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 and 'Scale Numbers'

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? Consider the modes of the diatonic scale:

Treble Clef cn dn en fn gn an bn
  Ionian
2773		 Dorian
2902		 Phrygian
3418		 Lydian
2741		 Mixolydian
2774		 Aeolian
2906		 Locrian
3434		  

Suppose we want to uniquely identify each mode by a number (handy for computers!). 'Binary' is a way of counting that only uses the digits '0' and '1'. So we can use a 12-digit binary number to represent a mode, where the twelve digits correspond to the twelve steps of the chromatic scale, and '1' means 'include the note' and '0' means 'don't'. Then, for the Ionian (Major) mode we would have:

C C# D D# E F F# G G# A A# B
1 0 1 0 1 1 0 1 0 1 0 1

Or:

    101011010101 (binary)

If one converts this binary number to our usual base-10, the result is 2773. So in our example, the number listed under the mode label 'Ionian' is 2773.

Or one can express it in 'hexadecimal' — base 16, a convenient way of working with binary numbers. (See Wikipedia). A common way to indicate a hexadecimal number is to prefix it with '0x'. So, using our example of the major scale, 2773 (base 10) equals 0xAD5 (base 16).

You'll notice, however, that we're deriving the 'canonic mode number' by assigning the lowest note in the scale the most significant place in the number. (In the decimal number '123', '1' is in the most significant place — the 100's place whereas '3' is in the least significant place — the 1's place.) Instead of assigning the lowest note to the most significant binary place, one could instead assign the lowest note to the least significant binary place. Using the major scale example again, that would like:

C B Bb A G# G F# F E Eb D Db
1 1 0 1 0 1 0 1 1 0 1 0

Or:

    110101011010 (binary)

This is just like the issue in computer engineering of 'big-endian' vs. 'little-endian' (see Wikipedia). So let's agree:

For Big-endian the highest notes are in the least significant places of our binary representation. If one reads the binary representation of the scale left-to-right, this amounts to using an ascending scale when determining 1's and 0's.

For Little-endian the Lowest notes are in the Least significant places (notice the alliteration of L's). If one reads the binary representation of the scale left-to-right, this amounts to using a descending scale when determining 1's and 0's.

Perfect/Imperfect Modes

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 perfect modes in a scale is its perfections, and the number of imperfect modes is its imperfections. Obviously "perfections + imperfections = N–notes".

Thus, the major scale has 6 perfections and 1 imperfection.

And the pentatonic scale has 4 perfections:

Treble Clef C D E G A C

There's only one scale with 0 imperfections (that is, perfections = N–notes) — 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 are tables listing the number of scales (with a leap of 4) having a given number of perfections and imperfections:

  Number of Perfections  
     0    1    2    3    4    5    6    7    8    9    10    11    12 Total Scales
3-note scale 1                         0
4-note scale 3 4 2                     9
5-note scale 1 8 15 6 1                 31
6-note scale 1 2 18 25 12 1               59
7-note scale     2 16 28 12 1             59
8-note scale         9 21 11 1           42
9-note scale             10 8 1         19
10-note scale                 5 1       6
11-note scale                     1     1
12-note scale                         1 1
Total Scales 6 14 37 47 50 34 22 9 6 1 1 0 1 228
Total Modes 0 14 74 141 200 170 28 63 48 9 10 0 12 769

  Number of 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 Scales 1 7 56 104 54 5 1 228
Total Modes 0 7 112 312 216 25 6 678

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 intervallically 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 intervallically 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 intervallically 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 this website, the scales are listed in order of number of perfections (most 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).

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"

This naming convention was designed to preserve and be consistent with the common names "pentatonic" and the names of the ecclesiastical modes.

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 all intervals including the octave, 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).

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