# How many natural scales are possible?

Recently I have increased my study of musical scales with ever more detail. One thing that confuses me a certain term I use "natural." Most scales have the same pattern of intervals ascending as it is descending (Diatonic, for example.) if they have the same pattern ascending and descending, is this what is called "natural?"

Also if so, can all twelve tonic scales have a "natural" form with its pattern of tones and semitones? What are they? And is the amount of modes in each scale the same number of tonics(like pentatonic has 5, hexatonic, 6 modes)?

• I think you ought to make the second question a separate question, since it has little to do with the term natural besides being heavily dependent on information that you hope to gain from the question. In particular, the question about the amount of modes seems unrelated. Jul 4 '20 at 2:55

Technically there are infinite number of scales, but for the sake of brevity we'll use the equal tempered, 12 tone scale. Here's the (technical) amount of unique combinations for each number of degrees in the scale.

• Monotonic(1): Only one note per octave, so there's only 12 of these scales. 1 mode.
• Ditonic(2): 66 scales. 11 modes.
• Tritonic(3): 220 scales. 55 modes.
• Tetratonic(4): 495 scales. 165 modes.
• Pentatonic(5): 792 scales. 330 modes.
• Hexatonic(6): 924 scales. 462 modes.
• Heptatonic(7): 792 scales. 462 modes.
• Octatonic(8): 495 scales. 330 modes.
• Nonatonic(9): 220 scales. 165 modes.
• Decatonic(10): 66 scales. 11 modes.
• Hendecatonic(11): Since only one note is missing there are 12 scales, or 11 modes.
• Chromatic(12): All 12 notes are used so there's only 1 of this scale, and 1 mode.

In total there are (12 + 66 + 220 + 495 + 792 + 924 + 792 + 495 + 220 + 66 + 12 + 1) = 4095 unique scales to be derived from the 12 tone ET set, and (1 + 11 + 55 + 165 + 330 + 462 + 462 + 330 + 165 + 11 + 11 + 1) = 2004 modes. To get the unique amount of modes use this formula: [s/(12/t)] where t is tonic and s is scale total. "Natural" is usually used to differentiate the natural minor and harmonic/melodic minor scales. If you did that for any scale among these the possibilities are pretty expansive. In common practice most scales are based on the major scale, and thus most scales are related back to it, which is why the natural minor is simply the major scale starting on its 6th degree.

• +100 for this answer. I would just add that the chromatic system of 12 tones on which all these calculations are based is "naturally occurring" (putting aside the fudging we do for equal temperament, etc) as we move through the circle of 5ths using Pythagoras's methodology on his monochord. (or whoever figured it out first...) Oct 26 '17 at 6:08
• I'm for real thinking about writing down all the modes by hand...
– Tama
Oct 26 '17 at 6:26
• @JaredSchlosser - the real word I was looking for is "equally-tempered" How did you get that? Equal Tempered Scales is something else entirely - it means a tuning system that spreads the out the ragged mathematical edges of Just Temperament across all scales/keys so that instruments with fixed pitches, such as the piano and the guitar can play in tune in all keys. (Regardless of your terms, Tama's answer seems to be what you are looking for.) Oct 26 '17 at 19:44
• Could you provide a source for the numbers you posted? I enumerated all the modes for 12 tones a while back, and counted fewer than you mention. (maybe due to symmetrical scales?) My calculations match up with this, maybe worth investigating? oeis.org/A035495 p.s. sorry for being a besserwisser.
– user43681
Oct 29 '17 at 21:50
• > In total there are 5811 unique scales to be derived from the 12 tone ET set 5811 doesn't seem accurate. There are 12 notes, which can either be played (included in the scale) or skipped (excluded from the scale), so essentially it's a binary problem, and the calculation is 2^12, or 4096. If you subtract the one with all notes excluded, that leaves 4095 possible combinations of notes in a 12-note system. Since the difference between 5811 and 4095 is 1716, and that equals 924+792, my guess is the poster double-counted 924 and one group of 792. Apr 11 '19 at 8:46

It's hard to answer a question that is based on having read something, without being able to read the same thing you read, but I will try.

I know two uses of the word "natural" in the context of scales.

1. Natural minor, as opposed to melodic minor or harmonic minor. Let's compare the three flavors of A minor. Natural minor would be all white notes. Melodic minor would be the same, on the way down, but on the way up the 6th and 7th notes would be each raised a half-tone. Harmonic minor raises the 7th note a half-tone, both on the way up and on the way down.

2. Let's compare C major and G major. C major has all white keys, whereas G major has one sharp, or black key. Then we can say that C major is natural but G major isn't. Similarly, compare D Dorian and E Dorian. D Dorian is natural but E Dorian isn't. D Dorian uses all white keys but E Dorian doesn't.

Do you see that 1 and 2 are related to each other, in a way? A natural minor doesn't depart from the white keys.