# Questions tagged [music-theory]

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### Can we place the 12 musical notes in the chromatic scale on a dodecahedron's faces such that the set of musical intervals at each vertex is the same?

Can we distribute the 12 musical notes in the chromatic scale on the faces of a dodecahedron such that the set of musical intervals at each vertex is the same? (Assumes treating inverted invervals as ...
1 vote
58 views

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### Geometric relationships between musical notes and sequences of them

This question is inspired by the concept of the circle of fifths in music. I was watching this video where chords are certain subsets of the points on the circle of fifths. There are interesting (...
1 vote
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### Question Math of a Non Linear Semi Whole Tone of a Musical Scale Intervals as Rationals

Peculiar Question Irregular semi-tone of scale. Hi folks, am hoping to achieve a formula or function that can be applied to the general following problem - any feedback appreciated ; Given two values ...
208 views

### Quality and degree of a musical interval as linear combinations: Is this a known construction?

At some point I discovered that the 'degree' of an interval (unison, sixth, ...) is simply determined by writing it as $\;t\text{ major seconds} + s\text{ sharps}\;$ (both $\;t\;$ and $\;s\;$ can be ...
113 views

### Probability that notes are in the same scale

What is the probability that a random sequence of notes (on the 12-note chromatic scale) of length n is in the same major scale? Quite some time ago, I came across a song written by converting pi to ...
1 vote
384 views

### Can someone explain these groups of linear patterns in the dropping times of Collatz Sequences? Could this lead to a proof?

Please buckle in because this may be a long post, but I think it will be necessary to help the reader understand three things: How this data was generated. How the data is grouped into different '...
24 views

### Quantitative measure of a good (musical) temperament

Is there actually a quantitative measure to know if a temperament is "good" or not? The motivation for my question is that there are quite a few temperament for the 12-tone systems (which ...
1 vote
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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,...
230 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
39 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 ...
153 views

### 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 ...
201 views

### 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 ...
1 vote
104 views

### 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, ...
1 vote
216 views

### 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 ...
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 ...
62 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 ...
191 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/...
469 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 ...
156 views

### 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, ...
1k views

### 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 ...
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 $P(C-major) = c-minor,$ $L(C-major) = e-minor,$ ...
163 views

### 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 ...
225 views

### 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 ...
1 vote
146 views

### 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 ...
1 vote
89 views

### Is there a standard name for "signed distance to the nearest integer" that's reasonably succinct?

Notation. Write $\mathbb{R}_\mathbb{Z}$ for the set of all real numbers for which there exists a unique nearest integer. Explicitly: \mathbb{R}_\mathbb{Z} = \mathbb{R} \setminus(0.5 + \mathbb{Z}...
47 views

### Audio - Compute samples for a given length with a start and end tempo

I'm working on some audio analysis code and I'm currently trying to determine the number of audio samples that would be needed in a segment of audio in which the tempo changes linearly. I know start ...
223 views

### Question about 12-tone musical scale and rational approximations

On a modern tuned instrument, an octave has twelve notes with a common frequency ratio of $2^{\frac{1}{12}}$ Of course, twelve is a very good choice for the number of notes, as $2^\frac{12}{12}=1$ ...
35 views

### Rational Roots Theorem corollary and piano tuning: $\big(\frac{a}{b}\big)^n \neq 2$ [duplicate]

I'm trying to understand why pianos "can't be tuned" and am looking for a proof of a corollary of the rational roots theorem found here (looking for proof not by contradiction): https://youtu.be/...
1 vote
39 views

### Are the roots of Bessel functions closed under multiplication?

In terms of eigenfunctions, a circular drum vibrates in angular velocity of $\lambda _{mn}$, the nth positive root of the Bessel function $J_m(x)$, $m = 0, 1, \cdots$. If they are closed under ...
1 vote
220 views

### Wedge Product: Vectors wedged with Multivectors? (Disclaimer: Music theory ahead)

The Xenharmonic Wiki is a great resource to start understanding and, if enough determination is available, constructing temperaments as well as scales. As some of you may know, this Wiki is not ...
143 views

### Determine the shift in tonal center of a piece of music.

Starting with a sampled audio signal of acapella vocals, I am interested in determining the shift in the tonal center of the music through the performance. As a choir progresses through a ...
1 vote
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### Mathematical theory for the stability of notes in a musical scale

Most mathematical theories for music consider the issue of consonance/dissonance, but in music, we actually care more about the stability of notes in a scale. For example, the subdominant is unstable ...
159 views

### Mathematical music theory concerning melodic intervals and chord progressions

There are many books exploring musical theory with maths. However, so far I have only seen discussions about the consonance/dissonance of two notes played simultaneously (intervals) -- this is the ...
1 vote
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### Integer sequence that "fills a range in a balanced way"

I am writing a piece of software to generate musical sequences, and I would like a way to slowly introduce notes in a chord in a balanced way across the range of the chord. I want it to be more ...
1 vote
108 views

### Why is the prime of the Forte number 5-20?

See Forte number in Wikipedia. First of all, let me say that I know very little about music set theory... I am just curious why a set containing an obvious non-prime (8) is considered prime? Also, ...
107 views

### "Fragmentation" of a distribution (from paper)

I've been reading a paper by Robert Morris ("Sets, Scales and Rhythmic Cycles; A Classification of Talas in Indian Music") and came across a formula that I've found a bit tricky. He is referring to ...
732 views

### Finding the envelope frequency of a sinusoid (From a musical major triad) [closed]

[Editor 2’s introduction intended to address votes to close because the question wasn’t mathematical.] The trigonometric formula $\sin{(at)}+\sin{(bt)}=2\cos({a-b\over2}t)\sin({a+b\over2}t)$ can be ...
1k views

### The Complexity of "The Baby Shark Song".

This question is just for fun. I hope it's received in the same goofy spirit in which I wrote it. I just had the pleasure of reading Knuth's "The Complexity of Songs" and I thought it'd be ...
1 vote
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### How many ways can you generate 12 who's pairwise modulus difference is also a unique rank?

I want to generate a special 12-tone row starting with zero, with the remaining 11 columns being allowed any number from 1-11. I'll call this the original row. So rule one is every number must be ...
1 vote