Questions tagged [music-theory]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
1 vote
0 answers
51 views

(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,...
user avatar
10 votes
1 answer
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 ...
user avatar
1 vote
0 answers
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 ...
user avatar
2 votes
1 answer
50 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 ...
user avatar
  • 1,666
2 votes
1 answer
120 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 ...
user avatar
  • 1,666
1 vote
1 answer
84 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, ...
user avatar
  • 23
1 vote
1 answer
48 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 ...
user avatar
  • 1,666
1 vote
0 answers
45 views

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 ...
user avatar
0 votes
2 answers
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 ...
user avatar
2 votes
0 answers
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/...
user avatar
8 votes
1 answer
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 ...
user avatar
  • 643
5 votes
1 answer
117 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, ...
user avatar
  • 357
3 votes
1 answer
219 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 ...
user avatar
1 vote
0 answers
56 views

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 $P(C-major) = c-minor,$ $L(C-major) = e-minor,$ ...
user avatar
  • 518
0 votes
1 answer
108 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 ...
user avatar
  • 341
1 vote
2 answers
91 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 ...
user avatar
1 vote
1 answer
117 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 ...
user avatar
  • 1,297
1 vote
0 answers
63 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}...
user avatar
  • 65.1k
2 votes
1 answer
46 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 ...
user avatar
4 votes
1 answer
157 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 $2^\frac{12}{12}=1$ ...
user avatar
  • 1,192
0 votes
0 answers
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/...
user avatar
1 vote
0 answers
35 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 ...
user avatar
  • 1,297
1 vote
1 answer
167 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 ...
user avatar
  • 356
5 votes
0 answers
137 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 ...
user avatar
  • 151
1 vote
1 answer
57 views

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 ...
user avatar
  • 6,718
4 votes
1 answer
127 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 ...
user avatar
  • 6,718
1 vote
0 answers
47 views

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 ...
user avatar
  • 111
1 vote
3 answers
85 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, ...
user avatar
  • 145
0 votes
2 answers
1k views

What is the probability that a song will be played twice in a row when the music player is set to random?

Problem : In a music player, if it is set to shuffle (random), will a song be played twice in a row? If so, what would the probability be? Example: Take Spotify. Suppose I have $100$ songs in my ...
user avatar
2 votes
0 answers
103 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 ...
user avatar
4 votes
0 answers
539 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 ...
user avatar
  • 49
54 votes
1 answer
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 ...
user avatar
  • 38k
1 vote
1 answer
35 views

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 ...
user avatar
1 vote
3 answers
314 views

(Music) List of all possible "types of set" of 12 musical notes

I have looked into trying to figure what are all the possible "types" of note set combinations there are and how I would go by listing them if possible. It turns out this is harder than I thought. The ...
user avatar
1 vote
0 answers
159 views

Fourier Transform on Musical Notes

I am trying to apply the Fourier transform analysis on music. So far, I am aware that the Fourier transform is essentially the breaking down of superposed sine waves, into its individual frequencies. ...
user avatar
0 votes
1 answer
128 views

Is there an arithmetic mean limit on the symmetrical items of the harmonic series?

Is it possible to determine what is the arithmetic mean of the harmonic series where $n$ and $-n$ are added and divided by two in this manner: $$f(n) = \frac{ (\frac{3}{4})^n \times 2^{⌈-n \times ...
user avatar
  • 673
0 votes
2 answers
82 views

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 ...
user avatar
  • 673
1 vote
2 answers
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 ...
user avatar
2 votes
0 answers
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, ...
user avatar
  • 167
6 votes
2 answers
309 views

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 ...
user avatar
1 vote
0 answers
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 ...
user avatar
3 votes
2 answers
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...
user avatar
  • 133
0 votes
0 answers
81 views

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 ...
user avatar
  • 700
2 votes
1 answer
167 views

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 ...
user avatar
2 votes
3 answers
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 ...
user avatar
  • 269
1 vote
0 answers
26 views

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 ...
user avatar
1 vote
1 answer
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 ...
user avatar
  • 11
1 vote
1 answer
79 views

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 ...
user avatar
3 votes
1 answer
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 ...
user avatar
  • 2,625
2 votes
1 answer
42 views

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 $4375$, ...
user avatar
  • 21