The "major" and the "minor" scales, and to a far lesser extent about a dozen 1 or so other scales have kept composers in the West busy for about 500 years now. Let's be generous and say we've really used about 20 different scales/modes.
But based on the Western system of notes (as exemplified by the piano keyboard), there are 1490 possible scales. 2 In other words, with the combined genius of medieval monks through Beethoven through Stravinsky and Coltrane, we've managed to explore roughly 1% of this musical terrain. (Admittedly not all of this terrain is equally musically fertile — or is that assertion just a lack of imagination?)
Even limiting ourselves to the Western system, that's lot of musical territory yet to explore.
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The intent is to present a complete list of all possible scales/modes in a compact form that is composer-friendly.
For the purposes of this website, a scale is a set of equal-tempered notes arranged in ascending order in which no two are more than a major third apart. This is admittedly arbitrary, but one of the intents of this website is to show that even with these constraints, there is an enormous amount of musical territory to explore. Want to allow tuning systems other than equal-tempered? The mathematical possibilites explode! Limiting step sizes to a major third is motivated by common practice — want to allow steps of greater than a major third? The mathematical possibilities explode!
Instead of listing each of the ecclesiastical modes separately (for example), it is only necessary to list the major scale — each of the modes is represented by starting on a different degree of the scale. For the various modes you'll find the mode name under each step of the scale:
So in this example, if you start your mode at the first step, you'll get the Lydian mode. Imagine the scale continues up another octave, and begin at the second step labeled 'Mixolydian' — you'll get the Mixolydian mode.
Of course each of these modes can be transposed to the other 12 steps of the chromatic scale. For example, here the Mixolydian mode starts on D — it is left as an exercise for the reader to transpose it to the remaining 11 steps (D#, E, F, etc.) for themselves!
So we'll use the word 'scale' to mean some pitch-set of N notes (subject to our constraints) that represents N distinct 'modes' (usually N distinct modes — see 'Symmetric Modes' below).
The interval of a perfect fifth occupies a special position in almost all human music — second only to the octave/unison. If a mode includes the perfect fifth above the Tonic (the first note of the scale), let's call that a 'perfect mode'. If it doesn't, it's an 'imperfect mode'. Perfect modes have a tonal anchoring effect for the Tonic that imperfect modes do not. In the case of the major scale example, all the modes are perfect except for the Locrian, and indeed in ancient times composers were forbidden from using the Locrian mode because it was 'imperfect' in this way (it was called the "Devil's mode"). For convenience, perfect modes are indicated with hollow whole notes, imperfect ones with solid notes.
Since a scale of N notes compactly represents N modes (except for symmetric scales, see below) we say that a scale has 'N Perfections' if it contains N perfect modes, and 'N Imperfections' if it contains N imperfects modes. "Perfections + imperfections = number_of_notes."
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:
If one converts this binary number to our usual base-10, the result is 2773. You'll notice that in our example, '2773' is the number listed under the mode label 'Ionian'.
Since in our example the 7-note scale represents 7 different modes, just for consistency one might want some rule for choosing the mode which starts each list. So on this website the mode with the smallest numerical (binary) representation is first.
There are some scales in which starting from more than one step gives you the same mode intervallically speaking:
In this example, the mode starting on C and the mode starting on F# are intervallically identical. We call those 'symmetric'. No need to name them twice, so the duplicates are indicated with a 'hat' over the note and no label underneath.
2A scale being defined as a series of notes, each of higher pitch than the previous, the entire series of pitches less than an octave (so the sequence can repeat itself at each octave), where the step between successive notes in the scale is no greater than a major third. The requirement of no step greater than a major third is somewhat arbitrary, but does allow us to include commonly used scales like the Pentatonic. Allowing steps greater than a major third significantly increases the number of possible scales.
Hopefully this little opus shows that even with a fairly restrictive definition of a scale uses the Western equal-tempered chromatic scale, no steps greater than a major third that there is still an enormous musical territory to explore. Making that musical territory even larger by relaxing any of these restrictions only emphasizes my thesis that we haven't even scratched the surface of all possible music.