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Music listeners are not consciously aware of note identity
In my experience, as a music listener, and also as an amateur musician and composer, I do not readily identify the notes in a scale that occur in a melody.
I will caveat this pronouncement by stating that I do not have absolute pitch, and obviously if a music listener has absolute pitch then they can identify notes in the absolute sense, and presumably they could then easily identify those notes according to their position in the scale of the melody.
So, what exactly do I mean by identifying the "notes in a scale"?
For most modern western popular music, melodies are constructed from notes in a scale. The scale consists of a finite set of notes, usually 5, 6 or 7 notes, that form a pattern that repeats every octave.
In music theory it is normal practice to define a scale relative to a given "tonic" note, such as C major, where the scale consists of the notes C, D, E, F, G, A and B, and C is the tonic note.
We could also define a looser notion of scale, which is just a set of notes, without specifying the tonic note. So we might define C, D, E, F, G, A and B as a scale which we could label as the "diatonic scale". This "tonic-free" scale-as-a-set-of-notes could then correspond to C major, or A minor, or one of various other modes such as E phrygian.
The other thing about scales is that they are defined relatively. In a mathematical sense we can say that a scale is invariant under a transposition, where a transposition consists of adding x semitones to all the notes in the scale for some number x. So we could add 3 semitones to the C major scale to get the E♭ major scale which is E♭, F, G, A♭, B♭, C, D, which without a defined tonic would be the diatonic scale consisting of the notes Eb, F, G, A♭, B♭, C, D. Musically, the C major scale and the E♭ major scale are essentially the same scale.
In modern Western popular music there are only about 4 distinct common tonic-free scales used (at least with most of the music that I am familiar with personally). These are:
diatonic eg C, D, E, F, G, A, B
pentatonic eg C, D, E, G, A
blues eg C, D, D♭, E, G, A
harmonic minor eg C, D, E, F, G♯, A
Of these, the diatonic has the largest set of choices of "mode", ie choice of tonic note. The pentatonic and blues are both mainly limited to choice of C (major) or A (minor) in the above examples, and harmonic minor only ever has A as a tonic.
All of these four scales are fully uneven in the sense that they cannot be divided into smaller repeating sets of notes than the repeated set of notes within one octave (as a counter-example, the hypothetical scale C D Eb E F♯ G♯ A B♭ is not fully uneven because it repeats every 6 semitones - ie the steps are 2,1,1,2,2,1,1,2).
If we consider the relations between the notes within such an uneven scale, then each note is defined uniquely by the set of relationships it has with the other notes in the same scale.
For example, in the pentatonic scale, the intervals from each note to the other notes above it are (in semitones):
* C: 2,4,7,9
* D: 2,5,7,10
* E: 3,5,8,10
* G: 2,5,7,9
* A: 3,5,7,10
Based on these relationships each note in the scale has a unique identity as determined by those relationships. For example, A is the only note where the next 4 notes above it are 3, 5, 7 and 10 semitones higher. Also, these relationships are based on intervals which are invariant under transposition, so the unique identity of each note is invariant under transposition.
(Side note: In the above explanation, I have chosen to use absolute notes in my examples, ie C, D, E etc. Indeed traditional music theory does not have any standard naming scheme for notes that is both relative and independent of the choice of tonic, so it can be easier just to use note names themselves, while making it clear in the context which specific scale is being considered.)
At this point I think I have laid down enough theory to return to my psychological observation, which is:
Even though the notes in a typical uneven musical scale each have a well-defined identity as determined by their relationships with other notes in the same scale, music listeners do not reliably identify the identity of those notes when listening to music.
Which raises the question - how hard can it be? After all, there are only 5, 6 or 7 distinct notes in the scales used in most melodies, and we're talking about only four different scales most of the time.
How can it be that most of the music that we listen to is constructed from notes from one of these four 5, 6 or 7 note scales, and yet our brains never learn to know “which note is which”, and therefore we remain unable to identify those notes as we listen to the music?
We could compare this to spoken language.
Spoken language consists of sentences which are constructed from words, and the words are constructed from a limited finite set of consonant and vowel sounds (the exact number of each depending on the specific language). It is not that hard to consciously identify the specific consonant and vowel sounds that occur in any particular word.
In as much as notes in a scale are the irreducible minimal components of melody - if we ignore for the moment the question of note timing - and consonants and vowels are the irreducible minimal components of words, we might be surprised that in the speech case we can readily consciously identify those minimal components, and in the musical case we cannot easily consciously identify the minimal components.
A tentative conclusion one can draw is that, for some reason, it is very important for music listeners not to be consciously aware of the identities of the notes in the scales of the melodies of the music that they listen to.
Or perhaps there is some other reason.
What if we don't ignore the question of note timing?
In a musical melody, the different notes are related to each other by both pitch interval and time interval.
We should consider the possibility that notes are assigned unique identities based on the sum total of the combined pitch & temporal relationships with all the other notes that occur in the melody.
Under this analysis, each note in a melody will be assigned a unique identity as a note within that melody.
The same note with the same unique identity will only re-occur when the whole melody is repeated.
This is somewhat consistent with the actual subjective experience we have of listening to a melody (or at least it is consistent with my subjective experience).
It is also consistent with our experience of listening to notes in scales with specified tonic notes, because when one plays a scale, one is actually playing a very simple melody, for example the scale of C major is typically demonstrated by playing the melody C D E F G A B C, and the scale of A minor corresponds to A B C D E F G A, which is a different melody. So it will follow that the identities of the "same" notes within those different melodies will be different, because each melody has its own distinct set of note identities (although we might find that very similar melodies will have similar sets of note identities).
Extra Technical Notes
1. In modern western music, the 4 scales I listed above are all subsets of the (very even) 12 note chromatic scale. And, in practice, transpositions are limited to exact multiples of semitones as a consequence of the standardisation of musical instrument pitch values so that different musical instruments are consistent with each other. But the invariance of musical quality under transposition applies equally if the transposition is not a whole number of semitones, for example it could be 2 1/2 semitones. Also the general logic of note identity being a function of relative note positions does not depend on the set of notes in a scale being a subset of the chromatic scale - it applies to any scale that full even in the manner described previously.
2. The four scales listed do not cover every known popular musical item, in particular there are many tunes that are typically notated using extra accidentals, or, in the minor case, they may be 'natural' (ie diatonic) minor in one part of the melody, and harmonic minor in another part. However the existence of these variations does not affect the general logic of note identity, whether one considers accidentals to be identifiable exceptions to the default scale, or whether one chooses to expand the deemed scale of the melody to include those accidentals.