Periodic Scales as Sets of Complex Numbers on the Unit Circle

I am sure that this has probably been noticed by someone before, but I would like to introduce a special type of isomorphism from cents-notation of scales (the set of reals modulo 1200) to complex numbers on the unit circle.

If an interval can be expressed in cents-notation as C, then it can also be represented as exp(pi*i*C/600). In other words it is an isomorphism from (R_n, +) to ( {C: |C|=1}, *).

This bijective relation can also be extended from elements to sets of elements. For instance, the scale (0, 300, 600, 900) can be rewritten as (1, i, -1, -i). However, because this notation is likely to get messy for complicated fractions of an octave, I would recommend using phasor angles. In degrees this would give us (0, 90, 180, 270), and in radians we would have (0, pi/2, pi, 3pi/2).

Here are the basic results of the isomorphism:
-Addition becomes multiplication
-Multiplication becomes exponentiation
-Octave inversions become complex conjugates
-Interval span corresponds to the subtended angle (i.e. division)

In addition, I believe that the polar representation (e.g. like an analog clock) is more visually intuitive than a representation in cents (e.g. like a digital clock or a ruler). When looking at a scale represented as points around a circle, you are less likely to judge a scale with a root bias (the tendency to think of a scale as having an objective root).

Ryan Avella

The famous scientist Johannes Kepler also viewed the unit circle as a picture for music, such that the arcs were akin to string lengths of equal tension. He discusses this in his book "The Harmony of the World". His argument was that all concordant intervals could be constructed from inscribed regular polygons, while noting that since the heptagon (7-sides) is not constructable, neither are just intervals with factors of 7 considered concordances. It's essentially an argument for just intonation as the only true set of concordances.

On 5/24/2013 11:56 AM, Ryan Avella wrote:
>
> I am sure that this has probably been noticed by someone before, but I > would like to introduce a special type of isomorphism from > cents-notation of scales (the set of reals modulo 1200) to complex > numbers on the unit circle.
>
> If an interval can be expressed in cents-notation as C, then it can > also be represented as exp(pi*i*C/600). In other words it is an > isomorphism from (R_n, +) to ( {C: |C|=1}, *).
>
> This bijective relation can also be extended from elements to sets of > elements. For instance, the scale (0, 300, 600, 900) can be rewritten > as (1, i, -1, -i). However, because this notation is likely to get > messy for complicated fractions of an octave, I would recommend using > phasor angles. In degrees this would give us (0, 90, 180, 270), and in > radians we would have (0, pi/2, pi, 3pi/2).
>
> Here are the basic results of the isomorphism:
> -Addition becomes multiplication
> -Multiplication becomes exponentiation
> -Octave inversions become complex conjugates
> -Interval span corresponds to the subtended angle (i.e. division)
>
> In addition, I believe that the polar representation (e.g. like an > analog clock) is more visually intuitive than a representation in > cents (e.g. like a digital clock or a ruler). When looking at a scale > represented as points around a circle, you are less likely to judge a > scale with a root bias (the tendency to think of a scale as having an > objective root).
>
> Ryan Avella
>
>

--- In tuning-math@yahoogroups.com, "Ryan Avella" <domeofatonement@...> wrote:
>
> I am sure that this has probably been noticed by someone before, but I would like to introduce a special type of isomorphism from cents-notation of scales (the set of reals modulo 1200) to complex numbers on the unit circle.
>
> If an interval can be expressed in cents-notation as C, then it can also be represented as exp(pi*i*C/600). In other words it is an isomorphism from (R_n, +) to ( {C: |C|=1}, *).
>
> This bijective relation can also be extended from elements to sets of elements. For instance, the scale (0, 300, 600, 900) can be rewritten as (1, i, -1, -i). However, because this notation is likely to get messy for complicated fractions of an octave, I would recommend using phasor angles. In degrees this would give us (0, 90, 180, 270), and in radians we would have (0, pi/2, pi, 3pi/2).
>
> Here are the basic results of the isomorphism:
> -Addition becomes multiplication
> -Multiplication becomes exponentiation
> -Octave inversions become complex conjugates
> -Interval span corresponds to the subtended angle (i.e. division)
>
> In addition, I believe that the polar representation (e.g. like an analog clock) is more visually intuitive than a representation in cents (e.g. like a digital clock or a ruler). When looking at a scale represented as points around a circle, you are less likely to judge a scale with a root bias (the tendency to think of a scale as having an objective root).
>
>
> Ryan Avella
>
I like this, I have thought of it, and it's great also because it works with group theory (like C4 x C3), and the whole modular clock idea, cyclotomic math, etc. I wonder why it hasn't been implemented before, more often, by theorists. It kind of makes 2 the identity, assuming octave-equivalence. If you are interested in group theory, study character tables, which make use of the cyclotomics. This is also nice because it brings in complex analysis, which is on the analysis side so you have both sides, analysis and algebra.

What do you mean by objective root?

PGH

On Friday 24 May 2013 17:56:01 Ryan Avella wrote:
> I am sure that this has probably been noticed by someone
> before, but I would like to introduce a special type of
> isomorphism from cents-notation of scales (the set of
> reals modulo 1200) to complex numbers on the unit
> circle.

Charles Lucy has a model of the octave as a circle, and it
ties in with his tuning, from John Harrison, probably from
Halley, of pi as the ratio of the whole tone to the octave:

http://www.harmonics.com/lucy/lsd/chap2.html

We discussed complex numbers in relation to this but that
page isn't up any more.

More recently, I've been using 60 tones to the octave as a
Magic tuning. It's a not-bad division of the equally
tempered semitone and has application to middle-eastern
music in that it has unequal neutral thirds. 72 is a better
match for 11-limit harmony so it gets all the attention but
60 has its applications.

60 is a highly composite number. That means it supports the
maximal number of equal temperament subsets.

60, then, as a next-to-universal octave division, would be
the equivalent of dividing the octave into the minutes
division of a clock face.

You can also combine 60 and 72. Have an accidental set to
divide the semitone into 5 parts, and another to divide it
into 6 parts, and using them in combination gives 360
degrees to the octave. 360 is also a highly composite
number.

Graham

--- In tuning-math@yahoogroups.com, "Paul" <phjelmstad@...> wrote:
> I like this, I have thought of it, and it's great also because it works with group theory (like C4 x C3), and the whole modular clock idea, cyclotomic math, etc. I wonder why it hasn't been implemented before, more often, by theorists. It kind of makes 2 the identity, assuming octave-equivalence. If you are interested in group theory, study character tables, which make use of the cyclotomics. This is also nice because it brings in complex analysis, which is on the analysis side so you have both sides, analysis and algebra.
>
> What do you mean by objective root?
>
> PGH
>

To explain what I mean by "objective root bias," take a look at the following example.

After looking at both of these scales for about 5 seconds each, which one would you prefer to write music in?
A.) 0.0 - 386.0 - 498.0 - 702.0 - 969.0 - 1143.0 - 1200.0
B.) 0.0 - 57.0 - 443.0 - 555.0 - 759.0 - 1026.0 - 1200.0

Chances are, you probably preferred scale A over scale B. It appears to have much simpler ratios (5/4, 4/3, 3/2, 7/4) as compared to scale B, which looks like it has gross approximations of not-so-simple ratios (9/7, 11/8, 14/9, 9/5). However, scale A and scale B are in fact the same scale, but expressed over different roots.

This is what I call "objective root bias." It is the tendency to look at a scale and almost immediately characterize it as a "good" scale or a "bad" scale based on a single choice of root.

Ryan Avella

I don't know if this will be useful, but here's a possible extension of your idea:

Complex points p = (P/1200)*exp(pi*i*C/P) represent pairs (C, P), where C is an interval, and P>0 the period, both given in cents.

That means for P = 1200, all intervals C will be represented on the unit circle. For P = 1901.95..., they will be on the circle with circumference log_2(3/1) that is centered at 0. And so on...

Specifying a scale with intervals C, and drawing it for all P>0 might produce interesting results, especially if the scale is based on an equal division of an initial period P_0.

Another idea would be too experiment with golden MOS's and periods P_0*phi^z derived from an initial period P_0, where z is a whole number, and phi the golden ratio (1+sqrt5)/2.

- Gedankenwelt

--- In tuning-math@yahoogroups.com, "Ryan Avella" <domeofatonement@...> wrote:
>
> I am sure that this has probably been noticed by someone before, but I would like to introduce a special type of isomorphism from cents-notation of scales (the set of reals modulo 1200) to complex numbers on the unit circle.
>
> If an interval can be expressed in cents-notation as C, then it can also be represented as exp(pi*i*C/600). In other words it is an isomorphism from (R_n, +) to ( {C: |C|=1}, *).
>
> This bijective relation can also be extended from elements to sets of elements. For instance, the scale (0, 300, 600, 900) can be rewritten as (1, i, -1, -i). However, because this notation is likely to get messy for complicated fractions of an octave, I would recommend using phasor angles. In degrees this would give us (0, 90, 180, 270), and in radians we would have (0, pi/2, pi, 3pi/2).
>
> Here are the basic results of the isomorphism:
> -Addition becomes multiplication
> -Multiplication becomes exponentiation
> -Octave inversions become complex conjugates
> -Interval span corresponds to the subtended angle (i.e. division)
>
> In addition, I believe that the polar representation (e.g. like an analog clock) is more visually intuitive than a representation in cents (e.g. like a digital clock or a ruler). When looking at a scale represented as points around a circle, you are less likely to judge a scale with a root bias (the tendency to think of a scale as having an objective root).
>
>
> Ryan Avella