(2020-07-08) Four heptatonic scales are special in Western notation.
They have key signatures without any double alteration in all 12 keys.
In Western music, there are exactly four heptatonic scales which can be specified
in all 12 keys by a key signature without double alterations
(no double-flats or double-sharps).
Those four scales and their 28 modes (including some fairly exotic ones)
form the basis of almost all extant Western music.
They have already been presented in our introductory page
and are tabulated again below:
2212221 : The diatonic scale
(modes include major and natural minor).
Prominent college-level textbooks
(including Tonal Harmony)
don't even cover this, which leaves the curiosity of many practitioners unsatisfied.
Let's first prove that only the above four
basic scales satisfy the claim:
A satisfactory heptatonic scale must be non-cohemitonic
(which is to say that half-steps never occur consecutively).
Otherwise, there would exist a
mode containing the single-named notes G and A and the
the tone halfway between them. That tone can't be named without repeating either A or G.
Now, it's not difficult to see that only six heptatonic scales verify this necessary condition.
To establish that, let's remark that
a sequence of seven nonzero steps cannot avoid consecutive "1" steps ("1" means one semitone)
unless there are fewer than four of these. That leaves only two possibilities
for seven steps adding up to a total of 12:
Two "1" and five "2": 2212221 or 2122221 up to a modal rotation.
A 1x1x1xx pattern where one "x" is "3" and the others are "2".
As the latter leaves four choices, we're left with six possibilities,
including the four which have already been shown to be satisfactory.
The other two are not, as we shall soon find out (each fails for two keys).
Most professional musicians are unfazed by double-flats or double-sharps.
Some composers will use them even when they are not absolutely necessary.
On this page at least, I'm adamant about not doing so.
For example, Ultralocrian is mode VII of
Harmonic minor (one of the four golden scales).
While C-Ultralocrian does requires a double-flat (and 5 single flats)
the enharmonic B#-Utralocrian doesn't (it just calls for 5 single sharps).
Therefore, I only tabulate the latter
(also avoiding calling Ultralocrian "Super Locrian bb7", because that only
applies to some keys, whereas generic names should apply to all of them.)
That's the mission. That's the pledge. I'm not trying to convince anybody.
Music Theory Distilled:
(10:33,
11:32,
4:08) by Casey Connor (2018).
(2018-02-26) The Seven
Diatonic
Modes (modes of the major scale)
A key is given by a
tonic and a mode
(e.g., C-major or F-Lydian).
The modern diatonic modes described below are named after ancient Greek
harmoniai
(or tonoi) and/or the medieval church modes. However, those concepts are only loosely related.
By definition, a diatonic scale consists of seven steps; two half-tones (H) separated by alternating groups
of two or three whole-tones (W). There are seven ways to choose a starting point in such
a progression. Each such way is called a mode.
The most common ones are the aforementioned major and minor modes, also called
Ionian and Aeolian. Here's the complete list:
105 names for the 84 diatonic scales:
7 modes in 12 keys (15 allowed names). (For
each mode, the tonic can be one of the 12 choices in column "1".)
1
IV: Lydian
7
18
1
I: Ionian (= Major)
7
17
1
V: Mixolydian
7
16
1
II: Dorian
7
15
1
VI: Aeolian (= Minor)
7
14
1
III: Phrygian
7
13
1
VII: Locrian
7
12
F
G
A
Bb
C
D
E
F
G
A
Bb
C
D
E
C
D
E
F
G
A
B
C
D
E
F
G
A
B
G
A
B
C
D
E
F#
G
A
B
C
D
E
F#
D
E
F#
G
A
B
C#
D
E
F#
G
A
B
C#
A
B
C#
D
E
F#
G#
A
B
C#
D
E
F#
G#
E
F#
G#
A
B
C#
D#
E
F#
G#
A
B
C#
D#
B Cb
C# Db
D# Eb
E Fb
F# Gb
G# Ab
A# Bb
B Cb
C# Db
D# Eb
E Fb
F# Gb
G# Ab
A# Bb
F# Gb
G# Ab
A# Bb
B Cb
C# Db
D# Eb
E# F
F# Gb
G# Ab
A# Bb
B Cb
C# Db
D# Eb
E# F
C# Db
D# Eb
E# F
F# Gb
G# Ab
A# Bb
B# C
C# Db
D# Eb
E# F
F# Gb
G# Ab
A# Bb
B# C
Ab
Bb
C
Db
Eb
F
G
Ab
Bb
C
Db
Eb
F
G
Eb
F
G
Ab
Bb
C
D
Eb
F
G
Ab
Bb
C
D
Bb
C
D
Eb
F
G
A
Bb
C
D
Eb
F
G
A
M
m
m
M
M
m
mo
M
m
m
M
M
m
mo
1st
2nd
3rd
4th
5th
6th
7th
<<< mode of the major scale.
In this table and others tables like it, the next-to-last line gives the
quality (major, minor, diminished, augmented) of
the in-scale triad rooted in the corresponding column.
The last line repeats the rank of the mode named and numbered as a 1-7 span at the top of the table.
Two scales which share the same diatonic mode are said to be modes of each other.
For example, F-Lydian is the 4-th mode of C-major (i.e., C-Ionian) because
its tonic (F) is the 4-th note in the C-major scale. Likewise,
G-Mixolydian is the fifth mode of C-major or the second mode of F-Lydian.
In modern practice, this type of reference is most commonly used with respect to the
relevant major scale only, as indicated in the last line of the above table.
Dorian is the only palindromic mode (2122212).
In all other cases, a mirror mode is obtained by inversion:
Mixolydian (2212212) and Aeolian (2122122).
Ionian (2212221) and Phrygian (1222122).
Lydian (2221221) and Locrian (1221222). Brightest and darkest.
For each of those mirror modal pairs, the ascending version of one
is the same as the descending version of the other.
Béla Bartók (1881-1945)
remarked that all 12 chromatic tones are obtained by mixing
some pairs of diatonic modes on the same tonic (e.g., C-Lydian with C-Phrygian).
More generally, the table below can be used to count how many tones are
obtained when mixing two diatonic modes on the same tonic.
Number of tones obtained by mixing two modes
IV
I
V
II
VI
III
VII
F
Lydian
IV
7
8
9
10
11
12
12
C
Ionian
I
8
7
8
9
10
11
12
G
Mixolydian
V
9
8
7
8
9
10
11
D
Dorian
II
10
9
8
7
8
9
10
A
Aeolian
VI
11
10
9
8
7
8
9
E
Phrygian
III
12
11
10
9
8
7
8
B
Locrian
VII
12
12
11
10
9
8
7
Among many other things, this provides a name for a half-dozen special octatonic scales,
which are actually the first 6 modes of the Bebop (dominant) scale:
Lydian-Ionian adds Bb to F-Lydian or F# to C-Ionian. Mode IV of Bebop scale
Ionian-Mixolydian adds Bb to C-Ionian or F# to G-Mixolydian. Mode I of Bebop scale.
Mixolydian-Dorian adds Bb to G-Mixolydian or F# to D-dorian. Mode V of Bebop scale
Dorian-Aeolian adds Bb to D-dorian or F# to A-Aeolian. Mode II of Bebop scale.
Aeolian-Phrygian adds Bb to A-Aeolian or F# to E-Phrygian. Mode VI of the Bebop scale.
Phrygian-Locrian adds Bb to E-Phrygian or F# to B-Locrian. Mode III of Bebop scale.
(2020-07-20) Naming Non-Diatonic Scales and Modes
A systematic nomenclature to define precisely lesser-known modes.
The diatonic scale (especially its Major and Minor modes)
is the backdrop for almost all classical music and a good chunk of modern tunes,
although 6 of the common diatonic modes are now in common use
(locrian is left out).
Only four heptatonic scales in the Western chromatic system can be expressed
canonically in all 12 keys, by naming all seven notes (letters)
once and only once, with at most one sharp or one flat each. Namely:
This property, which is taken for granted by most casual students of Western music,
is indeed a rare one. It fails whenever the scale includes two consecutive
half-steps (HINT: For every mode, there's a key where
A and G are included with the tone between them, which can be called neither
A# nor Gb without repeating a letter).
There are only two other heptatonic scales for which this
doesn't happen (Hungarian major and
Romanian major) but they both fail in two keys,
for less obvious reasons.
This fact is lost on most composers and almost all practicing musicians, with
little or no consequences: Transposing a piece in writing for all possible keys
is rarely required, if ever.
So, there's little or no obstacle to experimentation with a
huge number of exotic scales.
The most popular ones eventually get a colorful name.
For others, the standard practice is to use a known name
(preferably one of the diatonic modes
and indicate what modification(s), sharp of flat, is to be applied to what
degree(s).
Some are queasy about using degree 1 in this scheme.
I beg to differ but accomodate those concerns by putting such names
inside square brackets.
The systematic exploration of non-diatonic scales and their modes started in 1907
with the investigations of synthetic scales
by Ferruccio Busoni (1866-1924) who first considered all
scales which could be derived from a diatonic mode by lowering or raising a single degree by one semitone.
Each of the 7 diatonic modes features 5 skipped notes which
can be included in one of two ways. Therefore, 70 names can be attributed
to new scales this way (for a total of 77 names) However,
some of those scales may have more than one name (as is the case for any mode of the
melodic minor scale).
For example, the Assyrian scale (mode II
of the melodic minor scale) possesses two distinct systematic names:
either Dorian b2 or Phrygian #6. In the key of G, that's:
G Ab Bb C D E F
As G-Dorian is G A Bb C D E F and G-Phrygian is G Ab Bb C D Eb F.
Conversely, some of the 66 different heptatonic scales can't be named at all with just one modification
from a diatonic mode.
The scope was later expanded to include all possible scales
and their respective modes.
Dominant means the 3 is major and the 7 is flat.
The whole problem is of greater theoretical interest than of practical worth. J. Murray Barbour (1929)
(2020-06-22) The two most common non-diatonic minor scales.
Modes of the melodic minor scale. Modes of the harmonic minor scale.
The natural minor scale is diatonic.
As such, it's just another mode of the major scale
(namely, the 6th mode of the major scale, also called Aeolian).
Therefore, all modes of the natural minor scale would also be
modes of the major scale and they are not considered separately.
However, neither the harmonic minor scale,
nor the melodic minor scale are diatonic and they are
not modes of each other either. Therefore,
both give rise to a full set of 7 distinct modes in 12 possible keys.
We may tabulate them as we did the more common diatonic modes
(i.e, the modes of the major scale).
2 1 2 2 2 2 1 2 1 2 2 2 2 1 :
Melodic Minor Scale All 7 modes ranked by brightness, in 12 keys (tonic in mode's column "1")
1
III: Lydian augmented
7
19
1
IV: Lydian dominant
7
17
1
I: Jazz minor scale
7
16
1
V: Hindu scale
7
15
1
II: Phrygidorian, Assyrian
14
1
VI: Half diminished
7
13
1
VII: Super Locrian
7
11
F
G
Ab
Bb
C
D
E
F
G
Ab
Bb
C
D
E
C
D
Eb
F
G
A
B
C
D
Eb
F
G
A
B
G
A
Bb
C
D
E
F#
G
A
Bb
C
D
E
F#
D
E
F
G
A
B
C#
D
E
F
G
A
B
C#
A
B
C
D
E
F#
G#
A
B
C
D
E
F#
G#
E
F#
G
A
B
C#
D#
E
F#
G
A
B
C#
D#
B
C#
D
E
F#
G#
A#
B
C#
D
E
F#
G#
A#
F#
G#
A
B
C#
D#
E#
F#
G#
A
B
C#
D#
E#
C# Db
D# Eb
E Fb
F# Gb
G# Ab
A# Bb
B# C
C# Db
D# Eb
E Fb
F# Gb
G# Ab
A# Bb
B# C
Ab
Bb
Cb
Db
Eb
F
G
Ab
Bb
Cb
Db
Eb
F
G
Eb
F
Gb
Ab
Bb
C
D
Eb
F
Gb
Ab
Bb
C
D
Bb
C
Db
Eb
F
G
A
Bb
C
Db
Eb
F
G
A
m
m
M+
M
M
mo
mo
m
m
M+
M
M
mo
mo
1st
2nd
3rd
4th
5th
6th
7th
<<< mode of melodic minor.
The 7 modes of the melodic minor scale are known under various names:
Dorian #7. Ionian b3. (Ascending) melodic minor scale.
Jazz minor.
The Hindu scale is palindromic. The other six modes
come in three mirror pairs consisting of a bright mode and a dark mode, increasingly different:
Jazz minor scale and phrygidorian.
Acoustic scale and aeolocrian.
Lydian augmented (brightest) and altered scale (darkest mode).
With an extra chromatic passing tone between the 5th and 6th degrees ,
mode I becomes an octatonic scale known as Bebop melodic minor.
The other proper Bebop scales are all modes of two other scales,
unrelated to the melodic minor scale.
Nevertheless, we may consider Bebop modes obtained from the other heptatonic modes listed above
by allowing a passing tone at the
blue-shaded position .
Only mode I is widely accepted:
In the description of scales, the term harmonic
indicates the presence of at least one step consisting of an augmented second
(3 semitones). Such a step is best abbreviated "A" in the alphabetical version of
interval structures (where H is a half-step of 1 semitone
and W is a whole-step of 2 semitones.
The scientific name for that is sesquitone,
but the abbreviation "S" is unused.
Rick Beato
insists on calling Ultralocrian "Super Locrian bb7" because he always
presents modes in the key of C (although B# is called for here)
which leads to Bbb. That's a missed opportunity
to stress that double alterations are never needed for modes of the four
normative scales (diatonic, melodic minor, harmonic minor and harmonic major).
The harmonic minor scale has no axis of symmetry.
The mirror inverses of its modes are modes of the
harmonic major scale, described next...
(2020-06-27) Harmonic Major Scale
The mirror inverse of the harmonic minor scale.
This is arguably either the most exotic of the normative heptatonic Western scales
or the least exotic of the exceptional ones. It was introduced in 1853 by
Moritz Hauptmann (1792-1868).
The current name of the scale was popularized by
Nikolai Rimsky-Korsakov (1844-1908)
in his Practical Manual of Harmony (1885).
2 2 1 2 1 3 1 2 2 1 2 1 3 1 :
Harmonic Major Scale All 7 modes ranked by brightness, in 12 keys (tonic in mode's column "1")
1
VI : Lydian Augmented #2
20
1
IV : Lydian diminished
7
17
1
I: Harmonic major
7
16
1
V : Mixolydian b2
7
15
1
II : Blues heptatonic
7
14
1
III : Phrygian b4
7
12
1
VII : Locrian b7
7
11
F
G
A
Bb
C
Db
E
F
G
A
Bb
C
Db
E
C
D
E
F
G
Ab
B
C
D
E
F
G
Ab
B
G
A
B
C
D
Eb
F#
G
A
B
C
D
Eb
F#
D
E
F#
G
A
Bb
C#
D
E
F#
G
A
Bb
C#
A
B
C#
D
E
F
G#
A
B
C#
D
E
F
G#
E
F#
G#
A
B
C
D#
E
F#
G#
A
B
C
D#
B
C#
D#
E
F#
G
A#
B
C#
D#
E
F#
G
A#
F#
G#
A#
B
C#
D
E#
F#
G#
A#
B
C#
D
E#
C#
D#
E#
F#
G#
A
B#
C#
D#
E#
F#
G#
A
B#
Ab
Bb
C
Db
Eb
Fb
G
Ab
Bb
C
Db
Eb
Fb
G
Eb
F
G
Ab
Bb
Cb
D
Eb
F
G
Ab
Bb
Cb
D
Bb
Cb
D
Eb
F
Gb
A
Bb
Cb
D
Eb
F
Gb
A
1st
2nd
3rd
4th
5th
6th
7th
...mode of the harmonic major scale.
The 7 modes of the harmonic major scale are called:
(2020-06-25) Double-Harmonic Major and Minor Scales
The latter is the fourth mode of the former. Both have one cursed root.
Like the diatonic scales (major and minor) the double-harmonic
scales are just modes of each other. Tne double-harmonic minor scale
is simply the fourth mode of the double harmonic major scale.
(Just as the minor scale is the sixth mode of the major scale.)
So, they have the same seven modes, which we'll examine shortly.
All four classical scales (major, natural minor, harmonic minor and melodic minor)
all retain the first, second, fourth and fith degree from the root.
The double-harmonic scale preserves only the first, fourth and fifth.
Double-harmonic major and minor scales.
Double Harmonic Major Scale
Double Harmonic Minor Scale
1
2
3
4
5
6
7
1
2
3
4
5
6
7
F
Gb
A
Bb
C
Db
E
F
G
Ab
B
C
Db
E
C
Db
E
F
G
Ab
B
C
D
Eb
F#
G
Ab
B
G
Ab
B
C
D
Eb
F#
G
A
Bb
C#
D
Eb
F#
D
Eb
F#
G
A
Bb
C#
D
E
F
G#
A
Bb
C#
A
Bb
C#
D
E
F
G#
A
B
C
D#
E
F
G#
E
F
G#
A
B
C
D#
E
F#
G
A#
B
C
D#
B
C
D#
E
F#
G
A#
B
C#
D
E#
F#
G
A#
F#
G
A#
B
C#
D
E#
F#
G#
A
B#
C#
D
E#
C#
D
E#
F#
G#
A
B#
C# Db
D# Eb
E Fb
F## G
G# Ab
A Bbb
B# C
G# Ab
A Bbb
B# C
C# Db
D# Eb
E Fb
F## G
Ab
Bb
Cb
D
Eb
Fb
G
Eb
Fb
G
Ab
Bb
Cb
D
Eb
F
Gb
A
Bb
Cb
D
Bb
Cb
D
Eb
F
Gb
A
Bb
C
Db
E
F
Gb
A
For both of these scales, there's a key which can't be transcribed with a
key signature in standard Western notation.
That problem is signaled with
pink highlighting
wherever it arises.
The same flaw is present in all heptatonic scales except the four standard ones,
previously presented
(diatonic scale, melodic minor, harmonic minor, harmonic major)
and two more exotic scales which will be presented
later
(Hungarian Major and Romanian Major).
The problem can often be fixed by using a double-flat or a double-sharp to name a natural tone,
which we'll a pivot) and show underscored in a
pink row
with a live link to the above discussion.
Our first example is in the following table, where A
stands for Bbb. More generally:
(2020-07-08) Hungarian Major & Romanian Major
Two non-cohemitonic heptatonic scales which are all but forgotten.
As explained above,
being non-cohemitonic is a necessary condition
for a heptatonic scale to have a simple key signature in all 12 keys
(as happens in only four cases:
diatonic, melodic minor, harmonic minor and harmonic major).
However, the two non-cohemitonic scales presented in this section
don't have a simple key signature in two of their 12 possible keys.
3 1 2 1 2 1 2 3 1 2 1 2 1 2 :
Hungarian Major Scale All 7 modes ranked by brightness, in 10+2 keys (tonic in mode's column "1")
The 7 modes of the Neapolitan major scale are called:
Neapolitan major scale.
Leading whole-tone.
Lydian augmented dominant.
Lydian minor. Lydian dominant b6.
Major Locrian.
Semilocrian b4.
Superlocrian bb3. Leading Whole-Tone Inverse.
The Neapolitan Major mode itself is palindromic (so, it has brightness 15).
The other modes can be grouped in pairs which are mirror-inverses of each other
(whose respective brigntnesses add up to 30).
Neapolitan Minor Scale :
1 2 2 2 1 3 1 1 2 2 2 1 3 1 :
Neapolitan Minor Scale All 7 modes ranked by brightness, in 10+2 keys (tonic in mode's column "1")
Harmonic Lydian has been compared to
Pelog, one of the two basic scales
in gamelan music from Java & Bali
(concurrently with the older Slendro).
Expressed in cents, the seven tones of Pelog are
reportedly:
0, 122, 271, 571, 677, 785, 947
This is somewhat close to a 7-tone scale (1121112) in 9-tone equal temperament:
0, 133, 267, 533, 667, 800, 933
Harmonic Lydian is pretty far from either: 0, 200, 400, 600, 700, 800, 1100 (222113).
An awkward match in 12-tone equal temperament would be 1231121.
What gives?
(2020-06-27) Blues leading-tone scale & Persian scale:
These heptatonic scales are mirror-inverses of each other.
The Blues Leading-Tone Scale :
This scale is obtained by adding a major seventh to the usual hexatonic Blues scale.
For every mode, a canonical key signature can be given for 10 of the 12 keys,
as tabulated. For the other two keys ,
a key signature using one double-flat or one double-sharp can be obtained from the table by replacing the
underscored pivot, G or A by F## or Bbb, respectively.
3 2 1 1 3 1 1 3 2 1 1 3 1 1 :
Blues Leading-Tone Scale All 7 modes ranked by brightness, in 10+2 keys (tonic in mode's column "1")
As the names imply, those modes are mirror inverses of the above modes of the
Blues Leading Tone scale.
The linear brightnesses
(rightmost in white) of two heptatonic modes so related always add up to 30.
(2020-07-18) The Bebop octatonic scales.
An eighth chromatic passing tone is added to some heptatonic scales.
Musicologist David N. Baker (1931-2016)
coined the term Bebop scales
for the octatonic scales obtained by adding an extra chromatic note for use as
a recurring passing tone (PT)
with certain heptatonic scales.
The first mode of the melodic minor scale
is called Bebop melodic minor when used this way with
a PT between the 5th degree and the 6th degree.
The Bebop scale itself (a.k.a. Bebop dominant) is obtained by
merging Ionian and Mixolydian on the same key.
More precisely, a passing tone is added to the Mixolydian scale
just before its root
( shaded in blue ).
2 2 1 2 2 1 1 1 2 2 1 2 2 1 1 1 :
Bebop (dominant) scale All 8 modes ranked by brightness, in 12 keys (tonic in mode's column "1")
1
I: Bebop (dominant)
8
20
1
II: Dorian-Aeolian
8
16
1
IV: Ionian-Lydian
8
16
1
VIII: Prokofiev
8
16
1
III: Bebop Locrian
8
12
1
V: Bebop Dorian
8
12
1
VII
8
12
1
VI: Phrygian-Aeolian
8
8
F
G
A
Bb
C
D
Eb
E
F
G
A
Bb
C
D
Eb
E
C
D
E
F
G
A
Bb
B
C
D
E
F
G
A
Bb
B
G
A
B
C
D
E
F
F#
G
A
B
C
D
E
F
F#
D
E
F#
G
A
B
C
C#
D
E
F#
G
A
B
C
C#
A
B
C#
D
E
F#
G
G#
A
B
C#
D
E
F#
G
G#
E
F#
G#
A
B
C#
D
D#
E
F#
G#
A
B
C#
D
D#
B
C#
D#
E
F#
G#
A
A#
B
C#
D#
E
F#
G#
A
A#
Gb
Ab
Bb
Cb
Db
Eb
Fb
F
Gb
Ab
Bb
Cb
Db
Eb
Fb
F
Db
Eb
F
Gb
Ab
Bb
Cb
C
Db
Eb
F
Gb
Ab
Bb
Cb
C
Ab
Bb
C
Db
Eb
F
Gb
G
Ab
Bb
C
Db
Eb
F
Gb
G
Eb
F
G
Ab
Bb
C
Db
D
Eb
F
G
Ab
Bb
C
Db
D
Bb
C
D
Eb
F
G
Ab
A
Bb
C
D
Eb
F
G
Ab
A
1st
2nd
3rd
4th
5th
6th
7th
8th
...mode of the Bebop scale.
The above blue shading also
indicates what note is used as a passing note in other Bebop modes.
For example, the PT is between the minor third and the perfect fourth in mode V
(Bebop Dorian, also called Bebop Minor).
The first six modes are precisely the octatononic scales which can be obtained
by mixing two adjacent diatonic modes,
which provides a name for them...
The remaining two Bebop scales are modes of the following scale,
obtained by adding a passing tone
after the fifth degree of the Major scale.
For mode VII (Bebop Aeolian) this is equivalent to inserting the
passing tone just before the root
(consider the highlighted row ,
where the base scale of mode VII is A-Minor).
2 2 1 2 1 1 2 1 2 2 1 2 1 1 2 1 :
Bebop Major All 8 modes ranked by brightness, in 12 keys (tonic in mode's column "1")
Mode VII combines all the notes from the natural minor scale (the Aeolian mode of the diatonic scale)
and the harmonic minor scale on the same root. Hence the various names it bears.
Principles of Melody: Bebop Lines
(15:06,
18:22) by Rick Beato (Oct. 2016).
(2020-06-27) Verdi's Enigmatic Scale
Composing good music in spite of a lousy scale.
Giuseppe Verdi (1813-1901)
didn't invent this scale. He just used it to compose his Ave Maria (1889)
in response to a musical conundrum (enigma) published (1888-08-05)
by the Gazzetta musicale di Milano,
(the magazine of Casa Ricordi)
challenging readers to harmonize the unnatural
scala enigmatica
proposed to them by Adolfo Crescentini (1854-1921) a professor of music at the
Bologna Conservatory.
Namely:
C Db E F# G# A# B C (1322211)
That's the first mode. in the key of C, of the scale tabulated next.
For each mode, half of the keys can't be expressed with a simple key-signature in Western notation.
In two cases ( darkened in red )
this can't even be done with a double-flat or a double-sharp.
(There's no pivot.)
1 3 2 2 2 1 1 1 3 2 2 2 1 1 :
Verdi's Enigmatic scale (ascending). All 7 modes ranked by brightness, in 6+6 keys (tonic in mode's column "1")
(2020-08-15) Petra's Crater Scale
[3351]
The most imperfect heptatonic scale (5 imperfections).
Cory Arnold (12tone) stumbled on this one while considering
heptatonic scales with the largest possible number of imperfections
(namely 5) defining an imperfection as an in-scale note whose perfect fifth isn't.
The scale had previously been brought to his
attention on Twitter by an autistic composer named
Petra who dubbed
it the Crater scale2018-12-15)
because of the prominent gap in it.
Cory quickly adopted Petra's name.
So did Ian Ring.
I'm happy to join this burgeoning consensus...
My own curiosity had led me to
Cory's video (link in the footnotes below) via a comment from
Mickey Rube, pointing out that the cleanest
way to present the Crater scale is in the key of D
(always a good choice for symmetrical scales) which uniquely
avoids dreaded double alterations:
D Eb Fb Gb A# B# C# D
I'm now using the word crater by itself to indicate the (rare) presence
of 4-semitone jump in any scale.
I'll call this one Petra's Crater Scale, following
Chas Belov who used the expression for his own experimental modulation
(2019-12-22, see footnote below).
That term was acknowledged by Petra herself the next day
(Twitter, 2019-12-23).
1 1 2 4 2 1 1 1 1 2 4 2 1 1 :
Petra's Crater Scale. All 7 modes ranked by brightness, in 1+11 keys (tonic in mode's column "1")
