Music and Musical Scales
Most music that most people listen to is constructed from pitch values which come from a musical scale.
A musical scale is a fixed subset of possible pitch values.
Most musical scales can be described as follows:
- There is a finite set of pitch values within a single octave.
- The scale repeats itself every octave.
- If all notes in the scale are translated by a fixed pitch interval, the result is effectively the same scale.
Scales can also be considered to a set of possible pitch values and a special note considered to be the "root" note, or "home" note.
So, for example, the set of white notes on a keyboard (C, D, E, F, G, A, B) can be considered to be:
- The C major scale, if C is the home note.
- The A minor scale, if A is the home note.
If we decide not to care about which note is the "home" note, then this scale could be described as the diatonic scale. (Note that traditional music terminology prefers to talk about a diatonic scale, of which C major or A minor are examples.)
For the purpose of this article, it doesn't actually matter so much which concept of scale is used, because my thesis will be that the music listener does not actually perceive either type of scale when listening to music.
Also, for the purpose of this article, I will only discuss melodies and music defined on the diatonic scale. However the gist of the article would apply to music constructed from other musical scales.
The identities of notes in a scale
If we treat notes differing by octaves as equivalent, then there are 7 distinct notes in the diatonic scale.
Because of the pitch-translation invariance of the scale, we cannot characterise these notes by their absolute pitch values.
Therefore we have to characterise the notes according to their relationships with each other.
For example, we can characterise each note by the set of intervals between that note and all notes less than one octave higher, as follows:
- A: 2, 3, 5, 7, 8, 10
- B: 1, 3, 5, 6, 8, 10
- C: 2, 4, 5, 7, 9, 11
- D: 2, 3, 5, 7, 9, 10
- E: 1, 3, 5, 7, 8, 10
- F: 2, 4, 6, 7, 9, 11
- G: 2, 4, 5, 7, 9, 10
(One could use the solfege notation do, re, mi, so, fa, la, ti to avoid references to absolute pitch values, however I find it easier to use alphabetic notation, and there should be no problem as long as it's always clear which scale I am talking about.)
We can see from the above list that each note in the scale is uniquely identified by the set of relationships it has with other notes in the scale.
Identities of notes in a scale, in the brain
Given that music is constructed from notes in scales, it seems plausible that, somewhere in the human brain, there is an area which identifies the set of notes occurring in a melody, and the sets of relationships between those notes, and from those sets of relationships, the brain uniquely identifies each note in the scale.
This seems plausible, but the only problem is that it does not correspond to how our brain consciously perceives notes in strong music.
What actually happens is this:
- A strong musical item has its own unique perceived quality, which defines the identity of that particular musical item.
- The identity of a strong musical item is perceived more strongly than the identity of its components.
- In particular, the notes of a strong tune are identified not as being particular notes, rather they are identified as being the notes of the tune that they form part of
One consequence of this is that we do not readily identify the occurrences of the same note in different places in one tune.
Also, we do not readily identify the occurrences of the same note in different tunes, even if the tunes are both played in the same key and scale.
Actually, sometimes we can identify the repeated occurrence of the same note. However this is not a function of our ability to reliably attach labels to notes according to their positions in a scale. Rather it is a function of our conscious awareness of whether notes in a tune are going up, or down, or staying the same, and also our awareness of whether a sequence of phrases is going up, or down, or staying the same, and including our awareness of situations where parts of a phrase go up, or down, or stay the same.
In all these scenarios, our ability to recognise two notes as being the same is a function of our ability to recognise, in certain situations, that something has not changed – it is not a function of an ability to reliably label all occurrences of all notes in a melody based on their location in the scale from which the melody is constructed.
If the repeated occurrence of a note does not fall into one of those scenarios, then we can not reliably identify when two notes occurring within a tune are the same and when they are not the same.
The identities of notes, in a tune
If the brain identified notes from a melody as notes in scales, then, as noted above, the brain would do the following:
- identify which notes occur in the melody
- determine the set of relationships between those notes in the scale
- identify each note according to its relationships with the other notes
If, as suggested by our lack of conscious awareness of the identities of notes in scales, this doesn't happen, then we are still left with the problem of determining how the brain identifies a tune for what it is, in a manner invariant under pitch-translation.
A possible answer to this question is a minor tweak to the above strategy, which processes the tune itself, without any intermediate determination of the scale, ie:
- identify which notes occur in the melody
- determine the set of relationships between those notes, as they occur in the melody
- identify each note according to its relationships with the other notes
The main difference, in practice, is that the relationships between notes in a scale consist entirely of pitch relationships, whereas the relationships betweeen notes as they occur in a melody consist of a combination of pitch and time relationships.
Perception of "scaleishness"
Even if, as I have hypothesized, the brain does not perceive musical scales in a manner that allows the labelling of notes within the scale, it is still necessary to explain why music does actually have to be constructed from scales.
The universal use of scales suggests that the property of being constructed from a scale is an input into the computation of overall "musicality" or musical "strength".
(It follows that any theory of music that seems to permit the construction of music not based on a scale is probably wrong. The following model of computation is an attempt to determine how the music listener's brain could require that music be constructed from a scale, while at the same time not providing information into conscious awareness about which note is which within that scale. I give the model as an example of how this could happen, and I do not claim to know that this is necessarily how it does happen.)
A model of the computation of scaleishness
The property of 'scaleishness' can be defined in the following manner:
- Modulo octaves, all pitch values in the melody come from a finite set of values.
- These values are approximately evenly-spaced.
- Pitch values not in the chosen finite set do not occur.
We can surmise the existence of a cortical map which responds to the occurrence of pitch values (such that pitch value determines location of activity in the map), and which 'remembers' the occurrence of pitch values, in a sustained manner, so that it is active in response to the full set of pitch values that occur over the length of a tune. In effect the pattern of activity in this cortical map would represent a description of the musical scale (from which the tune was constructed), in a manner that was not invariant under pitch-translation.
Because the representation of the scale in this map is not invariant under pitch-translation, the information in the map would not be directly included in the computation of the identity of a music item. (The identity of musical items has to be computed in a pitch-translation invariant manner, to allow for different people with different voice ranges to sing the same musical melody.) However the pattern of activity in the map could be used to compute a single value of 'scaleishness', representing to what degree the pitch values in the melody do come from a single musical scale consisting of a finite set of approximately evenly-spaced pitch values. This computed value would be pitch-translation invariant, so it could contribute to the conscious perception of the musicality and identity of the tune.