Last week's lecture in my jazz theory course was on a number of esoteric, derived, 7- and 8-note scales. The professor demonstrated each at the piano and took us through various harmonic applications. Several of these came from Nicolas Slonimsky's Thesaurus of Scales and Melodic Patterns.
Written in the lecture notes was a claim that there are over 23000 possible 7-note scales, increasing exponentially for 8-note scales. I was skeptical, so I did a quick calculation in my head:
The total number of collections of notes within an octave, assuming there's a root, would allow each of the 11 remaining notes to be included in the collection or not. That's 2 to the 11th power (2^11), which is only 2048. So there must be fewer 7-note scales than that. And I remember from my probability classes that there will be even fewer 8-note scales.
This line of thinking got me asking and answering a multitude of questions about the number of scales of different types. I will assume for the rest of this post that every scale includes a root note and is confined to an octave.
How many n-note scales are there?
I established above that there are 2048 possible collections of any number of notes, so whatever I come up with here should total 2048. To get started, there's only one 1-note scale (the root, alone) and 11 2-note scales (the root plus one of 11 possible other notes). Similarly, there's only one 12-note scale (the whole chromatic scale) and 11 11-note scales (exclude one of the 11 non-root notes).
For everything in between, I'll need to calculate how many ways n - 1 notes can be chosen from 11 non-root notes. This calls for a binomial expansion! Instead of just looking it up, I chose to derive it while I was trying to sleep the other night.
If note order mattered, the first one could take any of the 11 available slots, the second would have 10 available, and so on. That's 11 * 10 * 9 * etc. for as many notes as you have beyond the root. More generally, that's 11! / (11 - (n - 1))! For a 3-note scale, you have the root, 11 possibilities for the next note, 10 possibilities for the third note. That's 11! / (11 - (3 - 1))! = 11! / 9! = 11 * 10, as above.
But order doesn't matter, so each of these calculations must be divided by the number of ways to order the collection. Luckily, that's easy. Any collection of n elements can be ordered n! different ways. (The first can take any of n slots, the second any of n - 1 slots, and so on.)
Now I have a formula. There are 11! / ((11 - (n - 1))! * (n - 1)!) possible n-note scales. The shorthand for this is 11C(n - 1), as in "11 choose (n - 1)". (See the Wikipedia entry if you're totally lost but still care.)
1-note scales: 11C0 = 11! / (11! * 0!) = 1
2-note: 11C1 = 11! / (10! * 1!) = 11
3-note: 11C2 = 11! / (9! * 2!) = 11 * 10 / 2 = 55
4-note: 11C3 = 11! / (8! * 3!) = 11 * 10 * 9 / (3 * 2) = 165
5-note: 11C4 = 11! / (7! * 4!) = 11 * 10 * 9 * 8 / (4 * 3 * 2) = 330
6-note: 11C5 = 11! / (6! * 5!) = 11 * 10 * 9 * 8 * 7 / (5 * 4 * 3 * 2) = 462
7-note: 11C6 = 11! / (5! * 6!) = 11C5 = 462
8-note: 11C7 = 11! / (4! * 7!) = 11C4 = 330
9-note: 11C8 = 11! / (3! * 8!) = 11C3 = 165
10-note: 11C9 = 11! / (2! * 9!) = 11C2 = 55
11-note: 11C10 = 11! / (1! * 10!) = 11C1 = 11
12-note: 11C11 = 11! / (0! * 11!) = 11C0 = 1
Add those all up, and it's the same total from before, 2048 possible scales of any length.
So now I know that there are only 462 possible 7-note scales. How else can I limit this?
How many 7-note scales are there with no more than three half steps between adjacent notes?
This can be approached backwards, as in how many are there with an interval of at least four half steps between adjacent notes? To answer, place the big interval at the beginning, and multiply by seven modes for each resulting possibility.
The root starts the scale again, and the first three chromatic notes are off limits, because that's where I want the big interval. Now there are eight remaining notes to choose from, and six notes left in the scale.
8C6 = 8! / (2! * 6!) = 8 * 7 / 2 = 28
Each of those 28 scales has six other modes unaccounted for, with the big interval somewhere other than directly above the root. That's 28 * 7 = 196 scales with an interval of at least four half steps between adjacent notes. Now, to answer the question, subtract that from the total 7-note scales: 462 - 196 = 266.
That's cool. That amounts to only 38 generative scales with 7 modes each. I wonder how many are usable and how many are rubbish?
How many 7-note scales are there with no more than two half steps between adjacent notes and no consecutive half steps?
I'm now limited to 1- and 2-step intervals. I'll call them h for half step, w for whole step. I need seven of these intervals for seven notes, and they must add up to 12 half steps. So there must be two half steps and five whole steps.
There are only 14 ways to order these intervals without adjacent half steps. And they happen to correspond exactly to the modes of the major and (Jazz) Melodic Minor scales:
hwhwwww = Altered Dominant
hwwhwww = Locrian
hwwwhww = Phrygian
hwwwwhw = Dorian b2
whwhwww = Locrian #2
whwwhww = Aeolian
whwwwhw = Dorian
whwwwwh = Melodic Minor
wwhwhww = Mixolydian b6
wwhwwhw = Mixolydian
wwhwwwh = Ionian (major)
wwwhwhw = Lydian Dominant
wwwhwwh = Lydian
wwwwhwh = Lydian Augmented
That's just two scales along with their modes. This is suddenly not so overwhelming.
How many 7-note scales are there with only major and minor thirds?
This will take some trickery. I'll extend the scale to two octaves, using thirds instead of seconds. (This orders the notes as 1 3 5 7 2 4 6.) There must be three major thirds and four minor thirds. Further, the scale needs to collapse back to one octave with no duplicate notes. Three major thirds add up to an octave, as do four minor thirds, so I'll keep them from all sticking together.
I'll spare you the long story on these possibilities. There are 28. That's four generative scales with seven modes each. Using m for minor third, M for major third, h for half step, w for whole step, a for augmented second, they are:
MmMmmMm = wwhwwwh = Ionian (major)
mMMmmMm = whwwwwh = Melodic Minor
mMMmmmM = whwwhah = Harmonic Minor
MmMmmmM = wwhwhah = Harmonic Major
That's fantastic! That's a very digestible chunk of knowledge.
How many 6-note scales are there with no half steps?
Only one: the Whole Tone Scale. Cool!
How many 6-note scales are there with no intervals greater than two half steps between adjacent notes?
Only one: the Whole Tone Scale. Cool!!
How many 5-note scales are there using only whole steps and minor thirds between adjacent notes?
Using w for whole step and m for minor third:
These are all modes of two pentatonic scales: the major/minor pentatonic everyone knows and the other pentatonic scale no one knows.
The theme in all of this is that scalar possibilities are not endless. The numbers may be daunting, but they are finite. When the scope is limited, as in the questions above, the resulting possibilities aren't so mind-boggling. It makes mastery very achievable. Then you get into applications of all these scales, and how they relate within the progression of time, and the possibilities truly are endless.
What other useful questions can you come up with? Leave some in the comments, with a solution if you're as nerdy as I am.