# Number of possible scales with musical potential - check my math or theorizing?

I was trying to figure out how many possible scales have musical potential versus how many are simply modes of other scales. I did not think there could be that many, particularly with these parameters:

1) 7 intervals only (like major scale)
2) no interval of more than 3 semitones
3) no more than three consecutive intervals of only 1 semitone

My result was 34 possible scales, each with 7 modes (238 possibilities).

3 of these use NO 3-semitone intervals (Heptanic prima, secunda, etc.)
20 of these use ONE-3 semitone interval
11 of these use TWO-3 semitone intervals

I guess my concern is three fold:

1) Maybe I'm just wrong in my calculations.
2) Maybe I'm wrong to rule out scales with four 1-semitone intervals in a row as lacking musical potential.
3) Maybe I'm missing a bigger picture in trying to break it down this way, missing forest for trees or something like that.

Gracias,

Ken Myers
Student of Music Theory & Guitar (one year or so)
Oh, in case anyone really has a lot of spare time, or is curious, here is how I worked this out:

How many scale combinations in 12 note octave
- using 7 intervals
- using only intervals of 1 or 2
+ Conclusion: Must have FIVE 2's and TWO 1's in order to add up to 12 using only seven 1's and 2's.
1 1122222 (7 MODES)HEPTONIA TERTIA / Neapolitan Major scale & modes
2 1212222 (7 MODES)HEPTONIA SECUNDA / Melodic Minor scale & modes
3 1221222 (7 MODES)HEPTONIA PRIMA / Major scale & modes
+ Conclusion: Only three unique scales with 7 modes each (21 total)

How many scale combinations in 12 note octave
- using 7 intervals
- using only one interval of 3 and no higher intervals (all other 1 or 2)
3 _ _ _ _ _ _
1 1 1 1 1 1 = 9 x
2 = 10x
2 2 = 11x
2 2 2 = 12
+ Conclusion: Must have three 1's and three 2's.
1 3 1 1 1 2 2 2
2 1 3 1 1 2 2 2
3 1 1 3 1 2 2 2
4 1 1 1 3 2 2 2
5 1 1 1 2 3 2 2
6 1 1 1 2 2 3 2
+ Conclusion: six possibilities keeping 1's grouped away from 2's
7 3 2 1 2 1 2 1
8 3 1 2 1 2 1 2
+ Conclusion: two possibilities keeping 1's and 2's alternating
Six remaining prefix options:
1st: 3 1 1 2 _ _ 2 (2 options)
2nd: 3 2 2 1 _ _ 1 (2 options)
3rd: 3 1 2 2 1 _ _ (2 options)
4th: 3 2 1 1 2 _ _ (2 options)
5th: 3 1 2 1 _ _ _ (2 unique options)
6th: 3 2 1 2 _ _ _ (2 unique options)
9 3 1 1 2 1 2 2
10 3 1 1 2 2 1 2
11 3 2 2 1 2 1 2
12 3 2 2 1 1 2 1
13 3 1 2 2 1 1 2
14 3 1 2 2 1 2 1
15 3 2 1 1 2 1 2
16 3 2 1 1 2 2 1
17 3 1 2 1 1 2 2
18 3 1 2 1 2 2 1
19 3 2 1 2 2 1 1
20 3 2 1 2 1 1 2
+ Conclusion 20 unique scales (each w/ 7 modes) for 7 intervals, none higher than 3, using exactly one 3, for a total of 140 possibilities.

How many scale combinations in 12 note octave
- using 7 intervals
- using two intervals of 3 and no higher intervals (all other 1 or 2)
3 3 _ _ _ _ _
1 1 1 1 1 = 11 x
2 = 12
+ Conclusion: Must have four 1's and one 2.
1) 3 3 1 1 1 1 2 - note four consecutive semitones
2) 3 3 1 1 1 2 1
3) 3 3 1 1 2 1 1
4) 3 3 1 2 1 1 1
5) 3 3 2 1 1 1 1 - note four consecutive semitones
+ Conclusion: five options keeping 3's together, 7 modes each.
6) 3 2 3 1 1 1 1 - note four consecutive semitones
+ Conclusion: one option putting just 2 between 3's
7) 3 1 1 1 3 1 2
8) 3 1 1 1 3 2 1
+ Conclusion: two options putting three 1's between 3's
3 1 1 3 _ _ _
9) 3 1 1 3 1 1 2
10) 3 1 1 3 1 2 1
11) 3 1 1 3 2 1 1
+ Conclusion: three options putting two 1's between 3's
3 1 3 _ _ _ _
12) 3 1 3 1 1 1 2
13) 3 1 3 1 1 2 1
14) 3 1 3 1 2 1 1
15) 3 1 3 2 1 1 1
+ Conclusion: four options putting one 1 between 3's
All others appear to be modes
+ Conclusion: FIFTEEN unique scales (each w/ 7 modes) for 7 intervals, none higher than 3, using exactly two 3's, for a total of 105 possibilities. It should be noted that three of these options include FOUR semitones in a row, and thus have less viability for melody / musicality, leaving ELEVEN unique scales (seven modes each) with seven intervals, two intervals of 3 semitones, without more than three consecutive intervals of 1 semitone, for a total of 77 possibilities.

How many scale combinations in 12 note octave
- using 7 intervals
- using three intervals of 3 and no higher intervals (all other 1 or 2)
3 3 3 _ _ _ _
1 1 1 1 = 13 x IMPOSSIBLE
+ Conclusion: NONE.

CONCLUSION:
Number of Scales with:
- exactly 7 interval notes
- no interval more than 3 semitones
- no more than three consecutive intervals of only 1 semitone
21 (w/ no 3)
140 (w/ one 3)
77 (w/ two 3's)
238 unique scales
You know, you could have used all that time you spent doing this math learning how to build chords, harmonize the major scale, and use chord tones and approach notes... You would have a far more practical and usable result.
Quote by Xiaoxi
The Byzantine scale was useful until the Ottoman scale came around and totally annihilated it.
There's just one: The major scale.

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Just remember that there are no boring scales, just boring players.

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Quote by Xiaoxi
The Byzantine scale was useful until the Ottoman scale came around and totally annihilated it.
Quote by mrkeka
You know, you could have used all that time you spent doing this math learning how to build chords, harmonize the major scale, and use chord tones and approach notes... You would have a far more practical and usable result.

Thanks, this is sort of the feedback I was hoping for, as I can go off on tangent way too far. And it gives me some thought as to what to focus on next. Though the above math took under an hour, so it's not like I could have done all the stuff you listed in that time. It all started as I was trying to learn chord building, which then led me to first trying to understand intervals better, which led to thinking about scales. Then it seemed like wrapping my head around scale potential was pre-cursor to wrapping my head around chord potential.

Ken
in the end, there is only one scale.
The scale of music.
With this scale, one can mold the energy of sound
into anything they want,
without limit,
and infinite potential.
Chromatic scale is the best.
Quote by krm27
Thanks, this is sort of the feedback I was hoping for, as I can go off on It all started as I was trying to learn chord building, which then led me to first trying to understand intervals better, which led to thinking about scales. Then it seemed like wrapping my head around scale potential was pre-cursor to wrapping my head around chord potential.

Ken

for the chord part..try ted greenes' chord chemestry...modern chord progressions (2 vols)

(who knew there were that many A chords !!)
Cool, now using the same criteria can you figure out how many unique scales there are in 53-TET?

Edit: Inquiring minds want to know!
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Enginear
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I'm Good at Math
Last edited by J-Dawg158 at Sep 19, 2012,
What you're really trying to accomplish is how much time you can possibly waste.

...modes and scales are still useless.

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Thanks
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But our Band is Listana
Quote by Xiaoxi
What you're really trying to accomplish is how much time you can possibly waste.

Xiaoxi, You probably don't even know who I am on this website, but i looked at your profile, and you're a young guy! I always imagined you to be an old black guy!
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Xiaoxi, You probably don't even know who I am on this website, but i looked at your profile, and you're a young guy! I always imagined you to be an old black guy!

That's because I am! You flatter me too much.

...modes and scales are still useless.

Quote by PhoenixGRM
Hey guys could you spare a minute to Vote for my band. Go to the site Search our band Listana with CTRL+F for quick and vote Thank you .
Quote by sam b
Voted for Patron Çıldırdı.

Thanks
Quote by PhoenixGRM
But our Band is Listana
The error in your theorizing TS concerns the application and harmonic context in which you intend to use these scales. I bet most of them will be heard as minor variations to the major or minor scale, because that's what they will function as.
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Quote by krm27
I was trying to figure out how many possible scales have musical potential versus how many are simply modes of other scales.

What everyone else said.

Plus: Musical potential has nothing to do with modes or scales. It has to do with the potential to make music.

You can make music on a snare drum.

How many notes does that have?
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oh shut up with that /mu/ bullshit. fidget house shouldn't even be a genre, why in the world would it deserve its own subgenres you twat
i literally have no idea what the point of this thread is, or what the premise is, or what the first line you wrote is supposed to mean. all that stuff you calculated does nothing in actually WRITING a good MELODY. and as you figured out, there are a lot of possibilities of note choices. and it's all subjective anyway so i'm sorry to say but, you kinda just wasted your time on something that will probably never help you.

not to mention you can just learn theory. that's what it's there for. so you don't waste time calculating things like this. just look up the scales, learn the intervals and theory that goes with them, and apply where you see fit.
The only thing I gathered out of this is that since you can clearly get better than a D in maths and therefore qualify for university entrance in new zealand and so I hate you vehemently, even more so than maths teachers who suck.