The key that unlocks understanding of music is the distance of bass to the other parts or melodies. In fact bass itself can be a melody. But it is the bass, base, βασις, foundation [Bach]. The reference.
The notes are seven, ie Do Re Mi Fa Sol La Si. This is one dimenssion, a set of seven notes. We can consider them as bass or basis or foundation. This can be thought of as the first variable or dimension of a 3D vector of the form (note, number, time). Time obviously means when the event takes place. Number, the second dimenssion is the usual figure above bass [Bach].
And so numbers. Or figure or number or interval above bass. The second dimenssion. Just seven numbers again. 1 2 3 4 5 6 7. A set of seven numbers. The significance is how many steps above from bass ie the definition of an interval. For example C,3 means C bass and E. (C,7) means C bass and B. (E,2) means E bass and F (but usually written E,9 as 9 steps above bass is the same note name as 2 steps above bass). So (E,9) means E bass and F. (This is for example the first chord or code of Hey You - Pink Floyd. F is sharp as the scale is the E minor scale or key).
Other example (A,9) means A bass and B. (The first interval or chord event or code of Just Hold Me - Maria Mena.
What is a Cartesian product of 2 sets? It is a new set containing all combinations of the 2 sets.
So to summarize, we have:
Set 1 (Do, Re, Mi, Fa, Sol, La, Si)
Set 2 (1. 3, 4, 5, 6, 7, 9)
Note both sets have 7 elements each.
So their marriage, combination set or Cartesain product has 7x7=49 elements. These 49 elements are the basic or basis sound colors of music.
(C,3) ie C and E can also be called: the 3rd of C is E.
Likewise the 7th of C is B.
The 9th of A is B. (The chord hence defined or derived can be A bass and C, E, B (if we also take the 3rd and 5th) [Bach].
The 9th of B is A. (This is the code when Sandra sings the first lyric word "You" on the well known 80's song Maria Magdalena - Sandra. Bass is B, "You" is sung by the not A, and some other parts played by other instruments, eg synths add the 3rd ie D and 5th ie F which is F#, [Bach]. F is F sharp as the key or scale is B minor where all F are sharp or the first black key to the right of the natural or white F).
In music everything is referenced to the seven white keys or the first mention set of seven elements or notes.
Some more examples:
The 7th of D is C
The 7th of F is E
....
Just 49 combinations. The Cartesian product of the 2 sets has 47 elements of the form (note,number above note) or (note,so many number steps above note).
So a musical composition such as Old and Wise - Alan Parsons Project which has the following figured bass at the introduction:
A
G6
F7
E7
F7
E7
D7
E7
A3#
(3# in this case means A# as the chord is A major)
So we may write the chord events as such:
(La,1,1)
(Sol,6,2)
(Fa,7,3)
(Mi,7,4)
etc
Or
(A,1,1)
(G,7,2)
(F,7,3)
Etc
And all we mean is:
1st chord A bass with C, E
2nd chord G bass with B, E (the 3rd is taken with the sixth) [Bach]
3rd chord F bass with A, C, E (the 3rd is taken with the fifth, [Bach]
etc
Note the 6th and the 7th is prepared to the listener's brain as the non jone E persists [Bach].
This makes the code (G,6) and (F,7) the second and the 3rd chord of the introduction to Old and Wise - Alal Parsons Project so beatitiful as out of this world.
It is important to realize that G6 (G,6) or F7 (F,7) etc is figured bass or code or short hand notation of GBE Sol Si Mi,
FACE Fa La Do Mi.
All the above chords can just be played by the left hand, the little finger playing the bass and the other fingers the rest. For example (F,7) little finger playing F, middle finger A, index finger C, thumb playing E. It sounds very nice in the middle of the keyboard and the righ hand can play melody or other voice or melody parts.
Notes:
The 3rd dimenssion, time is very important. A (A minor) to (F,7) F7 (F major 7th) sounds very different to (F,7) to A. The human brain likes the sevenths prepared. Time ie when an event happens is very important as now we have 3 sets. The Cartesian product of 3 sets is many more elements and this practically leads to infinite variety. Like just 4 elements of DNA lead to an almost infinite variety. Is there a 4th element or dimension in music like DNA and what is it? Is it melody?
Note 2: Iannis Xenakis is a pioneer of the use of vector spaces in music representation. It is not known at this time if what is described here is a rip off of Xenakis work but this post is certainly influenced by his idea of vector spaces or sets in music.
What is this post trying to say?
Notes by themselves do not have much meaning. Perhaps like letters in a contemporary alphabet. It is when 2 notes are combined together that something more meaningful takes place [Bernstein]. It may be said that the alphabet of music is not the notes but the intervals of notes and the order of presentation of these intervals to the human brain.
The Cartesian product of notes. First set, and code ie the 2nd set is a new set of 2 dimensional elements or vectors. The elements of this new set should look something like this:
Do with Do (C,1) or (C,C)
Do with Re (C,9) or (C,D) actually played C bass E G D [Bach]
Do with Mi
Do with Fa
Do with Sol
Do with La (C,6) or (C,A) actually played C bass and E and A [Bach]
Do with Si (C,7) or (C,B) actually played C bass and E and B [Bach]
Then
Re with Re
Re with Mi (D,9) or (D,E) actually played D and F and E (Re Fa Mi) [Bach]
Re with Fa
Re with Sol
Re with La
Re with Si same as above
Re with Do same as above
Then
Mi with Mi
Mi with Fa
Mi with Sol
Etc
All these 49 combinations are derived from the C major scale.
We do not need to remember anything else when we apply these ingredients to other scales. We can play or compose all music with theses ingredients or elements or sound colors. Just knowing which of the notes are sharp or flat, ie usually knowing when to hit usually a black key if the scale is not the C major or A minor which have for all steps the white or natural keys.
Note 4: The elements of the 4 sets defining the 4 dimensional vector space (including melody) may be interacting. For example the need to prepare 6ths and 7ths or 9th etc to the human brain makes chord progressions such as A, G6, F7 quite common [Bach] in many genres of music. The melody itself may be used as a means to prepare the interval of 6, 7, 9 etc to the human brain. For example Mass in B minor - Bach.
Much of this comes from reflecting on Vicky's questions.
References, further reading, playing:
Bach bass rules:
https://normanschmidt.net/scores/bachjs-general_bass_rules.pdf
Young people's concerts - melody - Bernstein - youtube
For example (C,9) is the 1st element of Close to You - Bacharach
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