>
>1:3:5 35
>1:3:7 59
>1:5:7 75
>3:5:7 83
>1:3:9 91
>1:5:9 107
>3:5:9 115
>1:7:9 131
>1:3:11 131
>3:7:9 139
>1:5:11 147
>5:7:9 155
>3:5:11 155

It would help to be able to see the list you prefer this to.

>This is much better, but it is not taking into consideration how
>evenly spaced the chord elements are within the octave.

This sounds like your 3-equal idea again... your catalog has other
goals than a ranking by concordance?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> This sounds like your 3-equal idea again... your catalog has other
> goals than a ranking by concordance?

Unequal spacing is likely to be dissonant just because of that,
because of the smaller intervals, so we could simply add a
critical-band dissonance multiplier.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> >That sounds like a good place to start. I'd add that a, b, and c are
> >odd numbers
>
> Why?

What use is there in the even numbers if we are classifying up to
octave equivalence?

> >Of course, I assume that in every case we also will add 1/a:1/b:1/
> >to the catalog,
>
> Why assume that? There's little empirical support for such an
> idea.

It works for 1-3-5 and 1-3-5-7, it seems to me. Do you not think minor
triads are soundly based?

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> It would help to be able to see the list you prefer this to.

If you take products, the first which is not of the form 1:a:b is
3:5:7, with a product of 105. Before you hit this, you will go through
1:3:x up to 1:3:35, and 1:5:x up to 1:5:21, which I think shows this
is not a good system.

>> This sounds like your 3-equal idea again... your catalog has
>> other goals than a ranking by concordance?
>
>Unequal spacing is likely to be dissonant just because of that,
>because of the smaller intervals, so we could simply add a
>critical-band dissonance multiplier.

A critical band penalty is one thing, though your original proposal
was hardly that. However, the product rule takes care of the smaller
intervals because the numbers get big, and harmonic entropy takes
care of them for more or less the same reason. So critical band
considerations seem unnecessary, at least for typical timbres. In
fact, the harmonic entropy concept has already been found to match
subjective listening experience fairly well, so we can safely
concentrate on implementing it.

-Carl

>> >That sounds like a good place to start. I'd add that a, b, and
>> >c are odd numbers
>>
>> Why?
>
>What use is there in the even numbers if we are classifying up to
>octave equivalence?

Oh, did you say you wanted octave equivalence? That's a different
kind of inquiry. The product rule fails because it doesn't return
the same values for octave equivalent pairs, or even assign them
contiguous swaths of a ranking.

My own listening tells me that octave-equivalent formulations are
of limited use beyond the 7-limit. Even in the 5-limit, it's barely
a passable convenience.

>> >Of course, I assume that in every case we also will add 1/a:1/b:1/
>> >to the catalog,
>>
>> Why assume that? There's little empirical support for such an
>> idea.
>
>It works for 1-3-5 and 1-3-5-7, it seems to me.

It works reasonably well in the 7-limit (but have you listened
to all voicings?) but not beyond.

>Do you not think minor triads are soundly based?

I don't know what "minor" is, but it's not equiv. to "utonal".
I've proposed that minor = 'smooth but weakly rooted', and to
some extent, utonal chords deliver that, at least through the
7-limit.

-Carl

>> It would help to be able to see the list you prefer this to.
>
>If you take products, the first which is not of the form 1:a:b is
>3:5:7, with a product of 105. Before you hit this, you will go through
>1:3:x up to 1:3:35, and 1:5:x up to 1:5:21, which I think shows this
>is not a good system.

I usually use it with a width bound on the chord.

-Carl

Hi Gene and Carl (and anyone else paying attention),

Carl proposed product-limited chords, ie for a:b:c, a*b*c is <= k.

Gene countered with a^2+b^2+c^2 <= k.

May I suggest you try instead the following function -

For each chord element a_i, with i = 1, 2, 3, ... (corresponding to your a,
b, c,
and allowing tetrads etc.) write a_i = Product_j p_j^k_i_j as the unique
factorisation of a_i into powers k_i_j of primes p_j, for j = 1, 2, 3, ... .
For each j, take m_j to be the max power k_i_j for all i: m_j = max_i
(k_i_j).
Then PMPP, the product_j p_j^m_j is well defined for all chords
a_1:a_2:a_3:...
with integer a_i. (I'm sure there's a name for this product in the theory
of
numbers - didn't Euler use it?)

Anyway, bounding this product of maximum prime powers (PMPP) that divide
the chord elements will change your ranking. For distinct prime powers, the
result is the same as Carl's product, eg
PMPP (1:3:5) = 15,
PMPP (3:5:7) = 105, etc
but
PMPP (1:3:7) = 21,
PMPP (1:7:9) = 63, and interestingly,
PMPP (3:7:9) = 63.

My motivation for choosing this measure is simple: all chord elements with
the same PMPP can be reached from unison in the same number of steps
in each prime dimension. By which I mean, when you've found your way to
the note 9, you've already traversed the distance to the note 3, so there
is nothing more "remote" about it. A simple analogy here is the way we
speak
of F# as being more remote from C than D is, simply because we need to
take more steps along the circle (or spiral) of fifths to reach it.

Regards,
Yahya

-----Original Message-----
________________________________________________________________________

Message: 10
Date: Thu, 10 Mar 2005 03:52:12 -0000
From: "Gene Ward Smith" <gwsmith@svpal.org>
Subject: Re: A catalog of chords

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> I like product-limited chords, ie for a:b:c, a*b*c is <= some bound.
> Paul believes this will agree with harmonic entropy for JI chords,
> but it'd be nice to have a working formulation of harmonic entropy
> to include irrational chords.

I'm looking at what you get in this way, and I don't like it. More is
needed than just bounding the product; as it is, 3:7:9 scores a lot
lower than a great many chords I would regard as of less significance.
One feature of 3:7:9 is that the top number is bounded at the rather
low bound of 9, and that should be worth something. I tried
a^2+b^2+c^2 instead, and the results seem much better

1:3:5 35
1:3:7 59
1:5:7 75
3:5:7 83
1:3:9 91
1:5:9 107
3:5:9 115
1:7:9 131
1:3:11 131
3:7:9 139
1:5:11 147
5:7:9 155
3:5:11 155

This is much better, but it is not taking into consideration how
evenly spaced the chord elements are within the octave.

________________________________________________________________________

--
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Checked by AVG Anti-Virus.
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--- In tuning-math@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:
>
> Hi Gene and Carl (and anyone else paying attention),

> Then PMPP, the product_j p_j^m_j is well defined for all chords
> a_1:a_2:a_3:...
> with integer a_i. (I'm sure there's a name for this product in the
theory
> of
> numbers - didn't Euler use it?)

Yes, it's the least common multiple, denoted LCM(a1, ..., an). I
thought about using the LCM somehow, but I figured I'd need to do more
than just use it as a metric.

>May I suggest you try instead the following function -
>
>For each chord element a_i, with i = 1, 2, 3, ... (corresponding to your a,
>b, c,
>and allowing tetrads etc.) write a_i = Product_j p_j^k_i_j as the unique
>factorisation of a_i into powers k_i_j of primes p_j, for j = 1, 2, 3, ... .
>For each j, take m_j to be the max power k_i_j for all i: m_j = max_i
>(k_i_j).
>Then PMPP, the product_j p_j^m_j is well defined for all chords
>a_1:a_2:a_3:...
>with integer a_i. (I'm sure there's a name for this product in the theory
>of
>numbers - didn't Euler use it?)

E-mail requires line breaks every 76 chars.

-Carl

>Hi Gene and Carl (and anyone else paying attention),
>
>Carl proposed product-limited chords, ie for a:b:c, a*b*c is <= k.
>
>Gene countered with a^2+b^2+c^2 <= k.
>
>May I suggest you try instead the following function -
>
>For each chord element a_i, with i = 1, 2, 3, ... (corresponding to
>your a, b, c, and allowing tetrads etc.) write a_i = Product_j p_j^k_i_j
>as the unique factorisation of a_i into powers k_i_j of primes p_j,
>for j = 1, 2, 3, ... .

You lost me. You want a product here (capital pi)?

>Anyway, bounding this product of maximum prime powers (PMPP) that
>divide the chord elements will change your ranking. For distinct
>prime powers, the result is the same as Carl's product, eg
>PMPP (1:3:5) = 15,
>PMPP (3:5:7) = 105, etc
>but
>PMPP (1:3:7) = 21,
>PMPP (1:7:9) = 63, and interestingly,
>PMPP (3:7:9) = 63.
>
>My motivation for choosing this measure is simple: all chord elements
>with the same PMPP can be reached from unison in the same number of
>steps in each prime dimension. By which I mean, when you've found
>your way to the note 9, you've already traversed the distance to the
>note 3, so there is nothing more "remote" about it.

Hmm... isn't 9 farther along the 3 dimension than 3 itself?

-Carl