# "Mathematical" Pieces?



## BenG (Aug 28, 2018)

I find myself all the time for amazed at a piece like the Goldberg Variations, where all of the 30 Variations have 32 bars (expect for some which split into two and have 16), all of them follow the same harmony structure, and all are based on only one or two themes (which are transformed ingeniously).

And in this structural limitation, Bach composes 9 three part canons (one every three bars), 2 mini-fugues, and many other brilliant variations, all of them having integrity, beauty, and elegance.

It seems that Maths (or Math) has some relationship to the integrity of music, but I can't seem to figure out how it works. Do you think imposing Mathematical limits adds value to composition? And what other pieces are so Mathematical in their nature?


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## Phil loves classical (Feb 8, 2017)

I don't feel the Goldberg Variations is mathematical as much as symmetrical. There are some theories that tonal music have certain math relationships, as to what sounds good stemming from the golden ratio and the Fibonacci series I recall. But for composition I doubt you can use math other than serial composition.

Here is a piece using matrices (with some modifications). Babbitt also uses math in his compositions to determine pitch, duration, relationships.


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## ORigel (May 7, 2020)

Xenakis's Pithoprakta is a musical representation of the movements of gas molecules.


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## hammeredklavier (Feb 18, 2018)

https://en.wikipedia.org/wiki/Golden_ratio
"Ancient Greek mathematicians first studied what we now call the golden ratio, because of its frequent appearance in geometry; the division of a line into "extreme and mean ratio" (the golden section) is important in the geometry of regular pentagrams and pentagons."










Interestingly, some speculate that Mozart's sonata movements _generally_ tend toward this "golden ratio" of 0.618.
For example, Mozart's piano sonatas:
E = number of measures in the exposition
D = number of measures in the development+recapitulation
No. 1, K.279 1st movement { E = 38 | D = 62 }
D/(D+E) = 0.620 *
No. 17, K.570 1st movement { E = 79 | D = 130 }
D/(D+E) = 0.622
No. 7, K.309 1st movement { E = 59 | D = 97 }
D/(D+E) = 0.622 
No. 16, K.545 1st movement { E = 28 | D = 45 }
D/(D+E) = 0.616
No. 10, K.330 1st movement { E = 57 | D = 92 }
D/(D+E) = 0.617
No. 2, K.280 1st movement { E = 52 | D = 88 }
D/(D+E) = 0.611

* the entire movement is 100 bars. [ 38 bars of exposition | 62 bars of development + recapitulation ] is the closest ratio you can get with whole numbers to 0.618.


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## RICK RIEKERT (Oct 9, 2017)

A good contemporary and perhaps unexpected example of the strict use of mathematics in music are the works of Arvo Pärt. Pärt uses mathematics and every sort of calculation at all levels of music composition - in general form structure, in formation of melodic patterns, in polyphonic relation between voices. His music uses a wide range of mathematical operations - from linear algebra (algebraic operations and combinatorics) to the elements of mathematical analysis (mathematical unity of sets). One well-known example is _Fratres_, where the apparent simplicity of the piece is governed by strict mathematical rules that determine the movement of voices, length of the melody and phrases, time signature alternations, etc. etc.


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## MarkW (Feb 16, 2015)

I tried to set the Quadratic Formula to music, but the factors kept coming out flat.


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## Open Lane (Nov 11, 2015)

Phil loves classical said:


> I don't feel the Goldberg Variations is mathematical as much as symmetrical. There are some theories that tonal music have certain math relationships, as to what sounds good stemming from the golden ratio and the Fibonacci series I recall. But for composition I doubt you can use math other than serial composition.
> 
> Here is a piece using matrices (with some modifications). Babbitt also uses math in his compositions to determine pitch, duration, relationships.


Bobbit is also who came to mind for me, when i saw this thread.


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## Captainnumber36 (Jan 19, 2017)

I think there are multitudes of ways of composing, and imposing Math is just one.


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## Phil loves classical (Feb 8, 2017)

Open Lane said:


> Bobbit is also who came to mind for me, when i saw this thread.


Is Bobbit a hobbit composer?


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## lisaclassical (Oct 17, 2020)

Really interesting observations! It is really incredible. .. Music as mathematical expositions over time. . .Certainly Bach was a master of this. Beethoven also transformed themes but to me seemed to lead with emotion, rather than logic. He loved to play surprises in his sonatas. . . to defy expectations. . The brain is time machine, some would say. It would be very interesting to explore music from a mathematical perspective, if your talents go in that direction...


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## lisaclassical (Oct 17, 2020)

really math alone may be sterile. Although symmetry, as noted in a post above, as in Beethoven, it is important to violate symmetry and therefore the unconscious expectations as to where the music leads. . . making it a mysterious work of art in the temporal sense. In playing the Goldberg Variations, I found what helped my playing was visualizing each variation as something in nature. .. For some reason, these visualizations helped me to learn and encode the piece. Variation 2 I imagine a marching of ants. . .Variation 4 is a trumpeting of Tulips. . .Variation 7 are teh butterflies. . .Variation 8 are squirrels chasing each other through the trees, around the trunks and limbs. . . It was fun to learn by using these metaphors. .. Variation 9 Redwood trees. . Variation 10 Rabbits. . .


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## Rangstrom (Sep 24, 2010)

Sound waves are the basis of all we hear, including music. Fourier transformations (or Fourier transforms as they seem to be called now--it has been many decades since my MA in Math) can analyze sound waves and their interactions in exquisite detail. And the math involved can be very beautiful, but using "math" to compose almost always involves sloppy approximations of real math and boring music. It takes a lot of imagination to do math and, I suspect, even more (from a different realm) to compose quality music.


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## BachIsBest (Feb 17, 2018)

Phil loves classical said:


> I don't feel the Goldberg Variations is mathematical as much as symmetrical.


In a deep sense, there are entire fields of mathematics devoted to studying symmetry. I feel this necessarily means you belive the Goldberg Variations are mathematical.

That being said, I think the emphasis on Bach as a mathematician has been overstated. As someone in mathematics, I can say his music does have a certain inevitability that is very akin to well constructed mathematical argumentation; at least in the emotional response it provokes. However, I'm not sure this is so much to do with mathematics, but more to do with the fact that both seem to be very logically consistent and 'right'.


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## Mandryka (Feb 22, 2013)

BenG said:


> And what other pieces are so Mathematical in their nature?


The composer to explore is Antoine Beuger, his Cantor Quartets and Dedekind Duos.


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## Ariasexta (Jul 3, 2010)

> It seems that Maths (or Math) has some relationship to the integrity of music, but I can't seem to figure out how it works. Do you think imposing Mathematical limits adds value to composition? And what other pieces are so Mathematical in their nature?


keyboard works with names like Aria or Balletto are meant to be variations, Sweelinck, Byrd, Frescobaldi composed some of the most famous variations.And most keyboard intabulatures of vocal music are also variations, there are a lot of pieces of this kind. Pachelbels Hexachordum Apollinis can be taken as a climax of the this tradition. JS Bachs Goldberg Variation is its last ray of twilight. The variation theme is certainly very tricky and demanding, but mathematical? yes. Like changing of beats, making contrasting melodies, especially the figuration are quite mathematical, like tossing coin problems in math(probability). It means there are endless way of changing a melody, isnt it wonderful? Why I use Ariasexta as my nickname, it tells the big magic. :angel:

I would say you can read some Probability, especially Montecarlo methods, probability theories have the most affinity to musical 
technics. I do not liek to overtly technically discuss music because there is an predominant trend of deconstructing music into mathematical dialectics. I strongly promote enjoyment over analysis of music. :tiphat:


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## Ariasexta (Jul 3, 2010)

I find Montecarlo experiment is like musical constructions, for example figuration is like tossing several coins, different combinations can be compared to the note embellishment, which is the element of melodic figuration. Listening to variations, is like watching elements dancing, spheres bouncing, the starry night rotating, ebbing through the millania. Aria is the magic.

The Fibonacci thing is already hidden within the tuning theories, I will only discuss technical issues with people who are open-minded, not hippies and lefties. The golden ratio is already incoporated into the meantone system, different musical instruments can have different application of the GR without much consciously knowing. It will be much a tricky topic to go on. You can search from this perspective and find insteresting thingies.


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## consuono (Mar 27, 2020)

The Musical Offering, and also this:


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## Simon Moon (Oct 10, 2013)

Gyorgi Ligeti's piano etude 13, "the Devil's Staircase" is based on the mathematical function of the same name. Also known as "the Cantor function".

The structure of the piece adheres to the properties of the function both on the macro and micro levels.

https://en.wikipedia.org/wiki/Cantor_function


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