Are there any scales (in any musical culture) that can contain non-fixed members (ones that can be sharped or flattened at will)?

Question

I am wondering if there are any scales (that are in actual use) that also contain non-fixed pitches. By that I mean that in addition to having several fixed pitches, also have some "flexible" pitches that can be raised or lowered at will?

For example, are there any scales that can have either a raised or lowered scale-degree 3, where the descision whether to play a raised or a lowered 3rd is left at the discretion of the performer, and not prescripted by the scale itself?

Some examples of what I AM NOT after:

• In the melodic minor, scale-degrees 6 and 7 are raised on ascent and lowered on descent. Yet, in the context of either ascending or descending they DO have a fixed pitch.
• Similarly, in Indian music, the Arohana and Avarohana often contain different pitches, but those pitches are fixed in either context.
• Chromatic passing notes, which by definition are not intrinsically part of the scale.
• The use of modal mixture, that through some harmonic alteration may impact the notes of a melody.
• In the Klezmer scale Adonai malach (on C), there's an E natural when playing within the 1st octave of the scale, and an Eb when playing in the 2nd octave of the scale (see below). So the scale sometimes has E natural and sometimes Eb, but those too have very clear contexts, and in that sense those pitches are fixed as well.

Purely, to illustrate my point, I am attaching an example (below) of such scale (which I invented for the sake of explaining my question). This C scale, can make use of either E natural or Eb, as well as A natural and Ab interchangeably (at the discretion of the performer). In that sense, the scale itself contains some non-fixed pitches.

Are there such scales in wide use anywhere?

UPDATE The Hindustani Raag Pilu (or Piloo) seems to fit the bill. Any other examples?

Answer 1

I would be willing to bet that there are many throughout different cultures but what immediately comes to mind for me is the way I sometimes improvise over blues. I use an unofficial composite scale made up of combining parallel major and minor pentatonic scales and adding the #4/b5 blues note as well. It’s possible or even likely this scale has a name, I’ve never researched it. It looks like this in the key of C:

This scale allows for playing either major or minor pentatonic lines and also chromatic lines utilizing notes from both scales combined. It is completely chromatic from the D to the G. I even sometimes add extra chromatic passing notes between the ^5 and ^6 and also the ^b7 and ^1. The bottom line is you can use whichever notes you like at your discretion depending on whether you want a major or minor tonality or a more chromatic style.

Answer 2

A scale, by definition, is fixed. But there's this thing called 'chromatic notes'. A jazz player may interpret the chord 'C major' by basically using notes of the C major scale, but making frequent additional use of E♭ and G♭. It isn't a new scale, E♮ and G♮ are still used, but other notes are too.