Yahoo Tuning Groups Ultimate Backup tuning-math MOS Consonance Sets; Consonance Saturation

Let's suppose we have a graph of a scale (reduced within a single period) with each note represented by a vertice, and edges connecting vertices which form a consonance. In other words, this is what I'm talking about:
http://xenharmonic.wikispaces.com/Graph-theoretic+properties+of+scales

Let's also restrict ourselves to MOS scales. For starters, lets look at meantone[7].

Here is a very basic consonance set for meantone[7]:
C = (8/5, 6/5, 4/3, 3/2, 5/3, 5/4)

Using the val <0 1 4|, we can map these ratios to points on a generator chain, resulting in the following consonance set:
C = (-4, -3, -1, 1, 3, 4)

The amount of consonances in meantone[7] can therefore be represented as the following:

ConsonanceCount = (7-|-4|)+(7-|-3|)+(7-|-1|)+(7-|1|)+(7-|3|)+(7-|4|)
ConsonanceCount = 3+4+6+3+4+6
ConsonanceCount = 26

For a N-note MOS with consonance set C = (-x, -y, -z, z, y, x), where no member of the set C has magnitude greater than N-1, we can generalize this to the following expression:

ConsonanceCount = N*|C|-2*(x+y+z)

Where |C| is the magnitude of the consonance set C. We can also determine the maximum number of consonances in a given MOS:

ConsonanceCountMax = N*|C|-2*(x+y+z)
ConsonanceCountMax = N*2*(N-1)-2*(N choose 2)
ConsonanceCountMax = 2*N*(N-1)-N*(N-1)
ConsonanceCountMax = N*(N-1)

Using the above, we can establish a metric which relates the number of consonances of a MOS to the maximum possible. I will call this the "consonance saturation." This is the equation for the saturation of a MOS:

Saturation = ConsonanceCount/ConsonanceCountMax
Saturation = |C|/(N-1) - (x+y+z)/(N choose 2)

Ryan Avella

🔗domeofatonement <domeofatonement@yahoo.com>

--- In tuning-math@yahoogroups.com, "Ryan Avella" <domeofatonement@...> wrote:
>
> For a N-note MOS with consonance set C = (-x, -y, -z, z, y, x), where no member of the set C has magnitude greater than N-1, we can generalize this to the following expression:
>
> ConsonanceCount = N*|C|-2*(x+y+z)

After posting this, I realized that some non-MOS scales also qualify. As long as a scale is generated (every note is adjacent to at least one other note in the generator chain), then the above expression holds.

Ryan

🔗Mike Battaglia <battaglia01@gmail.com>

On Mon, Sep 24, 2012 at 9:14 PM, Ryan Avella <domeofatonement@yahoo.com>
wrote:
>
>
> For a N-note MOS with consonance set C = (-x, -y, -z, … z, y, x), where no
> member of the set C has magnitude greater than N-1, we can generalize this
> to the following expression:
>
> ConsonanceCount = N*|C|-2*(x+y+z…)
>
> Where |C| is the magnitude of the consonance set C.

By "magnitude" here, do you mean the cardinality of the set? i.e. the
number of things in it?

> We can also determine the maximum number of consonances in a given MOS:
>
> ConsonanceCountMax = N*|C|-2*(x+y+z…)
> ConsonanceCountMax = N*2*(N-1)-2*(N choose 2)

Is there a mistake here? The line "ConsonanceCountMax =
N*|C|-2*(x+y+z…)" is the same as the line "ConsonanceCount =
N*|C|-2*(x+y+z…)", and I don't see how the line after (with the first
appearance of the N choose 2 term) follows from it.

-Mike

🔗Ryan Avella <domeofatonement@yahoo.com>

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Mon, Sep 24, 2012 at 9:14 PM, Ryan Avella <domeofatonement@...>
> wrote:
> >
> >
> > For a N-note MOS with consonance set C = (-x, -y, -z, z, y, x), where no
> > member of the set C has magnitude greater than N-1, we can generalize this
> > to the following expression:
> >
> > ConsonanceCount = N*|C|-2*(x+y+z)
> >
> > Where |C| is the magnitude of the consonance set C.
>
> By "magnitude" here, do you mean the cardinality of the set? i.e. the
> number of things in it?

Yes, I mean the cardinality.

> > We can also determine the maximum number of consonances in a given MOS:
> >
> > ConsonanceCountMax = N*|C|-2*(x+y+z)
> > ConsonanceCountMax = N*2*(N-1)-2*(N choose 2)
>
> Is there a mistake here? The line "ConsonanceCountMax =
> N*|C|-2*(x+y+z)" is the same as the line "ConsonanceCount =
> N*|C|-2*(x+y+z)", and I don't see how the line after (with the first
> appearance of the N choose 2 term) follows from it.

I could see how that is confusing. It might make more sense if the first line read "ConsonanceCount" instead of "ConsonanceCountMax."

In order to get from the first line to the second, I made some substitutions. Specifically, |C| becomes 2*(N-1) when the consonances are at a maximum, and 2*(x+y+z) becomes 2*(N choose 2). If you need a proof of this, I'd be happy to provide one.

Ryan