Yahoo Tuning Groups Ultimate Backup tuning-math Generalized notion of complexity for chords with consonant subsets

Utonal chords are chords in which each individual dyad is of low
complexity, but in which the overall chord is of high complexity.
Likewise, essentially tempered dyadic chords are chords in which each
individual dyad is of low complexity, but in which the overall chord
is of high complexity, and in a way that doesn't make sense as a
single JI sonority.

Both of these chords share the property that despite being complex on
the whole, they have subsets that are of low complexity; this is the
"sound" of those chords in a sense. Psychoacoustically, they contain
subsets which are concordant, whereas the entire chord on a whole is
less concordant. We can come up with a metric for chords that measures
this property by considering the complexity of each subset of each
chord and taking an average. By doing so, we can search for chords
with consonant subsets, rather than just consonant dyads. We can also,
if we want, relax the property that -every- dyad in a chord has to be
consonant, and look for chords that enough consonant subsets to
perhaps "make up" for one dissonant dyad.

Here's the algorithm I'm using:

1) Choose a chord C
2) Look at all the subsets of C; compute its power set.
3) For each subset, come up with a transversal generating the simplest
JI representation possible of the overall chord
4) Compute the complexity of this representation
5) Take an average of the above for all subsets of C

#4 and #5 are parameters to tweak. As we will see, the type of choices
we will use for these will determine how much emphasis we place on the
subchords of the chord being concordant, vs the entire chord being
concordant. On one extreme, we can have something like odd-limit, and
on the other extreme we'll care much more about the subchords of the
chord, not caring at all if the chord makes sense triadically or
____adically. The type of average we pick will allow us to draw a
gradient between the two.

As a first pass, I've calculated the above using geomean(chord) for
#4, and the simple arithmetic mean for #5. for all triads, tetrads,
pentads, and hexads within the 31-odd-limit. All of my examples can be
found here: /tuning-math/files/MikeBattaglia/Chord%20Subset%20Complexity/
. This approach can easily be adapted to work with essentially
tempered triads, which is where it'll really shine, but as a first
pass I've stuck to JI chords for now.

Before we look at odd-limit, though, let's take a peek at integer
limit with octaves thrown in. Here's the top 10 triads in the
31-integer-limit:

TRIADS - 31-INTEGER-LIMIT, ARITHMETIC MEAN OF GEOMETRIC MEAN COMPLEXITY
1.4041 - 1:2:4
1.4876 - 1:2:3
1.5550 - 1:2:6
1.6024 - 1:3:6
1.6804 - 1:2:8
1.6997 - 2:3:6
1.7096 - 1:2:5
1.7739 - 1:4:8
1.7806 - 1:3:9
1.7837 - 1:3:4

Note that this metric rates 1:2:4 higher than 1:2:3, because even
though 1:2:3 is more triadically simple, the subchords of 1:2:4,
counting 1:2:4 itself, are more simple on average. This is going to be
a common theme in these results. A lot of these are octave-equivalent
to the same thing, so we might look at triads in the 31-odd-limit
instead. Here are the top 10:

TRIADS - 31-ODD-LIMIT, ARITHMETIC MEAN OF GEOMETRIC MEAN COMPLEXITY
1.7806 - 1:3:9
1.9010 - 1:3:5
2.0569 - 1:3:15
2.1028 - 1:3:7
2.1512 - 1:5:15
2.2771 - 1:3:21
2.4176 - 3:5:15
2.4287 - 1:3:11
2.4384 - 1:5:7
2.4626 - 1:7:21

Note how in this case 1:3:9 beats 1:3:5. Here we also have the top 10
tetrads in the 31-odd-limit:

TETRADS - 31-ODD-LIMIT, ARITHMETIC MEAN OF GEOMETRIC MEAN COMPLEXITY
2.6057 - 1:3:9:15
2.6585 - 1:3:5:15
2.7229 - 1:3:5:9
2.7600 - 1:3:5:7
2.8104 - 1:3:9:27
2.8421 - 1:3:9:21
2.9876 - 1:3:7:9
3.0754 - 1:3:7:21
3.1243 - 1:3:5:11
3.2094 - 1:3:15:21

A few stacked 5-limit chords beat out 1:3:5:7. Here's the top 10
pentads in the 31-odd-limit:

PENTADS - 31-ODD-LIMIT, ARITHMETIC MEAN OF GEOMETRIC MEAN COMPLEXITY
3.5010 - 1:3:5:9:15
3.5741 - 1:3:5:7:9
3.6165 - 1:3:9:15:21
3.7121 - 1:3:9:15:27
3.7892 - 1:3:5:7:15
3.8598 - 1:3:5:7:11
3.9659 - 1:3:5:9:11
3.9906 - 1:3:9:21:27
4.0121 - 1:3:7:9:21
4.0153 - 1:3:5:7:13

The 1:3:5:7:9 otonality does a lot better here than 1:3:5:7 in the
last one; it's only beaten by 1:3:5:9:15. Here's the top 10 hexads in
the 31-odd-limit:

HEXADS - 31-ODD-LIMIT, ARITHMETIC MEAN OF GEOMETRIC MEAN COMPLEXITY
4.4594 - 1:3:5:7:9:15
4.5514 - 1:3:5:7:9:11
4.6747 - 1:3:9:15:21:27
4.7038 - 1:3:5:7:9:13
4.7129 - 1:3:5:7:9:21
4.7423 - 1:3:5:9:15:21
4.8769 - 1:3:5:9:11:15
4.9022 - 1:3:5:7:11:15
4.9256 - 1:3:5:9:15:27
4.9408 - 1:3:5:7:11:13

As you can see, there's a marked tendency with these parameters for
things like major 9 chords to beat out 4:5:6:7:9. Although I tend to
like this result, I've noticed in the past that people have sometimes
disagreed on this point, sometimes preferring the latter and eschewing
chords that have even a single dissonant dyad, in this case the major
7. If this is how you feel, you might want to change the type of
average that's being computed in step #5. Moving from a simple
arithmetic mean to an RMS can change the balance of this - see here
the top 10 tetrads in the 31-odd-limit, using RMS instead of the usual
arithmetic mean to figure out the average complexity:

TETRADS - 31-ODD-LIMIT, RMS OF GEOMETRIC MEAN COMPLEXITY
2.9232 - 1:3:9:15
3.0388 - 1:3:5:15
3.1294 - 1:3:5:7
3.1588 - 1:3:5:9
3.2372 - 1:3:9:21
3.2727 - 1:3:9:27
3.5508 - 1:3:7:9
3.6170 - 1:3:7:21
3.6493 - 1:3:5:11
3.7078 - 1:3:15:21

You can now see that 1:3:5:7 is greater than 1:3:5:9, and 1:3:9:21 is
greater than 1:3:9:27. This may still not be enough for you, however.

Consider that the shift from the arithmetic mean to RMS is somewhat
analogous to the shift from using an L1 to an L2 norm in a space.
Hence, we might consider something analogous to the Linfty norm, and
take as the "average" the subchord that is highest in complexity. But
here's what we get for tetrads:

TETRADS - 31-ODD-LIMIT, MAX OF GEOMETRIC MEAN COMPLEXITY
0.3420 - 1:3:9:15
0.3826 - 1:3:9:21
0.3944 - 1:3:5:7
0.4055 - 1:3:5:15
0.4160 - 1:3:9:27
0.4472 - 1:3:5:9
0.4536 - 1:3:15:21
0.4807 - 1:5:15:25
0.4865 - 1:9:15:21
0.4932 - 1:3:15:27

Now all of the chords with a lot of 3-limit ratios are jumping up to
the top. This may mean that geomean(...) has its limits. Of course,
you may not think it desirable for us to come up with a property
mimicking odd-limit at all here, since we already have odd-limit. But
next time, just to see what happens, I'll experiment with 1/(1/a^2 +
1/b^2 + 1/c^2 + ...) complexity, and we'll see what comes out.

-Mike

By the way, all of these lists are uploaded in full to my folder here

/tuning-math/files/MikeBattaglia/Chord%20Subset%20Complexity/

Feel free to check them out. Also, feel free to suggest tips on how to
code up some of this for essentially tempered chords, which is the
only thing I really care about in this business.

-Mike

On Wed, Jan 11, 2012 at 5:25 AM, Mike Battaglia <battaglia01@gmail.com> wrote:
> Utonal chords are chords in which each individual dyad is of low
> complexity, but in which the overall chord is of high complexity.
> Likewise, essentially tempered dyadic chords are chords in which each
> individual dyad is of low complexity, but in which the overall chord
> is of high complexity, and in a way that doesn't make sense as a
> single JI sonority.
>
> Both of these chords share the property that despite being complex on
> the whole, they have subsets that are of low complexity; this is the
> "sound" of those chords in a sense. Psychoacoustically, they contain
> subsets which are concordant, whereas the entire chord on a whole is
> less concordant. We can come up with a metric for chords that measures
> this property by considering the complexity of each subset of each
> chord and taking an average. By doing so, we can search for chords
> with consonant subsets, rather than just consonant dyads. We can also,
> if we want, relax the property that -every- dyad in a chord has to be
> consonant, and look for chords that enough consonant subsets to
> perhaps "make up" for one dissonant dyad.
>
> Here's the algorithm I'm using:
>
> 1) Choose a chord C
> 2) Look at all the subsets of C; compute its power set.
> 3) For each subset, come up with a transversal generating the simplest
> JI representation possible of the overall chord
> 4) Compute the complexity of this representation
> 5) Take an average of the above for all subsets of C
>
> #4 and #5 are parameters to tweak. As we will see, the type of choices
> we will use for these will determine how much emphasis we place on the
> subchords of the chord being concordant, vs the entire chord being
> concordant. On one extreme, we can have something like odd-limit, and
> on the other extreme we'll care much more about the subchords of the
> chord, not caring at all if the chord makes sense triadically or
> ____adically. The type of average we pick will allow us to draw a
> gradient between the two.
>
> As a first pass, I've calculated the above using geomean(chord) for
> #4, and the simple arithmetic mean for #5. for all triads, tetrads,
> pentads, and hexads within the 31-odd-limit. All of my examples can be
> found here: /tuning-math/files/MikeBattaglia/Chord%20Subset%20Complexity/
> . This approach can easily be adapted to work with essentially
> tempered triads, which is where it'll really shine, but as a first
> pass I've stuck to JI chords for now.
>
> Before we look at odd-limit, though, let's take a peek at integer
> limit with octaves thrown in. Here's the top 10 triads in the
> 31-integer-limit:
>
> TRIADS - 31-INTEGER-LIMIT, ARITHMETIC MEAN OF GEOMETRIC MEAN COMPLEXITY
> 1.4041 - 1:2:4
> 1.4876 - 1:2:3
> 1.5550 - 1:2:6
> 1.6024 - 1:3:6
> 1.6804 - 1:2:8
> 1.6997 - 2:3:6
> 1.7096 - 1:2:5
> 1.7739 - 1:4:8
> 1.7806 - 1:3:9
> 1.7837 - 1:3:4
>
> Note that this metric rates 1:2:4 higher than 1:2:3, because even
> though 1:2:3 is more triadically simple, the subchords of 1:2:4,
> counting 1:2:4 itself, are more simple on average. This is going to be
> a common theme in these results. A lot of these are octave-equivalent
> to the same thing, so we might look at triads in the 31-odd-limit
> instead. Here are the top 10:
>
> TRIADS - 31-ODD-LIMIT, ARITHMETIC MEAN OF GEOMETRIC MEAN COMPLEXITY
> 1.7806 - 1:3:9
> 1.9010 - 1:3:5
> 2.0569 - 1:3:15
> 2.1028 - 1:3:7
> 2.1512 - 1:5:15
> 2.2771 - 1:3:21
> 2.4176 - 3:5:15
> 2.4287 - 1:3:11
> 2.4384 - 1:5:7
> 2.4626 - 1:7:21
>
> Note how in this case 1:3:9 beats 1:3:5. Here we also have the top 10
> tetrads in the 31-odd-limit:
>
> TETRADS - 31-ODD-LIMIT, ARITHMETIC MEAN OF GEOMETRIC MEAN COMPLEXITY
> 2.6057 - 1:3:9:15
> 2.6585 - 1:3:5:15
> 2.7229 - 1:3:5:9
> 2.7600 - 1:3:5:7
> 2.8104 - 1:3:9:27
> 2.8421 - 1:3:9:21
> 2.9876 - 1:3:7:9
> 3.0754 - 1:3:7:21
> 3.1243 - 1:3:5:11
> 3.2094 - 1:3:15:21
>
> A few stacked 5-limit chords beat out 1:3:5:7. Here's the top 10
> pentads in the 31-odd-limit:
>
> PENTADS - 31-ODD-LIMIT, ARITHMETIC MEAN OF GEOMETRIC MEAN COMPLEXITY
> 3.5010 - 1:3:5:9:15
> 3.5741 - 1:3:5:7:9
> 3.6165 - 1:3:9:15:21
> 3.7121 - 1:3:9:15:27
> 3.7892 - 1:3:5:7:15
> 3.8598 - 1:3:5:7:11
> 3.9659 - 1:3:5:9:11
> 3.9906 - 1:3:9:21:27
> 4.0121 - 1:3:7:9:21
> 4.0153 - 1:3:5:7:13
>
> The 1:3:5:7:9 otonality does a lot better here than 1:3:5:7 in the
> last one; it's only beaten by 1:3:5:9:15. Here's the top 10 hexads in
> the 31-odd-limit:
>
> HEXADS - 31-ODD-LIMIT, ARITHMETIC MEAN OF GEOMETRIC MEAN COMPLEXITY
> 4.4594 - 1:3:5:7:9:15
> 4.5514 - 1:3:5:7:9:11
> 4.6747 - 1:3:9:15:21:27
> 4.7038 - 1:3:5:7:9:13
> 4.7129 - 1:3:5:7:9:21
> 4.7423 - 1:3:5:9:15:21
> 4.8769 - 1:3:5:9:11:15
> 4.9022 - 1:3:5:7:11:15
> 4.9256 - 1:3:5:9:15:27
> 4.9408 - 1:3:5:7:11:13
>
> As you can see, there's a marked tendency with these parameters for
> things like major 9 chords to beat out 4:5:6:7:9. Although I tend to
> like this result, I've noticed in the past that people have sometimes
> disagreed on this point, sometimes preferring the latter and eschewing
> chords that have even a single dissonant dyad, in this case the major
> 7. If this is how you feel, you might want to change the type of
> average that's being computed in step #5. Moving from a simple
> arithmetic mean to an RMS can change the balance of this - see here
> the top 10 tetrads in the 31-odd-limit, using RMS instead of the usual
> arithmetic mean to figure out the average complexity:
>
> TETRADS - 31-ODD-LIMIT, RMS OF GEOMETRIC MEAN COMPLEXITY
> 2.9232 - 1:3:9:15
> 3.0388 - 1:3:5:15
> 3.1294 - 1:3:5:7
> 3.1588 - 1:3:5:9
> 3.2372 - 1:3:9:21
> 3.2727 - 1:3:9:27
> 3.5508 - 1:3:7:9
> 3.6170 - 1:3:7:21
> 3.6493 - 1:3:5:11
> 3.7078 - 1:3:15:21
>
> You can now see that 1:3:5:7 is greater than 1:3:5:9, and 1:3:9:21 is
> greater than 1:3:9:27. This may still not be enough for you, however.
>
> Consider that the shift from the arithmetic mean to RMS is somewhat
> analogous to the shift from using an L1 to an L2 norm in a space.
> Hence, we might consider something analogous to the Linfty norm, and
> take as the "average" the subchord that is highest in complexity. But
> here's what we get for tetrads:
>
> TETRADS - 31-ODD-LIMIT, MAX OF GEOMETRIC MEAN COMPLEXITY
> 0.3420 - 1:3:9:15
> 0.3826 - 1:3:9:21
> 0.3944 - 1:3:5:7
> 0.4055 - 1:3:5:15
> 0.4160 - 1:3:9:27
> 0.4472 - 1:3:5:9
> 0.4536 - 1:3:15:21
> 0.4807 - 1:5:15:25
> 0.4865 - 1:9:15:21
> 0.4932 - 1:3:15:27
>
> Now all of the chords with a lot of 3-limit ratios are jumping up to
> the top. This may mean that geomean(...) has its limits. Of course,
> you may not think it desirable for us to come up with a property
> mimicking odd-limit at all here, since we already have odd-limit. But
> next time, just to see what happens, I'll experiment with 1/(1/a^2 +
> 1/b^2 + 1/c^2 + ...) complexity, and we'll see what comes out.
>
> -Mike

Generalized notion of complexity for chords with consonant subsets

🔗Mike Battaglia <battaglia01@gmail.com>

🔗Mike Battaglia <battaglia01@gmail.com>

🔗Kalle Aho <kalleaho@mappi.helsinki.fi>