How many Eight Tone Scales are there? At first glance, this might seem like a simple CS GRE type question: there are 12 notes, so it would be 12 select 8 unordered.Â Not quite:

Why 8 notes:Â because in a 4/4 measure,Â there are 8 quarter notes,Â so if you haveÂ an 8 note scale,Â you have one distinct pitch to play per note.

- The bebopÂ scale is probably the best example of this:Â add a #5 to a major scale and not only do you have 8 notes,Â but each of the chord tones falls on the down beat,Â not just for the Major Triad,Â but for the Minor and Dominant seventh as well.
- The Harmonic minor scale can be viewed as the relative minorÂ plus a flat 9,Â with the distinctive minor third just skippingÂ the minor seventh.Â If you add back in the seventh, you get an 8 note scaleÂ that works well for most Eastern European style music.
- The Diminished scale,Â consisting of alternating half and whole stepsÂ has 8 notes, and makes for some great runs as well.

So with those in mind,Â I was wondering how many other 8 note scale there are.Â I will ignore the other qualities that make an 8 note scale interesting for now:Â I’m sure each will have some novelty to it.

So 12 choose 8Â would give us (12!)/(8!)(4!)Â orÂ 495 scales,Â the real number is much lower.Â In the case of the three examples I post above,Â note that the Harmonic Minor option really is just the same as the bebop scale,Â just a different mode:Â Played A to A instead of C to C.Â So I will count these two as one scale, and the diminished scale as one scale.Â Thus, all modes will be considered the same scale,Â as will all transpositions:Â C Bebop Major played from C to C is the same as F Bebop Major played from F to F.Â Â Â We want to ignore 11 transpositions,Â and 7 modes.Â That gives me an answer of 6.42…which tells me I did something wrong….hmmm. Really,Â we don’t just care about the number of scales,Â but we want to be able to generate all of the scales,Â and ignore the duplicates.Â Lets apply some heuristics to reduce the number of duplications we have.

Another way to look at it is to look at the intervals between each note.Â This is actually the more useful apporach,Â as it is the template a musician would use to compose a specific instance of a scale.Â We’ll use W for whole step and H for Half. For example,Â the Bebop major scale is W W H W HÂ H W H.Â Numerically,Â this is 2 2 1 2 1 1 2 1Â which sums to 12.Â So we could look at it as generating the ordered sequences of 8 digits that sum to 12.Â One nice feature of this is that we care about 8 distinct notes,Â and we still have 8 elements:Â the interval from the last note to looping around back to the first is also encapsulated in the sequence. Â However,Â we’ll have a couple scales unaccounted for with the W/HÂ approach, namely those that have an interval greater than 2.Â How many of those are there?

If we start with 7Â half steps,Â we have a chromatic scale followed by an interval of 5H,Â or a Major fourth in conventional intervals: H H H H H H H 5.

Note that any variation with a tritone in it is going to be a mode of this one.Â For example H H 5 H H H H H is just starting on the 5th H.

Running total:Â 1 Scale.

If we steal a half step fromÂ the 5,Â and add it to a H,Â we get a WÂ and an interval of 4H.Â HHH HHH W 4.

If we fix the 4 in the first position,Â It becomes a simple template like this:

4 XXX XXXX

Where we put the W in each position, and fill in the rest with Hs.

- 4 WHH HHHH
- 4 HWH HHHH
- 4 HHW HHHH
- 4 HHH WHHH
- 4 HHH HWHH
- 4 HHH HHWH
- 4 HHH HHHW

Running Total:Â 8 scales

If we steal a half step from the 4,Â we can either add it to the W or to one of the H.

- Â HHH HHH 3 3
- Â HHH HH WW 3

Note if we once again steal a half step,Â in order to generate distinct steps from this last 3,Â we have to add it to another H.

HHHH WWWW

Which I’ll enumerate later.

In the case of bothÂ Â HHH HHH 3 3Â andÂ HHH HH WW 3Â we can fix the first interval to be an HÂ and reduce the number of alternativesÂ by using the other remaining intervalsÂ for the last.

SoÂ in the case of Â HHH HHH 3 3 Â we make the template Â H XXX XXX 3 .Â The Remaining 3 can go into any of the Xs,Â and the rest are filled with Hs.

- H 3HH HHH 3
- H H3H HHH 3
- H HH3 HHH 3Â (This one is symetric, which we’ll have to account for in our modes.)
- H HHH 3HH 3
- H HHH H3H 3
- H HHH HH3 3

Running total 14 scales.

ForÂ HHH HH WW 3Â There are twoÂ variations:

- H XXX XXX W
- H XXX XXXÂ 3

Note thatÂ these cover all the combinations generated byÂ W XXX XXX 3.

First we’ll do all combinations where

H XXX XXXÂ 3Â Has the fewest permutations.Â The Two Ws can be in either order and produce the same scale.

- HÂ WWH HHH 3
- H WHW HHH 3
- H WHH WHH 3 (Not symmetric due to the 3)
- H WHH HWH 3
- H WHH HHW 3
- H HWW HHH 3
- H HWH WHH 3
- H HWH HWH 3
- H HWH HHW 3
- H HHW WHH 3
- H HHW HWH 3
- H HHW HHW 3
- H HHH WWH 3
- H HHH WHW 3
- H HHH HWW 3

Running totalÂ 29 Scales.

It might be tempting to say that the there are 2X15 or 30 variations ofÂ Â H XXX XXX W:Â those where WÂ precedes 3 and those where 3 precedes W.Â However,Â scales “wrap around.” So anything of the form 3H is covered by the above combinations.Â So we only need to account for scales with 3W in the middle.

- H 3WH HHH W
- H H3W HHH W
- H HH3 WHH W
- H HHH 3WH W
- H HHH H3W W

Running totalÂ 34 Scales.

The scale HHHHHH5Â is the variation with the most half steps.Â The one with the least?Â Lets start from a scale with None:Â The 6th note whole tone scaleÂ W W W W W W.Â (From A this would be A B C# D# F G).Â If we split any one of those whole tones,Â we get a seven note scale with two half steps and five whole steps.Â Thus,Â we have to split exactly two of them.Â Splitting the first two gives us:Â H H H H W W W W.Â How many variations of this are there?Â Again,Â lets use the technique of fixing the First,Â but now also the last, element of the Set.Â We will state that the first element is always HÂ and the last is alwaysÂ W.

H X X X X X X W.Â This will filter out most modes,Â but not all….back to that below. Simplest is all H and Ws together.

- H HHHÂ WWWW

Running Total 35 Scales

Thos middle 6 places now must get filled by Six intervals,Â 3 WsÂ andÂ Â 3 Hs.Â Another way to think of this is that we can put 0, 1, 2 or 3 Ws between each of the other intervals.Â Â Â W can haveÂ a run of 3Â Injected into H HHHH W like this.Â Note that we avoid the 4 in a row patterns from above.

- H WWW HHHH W
- HÂ HÂ WWW HH W
- HÂ HHÂ WWW H W

Running totalÂ 38 Scales.

Now we can either have WÂ followed byÂ WW.Â We willÂ avoid all combinations where Those bump up against another W.

- H WHW WHH W
- H WHH WWH W

Running Total 40 Scales

And the opposite pattern WW followed by W:

- H WWH WHH W (Bebop)
- H WWH HWH W

Running total 42 Scales.

Bebop Major is W W H W HÂ H W HÂ If we wrap this aroundÂ by one,Â moving the lastÂ Â interval into the firstÂ position, Â we getÂ H W W H W HÂ H W,Â which is variation 1 above.

The lastÂ variation is where we have alternating H and W.Â This is a very symmetric scale.Â There areÂ Only two modes:Â The one that starts with H WÂ and the one that starts with W H.

- H W H W H W H WÂ (Diminished)

Which gives a total of 43 scales.

Here’s the complete list:

- H H H H H H H 5.
- 4 WHH HHHH
- 4 HWH HHHH
- 4 HHW HHHH
- 4 HHH WHHH
- 4 HHH HWHH
- 4 HHH HHWH
- 4 HHH HHHW
- H 3HH HHH 3
- H H3H HHH 3
- H HH3 HHH 3Â (symmetric)
- H HHH 3HH 3
- H HHH H3H 3
- H HHH HH3 3
- HÂ WWH HHH 3
- H WHW HHH 3
- H WHH WHH 3
- H WHH HWH 3
- H WHH HHW 3
- H HWW HHH 3
- H HWH WHH 3
- H HWH HWH 3
- H HWH HHW 3
- H HHW WHH 3
- H HHW HWH 3
- H HHW HHW 3
- H HHH WWH 3
- H HHH WHW 3
- H HHH HWW 3
- H 3WH HHH W
- H H3W HHH W
- H HH3 WHH W
- H HHH 3WH W
- H HHH H3W W
- H HHHÂ WWWW
- H WWW HHH W (fixed, originally had HHHH in last block of H
- HÂ HÂ WWW HH W
- HÂ HHÂ WWW H W
- H WHW WHH W
- H WHH WWH W
- H WWH WHH W (Bebop)
- H WWH HWH W (Whoops this is a Duplicate of 39)
- H W H W H W H WÂ (Diminished,Â very symmetric)

So where do my Klezmer BebopÂ Â (Â A b C d D# e F g# A )Â and MyÂ 8 Tone ScaleÂ Â Â (E F# G G# A# C# D D# E.) fall in?

The Klezmer Bebop scale isÂ W H W H H H 3 HÂ rotatedÂ toÂ H W H W H H H 3 which is scale 16 above.

The other 8 tone scale is W H H W 3 H H H, rotated to H H H W H H W 3 which is scale 26 above.

The standard blues scaleÂ is usuallyÂ 3 W H H 3 W,Â a 6 tone scale.Â Starting on AÂ this is A C D D# E G A.Â This Could obviously be fitted into many of the above scales by splitting either the W or 3 intervals.Â If You we are playing over an A Dominant 7th chord,Â we can add in theÂ the Major 3Â and MajorÂ 6th:Â C# and F#.Â Â A C C# D D# E F# G AÂ Or 3 H H H H W H W.Â Rotated to H H H H W H W 3Â That is 28 above.

nice work

39) and 42) are the same though, right?

greets from berlin

Yup. Damnit. I need to update this with something we can hear.

Hi ..Quite a lot of work! Superb! I am writing a code for scales and chords. Please check it here https://www.scalechords.com . I want to get permutations of chords with legit names. Will see if I can. Will like to hear your opinion/suggestions on the scale generator..

You should generate the standard notation for the scales as well. Look in to ABC notation for an easy way to do that. I use an ABC plugin for music on this site.

I dig it! Nicely done/ An easy addition would be one the played the scale descending as well. Would be cool to hear the chords as well, maybe both all notes and arpeggios?