Questions tagged [music-theory]

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2
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1answer
34 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 ...
3
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1answer
57 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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0answers
29 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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0answers
19 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
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1answer
70 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 ...
5
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0answers
110 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
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1answer
37 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 ...
4
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1answer
105 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
43 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 ...
2
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2answers
44 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, ...
2
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0answers
80 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 ...
4
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0answers
86 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 ...
19
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0answers
470 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 ...
1
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1answer
27 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
166 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 ...
1
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0answers
126 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. ...
0
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1answer
94 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 ...
0
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2answers
35 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 ...
1
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2answers
59 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 ...
2
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0answers
195 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, ...
6
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2answers
233 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
51 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 ...
3
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2answers
34 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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0answers
57 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
122 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
139 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\} \...
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0answers
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
220 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
75 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 ...
3
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1answer
144 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 ...
2
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1answer
40 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
50 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
130 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 ...
56
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8answers
5k 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
1k 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 ...
5
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2answers
99 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 ...
2
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2answers
1k views

Which equation? Sin, math and music [closed]

I found it inside an italian music book: It represents the music scale. I think it is a series of trigonometry sin equation. With some difference in the amplitude ...
0
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2answers
929 views

Plotting points of Musical Note frequencies linearly on a Graph

I am trying to plot the frequencies of musical notes on a graph so that they are equally spaced apart. I have researched that the relationship between each note is $$f \times 2^\left( X / 12 \right)$$...
6
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0answers
595 views

Months and notes. Pure coincidence? [closed]

There are $12$ months in a year and $12$ notes in the chromatic scale. Moreover, there are $7$ long and $5$ short months and there are $7$ white and $5$ black keys in each octave on the piano ...
0
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3answers
188 views

In 12TET, is there a maximum number of musical scales possible?

For those who don't know, 12 Tone Equal Temperament (12TET) simply means that, for a given frequency $f$, each step between $f$ and $2f$ can be written as $2^{k/12}f$, where $k$ denotes the $k$th step ...
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0answers
62 views

What mathematical property does equal temperament have that lets it form keys?

Suppose I have the frequency $f$. According to just intonation, $\frac{3}{2} f$ is a perfect fifth. Now compare the following: $$ \left(\frac{3}{2}\right)^n \neq 2^m$$ for any integer $n,m$ But $$...
3
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0answers
142 views

Equation of the ellipse for musical notation (the quarter note/crotchet and shorter)

Note heads are often represented with a slightly rotated ellipse, as shown here for instance (first image). Does anyone happen to know the equation of the ellipsis, and the rotation that's applied to ...
5
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2answers
197 views

Sound of $\sin(x \cdot \sin(x))$ without accumulation

Playing around with the sine function, I noticed that when you plug the formula $y = \sin(x \cdot \sin(x))$ into your speakers, you can hear nice sequences of overtones. Especially if you add a ...
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1answer
214 views

How does a number correspond to a sound when listening to an OEIS sequence?

I noticed that the OEIS allows you to listen to a sequence. It converts a sequence of integers to sounds somehow, but I can't find any information on which numbers correspond to which sounds. The OEIS ...
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0answers
59 views

Help needed for statistical analysis of pitch class sets

Within Music Analysis, there is a quite mathematical type of analysis which looks at pitch class sets ($pcs$), not surprisingly known as pitch class set analysis. See http://en.wikipedia.org/wiki/...
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1answer
35 views

Number of series of this form whose product is 2

$I_n$ is the $n^\text{th}$ member of the series $I$ of length $k$. The first member of the series is of the form $2^\frac{m}{12}\mid m\in\Bbb Z\text{ and } 0\le m\le12$. If $k$ is larger than $1$ and $...
5
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1answer
160 views

How is it possible to change the pitch and the tempo of an audio track independently of each other?

If you slow down a turntable or cassette-player, both pitch and tempo are decreased. How is it possible to change one without affecting the other?
1
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1answer
151 views

Music of primes

In http://plus.maths.org/content/music-primes DuSatoy describes the relation between the prime number staircase and harmonics from music. So in the article he uses music as an analogy. But I wonder ...
19
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1answer
786 views

Revisiting “What is Mazzola's ”Topos of Music“ about?”

This question What is Mazzola's "Topos of Music" about? has already been asked, but I am dissatisfied with the response for several reasons and would like Math SE to revisit it. For ...
1
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1answer
196 views

wave equation on a square domain

I'm stuck on the following problem. Let $u(x, y, t)$ denote a solution to the linear wave equation $k^2(u_{xx}+u_{yy}) = u_{tt}$ with $k = 2$ on a square domain with corners at (0, 0), (0, 1), (...