Otonality Coefficient

This is a simple idea that will tell you "how otonal" a chord is. If it's
very positive, it's very otonal. If it's very negative, then it's very
utonal.

In intuitive terms, you compare the complexity of a chord expressed in
lowest terms, and the complexity of its inverse expressed in lowest terms,
and see which is greater. Whichever one has lower complexity (e.g. is
simpler) wins. For this model, I chose to represent complexity with Tenney
height, but you theoretically could use anything that you want.

For example: 10:12:15 has a Tenney height of 1800. Its inverse, 4:5:6, has a
Tenney height of 120. Since this chord's inverse has a lower Tenney height
than the chord itself, it is very utonal, which in fact 10:12:15 is.

The actual definition for the otonality coefficient, using Tenney height as
a complexity metric, is:

OC(a:b:c:...) = log(LCM(a,b,c,...)^N/(a*b*c*...)^2)

Where LCM is the least common multiple of a:b:c:d:e:..., and N is the number
of notes in the chord. This was derived from the following, more intuitive,
and non-Tenney specific definition:
log(complexity(reduce(1/chord))/complexity(reduce(chord))).

Here's the OC for some chords:

Otonal chords
4:5:6 = 2.7081
5:6:7 = 5.3471
4:5:6:7 = 10.6942

Utonal inverses of the above
10:12:15 = -2.7081
30:35:42 = -5.3471
60:70:84:105 = -10.6942

ASS's
3:5:9:15 = 0
3:7:9:21 = 0
3:5:7:15:21:35 = 0

Dyads
2:3 = 0
3:4 = 0
1:2 = 0

Random number pentads with numbers between 1 and 100
40:45:69:71:74 = 17.611
40:43:77:83:89 = 27.386
33:44:68:80:95 = 6.9878
16:20:52:59:89 = 31.687
9:12:14:54:68 = -2.0149
6:8:25:45:90 = 11.015

Random number pentads with numbers between 1 and 2^53-1
2.967e+014 4.8515e+014 4.9736e+015 7.2514e+015 8.1795e+015 14.13
1.2053e+015 1.8699e+015 2.8652e+015 6.0967e+015 7.4451e+015 8.1403
2.9606e+015 5.8557e+015 6.41e+015 7.8496e+015 8.7805e+015 20.383
1.2962e+015 2.2922e+015 2.9197e+015 3.619e+015 5.458e+015 16.816
8.5234e+014 2.1087e+015 2.911e+015 6.6686e+015 6.9319e+015 12.312
2.93e+015 4.711e+015 5.017e+015 7.2985e+015 7.4926e+015 0.96508
2.1147e+015 4.8513e+015 6.8024e+015 8.8312e+015 8.9325e+015 31.48

Weird Pathological Cases
1:1:2 = 0.6931
1:2:2 = -0.6931
1:2:3:4:5:... = Infinity
1/(1:2:3:4:5:...) = -Infinity

This assumes that the chord that you're putting into it is being expressed
in lowest terms - e.g. if you want the otonality coefficient for 10:12:15,
put in 10:12:15, not 1/6:1/5:1/4. You could figure something out that works
there, but it would require generalizing the GCF and the LCM for fractions.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> ASS's
> 3:5:9:15 = 0
> 3:7:9:21 = 0
> 3:5:7:15:21:35 = 0

This score, it should be noted, is not octave equivalent: the score depends on the root of the chord and spacing of the parts. The numbers you get for [4,5,6], [5,6,8] and [3,4,5] are not the same, for instance, and [2,3,5] is, surprisingly, the same as [5,6,8], whereas [1,3,5] is the same as [4,5,6]. Some chords work out as otonal in one position, and utonal in another: [36,45,56] is otonal, but [28,36,45] is very slightly utonal, and [45,56,72] is distinctly utonal.

> This assumes that the chord that you're putting into it is being expressed
> in lowest terms - e.g. if you want the otonality coefficient for 10:12:15,
> put in 10:12:15, not 1/6:1/5:1/4. You could figure something out that works
> there, but it would require generalizing the GCF and the LCM for fractions.

I just coded it in the obvious way: cleared denominators and converted the fraction form to your integer ratios.

On Sat, Feb 5, 2011 at 12:33 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > ASS's
> > 3:5:9:15 = 0
> > 3:7:9:21 = 0
> > 3:5:7:15:21:35 = 0
>
> This score, it should be noted, is not octave equivalent: the score depends on the root of the chord and spacing of the parts. The numbers you get for [4,5,6], [5,6,8] and [3,4,5] are not the same, for instance, and [2,3,5] is, surprisingly, the same as [5,6,8], whereas [1,3,5] is the same as [4,5,6]. Some chords work out as otonal in one position, and utonal in another: [36,45,56] is otonal, but [28,36,45] is very slightly utonal, and [45,56,72] is distinctly utonal.

If you wanted n-equivalence, I suppose the way to go is to eliminate n
by factoring out every n^p for every number in the chord then. So
every one of those chords would end up reducing to 1:3:5 for a
2-equivalent version.

I hadn't thought of that, but I was going to come up with a different
type of "octave-equivalent" version next that did 2/(chord) as the
inverse instead of 1/(chord). This would extend the concept of
otonality and utonality to dyads, such that 3/2 is otonal and 4/3 is
utonal. I'm not sure how to reconcile that with this here, because for
both of these dyads if you do the above you end up with 1:3.

> > This assumes that the chord that you're putting into it is being expressed
> > in lowest terms - e.g. if you want the otonality coefficient for 10:12:15,
> > put in 10:12:15, not 1/6:1/5:1/4. You could figure something out that works
> > there, but it would require generalizing the GCF and the LCM for fractions.
>
> I just coded it in the obvious way: cleared denominators and converted the fraction form to your integer ratios.

Yeah, but in terms of putting it into an equation like above, that
would have things a bit more complicated to type, and I didn't feel
like working it out anyway :P

-Mike

Great post! I like that you show results against a wide
variety of examples. -Carl

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> This is a simple idea that will tell you "how otonal" a chord is.
> If it's very positive, it's very otonal. If it's very negative,
> then it's very utonal.
[snip]

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> If you wanted n-equivalence, I suppose the way to go is to eliminate n
> by factoring out every n^p for every number in the chord then. So
> every one of those chords would end up reducing to 1:3:5 for a
> 2-equivalent version.

That gives

Otonal tetrad [1,3,5,7]: 9.308
Utonal tetrad [15,21,35,105]: -9.308
Supermajor tetrad [35,45,63,105]: -9.308
Subminor tetrad [3,5,7,9]: 9.308
Diminished seventh [15,21,25,35]: 0.0

It all seems to make sense!

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

> It all seems to make sense!

Here's some augmented triads:

[7,9,45]: 1.358
[7,35,45]: -1.358
[7,11,35]: 2.061
[11,35,55]: -2.061
[1,5,25]: 0.0

If we temper out 225/224 then both [7,9,45] and [7,35,45] merge with [1,5,25]. If we temper out 176/175, the same thing happens with [7,11,35] and [11,35,55].

On Sat, Feb 5, 2011 at 2:39 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > If you wanted n-equivalence, I suppose the way to go is to eliminate n
> > by factoring out every n^p for every number in the chord then. So
> > every one of those chords would end up reducing to 1:3:5 for a
> > 2-equivalent version.
>
> That gives
>
> Otonal tetrad [1,3,5,7]: 9.308
> Utonal tetrad [15,21,35,105]: -9.308
> Supermajor tetrad [35,45,63,105]: -9.308
> Subminor tetrad [3,5,7,9]: 9.308
> Diminished seventh [15,21,25,35]: 0.0

Huh. 3:5:7:9 and 1:3:5:7 end up being equivalent in otonalness. Weird.
The pattern breaks with 5:7:9:11, which gets you 16.301. I'm not sure
why that is.

If things are syncing up like that, then maybe a more sensible base
for the log could be picked that would make these all integers or
rationals or something, although I'm not sure quite why it worked out
like that.

-Mike