Scales - LUC Sakai - Loyola University Chicago

Scales
Aaron Greicius
Loyola University Chicago
c 2014 Aaron Greicius
All Rights Reserved
Scales
How should we model musical scales? Our first inclination, especially after
listening to our downstairs neighbor diligently pound through all 12 major
scales, is to treat these as sequences of pitch-classes. Order matters here,
right?
Scales
How should we model musical scales? Our first inclination, especially after
listening to our downstairs neighbor diligently pound through all 12 major
scales, is to treat these as sequences of pitch-classes. Order matters here,
right?
For example, we would represent the C major scale as the sequence
(0, 2, 4, 5, 7, 9, 11) (or (C, D, E, F, G, A, B), using pitch names) and the F] major
scale as (6, 8, 10, 11, 1, 3, 5) (or (F], G], A], B, C], D], E]), using pitch names).
Scales
How should we model musical scales? Our first inclination, especially after
listening to our downstairs neighbor diligently pound through all 12 major
scales, is to treat these as sequences of pitch-classes. Order matters here,
right?
For example, we would represent the C major scale as the sequence
(0, 2, 4, 5, 7, 9, 11) (or (C, D, E, F, G, A, B), using pitch names) and the F] major
scale as (6, 8, 10, 11, 1, 3, 5) (or (F], G], A], B, C], D], E]), using pitch names).
Looking at these two sequences carefully, however, we see that the particular
ordering of the chosen pitches here is not all that interesting: we have just
listed the given pitch-classes in their natural order around the pitch-class circle.
In other words, our sequences don’t contain much more information than the
corresponding sets {0, 2, 4, 5, 7, 9, 11} and {6, 8, 10, 11, 1, 3, 5}.
Scales
How should we model musical scales? Our first inclination, especially after
listening to our downstairs neighbor diligently pound through all 12 major
scales, is to treat these as sequences of pitch-classes. Order matters here,
right?
For example, we would represent the C major scale as the sequence
(0, 2, 4, 5, 7, 9, 11) (or (C, D, E, F, G, A, B), using pitch names) and the F] major
scale as (6, 8, 10, 11, 1, 3, 5) (or (F], G], A], B, C], D], E]), using pitch names).
Looking at these two sequences carefully, however, we see that the particular
ordering of the chosen pitches here is not all that interesting: we have just
listed the given pitch-classes in their natural order around the pitch-class circle.
In other words, our sequences don’t contain much more information than the
corresponding sets {0, 2, 4, 5, 7, 9, 11} and {6, 8, 10, 11, 1, 3, 5}. (The sequences
do in fact contain one more piece of information, namely who goes first, but
this is easily dealt with.)
Scales
How should we model musical scales? Our first inclination, especially after
listening to our downstairs neighbor diligently pound through all 12 major
scales, is to treat these as sequences of pitch-classes. Order matters here,
right?
For example, we would represent the C major scale as the sequence
(0, 2, 4, 5, 7, 9, 11) (or (C, D, E, F, G, A, B), using pitch names) and the F] major
scale as (6, 8, 10, 11, 1, 3, 5) (or (F], G], A], B, C], D], E]), using pitch names).
Looking at these two sequences carefully, however, we see that the particular
ordering of the chosen pitches here is not all that interesting: we have just
listed the given pitch-classes in their natural order around the pitch-class circle.
In other words, our sequences don’t contain much more information than the
corresponding sets {0, 2, 4, 5, 7, 9, 11} and {6, 8, 10, 11, 1, 3, 5}. (The sequences
do in fact contain one more piece of information, namely who goes first, but
this is easily dealt with.)
Furthermore, scales are often treated by composers as fixed collections of
pitches from which they draw subsets in order to build chords or melodies.
Scales
How should we model musical scales? Our first inclination, especially after
listening to our downstairs neighbor diligently pound through all 12 major
scales, is to treat these as sequences of pitch-classes. Order matters here,
right?
For example, we would represent the C major scale as the sequence
(0, 2, 4, 5, 7, 9, 11) (or (C, D, E, F, G, A, B), using pitch names) and the F] major
scale as (6, 8, 10, 11, 1, 3, 5) (or (F], G], A], B, C], D], E]), using pitch names).
Looking at these two sequences carefully, however, we see that the particular
ordering of the chosen pitches here is not all that interesting: we have just
listed the given pitch-classes in their natural order around the pitch-class circle.
In other words, our sequences don’t contain much more information than the
corresponding sets {0, 2, 4, 5, 7, 9, 11} and {6, 8, 10, 11, 1, 3, 5}. (The sequences
do in fact contain one more piece of information, namely who goes first, but
this is easily dealt with.)
Furthermore, scales are often treated by composers as fixed collections of
pitches from which they draw subsets in order to build chords or melodies.
Accordingly we will model scales as sets of pitch-classes, just as we did with
chords, but will develop some additional theory to reflect their particular
musical functions.
Note: for the rest of this section we will work exclusively in
pitch-class space. Accordingly we will drop the bracket notation,
and use modular arithmetic with impunity.
Note: for the rest of this section we will work exclusively in
pitch-class space. Accordingly we will drop the bracket notation,
and use modular arithmetic with impunity.
Definition
We will call a scale any subset X = {P1 , P2 , . . . , Pr } of r distinct
pitch-classes.
Note: for the rest of this section we will work exclusively in
pitch-class space. Accordingly we will drop the bracket notation,
and use modular arithmetic with impunity.
Definition
We will call a scale any subset X = {P1 , P2 , . . . , Pr } of r distinct
pitch-classes.
Most of the scales we will consider will be equal-tempered, which
means as usual that the Pi ∈ {0, 1, 2, . . . , 11} are taken from our
set of 12 equal-tempered pitch-classes.
Note: for the rest of this section we will work exclusively in
pitch-class space. Accordingly we will drop the bracket notation,
and use modular arithmetic with impunity.
Definition
We will call a scale any subset X = {P1 , P2 , . . . , Pr } of r distinct
pitch-classes.
Most of the scales we will consider will be equal-tempered, which
means as usual that the Pi ∈ {0, 1, 2, . . . , 11} are taken from our
set of 12 equal-tempered pitch-classes.
Furthermore, we typically will insist that r ≥ 5; i.e., you should
have at least 5 pitches to be considered a scale.
Note: for the rest of this section we will work exclusively in
pitch-class space. Accordingly we will drop the bracket notation,
and use modular arithmetic with impunity.
Definition
We will call a scale any subset X = {P1 , P2 , . . . , Pr } of r distinct
pitch-classes.
Most of the scales we will consider will be equal-tempered, which
means as usual that the Pi ∈ {0, 1, 2, . . . , 11} are taken from our
set of 12 equal-tempered pitch-classes.
Furthermore, we typically will insist that r ≥ 5; i.e., you should
have at least 5 pitches to be considered a scale.
Lastly, as with chords we say that two equal-tempered scales X
and Y are of the same type if there is a transposition tj ∈ T12
such that tj (X ) = Y (i.e., the one is a transposition of the other).
The equivalence classes determined by this relation are called
scale-types.
Example
The scale {0, 2, 4, 5, 7, 9, 11} is called the C diatonic scale. It can
be described as starting with the pitch C and applying in order the
following sequence of whole (W = M2) and half step (H = m2)
transpositions: (W , W , H, W , W , W , H).
Example
The scale {0, 2, 4, 5, 7, 9, 11} is called the C diatonic scale. It can
be described as starting with the pitch C and applying in order the
following sequence of whole (W = M2) and half step (H = m2)
transpositions: (W , W , H, W , W , W , H).
The same description in terms of W and H allows us to define
diatonic scales starting with any pitch. Thus the G diatonic scale is
just {7, 9, 11, 0, 2, 4, 6} = {0, 2, 4, 6, 7, 9, 11}.
Example
The scale {0, 2, 4, 5, 7, 9, 11} is called the C diatonic scale. It can
be described as starting with the pitch C and applying in order the
following sequence of whole (W = M2) and half step (H = m2)
transpositions: (W , W , H, W , W , W , H).
The same description in terms of W and H allows us to define
diatonic scales starting with any pitch. Thus the G diatonic scale is
just {7, 9, 11, 0, 2, 4, 6} = {0, 2, 4, 6, 7, 9, 11}.
It is clear each such diatonic scale is just a transposition of the C
diatonic scale, and thus together these comprise a single
scale-type, which we call diatonic.
Example
The scale {0, 2, 4, 5, 7, 9, 11} is called the C diatonic scale. It can
be described as starting with the pitch C and applying in order the
following sequence of whole (W = M2) and half step (H = m2)
transpositions: (W , W , H, W , W , W , H).
The same description in terms of W and H allows us to define
diatonic scales starting with any pitch. Thus the G diatonic scale is
just {7, 9, 11, 0, 2, 4, 6} = {0, 2, 4, 6, 7, 9, 11}.
It is clear each such diatonic scale is just a transposition of the C
diatonic scale, and thus together these comprise a single
scale-type, which we call diatonic.
How many different diatonic scales are there? Is it possible, for
example, that the G[ diatonic scale is just the same thing as the
B[ diatonic scale written in a different order?
Example
The scale {0, 2, 4, 5, 7, 9, 11} is called the C diatonic scale. It can
be described as starting with the pitch C and applying in order the
following sequence of whole (W = M2) and half step (H = m2)
transpositions: (W , W , H, W , W , W , H).
The same description in terms of W and H allows us to define
diatonic scales starting with any pitch. Thus the G diatonic scale is
just {7, 9, 11, 0, 2, 4, 6} = {0, 2, 4, 6, 7, 9, 11}.
It is clear each such diatonic scale is just a transposition of the C
diatonic scale, and thus together these comprise a single
scale-type, which we call diatonic.
How many different diatonic scales are there? Is it possible, for
example, that the G[ diatonic scale is just the same thing as the
B[ diatonic scale written in a different order?
Use Lagrange’s theorem! The stabilizer of the C diatonic collection
is trivial, so its orbit has 12/1 = 12 different transpositions in it.
There are indeed 12 different diatonic scales!
Modes
Let’s return to our earnest piano student downstairs. Listening more closely we
hear he actually plays two different versions of the C diatonic scale: one that
begins with C, the sequence (0, 2, 4, 5, 7, 9, 11), and one that begins with A,
the sequence (9, 11, 0, 2, 4, 5, 7).
Modes
Let’s return to our earnest piano student downstairs. Listening more closely we
hear he actually plays two different versions of the C diatonic scale: one that
begins with C, the sequence (0, 2, 4, 5, 7, 9, 11), and one that begins with A,
the sequence (9, 11, 0, 2, 4, 5, 7).
This difference is not captured currently in our mathematical model of scales,
but we fix this easily with the notion of a mode.
Definition
Let X = {P1 , P2 , . . . , Pr } be a scale, and assume the pitch-classes are written
in a clockwise sequential order. A mode of X is the sequence
(Pj , Pj+1 , . . . , Pr , P1 , . . . , Pj−1 ) you get by starting with a pitch Pj and working
around the scale in clockwise fashion.
Modes
Let’s return to our earnest piano student downstairs. Listening more closely we
hear he actually plays two different versions of the C diatonic scale: one that
begins with C, the sequence (0, 2, 4, 5, 7, 9, 11), and one that begins with A,
the sequence (9, 11, 0, 2, 4, 5, 7).
This difference is not captured currently in our mathematical model of scales,
but we fix this easily with the notion of a mode.
Definition
Let X = {P1 , P2 , . . . , Pr } be a scale, and assume the pitch-classes are written
in a clockwise sequential order. A mode of X is the sequence
(Pj , Pj+1 , . . . , Pr , P1 , . . . , Pj−1 ) you get by starting with a pitch Pj and working
around the scale in clockwise fashion.
Given the a mode (Q1 , Q2 , . . . , Qr ), we call the i-th pitch in the mode the i-th
scale degree of the mode (or scale degree i), denoted bi. We will use the same
terminology when dealing with scales too, at least when their names indicate a
preferred “first” pitch. For example, in the D diatonic scale, scale degree 3 is
b
3 = F], and scale degree 7 is b
7 = C].
Diatonic modes
In general a scale containing r distinct pitches will have r different modes,
determined by who goes first.
Diatonic modes
In general a scale containing r distinct pitches will have r different modes,
determined by who goes first.
Mode
(C, D, E, F, G, A, B)
(D, E, F, G, A, B, C)
(E, F, G, A, B, C, D)
(F, G, A, B, C, D, E)
(G, A, B, C, D, E, F)
(A, B, C, D, E, F, G)
(B, C, D, E, F, G, A)
W -H sequence
(W , W , H, W , W , W , H)
(W , H, W , W , W , H, W )
(H, W , W , W , H, W , W )
(W , W , W , H, W , W , H)
(W , W , H, W , W , H, W )
(W , H, W , W , H, W , W )
(H, W , W , H, W , W , W )
Name
C ionian (or C major)
D dorian
E phrygian
F lydian
G mixolydian
A aeolian (or A natural minor)
B locrian
Figure : The seven modes of the C diatonic scale
Diatonic modes
In general a scale containing r distinct pitches will have r different modes,
determined by who goes first.
Mode
(C, D, E, F, G, A, B)
(D, E, F, G, A, B, C)
(E, F, G, A, B, C, D)
(F, G, A, B, C, D, E)
(G, A, B, C, D, E, F)
(A, B, C, D, E, F, G)
(B, C, D, E, F, G, A)
W -H sequence
(W , W , H, W , W , W , H)
(W , H, W , W , W , H, W )
(H, W , W , W , H, W , W )
(W , W , W , H, W , W , H)
(W , W , H, W , W , H, W )
(W , H, W , W , H, W , W )
(H, W , W , H, W , W , W )
Name
C ionian (or C major)
D dorian
E phrygian
F lydian
G mixolydian
A aeolian (or A natural minor)
B locrian
Figure : The seven modes of the C diatonic scale
As with chords, we transpose modes simply by transposing each pitch:
tj ((P1 , P2 , . . . , Pr )) := (tj (P1 ), tj (P2 ), . . . tj (Pr )).
Diatonic modes
In general a scale containing r distinct pitches will have r different modes,
determined by who goes first.
Mode
(C, D, E, F, G, A, B)
(D, E, F, G, A, B, C)
(E, F, G, A, B, C, D)
(F, G, A, B, C, D, E)
(G, A, B, C, D, E, F)
(A, B, C, D, E, F, G)
(B, C, D, E, F, G, A)
W -H sequence
(W , W , H, W , W , W , H)
(W , H, W , W , W , H, W )
(H, W , W , W , H, W , W )
(W , W , W , H, W , W , H)
(W , W , H, W , W , H, W )
(W , H, W , W , H, W , W )
(H, W , W , H, W , W , W )
Name
C ionian (or C major)
D dorian
E phrygian
F lydian
G mixolydian
A aeolian (or A natural minor)
B locrian
Figure : The seven modes of the C diatonic scale
As with chords, we transpose modes simply by transposing each pitch:
tj ((P1 , P2 , . . . , Pr )) := (tj (P1 ), tj (P2 ), . . . tj (Pr )).
Since transposition preserves the W -H sequences above, these define the
different mode-types for diatonic modes, and we can use them to generate any
mode starting with any pitch. For example the F dorian mode is
Diatonic modes
In general a scale containing r distinct pitches will have r different modes,
determined by who goes first.
Mode
(C, D, E, F, G, A, B)
(D, E, F, G, A, B, C)
(E, F, G, A, B, C, D)
(F, G, A, B, C, D, E)
(G, A, B, C, D, E, F)
(A, B, C, D, E, F, G)
(B, C, D, E, F, G, A)
W -H sequence
(W , W , H, W , W , W , H)
(W , H, W , W , W , H, W )
(H, W , W , W , H, W , W )
(W , W , W , H, W , W , H)
(W , W , H, W , W , H, W )
(W , H, W , W , H, W , W )
(H, W , W , H, W , W , W )
Name
C ionian (or C major)
D dorian
E phrygian
F lydian
G mixolydian
A aeolian (or A natural minor)
B locrian
Figure : The seven modes of the C diatonic scale
As with chords, we transpose modes simply by transposing each pitch:
tj ((P1 , P2 , . . . , Pr )) := (tj (P1 ), tj (P2 ), . . . tj (Pr )).
Since transposition preserves the W -H sequences above, these define the
different mode-types for diatonic modes, and we can use them to generate any
mode starting with any pitch. For example the F dorian mode is
(F, G, A[, B[, C, D, E[).
Diatonic modes
In general a scale containing r distinct pitches will have r different modes,
determined by who goes first.
Mode
(C, D, E, F, G, A, B)
(D, E, F, G, A, B, C)
(E, F, G, A, B, C, D)
(F, G, A, B, C, D, E)
(G, A, B, C, D, E, F)
(A, B, C, D, E, F, G)
(B, C, D, E, F, G, A)
W -H sequence
(W , W , H, W , W , W , H)
(W , H, W , W , W , H, W )
(H, W , W , W , H, W , W )
(W , W , W , H, W , W , H)
(W , W , H, W , W , H, W )
(W , H, W , W , H, W , W )
(H, W , W , H, W , W , W )
Name
C ionian (or C major)
D dorian
E phrygian
F lydian
G mixolydian
A aeolian (or A natural minor)
B locrian
Figure : The seven modes of the C diatonic scale
As with chords, we transpose modes simply by transposing each pitch:
tj ((P1 , P2 , . . . , Pr )) := (tj (P1 ), tj (P2 ), . . . tj (Pr )).
Since transposition preserves the W -H sequences above, these define the
different mode-types for diatonic modes, and we can use them to generate any
mode starting with any pitch. For example the F dorian mode is
(F, G, A[, B[, C, D, E[). However, it is perhaps easier to just remember the
white note modes and transpose these accordingly.
Comment
It is not just scalar runs played by our downstairs neighbor that we
identify with a particular mode. When analyzing music, we often
describe entire passages as being written in a particular mode:
e.g., “this passage is in F lydian”, or “here the composer switches
to a G dorian mode”.
Comment
It is not just scalar runs played by our downstairs neighbor that we
identify with a particular mode. When analyzing music, we often
describe entire passages as being written in a particular mode:
e.g., “this passage is in F lydian”, or “here the composer switches
to a G dorian mode”.
Such assertions indicate two musical properties:
Comment
It is not just scalar runs played by our downstairs neighbor that we
identify with a particular mode. When analyzing music, we often
describe entire passages as being written in a particular mode:
e.g., “this passage is in F lydian”, or “here the composer switches
to a G dorian mode”.
Such assertions indicate two musical properties:
1. the underlying scalar collection the composer is using, and
Comment
It is not just scalar runs played by our downstairs neighbor that we
identify with a particular mode. When analyzing music, we often
describe entire passages as being written in a particular mode:
e.g., “this passage is in F lydian”, or “here the composer switches
to a G dorian mode”.
Such assertions indicate two musical properties:
1. the underlying scalar collection the composer is using, and
2. a particular pitch that is given special emphasis, sometimes
called the tonal center of the passage.
Comment
It is not just scalar runs played by our downstairs neighbor that we
identify with a particular mode. When analyzing music, we often
describe entire passages as being written in a particular mode:
e.g., “this passage is in F lydian”, or “here the composer switches
to a G dorian mode”.
Such assertions indicate two musical properties:
1. the underlying scalar collection the composer is using, and
2. a particular pitch that is given special emphasis, sometimes
called the tonal center of the passage.
For example, a passage written in G dorian makes use of the F
diatonic collection = {F, G, A, B[, C, D, E} = {0, 2, 4, 5, 7, 9, 10},
and gives special emphasis to G somehow: perhaps the melody
begins and ends on G, for example.
Comment
It is not just scalar runs played by our downstairs neighbor that we
identify with a particular mode. When analyzing music, we often
describe entire passages as being written in a particular mode:
e.g., “this passage is in F lydian”, or “here the composer switches
to a G dorian mode”.
Such assertions indicate two musical properties:
1. the underlying scalar collection the composer is using, and
2. a particular pitch that is given special emphasis, sometimes
called the tonal center of the passage.
For example, a passage written in G dorian makes use of the F
diatonic collection = {F, G, A, B[, C, D, E} = {0, 2, 4, 5, 7, 9, 10},
and gives special emphasis to G somehow: perhaps the melody
begins and ends on G, for example.
In general the scalar collection (1) can be indicated fairly
unambiguously, whereas the tonal center (2) can be trickier to
identify.
Example (“Paddy’s Green Shamrock Shore”, performed by
Paul Brady)
Example (“Paddy’s Green Shamrock Shore”, performed by
Paul Brady)
All pitches here are white notes (note the F] in the signature is always made
natural!), making the scalar collection here C diatonic. As G is clearly the tonal
center of the piece, we conclude it is written in G mixolydian.
Example (“I can’t explain”, The Who)
(I lifted this example from Tymoczko’s Music 105 lecture notes.)
The piece opens with the repeated sequence of major triads
E -D-A-E . Collecting all the pitches in these chords yields the A
diatonic collection {A, B, C], D, E, F], G]}. As E is clearly the
preferred pitch, this is E mixolydian.
Generated scales
Recall that we can generate the entire 12-tone scale by starting
with a pitch and transposing up repeatedly by a perfect fifth.
Generated scales
Recall that we can generate the entire 12-tone scale by starting
with a pitch and transposing up repeatedly by a perfect fifth.
If we start with F, then the first seven pitches in this sequence are
precisely the pitches of the C diatonic scale:
{F, C, G, D, A, E, B}.
We say in this case that the scale is generated.
Generated scales
Recall that we can generate the entire 12-tone scale by starting
with a pitch and transposing up repeatedly by a perfect fifth.
If we start with F, then the first seven pitches in this sequence are
precisely the pitches of the C diatonic scale:
{F, C, G, D, A, E, B}.
We say in this case that the scale is generated.
Definition
A scale X is generated, if there is a pitch P1 and a transposition t
such that
X = {P1 , t(P1 ), t 2 (P1 ), . . . , t r −1 (P1 )}.
(As usual, t j (P) means transpose by t a total of j times. )
Pentatonic scale
The first five pitches of the sequence of fifths starting on F
comprise what is called a pentatonic scale: {D, F , G , A, C }. By
definition it is a generated scale, and a subscale of the diatonic
scale.
Pentatonic scale
The first five pitches of the sequence of fifths starting on F
comprise what is called a pentatonic scale: {D, F , G , A, C }. By
definition it is a generated scale, and a subscale of the diatonic
scale.
The prime form of the pentatonic scale is {0, 2, 4, 7, 9}. I will
name a pentatonic scale according to the unique pitch that
functions as 0 in the prime form. Thus {F , G , A, C , D} is the F
pentatonic scale.
Pentatonic scale
The first five pitches of the sequence of fifths starting on F
comprise what is called a pentatonic scale: {D, F , G , A, C }. By
definition it is a generated scale, and a subscale of the diatonic
scale.
The prime form of the pentatonic scale is {0, 2, 4, 7, 9}. I will
name a pentatonic scale according to the unique pitch that
functions as 0 in the prime form. Thus {F , G , A, C , D} is the F
pentatonic scale. As the naming scheme suggests, there are 12
different pentatonic scales. (Use Lagrange’s theorem!)
Pentatonic scale
The first five pitches of the sequence of fifths starting on F
comprise what is called a pentatonic scale: {D, F , G , A, C }. By
definition it is a generated scale, and a subscale of the diatonic
scale.
The prime form of the pentatonic scale is {0, 2, 4, 7, 9}. I will
name a pentatonic scale according to the unique pitch that
functions as 0 in the prime form. Thus {F , G , A, C , D} is the F
pentatonic scale. As the naming scheme suggests, there are 12
different pentatonic scales. (Use Lagrange’s theorem!)
Note that the black notes of the keyboard {6, 8, 10, 1, 3} also form
a pentatonic scale: the G[ pentatonic scale.
Pentatonic scale
The first five pitches of the sequence of fifths starting on F
comprise what is called a pentatonic scale: {D, F , G , A, C }. By
definition it is a generated scale, and a subscale of the diatonic
scale.
The prime form of the pentatonic scale is {0, 2, 4, 7, 9}. I will
name a pentatonic scale according to the unique pitch that
functions as 0 in the prime form. Thus {F , G , A, C , D} is the F
pentatonic scale. As the naming scheme suggests, there are 12
different pentatonic scales. (Use Lagrange’s theorem!)
Note that the black notes of the keyboard {6, 8, 10, 1, 3} also form
a pentatonic scale: the G[ pentatonic scale. This should come as
no surprise. If we pick up our sequence of fifths where we left off
after generating the white notes, we get precisely the five black
notes {G[, D[, A[, E[, B[}.
Pentatonic scale
The first five pitches of the sequence of fifths starting on F
comprise what is called a pentatonic scale: {D, F , G , A, C }. By
definition it is a generated scale, and a subscale of the diatonic
scale.
The prime form of the pentatonic scale is {0, 2, 4, 7, 9}. I will
name a pentatonic scale according to the unique pitch that
functions as 0 in the prime form. Thus {F , G , A, C , D} is the F
pentatonic scale. As the naming scheme suggests, there are 12
different pentatonic scales. (Use Lagrange’s theorem!)
Note that the black notes of the keyboard {6, 8, 10, 1, 3} also form
a pentatonic scale: the G[ pentatonic scale. This should come as
no surprise. If we pick up our sequence of fifths where we left off
after generating the white notes, we get precisely the five black
notes {G[, D[, A[, E[, B[}. This is most often the first pentatonic
scale we meet in our musical life, and the black note pattern is the
best way of remembering the intervallic content of the pentatonic
scale.
Stacks of stacks
If you try creating generated scales with smaller intervals, thirds for examples,
you get “scales” with less than five notes, and with sizable gaps between
pitches.
Stacks of stacks
If you try creating generated scales with smaller intervals, thirds for examples,
you get “scales” with less than five notes, and with sizable gaps between
pitches.
For example, if we start with C = 0 and use a major third as our generating
interval, we get the “scale” {0, 4, 8}, which is none other than our augmented
triad. Though this stack of thirds is not enough to form a scale, we can
combine it with other augmented triads to get various 6-note scales:
{0, 4, 8} + {1, 5, 9}
=
{0, 1, 4, 5, 8, 9} the hexatonic scale (or augmented scale)
Stacks of stacks
If you try creating generated scales with smaller intervals, thirds for examples,
you get “scales” with less than five notes, and with sizable gaps between
pitches.
For example, if we start with C = 0 and use a major third as our generating
interval, we get the “scale” {0, 4, 8}, which is none other than our augmented
triad. Though this stack of thirds is not enough to form a scale, we can
combine it with other augmented triads to get various 6-note scales:
{0, 4, 8} + {1, 5, 9}
=
{0, 1, 4, 5, 8, 9} the hexatonic scale (or augmented scale)
{0, 4, 8} + {2, 6, 10}
=
{0, 2, 4, 6, 8, 10} the whole-tone scale
Stacks of stacks
If you try creating generated scales with smaller intervals, thirds for examples,
you get “scales” with less than five notes, and with sizable gaps between
pitches.
For example, if we start with C = 0 and use a major third as our generating
interval, we get the “scale” {0, 4, 8}, which is none other than our augmented
triad. Though this stack of thirds is not enough to form a scale, we can
combine it with other augmented triads to get various 6-note scales:
{0, 4, 8} + {1, 5, 9}
=
{0, 1, 4, 5, 8, 9} the hexatonic scale (or augmented scale)
{0, 4, 8} + {2, 6, 10}
=
{0, 2, 4, 6, 8, 10} the whole-tone scale
{0, 4, 8} + {3, 7, 11}
=
{0, 3, 4, 7, 8, 11}
Stacks of stacks
If you try creating generated scales with smaller intervals, thirds for examples,
you get “scales” with less than five notes, and with sizable gaps between
pitches.
For example, if we start with C = 0 and use a major third as our generating
interval, we get the “scale” {0, 4, 8}, which is none other than our augmented
triad. Though this stack of thirds is not enough to form a scale, we can
combine it with other augmented triads to get various 6-note scales:
{0, 4, 8} + {1, 5, 9}
=
{0, 1, 4, 5, 8, 9} the hexatonic scale (or augmented scale)
{0, 4, 8} + {2, 6, 10}
=
{0, 2, 4, 6, 8, 10} the whole-tone scale
{0, 4, 8} + {3, 7, 11}
=
{0, 3, 4, 7, 8, 11} = t3 ({0, 1, 4, 5, 8, 9}).
Stacks of stacks
If you try creating generated scales with smaller intervals, thirds for examples,
you get “scales” with less than five notes, and with sizable gaps between
pitches.
For example, if we start with C = 0 and use a major third as our generating
interval, we get the “scale” {0, 4, 8}, which is none other than our augmented
triad. Though this stack of thirds is not enough to form a scale, we can
combine it with other augmented triads to get various 6-note scales:
{0, 4, 8} + {1, 5, 9}
=
{0, 1, 4, 5, 8, 9} the hexatonic scale (or augmented scale)
{0, 4, 8} + {2, 6, 10}
=
{0, 2, 4, 6, 8, 10} the whole-tone scale
{0, 4, 8} + {3, 7, 11}
=
{0, 3, 4, 7, 8, 11} = t3 ({0, 1, 4, 5, 8, 9}).
Note that the whole-tone scale is a generated scale, using transposition by a
whole tone. The stabilizer of the whole-tone scale is H = {t0 , t2 , t4 , t6 , t8 , t10 }.
Thus there are 12/6 = 2 whole-tone scales. We name them as follows:
WT-0
=
{0, 2, 4, 6, 8, 10} = {C, D, E, F], G], A]}
WT-1
=
{1, 3, 5, 7, 9, 11} = {D[, E[, F, G, A, B}
Octatonic scale
If we play the same game with a minor third, we begin with C = 0
and generate a diminished seventh chord {0, 3, 6, 9}. Up to
transposition, combining any two such diminished seventh chords
always produces the same scale-type, called the octatonic:
{0, 3, 6, 9} + {1, 4, 7, 10} = {0, 1, 3, 4, 6, 7, 9, 10}
Octatonic scale
If we play the same game with a minor third, we begin with C = 0
and generate a diminished seventh chord {0, 3, 6, 9}. Up to
transposition, combining any two such diminished seventh chords
always produces the same scale-type, called the octatonic:
{0, 3, 6, 9} + {1, 4, 7, 10} = {0, 1, 3, 4, 6, 7, 9, 10}
How many different octatonic scales are there?
Octatonic scale
If we play the same game with a minor third, we begin with C = 0
and generate a diminished seventh chord {0, 3, 6, 9}. Up to
transposition, combining any two such diminished seventh chords
always produces the same scale-type, called the octatonic:
{0, 3, 6, 9} + {1, 4, 7, 10} = {0, 1, 3, 4, 6, 7, 9, 10}
How many different octatonic scales are there? The stabilizer is
H = {t0 , t3 , t6 , t9 }. Thus there are 12/4 = 3 different octatonic
scales. We will denote them as follows:
Oct0,1 = {0, 1, 3, 4, 6, 7, 9, 10}
Oct0,2 = {0, 2, 3, 5, 6, 8, 9, 11}
Oct1,2 = {1, 2, 4, 5, 7, 8, 10, 11}
We pause here to collect information about our current list of
scale-types.
Name
diatonic
pentatonic
hexatonic
whole-tone
octatonic
Step sequence
(2, 2, 1, 2, 2, 2, 1)
(2, 2, 3, 2, 3)
(1, 3, 1, 3, 1, 3)
(2, 2, 2, 2, 2, 2)
(1, 2, 1, 2, 1, 2, 1, 2)
Prime form
{0, 1, 3, 5, 6, 8, 10}
{0, 2, 4, 7, 9}
{0, 1, 4, 5, 8, 9}
{0, 2, 4, 6, 8, 10}
{0, 1, 3, 4, 6, 7, 9, 10}
Stab
{t0 }
{t0 }
ht4 i
ht2 i
ht3 i
Interval vector
254361
032140
303630
060603
448444
We pause here to collect information about our current list of
scale-types.
Name
diatonic
pentatonic
hexatonic
whole-tone
octatonic
Step sequence
(2, 2, 1, 2, 2, 2, 1)
(2, 2, 3, 2, 3)
(1, 3, 1, 3, 1, 3)
(2, 2, 2, 2, 2, 2)
(1, 2, 1, 2, 1, 2, 1, 2)
Prime form
{0, 1, 3, 5, 6, 8, 10}
{0, 2, 4, 7, 9}
{0, 1, 4, 5, 8, 9}
{0, 2, 4, 6, 8, 10}
{0, 1, 3, 4, 6, 7, 9, 10}
Stab
{t0 }
{t0 }
ht4 i
ht2 i
ht3 i
Interval vector
254361
032140
303630
060603
448444
The interval vector a1 a2 a3 a4 a5 a6 of a scale gives the number ai
of intervals contained in the scale of length i half steps.
We pause here to collect information about our current list of
scale-types.
Name
diatonic
pentatonic
hexatonic
whole-tone
octatonic
Step sequence
(2, 2, 1, 2, 2, 2, 1)
(2, 2, 3, 2, 3)
(1, 3, 1, 3, 1, 3)
(2, 2, 2, 2, 2, 2)
(1, 2, 1, 2, 1, 2, 1, 2)
Prime form
{0, 1, 3, 5, 6, 8, 10}
{0, 2, 4, 7, 9}
{0, 1, 4, 5, 8, 9}
{0, 2, 4, 6, 8, 10}
{0, 1, 3, 4, 6, 7, 9, 10}
Stab
{t0 }
{t0 }
ht4 i
ht2 i
ht3 i
Interval vector
254361
032140
303630
060603
448444
The interval vector a1 a2 a3 a4 a5 a6 of a scale gives the number ai
of intervals contained in the scale of length i half steps.
The step sequence just indicates the number of half steps between
successive pitches in the scale. These sequences can be read
straight off of the prime form, though I have cycled the diatonic
sequence around to its most familiar form (viz., “whole, whole,
half, whole, whole...”).
We pause here to collect information about our current list of
scale-types.
Name
diatonic
pentatonic
hexatonic
whole-tone
octatonic
Step sequence
(2, 2, 1, 2, 2, 2, 1)
(2, 2, 3, 2, 3)
(1, 3, 1, 3, 1, 3)
(2, 2, 2, 2, 2, 2)
(1, 2, 1, 2, 1, 2, 1, 2)
Prime form
{0, 1, 3, 5, 6, 8, 10}
{0, 2, 4, 7, 9}
{0, 1, 4, 5, 8, 9}
{0, 2, 4, 6, 8, 10}
{0, 1, 3, 4, 6, 7, 9, 10}
Stab
{t0 }
{t0 }
ht4 i
ht2 i
ht3 i
Interval vector
254361
032140
303630
060603
448444
The interval vector a1 a2 a3 a4 a5 a6 of a scale gives the number ai
of intervals contained in the scale of length i half steps.
The step sequence just indicates the number of half steps between
successive pitches in the scale. These sequences can be read
straight off of the prime form, though I have cycled the diatonic
sequence around to its most familiar form (viz., “whole, whole,
half, whole, whole...”).
For the stabilizer column, the notation htj i denotes the subgroup
of T12 generated by tj . Thus ht2 i = {t0 , t2 , t4 , t6 , t8 , t10 }, and
ht3 i = {t0 , t3 , t6 , t9 }.
Geometric summary with inversional symmetry indicated
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Small-gap scales
One of the defining characteristics of the diatonic scale is that the
gaps between successive pitches are no more than 2 half steps, and
that there are never two consecutive gaps of size one half step.
Small-gap scales
One of the defining characteristics of the diatonic scale is that the
gaps between successive pitches are no more than 2 half steps, and
that there are never two consecutive gaps of size one half step. It
is natural then to consider all scale-types satisfying these two
properties, as they will be in some sense diatonic-like. It turns out
that, up to translation, there are not so many.
Small-gap scales
One of the defining characteristics of the diatonic scale is that the
gaps between successive pitches are no more than 2 half steps, and
that there are never two consecutive gaps of size one half step. It
is natural then to consider all scale-types satisfying these two
properties, as they will be in some sense diatonic-like. It turns out
that, up to translation, there are not so many.
whole-tone
{C, D, E, F], G], A]}
(2, 2, 2, 2, 2, 2)
diatonic
{C, D, E, F, G, A, B}
(2, 2, 1, 2, 2, 2, 1)
acoustic
{C, D, E, F], G, A, B[}
(2, 2, 2, 1, 2, 1, 2)
octatonic {C, C], D], E, F], G, A, B[} (1, 2, 1, 2, 1, 2, 1, 2)
(Note: The acoustic scale is so called, as these pitches are the
equal-tempered best approximation of the first 7 pitches of the
harmonic scale.)
In A Geometry of Music Dmitri Tymoczko defines an n-gap scale
to be one where the gap between successive pitches is at most n
half steps. He groups 2-gap and 3-gap scales under the general
heading of small-gap scales. Tymoczko adds two 3-gap seven
note scales (harmonic minor and harmonic major) to our list,
and we will follow suit here, yielding the following (final) table of
scale-types:
pentatonic
hexatonic
whole-tone
diatonic
acoustic
harmonic minor
harmonic major
octatonic
{C, D, E, G, A}
{C, C], E, F, G], A}
{C, D, E, F], G], A]}
{C, D, E, F, G, A, B}
{C, D, E, F], G, A, B[}
{C, D, E[, F, G, A[, B}
{C, D, E, F, G, A[, B}
{C, C], D], E, F], G, A, B[}
(2, 2, 3, 2, 3)
(1, 3, 1, 3, 1, 3)
(2, 2, 2, 2, 2, 2)
(2, 2, 1, 2, 2, 2, 1)
(2, 2, 2, 1, 2, 1, 2)
(2, 2, 1, 2, 1, 3, 1)
(2, 2, 1, 2, 1, 3, 1)
(1, 2, 1, 2, 1, 2, 1, 2)
In A Geometry of Music Dmitri Tymoczko defines an n-gap scale
to be one where the gap between successive pitches is at most n
half steps. He groups 2-gap and 3-gap scales under the general
heading of small-gap scales. Tymoczko adds two 3-gap seven
note scales (harmonic minor and harmonic major) to our list,
and we will follow suit here, yielding the following (final) table of
scale-types:
pentatonic
hexatonic
whole-tone
diatonic
acoustic
harmonic minor
harmonic major
octatonic
{C, D, E, G, A}
{C, C], E, F, G], A}
{C, D, E, F], G], A]}
{C, D, E, F, G, A, B}
{C, D, E, F], G, A, B[}
{C, D, E[, F, G, A[, B}
{C, D, E, F, G, A[, B}
{C, C], D], E, F], G, A, B[}
(2, 2, 3, 2, 3)
(1, 3, 1, 3, 1, 3)
(2, 2, 2, 2, 2, 2)
(2, 2, 1, 2, 2, 2, 1)
(2, 2, 2, 1, 2, 1, 2)
(2, 2, 1, 2, 1, 3, 1)
(2, 2, 1, 2, 1, 3, 1)
(1, 2, 1, 2, 1, 2, 1, 2)
As observed by Tymoczko, this collection of scales is “tonally
complete” in the following sense: any chord X which does not
contain a chromatic cluster (three or more consecutive pitches
separated by half step) is contained within one of these scales.
Claude Debussy, Préludes I, “Voiles”
Claude Debussy, Préludes I, “Voiles”
Igor Stravinsky, Petroushka, II.Chez Petroushka
Igor Stravinsky, Petroushka, II.Chez Petroushka
Igor Stravinsky, Petroushka, II.Chez Petroushka
Olivier Messiaen, Vingt Regards sur l’Enfant-Jésus, I.
Regard du Père
Scalar intervals, transpositions and inversions
Tymoczko likes to think of a scale as a ruler that measures
pitch-class space in a particular way, in terms of scalar steps.
Scalar intervals, transpositions and inversions
Tymoczko likes to think of a scale as a ruler that measures
pitch-class space in a particular way, in terms of scalar steps.
Definition
Given a scale X = {P1 , P2 , . . . , Pr }, where we assume the Pi are
listed in clockwise order, we say an interval of the form {Pi , Pi+k }
has a scalar length of k (scalar) steps.
Scalar intervals, transpositions and inversions
Tymoczko likes to think of a scale as a ruler that measures
pitch-class space in a particular way, in terms of scalar steps.
Definition
Given a scale X = {P1 , P2 , . . . , Pr }, where we assume the Pi are
listed in clockwise order, we say an interval of the form {Pi , Pi+k }
has a scalar length of k (scalar) steps.
Following the interval naming conventions of the diatonic scale, we
call {Pi , Pi+1 } a (scalar) second, {Pi , Pi+2 } a (scalar) third, etc.
Scalar intervals, transpositions and inversions
Tymoczko likes to think of a scale as a ruler that measures
pitch-class space in a particular way, in terms of scalar steps.
Definition
Given a scale X = {P1 , P2 , . . . , Pr }, where we assume the Pi are
listed in clockwise order, we say an interval of the form {Pi , Pi+k }
has a scalar length of k (scalar) steps.
Following the interval naming conventions of the diatonic scale, we
call {Pi , Pi+1 } a (scalar) second, {Pi , Pi+2 } a (scalar) third, etc.
Example
Let X = Oct0,1 = {0, 1, 3, 4, 6, 7, 9, 10}. Then {0, 4} is an
octatonic fourth, since 4 is three scalar steps up from 0. Similarly,
{1, 7} is an octatonic fifth.
Scalar intervals, transpositions and inversions
Tymoczko likes to think of a scale as a ruler that measures
pitch-class space in a particular way, in terms of scalar steps.
Definition
Given a scale X = {P1 , P2 , . . . , Pr }, where we assume the Pi are
listed in clockwise order, we say an interval of the form {Pi , Pi+k }
has a scalar length of k (scalar) steps.
Following the interval naming conventions of the diatonic scale, we
call {Pi , Pi+1 } a (scalar) second, {Pi , Pi+2 } a (scalar) third, etc.
Example
Let X = Oct0,1 = {0, 1, 3, 4, 6, 7, 9, 10}. Then {0, 4} is an
octatonic fourth, since 4 is three scalar steps up from 0. Similarly,
{1, 7} is an octatonic fifth.
Let X = {0, 2, 4, 7, 9}, the C pentatonic scale. Then X has two
different kinds of pentatonic seconds: those of chromatic length 2
({0, 2}, {2, 4}, {7, 9}), and those of chromatic length 3
({4, 7}, {9, 0}).
Scalar transposition
Definition
Given a scale X = {P1 , P2 , . . . , Pr } written in clockwise order, we
define scalar transposition by k steps to be the function
stk : X → X defined as stk (Pi ) = Pi+k , where we must take i + k
modulo r for this to make sense.
Scalar transposition
Definition
Given a scale X = {P1 , P2 , . . . , Pr } written in clockwise order, we
define scalar transposition by k steps to be the function
stk : X → X defined as stk (Pi ) = Pi+k , where we must take i + k
modulo r for this to make sense.
Intuitively, the scalar transposition stk shifts each pitch k places
“forward” in the scale.
Scalar transposition
Definition
Given a scale X = {P1 , P2 , . . . , Pr } written in clockwise order, we
define scalar transposition by k steps to be the function
stk : X → X defined as stk (Pi ) = Pi+k , where we must take i + k
modulo r for this to make sense.
Intuitively, the scalar transposition stk shifts each pitch k places
“forward” in the scale.
Example
Let X = {0, 2, 4, 5, 7, 9, 11}, the C diatonic scale, and consider the
scalar transposition st1 that shifts everything up by 1. Then we
have
st1 (0) = 2, st1 (2) = 4, st1 (4) = 5, . . . , st1 (9) = 11, st1 (11) = 0.
Scalar transposition
Definition
Given a scale X = {P1 , P2 , . . . , Pr } written in clockwise order, we
define scalar transposition by k steps to be the function
stk : X → X defined as stk (Pi ) = Pi+k , where we must take i + k
modulo r for this to make sense.
Intuitively, the scalar transposition stk shifts each pitch k places
“forward” in the scale.
Example
Let X = {0, 2, 4, 5, 7, 9, 11}, the C diatonic scale, and consider the
scalar transposition st1 that shifts everything up by 1. Then we
have
st1 (0) = 2, st1 (2) = 4, st1 (4) = 5, . . . , st1 (9) = 11, st1 (11) = 0.
Note that unlike normal transposition, scalar transpositions move
pitches by a varying amount. They do not preserve the
(chromatic) distance between scale pitches, but they do preserve
the scalar distances!
Comment
Once we know how to define scalar transpositions on the pitches of
a scale, we go on to define scalar transpositions of subsets (chords)
and sequences (modes, melodies) in the usual way.
Comment
Once we know how to define scalar transpositions on the pitches of
a scale, we go on to define scalar transpositions of subsets (chords)
and sequences (modes, melodies) in the usual way.
For example, let X = {0, 2, 4, 5, 7, 9, 11} again, and consider the
scalar melody “Do a deer”: (0, 2, 4).
Comment
Once we know how to define scalar transpositions on the pitches of
a scale, we go on to define scalar transpositions of subsets (chords)
and sequences (modes, melodies) in the usual way.
For example, let X = {0, 2, 4, 5, 7, 9, 11} again, and consider the
scalar melody “Do a deer”: (0, 2, 4). Transposing this up by 1
scalar step yields the new melody (2, 4, 5), which is “Re a drop (of
golden sun)”.
Comment
Once we know how to define scalar transpositions on the pitches of
a scale, we go on to define scalar transpositions of subsets (chords)
and sequences (modes, melodies) in the usual way.
For example, let X = {0, 2, 4, 5, 7, 9, 11} again, and consider the
scalar melody “Do a deer”: (0, 2, 4). Transposing this up by 1
scalar step yields the new melody (2, 4, 5), which is “Re a drop (of
golden sun)”.
The example is Tymoczko’s, and his point is that though
chromatically speaking the two sequences are different (W-W,
versus W-H), when measured by the C diatonic scale they are
somehow the same: namely, both melodies simply ascend two scale
steps.
Scalar inversion
We can also define a scalar version of inversion. Fix a scale
X = {P1 , P2 , . . . , Pr }, written as usual in clockwise order.
Scalar inversion
We can also define a scalar version of inversion. Fix a scale
X = {P1 , P2 , . . . , Pr }, written as usual in clockwise order. To
invert around a scalar pitch Pj , we take any pitch that is k scalar
steps above Pj and and send it to the pitch that is k scalar steps
below: that is, we want a map that sends
Pj+k 7→ Pj−k .
Scalar inversion
We can also define a scalar version of inversion. Fix a scale
X = {P1 , P2 , . . . , Pr }, written as usual in clockwise order. To
invert around a scalar pitch Pj , we take any pitch that is k scalar
steps above Pj and and send it to the pitch that is k scalar steps
below: that is, we want a map that sends
Pj+k 7→ Pj−k .
It is easy to see that the map
Pi 7→ Pi−2(i−j) = P−i+2j
does the trick. As with scalar transposition, we must compute
−i + 2j modulo r for this to make sense.
Scalar inversion
We can also define a scalar version of inversion. Fix a scale
X = {P1 , P2 , . . . , Pr }, written as usual in clockwise order. To
invert around a scalar pitch Pj , we take any pitch that is k scalar
steps above Pj and and send it to the pitch that is k scalar steps
below: that is, we want a map that sends
Pj+k 7→ Pj−k .
It is easy to see that the map
Pi 7→ Pi−2(i−j) = P−i+2j
does the trick. As with scalar transposition, we must compute
−i + 2j modulo r for this to make sense.
Definition
Given a scale X = {P1 , P2 , . . . , Pr } written in clockwise order, and
choice of pitch Pj in the scale, we define scalar inversion with
respect to Pj to be the function sij : X → X defined as
sij (Pi ) = P−i+2j , where we must take −i + 2j modulo r for this to
make sense.
Example
Return to our example from The Art of Fugue.
                      
Subject
Inversion
Example
Return to our example from The Art of Fugue.
                      
Subject
Inversion
Recall that the inversion is not a strict chromatic inversion of the theme. Can
we express this operation in terms of scalar operations?
Example
Return to our example from The Art of Fugue.
                      
Subject
Inversion
Recall that the inversion is not a strict chromatic inversion of the theme. Can
we express this operation in terms of scalar operations?
Yes, but to do so, we need to use the harmonic minor scale on D:
X = {D, E, F, G, A, B[, C]} = {P1 , P2 , . . . , P7 }.
Example
Return to our example from The Art of Fugue.
                      
Subject
Inversion
Recall that the inversion is not a strict chromatic inversion of the theme. Can
we express this operation in terms of scalar operations?
Yes, but to do so, we need to use the harmonic minor scale on D:
X = {D, E, F, G, A, B[, C]} = {P1 , P2 , . . . , P7 }.
Now to get the inverted form from the subject, first transpose up by 3 scale
steps (using st3 ) to make the A a D, then invert with respect to D (using si1 ).
The corresponding scalar operation is then
si1 ◦ st3 (Pi ) = si1 (Pi+3 ) = P−(i+3)+2 = P−i−1 = P−i+6
Example
Return to our example from The Art of Fugue.
                      
Subject
Inversion
Recall that the inversion is not a strict chromatic inversion of the theme. Can
we express this operation in terms of scalar operations?
Yes, but to do so, we need to use the harmonic minor scale on D:
X = {D, E, F, G, A, B[, C]} = {P1 , P2 , . . . , P7 }.
Now to get the inverted form from the subject, first transpose up by 3 scale
steps (using st3 ) to make the A a D, then invert with respect to D (using si1 ).
The corresponding scalar operation is then
si1 ◦ st3 (Pi ) = si1 (Pi+3 ) = P−(i+3)+2 = P−i−1 = P−i+6 = si3 (Pi )!
Example
Return to our example from The Art of Fugue.
                      
Subject
Inversion
Recall that the inversion is not a strict chromatic inversion of the theme. Can
we express this operation in terms of scalar operations?
Yes, but to do so, we need to use the harmonic minor scale on D:
X = {D, E, F, G, A, B[, C]} = {P1 , P2 , . . . , P7 }.
Now to get the inverted form from the subject, first transpose up by 3 scale
steps (using st3 ) to make the A a D, then invert with respect to D (using si1 ).
The corresponding scalar operation is then
si1 ◦ st3 (Pi ) = si1 (Pi+3 ) = P−(i+3)+2 = P−i−1 = P−i+6 = si3 (Pi )!
Let’s check that this operation exactly maps the subject onto the inverted form:
P
P5
P1
P2
P3
P4
P5
P6
P5
P4
P3
Example
Return to our example from The Art of Fugue.
                      
Subject
Inversion
Recall that the inversion is not a strict chromatic inversion of the theme. Can
we express this operation in terms of scalar operations?
Yes, but to do so, we need to use the harmonic minor scale on D:
X = {D, E, F, G, A, B[, C]} = {P1 , P2 , . . . , P7 }.
Now to get the inverted form from the subject, first transpose up by 3 scale
steps (using st3 ) to make the A a D, then invert with respect to D (using si1 ).
The corresponding scalar operation is then
si1 ◦ st3 (Pi ) = si1 (Pi+3 ) = P−(i+3)+2 = P−i−1 = P−i+6 = si3 (Pi )!
Let’s check that this operation exactly maps the subject onto the inverted form:
P
si1 ◦ st3 (P)
P5
P1
P1
P5
P2
P4
P3
P3
P4
P2
P5
P1
P6
P7
P5
P1
P4
P2
P3
P3 .
Maximally even scales
What makes the diatonic scale so special? We have seen already
that it is rich in intervallic content, as evidenced by its interval
vector 254361. This is also apparent in the following property:
each scalar interval of the diatonic scale comes in two chromatic
flavors (m2/M2, m3/M3, etc.). A scale satisfying this property is
called maximally even.
Maximally even scales
What makes the diatonic scale so special? We have seen already
that it is rich in intervallic content, as evidenced by its interval
vector 254361. This is also apparent in the following property:
each scalar interval of the diatonic scale comes in two chromatic
flavors (m2/M2, m3/M3, etc.). A scale satisfying this property is
called maximally even.
Definition
Given a scale X = {P1 , P2 , . . . , Pr } written in clockwise order, we
say X is maximally even if for every 1 ≤ k ≤ r − 1, the scalar
intervals of size k are either all of the same chromatic length, or
else come in exactly two consecutive chromatic lengths: that is,
one of size ` half steps, the other of size ` + 1 half steps.
Example
Take the C pentatonic scale
X = {0, 2, 4, 7, 9} = {P1 , P2 , P3 , P4 , P5 }. We investigate the
different chromatic flavors of each scalar interval of size k,
1 ≤ k ≤ 7.
Example
Take the C pentatonic scale
X = {0, 2, 4, 7, 9} = {P1 , P2 , P3 , P4 , P5 }. We investigate the
different chromatic flavors of each scalar interval of size k,
1 ≤ k ≤ 7.
Scalar size k Chromatic sizes
1
2, 3
2
4, 5
3
7, 8
4
9, 10
This shows the C pentatonic is maximally even, and hence that all
pentatonic scales are maximally even.
Example
Take
Oct0,1 = {0, 1, 3, 4, 6, 7, 9, 10} = {P1 , P2 , P3 , P4 , P5 , P6 , P7 , P8 }.
We investigate the different chromatic flavors of each scalar
interval of size k, 1 ≤ k ≤ 7.
Example
Take
Oct0,1 = {0, 1, 3, 4, 6, 7, 9, 10} = {P1 , P2 , P3 , P4 , P5 , P6 , P7 , P8 }.
We investigate the different chromatic flavors of each scalar
interval of size k, 1 ≤ k ≤ 7.
Scalar size k Chromatic sizes
1
1, 2
2
3
3
4, 5
4
6
5
7, 8
6
9
7
10, 11
This shows Oct0,1 is also maximally even, and thus the same is
true for all octatonic scales.
Example
Take the hexatonic scale
X = {0, 1, 4, 5, 8, 9} = {P1 , P2 , P3 , P4 , P5 , P6 }. We investigate the
different chromatic flavors of each scalar interval of size k,
1 ≤ k ≤ 5.
Example
Take the hexatonic scale
X = {0, 1, 4, 5, 8, 9} = {P1 , P2 , P3 , P4 , P5 , P6 }. We investigate the
different chromatic flavors of each scalar interval of size k,
1 ≤ k ≤ 5.
Scalar size k Chromatic sizes
1
1, 3
2
4
3
5, 7
4
8
5
9, 11
This shows the hexatonic scales are not maximally even: the scalar
seconds, for example, come in two chromatic lengths, 1 and 3,
which are not consecutive
Why does this property about the relation of scalar intervals to chromatic ones
deserve to be called maximally even?
Why does this property about the relation of scalar intervals to chromatic ones
deserve to be called maximally even? One would have guessed that a scale
containing n distinct pitches should be called maximally even if the pitches are
as evenly distributed around the pitch-class circle as possible.
Why does this property about the relation of scalar intervals to chromatic ones
deserve to be called maximally even? One would have guessed that a scale
containing n distinct pitches should be called maximally even if the pitches are
as evenly distributed around the pitch-class circle as possible.
Put another way, for any fixed n, we can always pick n pitches that divide the
circle evenly into n segments. However, when n - 12, these pitches will not be
equal-tempered! A maximally even collection should be the collection of
pitches that are the best equal-tempered approximation of this perfectly even
distribution.
Why does this property about the relation of scalar intervals to chromatic ones
deserve to be called maximally even? One would have guessed that a scale
containing n distinct pitches should be called maximally even if the pitches are
as evenly distributed around the pitch-class circle as possible.
Put another way, for any fixed n, we can always pick n pitches that divide the
circle evenly into n segments. However, when n - 12, these pitches will not be
equal-tempered! A maximally even collection should be the collection of
pitches that are the best equal-tempered approximation of this perfectly even
distribution.
Miraculously, it turns out that our definition of maximally even is equivalent to
this!
Theorem
Fix n. Let Xn = {0, 12/n, 2(12/n), . . . , 11(12/n)} be the collection of pitches
that divides the circle up into equal segments. (If n - 12, then some pitches of
Xn will not be equal-tempered.) Let Yn be the equal-tempered collection you
get by taking the integer points closest to each of the pitches i(12/n) in Xn .
Why does this property about the relation of scalar intervals to chromatic ones
deserve to be called maximally even? One would have guessed that a scale
containing n distinct pitches should be called maximally even if the pitches are
as evenly distributed around the pitch-class circle as possible.
Put another way, for any fixed n, we can always pick n pitches that divide the
circle evenly into n segments. However, when n - 12, these pitches will not be
equal-tempered! A maximally even collection should be the collection of
pitches that are the best equal-tempered approximation of this perfectly even
distribution.
Miraculously, it turns out that our definition of maximally even is equivalent to
this!
Theorem
Fix n. Let Xn = {0, 12/n, 2(12/n), . . . , 11(12/n)} be the collection of pitches
that divides the circle up into equal segments. (If n - 12, then some pitches of
Xn will not be equal-tempered.) Let Yn be the equal-tempered collection you
get by taking the integer points closest to each of the pitches i(12/n) in Xn .
Then Yn is maximally even. Furthermore, an equal-tempered n-pitch scale
X = {P1 , P2 , . . . , Pn } is maximally even if and only if it is a transposition of
Yn .
Why does this property about the relation of scalar intervals to chromatic ones
deserve to be called maximally even? One would have guessed that a scale
containing n distinct pitches should be called maximally even if the pitches are
as evenly distributed around the pitch-class circle as possible.
Put another way, for any fixed n, we can always pick n pitches that divide the
circle evenly into n segments. However, when n - 12, these pitches will not be
equal-tempered! A maximally even collection should be the collection of
pitches that are the best equal-tempered approximation of this perfectly even
distribution.
Miraculously, it turns out that our definition of maximally even is equivalent to
this!
Theorem
Fix n. Let Xn = {0, 12/n, 2(12/n), . . . , 11(12/n)} be the collection of pitches
that divides the circle up into equal segments. (If n - 12, then some pitches of
Xn will not be equal-tempered.) Let Yn be the equal-tempered collection you
get by taking the integer points closest to each of the pitches i(12/n) in Xn .
Then Yn is maximally even. Furthermore, an equal-tempered n-pitch scale
X = {P1 , P2 , . . . , Pn } is maximally even if and only if it is a transposition of
Yn .
Thus for each n there is a unique maximally even scale-type!
Maximally even: n = 5
Maximally even: n = 5
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Maximally even: n = 7
Maximally even: n = 7
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Maximally even: n = 8
Maximally even: n = 8
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