A First Operation on Musical Objects
by Ryan Vitale
In the last article I introduced pre-musical objects, which are the raw information that everything else can be built from. They were defined to be cyclic ordered lists, for example [: A, C, E :]. This example could be read as an ordered list of the pitch classes A, C, and E which repeats cyclically, so really would be the ordered set [A, C, E, A, C, E, A, C, E, …]. [pullquote]The particular pre-musical object [: A, C, E :] could be used as an A-minor triad, A minor arpeggio, or a bunch of other things based on how we decide to express the data. [/pullquote]
The particular pre-musical object [: A, C, E :] could be used as an A-minor triad, A minor arpeggio, or a bunch of other things based on how we decide to express the data.
The example from last time showed that we can take some pre-musical object like the C-major scale [: C , D , E , F , G , A , B :] and then operate on it to create the harmonization of C-major with a single rule:
T[: C , D , E , F , G , A , B :] =
[: ([: C , E , G :]) , ([: D , F , A :]) , ([: E , G , B :]), ([: F , A , C :]), ([: G , B , D :]), ([: A , C , E :]) , ([: B , D , F :]) :]
This operation T is what we call a musical object. The musical object is referencing both the rule T and its output. Musical objects are essentially little chunks of music that we can ‘add’ together, associate to one another, and finally print out as MIDI information or onto a score. What structure should we add to a pre-musical object to get a musical object? A good way to define a musical object will be as a list of pre-musical objects (possibly just a single one) where each pre-musical object in the list is determined by a common rule. We’ll distinguish rhythmic objects from pitch objects since they’ll both have to interact with one another to create music.
To start, here’s an example of a pitch object.
First, take the notes of the chromatic scale [: C, C#, D, D#, E, F, F#, G, G#, A, A#, B :].
Then, we’ll label those respective notes as [: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 , 11 :].
We’ll again use the rule T, which when given a pitch class p outputs [: (p), (p + 4), (p + 7) :].
Now we reduce the values in the output using a cyclic system of 12. That means that 0=12=24=…, 1=13=25=…, and so on. In short, the cycle repeats every 12 steps.
That means, for example, that T(4) = [: 4, 8, 11 :], T(5) = [: 5, 9, 12:] = [: 5, 9, 0:]. And if we plug in the whole pre-musical object we write:
T([: 0,1,2,3,4,5,6,7,8,9,10,11 :]) =
[:T(0), T(1), T(2), T(3), T(4), T(5), T(6), T(7), T(8), T(9), T(10), T(11) :]=
[: ([:0,4,7:]), ([:1,5,8:]), ([:2,6,9:]), ([:3,7,10:]), ([:4,8,11:]), ([:5,9,0:]), ([:6,10,1:]), ([:7,11,2:]), ([:8,0,3:]), ([:9,1,4:]), ([:10,2,5:]), ([:11,3,6:]) :].
T takes any pre-musical object and outputs the triads built off of each entry of the list. Since we plugged in the whole chromatic scale above we got every (at least in 12-tone music) possible major triad in the output. Now how about working with these musical bits? When we express the pitch classes using numbers as above we can really add them together, and addition means something musically. Adding a common value to everything corresponds to translating the triads above. If we add 2 to everything we see the list would be the same but shifted cyclically by two, so that C major would become D major. Adding 2 is the same as adding two semitones, so the same as adding a whole tone to everything. If we look back at the picture from before, addition by a number would be rotation by that number of positions.
In this case we have T[: C, D, E, F, G, A, B :] = T[: 0, 1, 2, 3, 4, 5, 6 :] as our object and adding 2 scale tones to everything (and reducing by 7) would rotate the input/output replacing each triad with the triad 2 scale degrees higher. One main difference between inputting the C-major pre-musical object and inputting the chromatic scale pre-musical object is that in the case of the chromatic scale each output triad had the same triad type, major, where when we input the C-major scale we were adding scale degrees instead of semitones, so that the output had major, minor, and diminished triads. Both cases were achieved by the same idea, and many objects in music theory can be constructed in this way.
This is the construction of musical objects, and now in the next article we can turn to pictures and manipulations of these musical objects to create higher order musical lists, which will be instructions to print out MIDI information. Also, I’ll give an example of a rhythmic object and see how it can interact with pitch objects in the process of printing out music. I promise I’ll finally make a short piece of music with these ideas in the next article (and with diagrams/illustrations instead of lists)!