Originally Answered: How many music scales are there?
or instance the Japanese Koto scales, the Indian Rajas and the Maqams have alternative pitches for some of the notes and your choice depends on the musical context. Often with complex rules not just ascending and descending.
Then if you play with strict just intonation - so keeping all the intervals pure - then you find it impossible again to fit it within a single scale. The most famous example is the comma pump
C G D A E C
If you do all of those as pure harmonic series based musical intervals it goes
1/1 3/2 9/8 27/16 81/64, 81/80 where you get back to 81/80 instead of 1/1 and every time you go around that cycle of intervals the pitch rises by 81/80 if you do all the intervals as just intonation. This is an infinite spiral and can't be fitted into a single scale - though you can approximate it with enough steps. E.g. one step of 53 equal is approximately 81/80, though still out by over a cent. See 53 equal temperament
Then also there's "adaptive just intonation" where for instance, singers micro-adjust the pitches as they sing depending on the other notes playing at the same time. So for instance if two vocalists sing a note a major third apart then even though singing in a pythagorean or equal temperament with pure or almost pure fifths, they might micro-adjust the major third so that it is more like a 5/4.
You also get expressive changes of pitch of a note as it plays so that it starts or ends at a different pitch or changes pitch in the middle, and again of course this can't be expressed using just a simple scale tuning - and is an important aspect of many types of music..
You also get scales that repeat at other intervals apart from the octave. Common examples are scales that repeat at 3/1 or 3/2.
And - another thing - you can explore scales that are not strictly ascending - where the normal way of going through the scale is a sequence of pitches that sometimes goes up and sometimes down in pitch.
Also,the interval of repetition doesn't have to be the interval furthest away from the 1/1 in the scale.
Here is one of my fractal tunes using a non strictly ascending scale repeating at 4/3 though the scale goes up to the octave.
The video first plays the scale, then a "seed phrase" played within the scale which is used to generate all the melodic material by a sort of musical fractal.
Tuning: is 1/1 7/6 3/2 2/1 5/4 4/3, continuing with repeat at the 4/3 as
4/3 14/9 2/1 8/3 5/3
16/9 56/27 8/3 32/9 20/9 64/27 ...
You can also continue it the other way below the 1/1 as
... 27/32 135/256 9/16 21/32 27/32 9/8 45/64 3/4 7/8 9/8 3/2 15/16 1/1 ...
There the previous copy of the tuning starts at the 3/4, a 4/3 below the 1/1.
So the 3/2 there is a 2/1 above the 3/4 that starts the previous repeat.
Like this:
3/4 = previous repeat of the tuning's 1/1. = scale degree -5
7/8 = previous repeat's 7/6. = scale degree -4
9/8 = previous 3/2 (i.e. 3/2 above the 3/4). = scale degree -3
3/2 = previous 2/1. = scale degree -2
15/16 = previous 5/4. = scale degree -1
1/1 = previous 4/3. = scale degree 0
(I also used that example in my answer here: Robert Walker's answer to Can temporal self-similarity done deliberately, yield music?)
You also get scales that never repeat - but continue to smaller and smaller musical intervals. Simplest example here is the harmonic series, which goes to smaller and smaller musical intervals as you go up in pitch.
Then there are other ways of organizing notes, that can't be fitted into any finite single musical scale. I have some examples in my Fibonacci Tone Scapes
Here it's based on a Fibonacci rhythm which is made up of two or more sizes of beat which are organized in a heirarchical but non repeating pattern - and the pitch goes up or down by different intervals depending on the size of the beat. There is no way to fit these notes into a single scale.
For musical examples of this see my Fibonacci Tone Scapes
About the Author
Robert Walker
Writer of articles on Mars and Space issues - Software Developer of Tune Smithy, Bounce Metronome etc.Studied at Wolfson College, Oxford
Lives in Isle of Mull
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