Introduction

A "scale" can be described as that set of pitches to which a musician limits themselves in the creation of a melody, in some given musical context, or even in an entire piece. That set of pitches is determined by a combination of the musical-culture in which the musician lives, the idiosyncrasies of the instruments at their disposal, and the musician's personal choices.

Even a cursory investigation of human music across space (different locales around the world) and time (down through history) shows that one way of giving a style of music a unique character is the use of different scales. For example, the "blues scale" is one important element of what gives the "blues" its unique character, and the unique scales of the gamelan are an important element of that genre.

Humans have used countless different scales in the creation of their music. The question naturally arises how many possible scales there might be. If we admit of any possible tuning system on any possible musical instrument, the number of possible scales is of course infinite. But if we limit ourselves to, say, the standard pitch set of Western music for the past couple centuries — the equal-tempered piano keyboard — it is possible to determine a finite number, and to enumerate all the possible scales. That is the purpose of this website.

"Octave equivalence"1 being a virtually universal principal of human music, it is common practice to assume that a scale is a set of pitches within an octave span to which a musician limits themselves, the tacit assumption being that the same intervalic pattern of pitches is repeated at the octaves as desired. Thus the "C-major scale" can start on a "C" in any octave you choose — the intervalic pattern of whole and half steps is always the same. 2

On this website I limit myself to the Western equal-tempered chromatic scale, not because there is something 'sacred' about that tuning system, but because it is so ubiquitous, and because the point of this website is to show that even with that 'severe' limitation it is easy to show that we haven't begun to explore the musical possibilities of even that 'limited' system.

So, we'll find it convenient to unambiguously describe a scale by the number of semitones required for each degree (step). For example, the major scale is 2,2,1,2,2,2,1 — starting on any given note (the first "degree"), the second degree is 2 semitones above the first, the third is 2 semitones above the second, the fourth is 1 semitone above the third, and so on.

Treble Clef C D E F G A B C C
  2 2 1 2 2 2 1  

Furthermore, we may want to consider (and possibly limit) the distance between any two successive degrees of a scale. In the major scale (2,2,1,2,2,2,1), that largest step is 2 semitones, or a whole step. In the pentatonic scale (2,3,2,2) the maximum step size is 3 semitones, or a minor third:

Treble Clef C D E G A C C
  2 2 3 2 3  

We will call this the "leap" of a scale (that is, a scale's "largest step" is its "leap"). So the leap of the pentatonic scale is 3.

Finally, many scales are related to each other by being modes of each other. For example, if our scale is the "white notes" on the piano keyboard (also known as the "diatonic scale"), seven modes can be obtained: the first by beginning on C (CDEFGABC), the second by beginning on D (DEFGABCD), and so forth. (This particular set of modes is known as the "ecclesiastical modes".) Thus seven modes can be unambiguously described by the one diatonic scale, with the understanding that the modes are derived by beginning each mode on each of the seven degrees of the scale. Modes which are related by being modes of each other are called "siblings" of each other. Thus the Dorian ecclesiastical mode is a sibling of the Locrian ecclesiastical mode.

In the list of scales, the name of the mode appears under the degree with which you'd start. Thus, for the ecclesiastical modes, we'd have:

Treble Clef C D E F# G A B C
  Lydian Mixolydian Aeolian Locrian Ionian Dorian Phyrgian  

There are scales containing modes that are identical to each other. For example, the "diminished" scale:

Treble Clef C D Eb F F# G# A B C C
  2 1 2 1 2 1 2 1  

This scale actually only two unique modes: 2,1,2,1,2,1,2,1 (if you start on an odd degree of the scale) and 1,2,1,2,1,2,1,2 (if you start on an even degree of the scale). These are called "symmetric" scales and will be discussed later.

Here is a table which lists the possible unique modes and scales, based on the equal-tempered Western chromatic scale:

Leap Scales Modes
1 1 1
2 31 232
3 132 926
4 228 1489

In other words, limiting ourselves to a leap of 4 semitones, and assuming the Western equal-tempered octave, there are 228 possible scales and 1489 possible modes based on those scales. If we relax any of our limitations — octave equivalence, Western equal-tempered tuning, a leap of 4 — the number of possible scales and modes is of course much greater. (This does not include transposing each of the modes to the other 11 "keys", which increases the number of possibilities by a factor of 12).

Canonical Form

Since any n-note scale has "n" modes (apart from symmetry considerations), we can reduce the physical list of our scales considerably by only including one scale to represent all the modes readily derived from that scale. But — how to choose the one mode to represent all of its siblings…

In this website the "canonical form" of a scale is that mode that has the greatest possible number of its larger steps at the beginning of the mode, and the greatest possible number of its smaller steps at the end of the mode. For example, considering the diatonic scale:

Ionian (aka "major") 2,2,1,2,2,2,1
Dorian 2,1,2,2,2,1,2
Phrygian 1,2,2,2,1,2,2
Lydian 2,2,2,1,2,2,1
Mixolydian 2,2,1,2,2,1,2
Aeolian (aka "minor") 2,1,2,2,1,2,2
Locrian 1,2,2,1,2,2,1

The mode with the greatest number of larger intervals first is the one beginning with the greatest sequence of whole tones — the Lydian mode.

By using this convention in the Compleat List of Scales, one can determine the leap of the scale at a glance by observing the size of the first step in the scale, since its largest step will occur first.

An index is included to eliminate the need to determine the canonical form by hand. Instead, list the size of each step in semitones, then look up the mode in the "index of Modes." For example:

Treble Clef C D E G A C C
  2 2 3 2 3  

The size of each step of this mode is 2,2,3,2,3. This is a 5-note scale – in the "index of modes" it is scale number 1. Thus 5-note scale number 1 is the canonical form of this mode.

Perfection

The perfect fifth plays a virtually universal role in human music almost as important as the octave. A musically suggestive assessment of a scale might be to count how many perfect fifths can be formed limiting yourself to the notes in the scale. For example, in the major scale, a degree for which the perfect fifth above it is also in the scale is indicated by a whole note, a degree for which the perfect fifth above is not also in the scale is indicated by a solid note:

Treble Clef C D E F# G A B C

Since the note a perfect fifth above the seventh degree of this scale is not also in the scale, we say that this degree is imperfect. The number of imperfect degrees of the scale is how many imperfections the scale has. Thus, the major scale has 1 imperfection.

The pentatonic scale also has 1 imperfection:

Treble Clef C D E G A C

There's only one scale with no imperfections: the complete 12-note chromatic scale:

Treble Clef C C# D D# E F F# G G# A A# B C

Interestingly, the diatonic scale is the only 7-note scale with only one imperfection (with a leap of 4 or less). The only other scales with only one imperfection are those that are super and subsets of the diatonic scale.

Of the 7-note scales with a leap of 2 semitones (like the diatonic scale), there are none with only 2 imperfections, only one with 3 imperfections – the "ascending melodic minor" scale:

Treble Clef C D Eb F G A B C

and only one with 4 imperfections – the "harmonic minor" scale:

Treble Clef C D Eb F G Ab B C

Here is a table listing the number of scales (with a leap of 4) having a given number of imperfections:

  Imperfections  
     0    1    2    3    4    5    6 Total
3-note scale       1       1
4-note scale     2 4 3     9
5-note scale   1 6 15 8 1   31
6-note scale   1 12 25 18 2 1 59
7-note scale   1 12 28 16 2   59
8-note scale   1 11 21 9     42
9-note scale   1 8 10       19
10-note scale   1 5         6
11-note scale   1           1
12-note scale 1             1
Total 1 7 56 104 54 5 1 228

In this website, the scales are listed in order of perfection (most perfect first), with the perfect degrees in each scale indicated by whole notes and the imperfect degrees indicated by sold notes (as in the examples above).

Symmetry

Usually all the modes of a scale are unique – that is, no two modes are intervalically identical. For example, from the diatonic scale we obtain the seven ecclesiastical modes — no two of which have the same intervallic sequence:

Ionian (aka "major") 2,2,1,2,2,2,1
Dorian 2,1,2,2,2,1,2
Phyrigain 1,2,2,2,1,2,2
Lydian 2,2,2,1,2,2,1
Mixolydian 2,2,1,2,2,1,2
Aeolian (aka "minor") 2,1,2,2,1,2,2
Locrian 1,2,2,1,2,2,1

But there are scales where more than one mode is intervalically identical. For example, consider the diminished scale:

Treble Clef C D Eb F F# G A B C C
  2 1 2 1 2 1 2 1  

All the modes starting on odd degrees are intervalically identical (1,3,5,7). Namely, if you start on degree 1, 3, 5 or 7, the intervals of the scale are 2,1,2,1,2,1,2. Likewise all the modes starting on even degrees are intervalically identical (2,4,6,8) – the intervals of the scale are 1,2,1,2,1,2,1. So the diminished scale only has 2 unique modes, even though it has 8 degrees.

Since the pattern repeats every 3 semitones, we say that the diminished scale is "symmetric at 3 semitones" or "symmetric at the minor third."

The whole tone scale only has one mode, even though it has 6 degrees in the scale. It is symmetric at 2 semitones (or a whole step).

Treble Clef C D E F# G# A# C
  2 2 2 2 2  

Clearly a symmetric scale can only symmetric at an even divisor of 12, so we can only have scales symmetric at 2, 3, 4 and 6 semitones. Following is a table showing the number of symmetric scales (with a leap of 4 or less):

Degrees
in scale
Symmetric
at 1 semitone
Symmetric
at 2 semitones
Symmetric
at 3 semitones
Symmetric
at 4 semitones
Symmetric
at 6 semitones
 Total 
3       1   1
4     1   1 2
5           0
6   1   1 3 5
7           0
8     1   2 3
9       1   1
10         1 1
11           0
12           1
Total 1 1 2 3 7 14

In the list of scales, if a scale is symmetric at n-semitones, a " ˆ " appears over every n-th semitones of the canonical scale. For example:

Treble Clef C D Eb F F# G# A B C

This scale is symmetric at the minor third, and the " ˆ " appears over every minor third from the root of the symmetric form of this scale. Thus one can see at a glance that this scale has two modes: one starting at the " ˆ ", and one starting at the degree following the " ˆ ".

The Names of the Modes

A curious feature of humans is that a thing seems to be less "real" until it has a name. One of the first things done in concentration camps to de-humanize the inmates is to expunge their names and replace them with numbers. And the only task God gave Adam in the Garden of Eden (apart from staying away from the Tree of Knowledge) was to name all the animals. And one of the first things human parents do is name their new child. The child isn't even officially born (as far as the government is concerned) until a name is filled in on the birth certificate. And somehow it seems much easier to compose music in the "aeolithic mode" than in "mode number 427".

So, it would seem that all the animals in our modal musical garden need names. Therefore I have named them. There is at least this pattern: all the names of the 5-note names end in "-atonic" (for example, the "Pentatonic"), all the names of the 7-note scales end in "-ian" (for example, "Dorian"), and so on:

  end in...
4-note modes  "-ic"
5-note modes  "-itonic"
6-note modes  "-imic"
7-note modes  "-ian"
8-note modes  "-yllic"
9-note modes  "-ygic"
10-note modes  "-yllian"
11-note modes  "-atic"

The common names "pentatonic" and the names of the ecclesiastic modes fit in this framework and are preserved.


1"Octave Equivalence" — the idea in Western music (at least) that notes an octave or multiples of an octave apart are functionally equivalent. We see this idea embedded in the nomenclature itself: C (261 hz, or middle C) and C (522 hz, an octave above) are both named 'C'.

2 Even "twelve-tone" music by Schoenberg and his disciples, who deliberately rejected the idea that any pitch interval was functionally different from any other (in order to reject "tonality" itself) – yet they still affirmed octave equivalence, and ascribed to the octave a functionally unique role. If one wanted to reject the unique functional characteristics of the octave (in addition to all the other intervals), what was really wanted was not "12-tone" serialism, but "88-tone" serialism (assuming the piano keyboard) or "120-tone" serialism (assuming a typical range of human hearing – 20 hz to 20 khz is approximately 120 semitones).