Measure for "Quasi-Equalness" of a scale

Sometimes it may be useful to find how much a scale deviates from being quasi-equal. By quasi-equal, I am referring to scales of N-notes where the size of each note is approximately P/N, where P is the period in cents.

Let us notate our scale of n-notes using the sequence A1, A2 ... , An. These represent the sizes of steps in cents. The diatonic scale in 12-equal we would write as [200, 200, 100, 200, 200, 200, 100], for example.

The formulas below tell us how to calculate the period (P) and the average step size (Q).

P = sum_ i (Ai)
Q = P/n

We can then find how much it deviates (on average) from being quasi-equal with the following summation equation:

D = 1- sum_ j abs(Aj/P-1/n)
D = 1- 1/(n*P) sum_ j abs(n*Aj-P)

Multiply by 100 to yield a percent:

D% = 100 - 100/(n*P) sum_ j abs(n*Aj-P)

Returning to the previous example, our diatonic scale in 12-equal is 76.2% quasi-equal. The diatonic scale in 26-equal is much closer to 7-equal, and is consequently 89.0% quasi-equal.

Ryan

--- In tuning-math@yahoogroups.com, "Ryan Avella" <domeofatonement@...> wrote:
> We can then find how much it deviates (on average) from being quasi-equal with the following summation equation:
>
> D = 1- sum_ j abs(Aj/P-1/n)
> D = 1- 1/(n*P) sum_ j abs(n*Aj-P)
>
> Multiply by 100 to yield a percent:
>
> D% = 100 - 100/(n*P) sum_ j abs(n*Aj-P)

Okay, I actually got something wrong here, so I'm going to correct my mistake. The above equations should actually read as follows.

D = 2 - sum_ j abs(Aj/P-1/n)
D = 2 - 1/(n*P) sum_ j abs(n*Aj-P)
D% = 100 - 50/(n*P) sum_ j abs(n*Aj-P)

This changes some of my results from above. The diatonic scale in 12-equal is actually 52.4% quasi-equal, and in 26-equal it is 78.0% quasi-equal.

Mike told me he is thinking of possible using RMS and/or the Standard Deviation to model the same thing, with negligible differences. He also had the cool idea of using the multiset of differences between notes in the scale (e.g. every interval) instead of just small steps. This would make more sense, since we also have to consider rare cases like Blackwood where half of it is perfectly equal and the other half isn't quite.

Ryan

--- In tuning-math@yahoogroups.com, "Ryan Avella" <domeofatonement@...> wrote:

> We can then find how much it deviates (on average) from being quasi-equal with the following summation equation:
>
> D = 1- sum_ j abs(Aj/P-1/n)
> D = 1- 1/(n*P) sum_ j abs(n*Aj-P)

Why not 1 - sum_j abs(Aj/P - j/n)?

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning-math@yahoogroups.com, "Ryan Avella" <domeofatonement@> wrote:
>
> > We can then find how much it deviates (on average) from being quasi-equal with the following summation equation:
> >
> > D = 1- sum_ j abs(Aj/P-1/n)
> > D = 1- 1/(n*P) sum_ j abs(n*Aj-P)
>
> Why not 1 - sum_j abs(Aj/P - j/n)?
>

It is 1/n as opposed to j/n, because Aj represents a sequence of steps. E.g. (2, 2, 1, 2, 2, 2, 1).

Also, in case if you missed my second post, I made a correction to the equation. It should actually be the following:

D = 2 - sum_ j abs(Aj/P-1/n)
D = 2 - 1/(n*P) sum_ j abs(n*Aj-P)

It is very important to replace the 1 with a 2, otherwise negative values can result for certain scales.

Ryan

--- In tuning-math@yahoogroups.com, "Ryan Avella" <domeofatonement@...> wrote:

> It is very important to replace the 1 with a 2, otherwise negative values can result for certain scales.

Isn't the obvious solution to leave off the subtraction?

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning-math@yahoogroups.com, "Ryan Avella" <domeofatonement@> wrote:
>
> > It is very important to replace the 1 with a 2, otherwise negative values can result for certain scales.
>
> Isn't the obvious solution to leave off the subtraction?
>

I suppose we could leave off subtraction, though I don't know what to call the resulting measure. Should it then be the measure of how unequal a scale is?

Ryan

--- In tuning-math@yahoogroups.com, "Ryan Avella" <domeofatonement@...> wrote:

> I suppose we could leave off subtraction, though I don't know what to call the resulting measure. Should it then be the measure of how unequal a scale is?

Right. Call it unevenness or something.

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning-math@yahoogroups.com, "Ryan Avella" <domeofatonement@> wrote:
>
> > I suppose we could leave off subtraction, though I don't know what to call the resulting measure. Should it then be the measure of how unequal a scale is?
>
> Right. Call it unevenness or something.
>

Alright, check out this file I've uploaded to tuning-math.

http://f1.grp.yahoofs.com/v1/gCRITxf_g8gFI-QR3sBLi7kLLmjWIzs5b-otIq_vn8hs2YfTBh_-Ha4PcQrok10vdYfMXfaVdGtEz6osvWm-Rc-oSFLUiNa6GBbtqik/Ryan/Measuring%20Scale%20Deviations%20from%20Optimal%20Spacing.pdf

I've improved the algorithm so that it looks at every possible interval, and not just the smallest steps.

If you take the decimal form of "quasi-equalness" and raise it to the power of ln(2)/ln(12/11), it normalizes the ratings so that 12-equal diatonic is exactly 50% quasi-equal. This could be useful for the purposes of comparing scales. Using this normalization, 19-equal diatonic is 65% quasi-equal, and Dicot[7] is about 71% quasi-equal.

ryan

--- In tuning-math@yahoogroups.com, "Ryan Avella" <domeofatonement@...> wrote:
> Alright, check out this file I've uploaded to tuning-math.
>
> http://f1.grp.yahoofs.com/v1/gCRITxf_g8gFI-QR3sBLi7kLLmjWIzs5b-otIq_vn8hs2YfTBh_-Ha4PcQrok10vdYfMXfaVdGtEz6osvWm-Rc-oSFLUiNa6GBbtqik/Ryan/Measuring%20Scale%20Deviations%20from%20Optimal%20Spacing.pdf

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Keenan