Ridge Tones

A fun little exploration of patterns and symmetries

Introduction

This exploration began with a facebook post by Kieran Ridge, in the Music Theory group on Facebook:

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So what's up with that? Why is "E" in all those scales, and no other note? And why the "E"? Is there something special about the major scale that determines this common tone?

Obviously these common tones can exist in other scales. Just consider the chromatic 12-tone scale, where obviously every note appears in every transposition of itself. But what is the pattern? Will there be common tones when a Bebop Locrian scale is built upon all the tones of a Bebop Locrian?

It will be shown that when a scale is built upon each member of itself, some of those tones may be present in all the scales. I propose these common tones be called "Ridge Tones", in homage to Kieran Ridge, their discoverer.

Background

Before we begin, let's take a remedial moment to define some of the concepts we'll be using.

Vector

A one-dimensional array of data. We can think of it as a list.

Matrix

A two-dimensional array of data. We would display a matrix as a table of rows and columns. A matrix is like a collection of vectors that all have the same length.

Pitch Class

The name for a pitch, disregarding its "height" such that all pitches with the same name in different octaves are considered identical. For example, E flat is a pitch class. In musical analysis, pitch classes are usually numbered 0 through 11.

Pitch Class Set

A set of pitch classes typically expressed as a set of integers between 0 and 11, and displayed like this: {0,2,4,5,7,9,11}.

Scale

A subclass of pitch class set. While a pitch class set can be any arbitrary collection of tones, a scale is something we typically think of as consisting of steps (not leaps), beginning with a root tone, and often (but not always) repeating at an octave.

First, let's model what is exactly going on. Kieran defined the pattern as building a major scale, with a root on each note of a major scale. Let's write them all out, expressed as pitch class sets. For those unfamiliar with PCS, it's a simplified numeric scale with C = 0, C# = 1, D = 2, and so on. For the sake of math, we assume octave equivalence and enharmonic equivalence, so e.g. C flat is the same as B. What we end up with is a set of integers from 0 to 11.

The pitch class set for the major scale is {0,2,4,5,7,9,11}. Let's call this vector S.

S = [0,2,4,5,7,9,11]

To build a major scale on each of those scale degrees, what we're doing is transposing S by an interval equal to each member of S. The result will be 7 different pitch class sets (vectors), that we will call a matrix M. M[0] is the same as S, because S[0] is zero and there is no change. M[1] is S, tranposed up by the value of S[1]. M[2] is the vector S transposed up by S[2], and so on.

M[n] = S + S[n]

Since we're talking about vectors and positions in a set, we need to make it clear that from this point on we will refer to degrees of a scale - and positions in a set - as a zero-based index. In simple terms, it's a list that starts with zero, instead of one. That means that what we'd call the first degree of the scale, the one that comes at the beginning, is the "zeroth" in our set or array. It will appear weird and grammatically incorrect, but to be certain we're in zero-land I will always append "-th" to a numeral, calling things the "0th" and "1th", instead of "first" and "second".

M[x][y] = S[x] + S[y] For our vector S, here is the matrix M:

0	2	4	5	7	9	11
2	4	6	7	9	11	1
4	6	8	9	11	1	3
5	7	9	10	0	2	4
7	9	11	0	2	4	6
9	11	1	2	4	6	8
11	1	3	4	6	8	10

The observation that Kieran made was that the note E, the mediant of the major scale, appears in all those transpositions. Here it is, highlighted:

0	2	4	5	7	9	11
2	4	6	7	9	11	1
4	6	8	9	11	1	3
5	7	9	10	0	2	4
7	9	11	0	2	4	6
9	11	1	2	4	6	8
11	1	3	4	6	8	10

At this point it's important to remark that because a scale continues up and down beyond the contraints of a single octave, this table is actually toroidal. That might seem bizarre, so instead we can envision it as a grid that extends infinitely, repeating in all directions. By accepting that the grid repeats infinitely, we can dispense with having to do modulo arithmetic in our equations.

To summarize, the distance between tones in the equal temperament system is a specific interval that loops at 12; we can say that for specific intervals, 12 is the equivalent to 0, 13 is equivalent to 1, and likewise -1 is equivalent to 11, and so on in the negative direction as well. Speaking of degrees in a scale, the generic interval, it loops at the number of tones in the scale. For a heptatonic scale with 7 notes, they are indexed zero through six; in this heptatonic universe the tone on the 7th degree is the same as the 0th, the 8th degree is the same as the 1th, etc.

Doing this exercise using numbers and pitch class sets shows, I hope, that this isn't a peculiar property of the white and black keys on a piano, or of C major in particular. This property will be true of the mediant of all major scales whether it's an "E" in C major, a "C" in A flat major, etc.

The curious and meaningful part of this matrix M happens when we read it diagonally, as in the highlighted cells. The note "E" is the third member of M[0], the second member of M[1], the first member of M2.

Looking at our set M with diagonal goggles on, we can see that the "4" (which in our set, represents the note "E") appears as the 2th note in the scale built on the 0th degree, the 1th note in the scale built on the 1th degree, the 0th note of a scale built on the 2th degree, and so on. When we get to the scale built on the 6th degree (which is B), our "4" is the 3th note of the scale.

So, let's define another matrix, which is a transformation of M, which we will call K. This new set contains the same pitches as M, but each column is slid down one position showing the relative position of sequential positions of a scale.

0	1	1	0	0	1	1
2	2	3	2	2	2	3
4	4	4	4	4	4	4
5	6	6	5	6	6	6
7	7	8	7	7	8	8
9	9	9	9	9	9	10
11	11	11	10	11	11	11

The tranformation of M into K is done like this:

K[x][y] = M[x - y][y]

And there we have it - that third row is all fours, showing that the pitch class 4 appears in all 7 of our scales. No other tones are shared between all S built upon the tones of S. The matrix M was a convenient step to show how the scales are built and then K is a transformation to make the data more visually apparent. There is a quicker way to arrive at K without building M first:

K[x][y] = S[x - y] + S[y]

So with this equation we can declare a rule that will determine the common-tones for scales S built upon the degrees of S:

So now we have a brute force way of building a matrix of scales built upon scales, providing a loopy and inefficient way to calculate the ridge tones. But is there another way? In other words, is there an elegant way to identify ridges, without brute force calculation using matrices?

As a matter of fact, there is. Let's explore that now.

Ridge Tones and Symmetry

Instead of trying to invent a formula for finding ridges, let's instead approach from the opposite direction: we'll choose a scale, then choose an arbitrary tone, and see if we can determine if that tone is a Ridge.

Since we've already figured out the Major scale, let's tackle a different one: Lydian Augmented, which has a pitch class set of {0,2,4,6,8,9,11}.

Now let's arbitrarily pick a note in this scale, and see if it is a Ridge. We'll pick the note at position 4 semitones above the root. This one was a ridge in the Major scale, so maybe we'll get lucky.

If our chosen pitch 4 is a ridge, then there will be a note in position 4 when we transpose up by each of the distances in the pitch class set {0,2,4,6,8,9,11} .

Transposing up by 0 semitones is simple, there is no change, and the 4th position does have a tone. So we can move on.

The next distance in our sequence is 2. For there to be a note at the 4th position when transposing up by 2, there would have to be a note 2 semitones below the 4th position in the original scale. As it happens, there is.

The next distance in our sequence {0,2,4,6,8,9,11} is 4. For there to be a note at the 4th position when transposing up by 4, there would have to be a note 4 semitones below the 4th position in the original scale. Of course, there is one: it's the root of the scale.

The next distance in our pitch class set {0,2,4,6,8,9,11} is 6. Is there a note six semitones below the 4th position in the original scale? No there is not.

Therefore, we can conclude that 4 is not a Ridge. We can stop now and try a different note, but this exercise illustrated an important pattern. Were we to continue, we would test for the presense of notes at 8, 9, and 11 notes below position 4. Essentially what we have done is build a Lydian Augmented scale backwards, with position 4 as our root. If position 4 was a Ridge, then every one of our backward notes would correspond to a note present in the original scale.

Clearly, this alignment of notes between a scale and its reverse can only happen if the scale is symmetrical. Furthermore we see the relationship between the original scale and the reflected version is that the original scale ascends upward from the root tone, and the reflection descends downward from what would be a Ridge Tone. Thus we can deduce that Ridge tones will appear across the axis of symmetry from the root.

We can illustrate this pattern with more examples:

Major {0,2,4,5,7,9,11}		With an axis of symmetry passing through 2, we see a ridge opposite the root at position 4.
Dorian {0,2,3,5,7,9,10}		With the axis of symmetry passing through 0, there is a ridge on the root itself.
Phrygian {0,1,3,5,7,8,10}		With an axis of symmetry passing through 4, we see a ridge opposite the root at position 8.
Lydian {0,2,4,6,7,9,11}		With an axis of symmetry passing through 3, we see a ridge opposite the root at position 6.
Mixolydian {0,2,4,5,7,9,10}		With an axis of symmetry passing through 1, we see a ridge opposite the root at position 2.
Aeolian {0,2,3,5,7,8,10}		With an axis of symmetry passing through 5, we see a ridge opposite the root at position 10.
Locrian {0,1,3,5,6,8,10}		With an axis of symmetry passing through 3, we see a ridge opposite the root at position 6.
Messiaen Mode 7 {0,1,2,3,5,6,7,8,9,11}		With two axes of symmetry passing through 1 and 4, we see a ridge opposite the root at position 2 and 8.
Octatonic {0,1,3,4,6,7,9,10}		Here we see four axes of reflective symmetry, at 0.5, 2, 3.5, and 5. Hence we have ridges at 1, 4, 7, and 10. This scale also has rotational symmetry at 3, which is evident because the ridges are all 3 semitones apart from each other.
Diminished {0,2,3,5,6,8,9,11}		This set has four axes of reflective symmetry, at 1, 2.5, 4, and 5.5. Hence we see ridges at 2, 5, 8, and 11. Like the Octatonic scale, this scale has rotational symmetry at 3, and so the ridges are all 3 semitones apart from each other.
Augmented {0,3,4,7,8,11}		This set has three axes of reflective symmetry, at 1.5, 3.5, and 5.5. Three ridges therefore appear as reflections of the root, at 3, 7 and 11. This set also has rotational symmetry at 4, which is evident because the ridges are also 4 semitones apart from each other.

Counting Ridges

As a final hurrah to this topic, let's calculate how many scales have ridges, aggregated by cardinality. For this we will ignore the Zeitler Definition Of A Scale, and allow all pitch class sets that have a root tone.

		Number of ridge tones
		0	1	2	3	4	5	6	7	8	9	10	11	12
Number of tones in the scale	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	1	0	1	0	0	0	0	0	0	0	0	0	0	0
	2	0	10	1	0	0	0	0	0	0	0	0	0	0
	3	42	12	0	1	0	0	0	0	0	0	0	0	0
	4	112	48	4	0	1	0	0	0	0	0	0	0	0
	5	280	50	0	0	0	0	0	0	0	0	0	0	0
	6	354	102	3	2	0	0	1	0	0	0	0	0	0
	7	392	70	0	0	0	0	0	0	0	0	0	0	0
	8	224	96	8	0	2	0	0	0	0	0	0	0	0
	9	126	36	0	3	0	0	0	0	0	0	0	0	0
	10	0	50	5	0	0	0	0	0	0	0	0	0	0
	11	0	11	0	0	0	0	0	0	0	0	0	0	0
	12	0	0	0	0	0	0	0	0	0	0	0	0	1

Observing the table above, we can glean some insight from it. The ones (literally, the 1's) that are arranged diagonally are all the scales that are evenly spaced; the unison, tritone, augmented triad, diminished seventh, whole tone, and way off in the bottom right corner, the 12 tone chromatic scale. We observe that all two-tone sets have a ridge, and consequently so do all the decatonics; intuitively we can imagine that every duotonic interval has an axis of symmetry half way between the two tones, and similarly every decatonic has an axis spaced equally between the two tones that are not in the scale; gadyllian illustrates this. All the scales with an odd number of tones can never have more than one ridge, because it is impossible for an odd number of notes to have more than one axis of symmetry. The number of scales with more than one ridge is rather small: there are only 32. The two octatonics with 4 ridges are already known to us: the octatonic and diminished. The three nonatonic scales with 3 ridges are the three modes of Messiaen's Third Mode of Limited Transposition: 3549, 1911, and 3003.

Gratitudes to Kieran Ridge for his permission to explore and publish this little essay, and to members of the Facebook Music Theory Group for a lively discussion which inspired me to investigate this interesting pattern. Thanks also to Eric Mendez for prompting questions about the ridge patterns of a set and its complement, and for a hypothesis about decatonics which turned out to be true. Cheers!