Questions tagged [music-theory]

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76 views

Music and Fourier Transform

I am trying to build a "program" to determine musical chords. I want to add that that it's not my field in mathematics so it should be considered an unrigorous project. Currently I am using ...
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69 views

How does the ABC-conjecture sound?

One version of the abc-conjecture can be stated in terms of the positive definite kernel over the natural numbers: $$K(a,b) = \frac{\gcd(a,b)^3}{ab(a+b)}$$ Based on this kernel, I wrote a small python ...
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14 views

Does every closed orientable surface contain all finite cyclic groups as subgroups?

TL;DR For every connected compact orientable surface (namely, the sphere, the torus, and conjoined tori) $X$, is there an $n \in \mathbb{N}$ such that there is an imbedding of $X$ into $\mathbb{R}^n$ ...
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1answer
55 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 ...
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8 views

Independence Check for Music Singing

The dataset is constructed by 1st interval, 2nd interval......to 24th interval. each interval is measured by the error between targeted cents and people actual singing. For example, if the 1st ...
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1answer
106 views

Some Way to Connect Music and Math [closed]

I wish to start a study on how I can craft music using math (I know there are books pertaining to the aforementioned topic, but I am looking forward to a different idea), and for this, I'd like to ...
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19 views

Can “dissonant chords” be more consonant than “consonant chords”?

Consonance and dissonance seems to have an objective mathematical definition because, as it says in this wiki page https://en.wikipedia.org/wiki/Consonance_and_dissonance, if you have defined ...
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48 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 $P(C-major) = c-minor,$ $L(C-major) = e-minor,$ ...
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1answer
53 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 ...
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12 views

harmonics and wave equation

I would like to ask if there how I could see from solution (by the method of separation) of 1-D wave equation that e.g. a string creates harmonics and not just overtones as e.g. a drumhead. What shall ...
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50 views

Mathematical music theory text that treats the concept of the “freestyle”

This question relates to mathematical music theory and linguistics. I am investigating and trying to understand a phenomena I call "pseudo-musical speech." That is, the artist just let's the words "...
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1answer
36 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 ...
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1answer
74 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 ...
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49 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}...
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1answer
42 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 ...
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1answer
103 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 $2^\frac{12}{12}=1$ ...
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32 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/...
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27 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 ...
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1answer
108 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 ...
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126 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 ...
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1answer
45 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 ...
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1answer
120 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 ...
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0answers
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 ...
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3answers
66 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, ...
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0answers
85 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 ...
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226 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 ...
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662 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 hilarious ...
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1answer
28 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 ...
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3answers
232 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 ...
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0answers
141 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. ...
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1answer
105 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 ...
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2answers
48 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 ...
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2answers
68 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 ...
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0answers
249 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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2answers
264 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 ...
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0answers
68 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 ...
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2answers
37 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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69 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 ...
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1answer
139 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 ...
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3answers
196 views

Example of a power of 3 which is close to a power of 2

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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24 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 ...
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1answer
239 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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1answer
78 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 ...
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1answer
172 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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1answer
41 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 $4375$, ...
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0answers
54 views

Analogous Mathematical term for extending beyond the elements in a set?

This question relates to projecting pitches in the context of music theory. However, I'm looking for the appropriate mathematical term for this particular concept. Here's the musical context: ...
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0answers
139 views

Fourier Transform: Musical Instruments cotd.

Upon analysing the Fourier Transform of a musical sound, are there any other applications of the Fourier Transform so obtained? Any ideas would be appreciated. Edit 1: To clarify the situation, I ...
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9answers
6k views

How many 7-note musical scales are possible within the 12-note system?

This combinatorial question has a musical motivation, which I provide below using as little musical jargon as I can. But first, I'll present a purely mathematical formulation for those not interested ...
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2answers
2k views

How would I determine all possible rhythmic combinations given certain limitations?

If I have: 1) kick drum 2) hihat 3) snare drum And I can play 1) One measure of 4/4 time 2) any note values between sixteenth and whole EDIT: (excluding tuplets) How many possible drum beats could ...
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2answers
102 views

How to determine if this musical exercise is valid: will the pattern complete?

I'm hoping that math has an answer to a question arising out of a musical exercise. In music terms, the exercise is: Choose two arpeggios (sets of notes) of equal (or roughly equal) span (number ...