# Questions tagged [music-theory]

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### (weird) Explanations in Xenakis' Formalized Music

First time posting on MSE, very nervous. Been lurking for a long time. I've been looking through Formalized Music (Xenakis) and building software around the ideas in the book. Upon thorough inspection,...
185 views

### Finding information about the combinatorial concepts (arriving from Music Theory)

In my music theory PhD work on scales, I've come across certain classes of musical scales, which I believe might have parallels within mathematics and I hope you can help me learn more about these ...
1 vote
29 views

### How to find a successive frequency from a series of frequencies

I am building a new instrument, and I need to find the sixth sequence in the given series, {659.25, 440.00, 293.66, 196.00, 130.81} that would be around 80 Hz. An exponential regression at wa, gives ...
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### Drum PDEs, Double Fourier Expansions, and Synthesis

In studying the $2$D Wave Equation, the application often encountered is the displacement of a drum. The main solution to the PDE is a double Fourier summation, either a double Fourier Sine Series, a ...
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### Violin String PDE Modeling

I have this exercise in my differential equations book... If you pluck a violin string, and then finger the string, fixing it precisely in the middle, the tone increases by one octave. In ...
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1 vote
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### Mathematical Representation of a two column (two channels, or stereo) audio signal

I am a computer science master's student but my experience in converting a code to mathematical operations is very very low. Now I am starting to write the master's thesis as well as writing a paper, ...
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1 vote
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### Translating Between Fourier Sine Series and Fourier Cosine Series

A mathematician can choose to represent a target function defined from $0$ to $L$ using a Fourier Sine Series or a Fourier Cosine Series at her discretion, by temporarily introducing either an odd ...
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1 vote
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### How to extract a component of integer vector / music interval? [duplicate]

(I will ask this question in musical terms, but this seems to be related to projecting integer vectors onto each other, which I'm unfamiliar with. Perhaps I'm just looking for some existing notation ...
51 views

### Is there a proof that no rational number splits the octave equally?

In music circles, when the topic of tuning comes up, it is said that there is no rational number that splits the octave (the interval between a musical pitch and another with double its frequency, for ...
168 views

### Twelfth root of two?

I have been experimenting in mathematically analysing and combining two melodies based on the twelfth root of two. Here is a mix of two known melodies: https://drive.google.com/file/d/...
289 views

### Music and Maths - Modes of Limited Transposition

In music, 'Modes of limited transposition' are modes that have a limited availability of transpositions. Unlike a major scale that has $12$ possible unique transpositions, the seven modes of limited ...
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### Understanding the symmetries of f(x)=7x (mod 12)

I found something interesting while studying the circle of fifths. Define a map $f(x) = 7x$ (mod 12), this models the circle of fifths as a table. The input is the location on the circle of fifths, ...
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### Measuring the Shannon Entropy of an ordered sequence

I have 927 unique sequences of the numbers 1, 2 and 3, all of which sum to 12 and represent every possible one-octave scale on the piano, with the numbers representing the intervals between notes in ...
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1 vote
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### How to identify the two copies of $D_{24}$ in the homomorphisms of the 2 musical actions?

Let $S$ be the set of minor and major triads. Two sets of actions are defined on the set: 1) Musical transposition and inversion 2) P, L, R actions $P(C-major) = c-minor,$ $L(C-major) = e-minor,$ ...
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### Optimal guitar fingerings

I want to find optimal frettings for guitar scales, played one note at a time. In particular let's assign a metric $d$ that assings a distance between pairs $(s_1,f_1)$, $(s_2,f_2)$ of integer numbers ...
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1 vote
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### Is there an "easy" formula for calculating the species and quality of the musical interval between two notes?

Let's number the scale steps of the major scale $1,2,\ldots 7$, i.e., label them from the tonic upward mod $7$ and then add $1$. With that numbering scheme, let the lowest note of a given diatonic ...
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1 vote
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### How to skew-stack tetrahedral-octahedral honeycombs?

In 1D, the densest packing of 0-sphere in a line is by apeirogon, placing their centre on the apeirogon's vertices. In 2D, the densest packing of 1-sphere in a plane is by triangular tiling, which ...
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1 vote
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### What is the solution of normalized harmonic series based on $4/3$ between one and two?

In music theory notes generated by the consequencing interval of $4/3$ generates harmonic series. Series can be normalized by multiplicating the fraction with a $2$ in power $n$. What is a formula ...
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1 vote
69 views

### Find three points on a scale using golden ratio

My question came from my musicial part: The note A4 on a piano is 440hz and the note A5 (one octave higher) is 880hz. On the piano there are 12 notes between A4 and A5 (include). Im trying to find ...
376 views

### Algorithm for converting complex sine wave to constituent simple sine waves

Since any real sound is by nature a complex sine wave based on the harmonic series, every sound is made up of many simple sine waves. Since a sound is constructed via the combination of these waves, ...
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### Musical and combinatorial proof

How many distinct rhythms can a musical measure have? Obviously the answer is not "$\infty$", so to answer this question we set a minimum rhythm $\frac{1}{4}$. We will consider both notes and rests ...
1 vote
88 views

### Is there a name for this exponential analog to modular arithmetic? (octave equivalency)

In music theory, there is a concept called octave equivalency: two pitches are said to have the same pitch class if the quotient of their frequencies is a power of 2, i.e. if they are an integer ...
47 views

### Constraining the results of a division between two "poles"

I often have to do calculations that pertain to musical intervals, i.e. the ratios between two pitches. E.g., the interval that commonly we call a "major third" (c-e...
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### Maths of chords

I have a few naive questions on music theory. Let us assume that I have two pitches A and C with certain frequencies. Then the corresponding sound waves are pure sinusoidal waves. But what happens if ...
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### Could Collatz rules $3x+b$ hold every possible song melody in existance?

Since Modulo seems to have a significant role in the Collatz Conjecture for one reason or another, I wondered what would happen if the Conjecture was put to music (I was inspired by this interesting ...
381 views

### Example of a power of 3 which is close to a power of 2 (Related to music theory and Superparticular ratios)

I'm looking for a power of 3 close to a power of 2. Let's say, what is $(n,m)$ such that \left|\frac{2^n}{3^m}-1\right| = \min\left \{\left|\frac{2^i}{3^j}-1 \right|, 1\leq i,j\leq 20\right\} \quad ...
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1 vote
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### Acoustical properties of a rectangular prism with one open end

I'm studying musical instrument design as a hobby, and could not find the answer to a question regarding instruments with a square-cross sectional bore. After seeing a design for a Paetzold square ...
1 vote
255 views

### How many ways can 6 pairs of 2 pegs be arranged in 12 holes… discarding duplicates read in opposite direction (re: music theory)

I hope it’s OK for a non-mathematician (a musician) to ask a question here. My question is actually about music, but I’ve dressed it in plain clothes. Suppose you have a line of 12 equally-spaced ...
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### Process of Elimination With Musical Keys

This comes from a discussion I was having with my music teacher regarding the quickest method to confirm that you are playing in the right key starting from a given note. This is complicated by ...
201 views

### Advanced mathematics in stringed instrument industry

This is a soft question. I play classical guitar and I find stringed instrument industry a very fascinating art. I know that, at least for classical guitar, this industry is still developing and ...
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### What are the largest pairs of $p$-smooth integers with a difference of one?
This is an interesting question because for very small $p$, we already know the answer: for $p = 2, 3, 5, 7$, the answers are $1$ and $2$, $8$ and $9$, $80$ and $81$, and $4374$ and $4375$, ...