Diatonic and heptatonic scales, and degrees

Question

I have been working on listing all the possible diatonic scales in order of degrees, the number of accidentals in the scale. The first is the ionian ( major ), since there are zero accidentals. The next would be the mixolydian (b7), then a scale with a b6th, etc., etc.

The harmonic minor doesn't fit the definition of a diatonic scale, because of the minor 3rd interval between the b6th and the natural 7th, but I decided to include it, since it does contain seven tones, and does not contain three or more simultaneous semitones ( 3rd, 4th and b5th, for example) . In doing so, I found two more scale families, each with seven new modes. Im sure that these scale sets have names, but I have not yet found them.

The first contains a scale with a b6. I thought it might be called Harmonic Major, but Im only guessing. The other contains a scale with a#2,#4,b7. I just call it a lydian dominant #2.

How do I determine what the primary scale is ( as is the ionian in the major scale ), and do these scale families have specific names, like, for instance, the Melodic minor, or Major scale?

Also, am I using the term "degrees" correctly? For example, how many degrees in a scale like this: C Db Eb Fb Gb Abb Bbb (h W h W h W mi3rd) Is it 3, since the Fb, Abb, and Bbb are enharmonically natural notes, or is it 6, since all six notes are altered?

Answer 1

I don't know what you mean by the primary scale. Do you have a reference which defines that term? Do you intend to find ways to divide the octave up into 7 intervals, and then somehow decide which of the pitches would have to be the tonic in order for the scale to be primary? If so, then I don't know of anyone else who does that, so I don't see the value of an answer to the question.

I decided to include it, since it [...] does not contain three or more simultaneous semitones

If you don't want to study scales that contain three or more simultaneous semitones, that's fine, but you could if you wanted to. Such scales exist and have been used and named. For example the Hungarian minor scale

C D Eb F# G Ab B C

has four minor seconds, as does the double harmonic major scale, which is a transposition of it:

C Db E F G Ab B C

These scales are transpositions of each other, e.g. G double harmonic major uses the same pitches as C Hungarian minor.

The Phrygian dominant scale

C Db E F G Ab Bb C

has three.

The Wikipedia pages linked to above in turn link to other pages you might find useful.

It seems to me that you are using the term "degrees" correctly. Your last example contains 7 degrees because it contains 7 pitch classes. (By "pitch class" I mean pitch as specified by name, though the octave doesn't matter. All Cs in all octaves are the same pitch class.)

In this context, it doesn't matter what is enharmonically equivalent to what. C Lydian's 4th degree F# is enharmonically equivalent (in 12-equal) to C Locrian's 5th degree Gb, but that is irrelevant to music theory.

Answer 2

Also, am I using the term "degrees" correctly? For example, how many degrees in a scale like this: C Db Eb Fb Gb Abb Bbb (h W h W h W mi3rd) Is it 3, since the Fb, Abb, and Bbb are enharmonically natural notes, or is it 6, since all six notes are altered?

Enharmonic notes do not imply the number of degrees in a scale. You have used all 7 letters, so this exotic scale has 7 degrees.

How do I determine what the primary scale is

You should be more clear what you mean with primary scale. In music set theory, a prime scale exists, but is this what you are looking for?