The fact that noise doesn't repeat makes it useful the way a paint brush is useful when painting. You use a particular paint brush because the bristles have a particular statistical quality - because of the size and spacing and stiffness of the bristles. You don't know, or want to know, about the arrangement of each particular bristle. In effect, oil painters use a controlled random process (centuries before John Cage made such a big deal about it).

Ken Perlin

Perlin Noise Visualizer (made by me)

Desmos

Resources

• Noise Generator

Let \(s\) be an irrational real number. Take the sequence of values \(\displaystyle m\) such that, given \(\displaystyle a_{k} =\sum _{n=0}^{k}( -1)^{\lfloor ns\rfloor }\),

\begin{equation} a_{k} =a_{2m-k} \end{equation}

For \(\displaystyle k=0,1,\dotsc , m\).

Alternatively, write \(\displaystyle k=m-j\) for \(\displaystyle j=0,1,\dotsc m\). Then (1) can also be written as: \begin{equation*} a_{m-j} =a_{m+j} \end{equation*} We call these \(\displaystyle m_{k}( s)\) the symmetries of \(\displaystyle s\).

Some symmetry formulas

\(\displaystyle m_{k}\left(\sqrt{5}\right) =\begin{cases} -\frac{1}{2} +\frac{\left( 9-4\sqrt{5}\right)^{\frac{k}{2} +1} +\left( 9+4\sqrt{5}\right)^{\frac{k}{2} +1}}{4} & k=0\bmod 2\\ -\frac{1}{2} +\frac{\left( 9-4\sqrt{5}\right)^{\frac{k-1}{2}}\left( 85-38\sqrt{5}\right) +\left( 9+4\sqrt{5}\right)^{\frac{k-1}{2}}\left( 85+38\sqrt{5}\right)}{20} & k=1\bmod 2 \end{cases}\)

\(\displaystyle m_{r}( s) =-\frac{1}{2} +\frac{\left( 1-ns^{*}\right)^{\left\lfloor \frac{r}{nk}\right\rfloor } -( 1-ns)^{\left\lfloor \frac{r}{nk}\right\rfloor }}{2( s+k)}\lceil m_{0}( r) s\rceil +\left[( 1-ns)^{\left\lfloor \frac{r}{nk}\right\rfloor }( 2-ns) +\left( 1-ns^{*}\right)^{\left\lfloor \frac{r}{nk}\right\rfloor }\left( 2-ns^{*}\right)\right] \cdotp \frac{m_{0}( r) +1/2}{2nk+4}\) where \(\displaystyle s=-k+\sqrt{k^{2} +\frac{2k}{n}} ;\ s^{*} =-k-\sqrt{k^{2} +\frac{2k}{n}} ;\ m_{0}( r) =n(\bmod( r,\ k) +1) \ =n,2n,...kn\)

\(\displaystyle \begin{array}{{>{\displaystyle}l}} m_{n}\left(\sqrt{19}\right) =-\frac{1}{2}\\ +\left( 57799-13260\sqrt{19}\right)^{\left\lfloor \frac{n}{10}\right\rfloor }\left[ -\frac{\left\lceil m_{0}( n)\left( -4+\sqrt{19}\right)\right\rceil }{2\sqrt{19}} +\left(\frac{1}{2} -\frac{2}{\sqrt{19}}\right)\Bigl[ m_{0}( n) +1/2\Bigr]\right]\\ +\left( 57799+13260\sqrt{19}\right)^{\left\lfloor \frac{n}{10}\right\rfloor }\left[\frac{\left\lceil m_{0}( n)\left( -4+\sqrt{19}\right)\right\rceil }{2\sqrt{19}} +\left(\frac{1}{2} +\frac{2}{\sqrt{19}}\right)\Bigl[ m_{0}( n) +1/2\Bigr]\right]\\ \\ m_{0}( n) =\begin{cases} 2 & n=0\bmod 10\\ 5 & n=1\bmod 10\\ 19 & n=2\bmod 10\\ 345 & n=3\bmod 10\\ 1362 & n=4\bmod 10\\ 2379 & n=5\bmod 10\\ 15639 & n=6\bmod 10\\ 28899 & n=7\bmod 10\\ 42159 & n=8\bmod 10\\ 55149 & n=9\bmod 10 \end{cases} \end{array}\)

Unorganized ideas (to be cleaned up!)

Check out Joseph Monzo's Tonalsoft Encyclopedia here!

Or check out the xenharmonics wiki!

Let \( f( x) =\left(\frac{3}{2}\right)^{x} 2^{-\lfloor x\log_{2}( 3/2)\rfloor }\).

Then:

• \( a_0 = 1 \)
• \( a_n \) is the smallest natural number such that \( f(a_n) < f(a_{n-1}) \)

Some values:

n	0	1	2	3	4	5
\(a_n\)	1	2	7	12	53	359

These correspond to \(a_n\)-ET (equal temperament) tunings that are each successively closer to a Pythagorean tuning.

Also see

• Topological Music Theory a paper by Dmitri Tymoczko

• A Collection of Algebraic Identites - (archived Mar 26 2023)
• Digital Library of Mathematical Functions
• Topological Counterexamples
• The L-functions and modular forms database
• Online Encyclopedia of Integer Sequences
• Online Math Editor
• Desmos - Online Graphing Calculator
• Graphical Linear Algebra
• Tilings of Riemann Surfaces
• Math Paper advice