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 ...
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Possible links between class 1 numbers, Lucas numbers, and 12-tone octave
Definitions
Class 1 numbers: 1, 2, 3, 7, 11, 19, 43, 67, 163
Lucas numbers: …2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199...
12-tone octave ratios: $\frac{16}{15}, \frac{9}{8}, \frac{6}{5}, \frac{5}{4}...
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How many musical notes does a scale need such that any integer intervals can be found in this scale?
We know that in the 12-tone equal temperament musical system, each octave is equally divided into 12 semitones. In a major scale, for example the C major scale, the notes are C, D, E, F, G, A, and B, ...
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Finding non-overlapping loops in a circular tripartite graph
I'm working on a music theory problem (see below the TL;DR for details), and came up with this graph problem.
My academic background hasn't much to do with graphs, so sorry if not using the correct ...
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Resonance/Acoustics equations.
I have a bit of a conundrum I'm trying to figure out.
I have some spare time on my hands and I decided I wanted to custom make a set of wind chimes for my parents, and have been doing some research on ...
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How to represent and compare 'subsets of a group with modulo'?
For a group $G$ with operation $+$, I'm interested in the set $\mathscr H$ of its subsets that can be constructed using only the following two rules:
$\{g\}$ (so a singleton set) is in $\mathscr H$ ...
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How is the pythagorean scale using fifths constructed?
I was trying to construct the pythagorean scale using fifths. You know by multiplying the tonic with 3:2 ratios and then that with 4:3 ratios and so on. Now in this way the ratios keep on getting ...
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Mathematics behind diatonic scales [closed]
I am looking for the mathematics that can be used to calculate a diatonic scale.
It is my understanding that a number of musical scales can be represented in logarithmic form using mathematical ...
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What kind of curve is used in Digital Audio Workstations for i.e. Automation?
I'm trying to recreate the curves used in automation clips of DAWs but I can't seem to find the right one. Here is an example of what i mean (specifically the first curve on the left; the right one ...
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(Continued) Xenakis' notation
\begin{equation}
P_x = c \ e^{-c\ x } dx
\end{equation}
Where c is the density of the points, x is the length of the section and $P_x$ is the probability to find a section of that length and $\...
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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 (...
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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 ...
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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 ...
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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 ...
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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 '...
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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 ...
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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,...
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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 ...
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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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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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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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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 ...
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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 ...
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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/...
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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"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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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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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 ...
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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 ...
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(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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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.
...
