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Mathematical music is music that you can attribute certain musical operations to that also exist in the musical realm. Mathematically: Imitation or translation of a basic shape with attendant symmetry operations. Musically: Transposition, augmentation, diminution or inversion.
Transposition is the change from one key to another. Say we have a melody that goes like: C D F E G. A transposition to G would make it G A C B D. We took the same distances between each note but started with a different note.
Augmentation or diminution include a change inside the intervals between each note themselves. So if we have C D F E G , it becomes C D F # E G for example or C D F E flat G
Inversion here means opposite imitation. Meaning if we have C D F E G , it will be G E F D C. It is different than harmonic inversion since we are here speaking in counterpoint (horizontally) not harmonically (vertically)
In mathematical music, we often compose in canon. If we take a melody in soprano that lasts say, 10 seconds, and after 2 seconds, we play the same melody in the bass while the same melody in soprano is still playing for another 8 seconds, this is called a canon.
Popular canons include frères jacques and pachelbel's canon in D
however, canons often are not that simple. In mathematical music, the themes are woven intricately into the fabric of the music that canons are not very obvious unless you are well versed in the form of the piece and are looking for them.
not only that, a variation on the theme usually is applied by the composer to add interest, or even inverted.
In Bach's mathematical music, one theme gets introduced first in one voice, then it is joined by another voice and another voice, up to 5 voices
In Bach's mathematical music, one theme gets introduced first in one voice, then it is joined by another voice and another voice, up to 5 voices
a "voice" is a line of melody in a specific musical range. imagine you are keyboardist playing 5 different melodic lines at the same time, with only 10 fingers, not only that, some notes are longer than others, so you have to sustain them while the other melodic lines keep moving
Now let us talk about the fugue which was Bach's favourite compositional style. It is a musical form, that means say, a formula by which one writes music. It is basically the most prestigious form of musical writing in counterpoint and canon.
It contains mathematical intricacy, formality, symmetry and variety. The fugue combines mathematical patters with musical freedom. It is freer than standard the canon system.
Fugues start slowly and methodically. One voice alone announces the fugue subject in the opening exposition. The continuation of the subject is called countersubject. In here, the musical operations are again applied.
Let us take an example from the first fugue of the well tempered clavier by Bach, in C major. The melody highlighted is the initial theme: the subject
Then another melodic line enters in the bass (red). This melody is separate from the initial theme as it is different.
In green, the main theme is repeated but we applied a mathematical operation on it: we transposed it one fifth up. It started in G instead of C and all the
In green, the main theme is repeated but we applied a mathematical operation on it: we transposed it one fifth up. It started in G instead of C and all the
the other notes followed suit, conserving the distances between them.
Now something interesting happens, the theme moves from the soprano, to the bass (green), and at the same time a 3rd voice is introduced (red) while a different melody is played in the soprano (yellow)
many players interpret this as the theme always being dominant audibly, that is the one we should play loudest at all times no matter what voice it is in. This is a technique i played as well even though theoretically all melodic lines are of equal importance but that is
a modern stylistic interpretation that has become the convention.
In reality, these fugues were played on the harpsichord that did not function based on how strong you play a key but by texture: meaning how many voices are played at the same time.
In reality, these fugues were played on the harpsichord that did not function based on how strong you play a key but by texture: meaning how many voices are played at the same time.
that means, for me as a modern player, the moment the 3rd voice is introduced, i have to play louder because 300 years ago, that is how the harpsichord would have reacted to the music played.
there are also some minor imitations in other voices separate from the theme , but for the time being we will move on and focus on the theme only.
ok now, not only did we play the theme again, but we introduced the fourth voice simultaneously. The theme however is varied, Bach doesn't let it play till the end but cuts it off at the last 4 notes. It is not transposed, it is still in C, but it is 1 octave lower than the first
time it was introduced.
This is basically the main idea. The further you go along any fugue, this thematic repetition becomes more intricate and more woven together as other themes of less importance get introduced and varied as well.
this is from the end of this fugue, the second is varied at the end and starts only 4 notes after the third one starts in the tenor voice
anyway hope you enjoyed this thread :) have a good night
