Eight Tone Scales

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.  Not quite:

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

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

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

So 12 choose 8  would give us (12!)/(8!)(4!)  or  495 scales,  the real number is much lower.  In the case of the three examples I post above,  note that the Harmonic Minor option really is just the same as the bebop scale,  just a different mode:  Played A to A instead of C to C.  So I will count these two as one scale, and the diminished scale as one scale.  Thus, all modes will be considered the same scale,  as will all transpositions:  C Bebop Major played from C to C is the same as F Bebop Major played from F to F.    We want to ignore 11 transpositions,  and 7 modes.  That gives me an answer of 6.42…which tells me I did something wrong….hmmm. Really,  we don’t just care about the number of scales,  but we want to be able to generate all of the scales,  and ignore the duplicates.  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.  This is actually the more useful apporach,  as it is the template a musician would use to compose a specific instance of a scale.  We’ll use W for whole step and H for Half. For example,  the Bebop major scale is W W H W H  H W H.  Numerically,  this is 2 2 1 2 1 1 2 1  which sums to 12.  So we could look at it as generating the ordered sequences of 8 digits that sum to 12.  One nice feature of this is that we care about 8 distinct notes,  and we still have 8 elements:  the interval from the last note to looping around back to the first is also encapsulated in the sequence.   However,  we’ll have a couple scales unaccounted for with the W/H  approach, namely those that have an interval greater than 2.  How many of those are there?

If we start with 7  half steps,  we have a chromatic scale followed by an interval of 5H,  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.  For example H H 5 H H H H H is just starting on the 5th H.

Running total:  1 Scale.

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

If we fix the 4 in the first position,  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.

  1. 4 WHH HHHH
  2. 4 HWH HHHH
  3. 4 HHW HHHH
  4. 4 HHH WHHH
  5. 4 HHH HWHH
  6. 4 HHH HHWH
  7. 4 HHH HHHW

Running Total:  8 scales

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

  •  HHH HHH 3 3
  •  HHH HH WW 3

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

HHHH WWWW

Which I’ll enumerate later.

In the case of both   HHH HHH 3 3  and  HHH HH WW 3  we can fix the first interval to be an H  and reduce the number of alternatives  by using the other remaining intervals  for the last.

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

  1. H 3HH HHH 3
  2. H H3H HHH 3
  3. H HH3 HHH 3  (This one is symetric, which we’ll have to account for in our modes.)
  4. H HHH 3HH 3
  5. H HHH H3H 3
  6. H HHH HH3 3

Running total 14 scales.

For  HHH HH WW 3  There are two  variations:

  • H XXX XXX W
  • H XXX XXX  3

Note that  these cover all the combinations generated by  W XXX XXX 3.

First we’ll do all combinations where

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

  1. H  WWH HHH 3
  2. H WHW HHH 3
  3. H WHH WHH 3 (Not symmetric due to the 3)
  4. H WHH HWH 3
  5. H WHH HHW 3
  6. H HWW HHH 3
  7. H HWH WHH 3
  8. H HWH HWH 3
  9. H HWH HHW 3
  10. H HHW WHH 3
  11. H HHW HWH 3
  12. H HHW HHW 3
  13. H HHH WWH 3
  14. H HHH WHW 3
  15. H HHH HWW 3

Running total  29 Scales.

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

  1. H 3WH HHH W
  2. H H3W HHH W
  3. H HH3 WHH W
  4. H HHH 3WH W
  5. H HHH H3W W

Running total  34 Scales.

The scale HHHHHH5  is the variation with the most half steps.  The one with the least?  Lets start from a scale with None:  The 6th note whole tone scale  W W W W W W.  (From A this would be A B C# D# F G).  If we split any one of those whole tones,  we get a seven note scale with two half steps and five whole steps.  Thus,  we have to split exactly two of them.  Splitting the first two gives us:  H H H H W W W W.  How many variations of this are there?  Again,  lets use the technique of fixing the First,  but now also the last, element of the Set.  We will state that the first element is always H  and the last is always  W.

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

  1. H HHH WWWW

Running Total 35 Scales

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

  1. H WWW HHHH W
  2. H  H WWW HH W
  3. H  HH WWW H W

Running total  38 Scales.

Now we can either have W  followed by  WW.  We will  avoid all combinations where Those bump up against another W.

  1. H WHW WHH W
  2. H WHH WWH W

Running Total 40 Scales

And the opposite pattern WW followed by W:

  1. H WWH WHH W (Bebop)
  2. H WWH HWH W

Running total 42 Scales.

Bebop Major is W W H W H  H W H  If we wrap this around  by one,  moving the last   interval into the first  position,   we get  H W W H W H  H W,  which is variation 1 above.

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

  1. H W H W H W H W  (Diminished)

Which gives a total of 43 scales.

Here’s the complete list:

  1. H H H H H H H 5.
  2. 4 WHH HHHH
  3. 4 HWH HHHH
  4. 4 HHW HHHH
  5. 4 HHH WHHH
  6. 4 HHH HWHH
  7. 4 HHH HHWH
  8. 4 HHH HHHW
  9. H 3HH HHH 3
  10. H H3H HHH 3
  11. H HH3 HHH 3  (symmetric)
  12. H HHH 3HH 3
  13. H HHH H3H 3
  14. H HHH HH3 3
  15. H  WWH HHH 3
  16. H WHW HHH 3
  17. H WHH WHH 3
  18. H WHH HWH 3
  19. H WHH HHW 3
  20. H HWW HHH 3
  21. H HWH WHH 3
  22. H HWH HWH 3
  23. H HWH HHW 3
  24. H HHW WHH 3
  25. H HHW HWH 3
  26. H HHW HHW 3
  27. H HHH WWH 3
  28. H HHH WHW 3
  29. H HHH HWW 3
  30. H 3WH HHH W
  31. H H3W HHH W
  32. H HH3 WHH W
  33. H HHH 3WH W
  34. H HHH H3W W
  35. H HHH WWWW
  36. H WWW HHH W (fixed, originally had HHHH in last block of H
  37. H  H WWW HH W
  38. H  HH WWW H W
  39. H WHW WHH W
  40. H WHH WWH W
  41. H WWH WHH W (Bebop)
  42. H WWH HWH W
  43. H W H W H W H W  (Diminished,  very symmetric)

 

So where do my Klezmer Bebop    ( A b C d D# e F g# A )  and My  8 Tone Scale    (E F# G G# A# C# D D# E.) fall in?

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

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