### How Many Scales Are There?

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:

wwwmm

wwmwm

wwmmw

wmwwm

wmwmw

wmmww

mwwwm

mwwmw

mwmww

mmwww

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.

Will (8 Feb 2010 at 10:41am)

Question - if one gets all these scales up to full speed say 250+ bpm in all keys, will he/she then be able to improvise over a jazz standard? :)

Joe (8 Feb 2010 at 10:48am)

Will, you underestimate the technical demands of the jazz idiom. A minimum of 300bpm is required.

Connor (8 Feb 2010 at 6:42pm)

Oh Lord. You've started reading Slonimsky's Thesaurus. Say goodbye to any practical music you love.

Joe (8 Feb 2010 at 10:05pm)

Connor, don't worry, I haven't touched it yet. I don't even own it, but it's on my wish list. I'm not ready to say goodbye yet.

Matt stevens (11 Feb 2010 at 4:48am)

Zappa was a big fan of Slonimsky's Thesaurus

Hucbald (26 Feb 2010 at 11:47am)

I did this exercise when I was at Berklee in the early 80's, and the number of USEFUL scales - those with no more than two augmented seconds and three consecutive semitones - was circa 140 if I recall correctly (Counting the modes of the resultants too, of course).

Congrats: This is a phase all musical explorers go through.

Joe (26 Feb 2010 at 11:54am)

Thanks, Hucbald. I'm still in it. I might write a followup post, specifically for seven-note scales, investigating by types of 2nds, 3rds, and 4ths. For example, if you limit 4ths to perfect and augmented, you get the seven diatonic modes, nothing else.

Ty (8 Mar 2010 at 2:02am)

Hey, dude, good work! I recommend writing out all 2048 scales/modes to see their relationships to one another, and also cross-referencing them with Slonimsky. The math is amazingly powerful, but I have to admit arriving at 2048 with no math, but just sweat and logic. Having to write these all out informed me about certain relationships (which shall remain nameless until my book is published) that are extremely interesting, relationships I doubt you will see using these formulae. Good luck, and I'll let you know when my book is done.

Ty (8 Mar 2010 at 2:03am)

By the way, 38 is extremely relevant and important, but not for the reasons you think! I'd be robbing you of the joy of discovery if I were to tell you why! Keep working, and good luck.

Jessie (8 Feb 2014 at 1:36am)

Actually, order does matter!

I think a more fitting calculation would be to count permutations rather than combinations.

People usually like to call scales that start on different note by different names, even though they are composed of the same exact notes.

For instance, the major scale C D E F G A B C

is not typically thought of as the same scale as

D E F G A B C D (we could call the C major scale the "parent" scale of this d minor scale)

I mean, they are the same notes, but you can think of them in a different order.

That means the actual number is actually quite a bit bigger than what you have calculated, if you think of scales in this way.

I might write it out in a post of my own in greater detail.

Still very interesting, thanks for sharing!

Joe (8 Feb 2014 at 11:08am)

Hi Jessie. I think there are a couple things we're understanding differently. Sorry, I wasn't very clear on either one in the original post.

1. When I wrote "order doesn't matter," I meant that I didn't want to double count the same collection of notes, with the same root, chosen in different orders. To illustrate, all of these should count as the same scale: C D E F G A B, C E D F G B A, C F B E A D G, C G F A D E B, etc.

If the root is fixed, then the order in which you add the rest of the notes to the scale doesn't matter.2. In my calculations, I'm not including all twelve roots. So C major, D major, and B major don't count as three different scales, but only one: the major scale. Multiply all my results by twelve to include all possible roots for every scale. The examples you cited, C major and D Dorian, were both included in my results, labeled without their roots under "Ionian (major)" and "Dorian."

Send me your post if you expand on your ideas; I'd love to read it!

jin johnson (9 Apr 2015 at 9:28pm)

Very well appreciated:

I have thought about all scales for years and have self published Basic Rhythms Graphs that shows all binary combos up to eight and trinary combos up to four - color coded, ordered and random.

My plan is to publish all scales in root C, scale intervals, and color coded.

I have started with 108 Melakartas (36 diminished), and Trinary Scale Modality -

that are in Word document I can email.

Thanks very much... Jin

Howard (5 Oct 2015 at 2:04pm)

Greetings Joe,

I'm glad to see see and I'd like to confirm that your calculations and the structures are exactly correct. I'd like to expand on your considerable effort.

Years ago (ca. 1978) we figured out that there's 2,047 scales within our Western octave. Strictly speaking, we didn't count the '1 note scale' as it's an interval - we thought a scale had to have at least two intervals to fit the definition. That's how we ended up with (2^11)-1 = 2,047.

Around 1982ish, I wrote a pascal program to print all these out, but only in one key :)) ... the list was quite large as you can imagine. I used the same set of scale groups (scale systems) based on {11, 55, 165 ... etc } that you did. (More on these in a second).

The person who told me about these structures - and deserves the credit - was a jazz guitarist named Richard Clooney. A magician in music theory, Richard painstakingly figuring out all these underlying structures without the use of math. He did it empirically, by brute force analysis on paper. Pretty impressive. He'd already determined these structures in the mid 70's, but never published anything. (Unfortunately, I lost touch with him years ago.) I can well imagine only a small handful of people like yourself have figured out the 2048 ! :))

The structures I used, based on Richard's work, were:

Large Structures:

LTC-UTC

LTC-D-UTC

where LTC = Lower Tetra Chord, UTC = Upper Tetra Chord, and D=Dividing interval.

The total intervallic distance of the tetra cord ranges from 1 to 6 (while the other tetra chord has to balance, ranging down from 6 to 1). So the tetra chords are similar, having a smaller, finer structure that looks like this:

x

xx

xxx

xxxx

xxxxx

xxxxxx

The x represents an interval.

The middle structure, "D" has only

x

(a single interval)

So for the system of scales that we use most frequently, the fine structure is

xxx-x-xxx

Filling the interval in each of those x placeholders, yields the scale. For instance, the major scale is:

221-2-221

the (Aeolian) minor scale is:

212-2-212

etc.

To repeat just to be clear, the actual notes are *between* the x's. So, for instance in the key of C major, the 221-2-221 structure is *spelled* out this way:

|x|x|x|-X|-x|x|x|

c2d2e1f-2g-2a2b1c

Going full circle, that 'large structure' I mentioned, makes the programming a sane project. The finer structure of a tetra chord is added together, so you get, e.g.,

221-2-221

^ ^

5 -2- 5

___ The LTC + D (if one) + UTC must always equal 12 ____

Those structures have to balance.

Another example is the chromatic scale.

It's large structure is merely

x-x

6-6

making the fine structure, the boring

111111-111111

:)