Evenness measure

An evenness measure I'm finding useful, for octave-equivalent chords or
scales, is the geometric mean of the ratio between step sizes and the
average step size. If we have an n-element chord or scale in Scala
format "a", in ascending order, this will be

(a[1] * product(a[i+1]-a[i], i=1..n-1))^(1/n) * n/a[n]

If "a" is exactly even, ie an equal temperament, including the
diminished seventh and augmented triads in 12-et, then this gives a
value of 1. Close to 1 is close to evenness, and the measure is
sensitive to small intervals in the mix, which brings it down.

Has someone proposed this already, and if so with what name? What else
is along these lines? This would be a good statistic to add to Scala,
it could maybe be called "mean evenness" or something.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:

> An evenness measure I'm finding useful, for octave-equivalent chords
or
> scales, is the geometric mean of the ratio between step sizes and the
> average step size.

If anyone is wondering why this is always less than or equal to one,
the answer is the arithmetic-geometric inequality, which says that the
geometric mean of any set of positive real numbers is always less than
or equal to the arithmetic mean. Since the arithmetic mean of the ratio
between step size and arithmetic mean step size is obviously exactly 1,
the geometric mean of these ratios is <= 1.

The arithmetic-geometric inequality can be extended to the arithmetic-
geometric-harmonic inequality, which says that the geometric mean is
always greater than or equal to the harmonic mean. This suggests using
the harmonic mean of the ratios of step sizes, which I haven't been
doing, but maybe I should. It is even more sensitive to small intervals
gumming up the works, and for scales or chords given in terms of equal
temperaments, will involve only rational number computations.

If "a" again is our scale or chord in Scala format (say, in cents,
ending with 1200) then the harmonic evenness, which is what we could
call this measure, will be n^2/(a[n]*s), where s is the sum

s = 1/a[i] + sum(1/(a[i+1]-a[i]),i=1..n-1)

I guess we could call the first geometric evenness and the second
harmonic evenness.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:

> s = 1/a[i] + sum(1/(a[i+1]-a[i]),i=1..n-1)

s = 1/a[1] + sum(1/(a[i+1]-a[i]),i=1..n-1)

Gene Ward Smith wrote:
> An evenness measure I'm finding useful, for octave-equivalent chords or > scales, is the geometric mean of the ratio between step sizes and the > average step size. If we have an n-element chord or scale in Scala > format "a", in ascending order, this will be
> > (a[1] * product(a[i+1]-a[i], i=1..n-1))^(1/n) * n/a[n]
> > If "a" is exactly even, ie an equal temperament, including the > diminished seventh and augmented triads in 12-et, then this gives a > value of 1. Close to 1 is close to evenness, and the measure is > sensitive to small intervals in the mix, which brings it down.
> > Has someone proposed this already, and if so with what name? What else > is along these lines? This would be a good statistic to add to Scala, > it could maybe be called "mean evenness" or something.

Sounds potentially useful; could you give some examples (like TOP meantone[7], hanson[11], magic[19], or familiar JI scales such as the Ellis Duodene or Partch tonality diamond)?

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:

> Sounds potentially useful; could you give some examples (like TOP
> meantone[7], hanson[11], magic[19], or familiar JI scales such as the
> Ellis Duodene or Partch tonality diamond)?

Meantone[7] in 31-et

geometric: .97571053337611930293
harmonic: .94838709677419354839

Hanson[11] in 53-et

geometric: .88947918834864209958
harmonic: .80577136514983351831

Werckmeister3

geometric: .99788786838187169000
harmonic: .99576953661240383825

Duodene

geometric: .97924320786164421908
harmonic: .95730414586760375464

Triad

geometric: .98266958356112087700
harmonic: .96592552022038379182

Tetrad

geometric: .98859990604452346811
harmonic: .97745561686447977552

7-limit diamond

geometric: .81619888500293970819
harmonic: .68983112481637305460

19-limit diamond

geometric: .81430268495155224368
harmonic: .71057152789384988951

At 02:14 PM 2/27/2007, you wrote:
>An evenness measure I'm finding useful, for octave-equivalent chords or
>scales, is the geometric mean of the ratio between step sizes and the
>average step size. If we have an n-element chord or scale in Scala
>format "a", in ascending order, this will be
>
>(a[1] * product(a[i+1]-a[i], i=1..n-1))^(1/n) * n/a[n]
>
>If "a" is exactly even, ie an equal temperament, including the
>diminished seventh and augmented triads in 12-et, then this gives a
>value of 1. Close to 1 is close to evenness, and the measure is
>sensitive to small intervals in the mix, which brings it down.
>
>Has someone proposed this already, and if so with what name? What else
>is along these lines? This would be a good statistic to add to Scala,
>it could maybe be called "mean evenness" or something.

What advantage does it have to something like smallest/largest
step?

-Carl

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

> What advantage does it have to something like smallest/largest
> step?

It's more like smallest/average, and the point of it, I suppose, is
that it isn't utterly dependent on outliers. Whether it's better or
worse depends on what you want it for.

If you look at the triad, the smallest step is 6/5, and cents(6/5)/400
is 78.91%. Compare that to the tetrad, and cents(8/7)/300, which is
77.06%. By this measure, the tetrad is slightly more irregular.
However, it has two intervals, 7/6 and 6/5, which are pretty close to
the average size, whereas the triad has only one. By the mean evenness
measures, the tetrad comes out on average slightly more even, and that
also makes sense.

>> What advantage does it have to something like smallest/largest
>> step?
>
>It's more like smallest/average, and the point of it, I suppose, is
>that it isn't utterly dependent on outliers. Whether it's better or
>worse depends on what you want it for.
>
>If you look at the triad, the smallest step is 6/5, and cents(6/5)/400
>is 78.91%. Compare that to the tetrad, and cents(8/7)/300, which is
>77.06%. By this measure, the tetrad is slightly more irregular.
>However, it has two intervals, 7/6 and 6/5, which are pretty close to
>the average size, whereas the triad has only one. By the mean evenness
>measures, the tetrad comes out on average slightly more even, and that
>also makes sense.

I suppose, but I generally prefer things that take into account
the order of the 2nds. For example, I like to justify maximal
evenness in terms of the fact that it reduces Rothenberg mean
variety.

-Carl