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<!DOCTYPE html>
<html>
<head>
<title>Triangle / Lambdoma</title>
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</head>
<body>
<div id="message"></div>
<div id="help">
<div class="content">
<h1>Triangle / Lambdoma</h1>
<p>
This page is a musical instrument made up of a grid of just intonation
intervals, all based on a root frequency.
</p>
<p>
The rows display the <b>overtone series</b> (1/1, 2/1, 3/1 ...). The
columns follow its inverse, the <b>undertone series</b> (1/1, 1/2,
1/3, ...). Multiplying everything out, the resulting intervals contain
an inherent music arising from simple, whole-number ratios.
</p>
<p>
<b>Color</b> indicates position in the octave, with red being the root
or unison interval 1/1. <b>Brightness</b> indicates octave, with white
and black tending toward the extremes of human hearing.
</p>
<ul>
<li>
The default instrument is a Hokema sansula, a type of kalimba.
</li>
<li>
Right-click notes to turn on sine waves matching the interval.
</li>
<li>
Resonator mode puts white noise through tuned bandpass filters.
</li>
<li>
Drag-and-drop samples into the window to play with your own sounds.
</li>
</ul>
<h2>keyboard shortcuts</h2>
<tableContainer
><table>
<tr>
<td>ESC</td>
<td>Stop all sound</td>
</tr>
<tr>
<td>ESC ESC</td>
<td>Return to home position</td>
</tr>
<tr>
<td>?</td>
<td>Show this help</td>
</tr>
<tr>
<td>+ -</td>
<td>Change scale</td>
</tr>
<tr>
<td>up<br />down<br />left<br />right</td>
<td>Scroll the grid</td>
</tr>
<tr>
<td>0-9 a-z</td>
<td>Keyboard mapped to the top-left 8x8 grid, sorted by pitch</td>
</tr>
<tr>
<td>~</td>
<td>Toggle sine and resonator mode</td>
</tr>
<tr>
<td>\</td>
<td>Detect MIDI device (listening on channel 1)</td>
</tr>
<tr>
<td>⌘ +<br />⌘ -</td>
<td>Change scale root by +/- 1 hz</td>
</tr>
<tr>
<td>⌘⇧ +<br />⌘⇧ -</td>
<td>Change scale root by +/- 10 hz</td>
</tr>
<tr>
<td>⌘ up<br />⌘⇧ up<br />⌘⇧⌃ up</td>
<td>Change pitch of sampler by +10 / +1 / -0.1 hz</td>
</tr>
<tr>
<td>⌘ down<br />⌘⇧ down<br />⌘⇧⌃ down</td>
<td>Change pitch of sampler by -10 / -1 / -0.1 hz</td>
</tr>
</table>
</tableContainer>
<h2>scales</h2>
<p>Alternate scales are accessed by pressing the +/- keys.</p>
<tableContainer>
<table>
<tr>
<td class="mode" name="natural">natural</td>
<td>Natural numbers: 1, 2, 3 ...</td>
</tr>
<tr>
<td class="mode" name="undertone">undertone</td>
<td>Subharmonic intervals under the line 1/1</td>
</tr>
<tr>
<td class="mode" name="overtone">overtone</td>
<td>Harmonic intervals above the line 1/1</td>
</tr>
<tr>
<td class="mode" name="primes">primes</td>
<td>Prime numbers only (most dissonant)</td>
</tr>
<tr>
<td class="mode" name="arithmetic">arithmetic</td>
<td>Multiply all cells by an interval rather than scrolling</td>
</tr>
<!--
<tr>
<td>equal</td>
<td>Equal-tempered intervals based on 1:<super>n</super>√2</td>
</tr>
-->
<tr>
<td class="mode" name="hyperbolic">hyperbolic</td>
<td>
Change stride rather than scrolling. Denominator magnifies along
the 1/1 line, numerator emphasizes the hyperbolic extremes.
</td>
</tr>
<tr>
<td class="mode" name="collatz">Collatz</td>
<td>
<a
href="https://en.wikipedia.org/wiki/Collatz_conjecture"
target="_blank"
>Hailstone numbers</a
>
of Lothar Collatz
</td>
</tr>
<tr>
<td class="mode" name="pythagorean">Pythagorean</td>
<td>
<a
href="https://en.wikipedia.org/wiki/Pythagorean_interval"
target="_blank"
>Pythagorean intervals</a
>
where each ratio is a power of 2<super>n</super> or 3<super
>n</super
>
</td>
</tr>
</table>
</tableContainer>
<h2>about this page</h2>
<p>
This webpage was inspired in part by
<a href="https://www.youtube.com/watch?v=4pdSYkI86go"
>Peter Neubäcker</a
>, inventor of the Melodyne software. In the short biographical
documentary <i>Wie klingt ein Stein?</i> (<i
>What does a stone sound like?</i
>), Neubäcker describes the basic principles of harmonic intervals. He
first demonstrates how one plays harmonics on a monochord. He then
shows it next to a grid of whole-number ratios, and demonstrates how
one can use these ratios to find specific intervals. I had never seen
just intonation demonstrated so elegantly, so I made this page to
explore the concept.
</p>
<p>
I later learned that I had constructed the "lambdoid diagram" or
<b>Lambdoma</b>, named for its resemblance to the Greek letter
<i>Lambda</i> Λ. The synergy of color and tone, linking the octave to
the color wheel, seemed intuitive, and revealed a beautiful pattern,
both visual and musical. This pattern had previously been uncovered by
artist and sound practitioner
<a href="https://www.lambdoma.com/">Barbara Hero</a>, who built an 8x8
electronic Lambdoma instrument for sound healing purposes, using the
same pattern of colors.
</p>
<p>
Hero learned of the Lambdoma from
<i>Tone: A Study in Musical Acoustics</i> (1968) by
<a
href="https://archive.org/details/tonestudyinmusic0000leva/page/n5/mode/2up"
>Levarie and Levy</a
>, who trace the Lambdoma back to Pythagoras (ca. 500 BCE) by way of
the <i>Introduction to Arithmetic</i> by
<a
href="https://archive.org/details/nicomachus-introduction-to-arithmetic/page/191/mode/1up"
>Nicomachus of Gerasa</a
>
(ca. 100 BCE) and the <i>Theologumena arithmeticae</i> of
<a
href="https://archive.org/details/astius-theologumena-arithmeticae-gr-1817/page/159/mode/1up"
>Iamblichus</a
>
(ca. 300 CE). The Lambdoma is also mentioned by
<a
href="https://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A2008.01.0385%3Astephpage%3D1027b"
>Plutarch</a
>
(ca. 100 CE) in his commentary on Plato's <i>Timaeus</i>. It was
<a
href="https://archive.org/details/bsb10527783/page/137/mode/1up"
target="_blank"
>depicted</a
>
in the 19th century by Albert von Thimus in the neo-Pythagorean
treatise
<i>Die harmonikale Symbolik des Alterthums</i> (1876) which connects
musical intervals to other harmonic relationships in nature. The
Lambdoma was also used by mathematician Georg Cantor in his theory of
transfinite sets (see below). More information can be gleaned from
Hero's
<a
href="https://lambdoma.com/pdfs/the-lambdoma-matrix-and-harmonic-intervals.pdf"
target="_blank"
>paper</a
>, <i>The Lambdoma Matrix and Harmonic Intervals</i> (1999).
</p>
<h2>the music of whole numbers</h2>
<p>
The smallest whole numbers have the greatest significance to our
understanding of music. With the root, fifth, fourth, and octave in
the top-left corner, the Lambdoma shows how the 3:2 proportion is the
foundation of tonality. These frequencies sound quite similar to each
other, which as any musician knows, can sometimes fool the ear.
</p>
<p>
The next prime number interval, 5:4, is a just major third, and its
inverse, 4:5 is a minor third. Thus the overtone series sounds
"major", and the undertone series sounds "minor". The next prime
number out, 7, extends the minor to diminished, and major to dominant.
</p>
<p>
At this point, we have just enough notes to form a fairly melodious
just-intonated scale, which you can play using your computer keyboard.
Within a just-intonated tuning system, each interval can sound highly
distinctive, since the notes are not distributed evenly across the
octave.
</p>
<h2>tuning systems</h2>
<p>
The musical Circle of Fifths, created by repeatedly multiplying a
frequency by 3:2, can be studied in more detail in this program's
<u class="mode" name="pythagorean">Pythagorean</u> setting, where each
ratio is a power of 2 or 3. Powers of 3 move by fifths, and powers of
2 by octaves. Using this principle, we can transpose any note back
down into the same octave and create a scale.
</p>
<p>
On this instrument, similar notes can be found by color and compared.
One can easily hear how stacking fifths does not bring you back to the
starting note: find two far-apart red notes of the same brightness,
and play both at once. These two frequencies are not quite the same,
and they will audibly vibrate or "beat" against each other.
</p>
<p>
The interval between this pseudo-unison is known as the
<i>ditonic comma</i>, and it is one of various "commas" that tuning
systems adjust for. Another important comma is the
<i>syntonic comma</i>, which is the difference between a just major
third (5:4) and the closest equivalent achieved from stacking fifths.
</p>
<p>
In a sense, 12-tone equal temperament "bends" all of the notes to make
the intervals evenly spaced. Fifths in equal temperament are all
nearly in tune, making it easy to modulate between keys. By
comparison, thirds are quite out of tune compared to a just major
third (5:4). An alternative is meantone temperament, which favors
harmonious major thirds over out-of-tune fifths, by distributing the
comma differently.
</p>
<h2>tone and transfinite sets</h2>
<p>
Intervals in just intonation are rational numbers, composed of a
numerator and denominator that are both whole numbers. Notes in equal
temperament, however, are separated by a semitone of 1 to the 12th
root of 2 (1:<super>12</super>√2), an irrational number, and can only
be approximated by rationals. While all rational numbers are real
numbers, not all real numbers are rational, and true equal-tempered
invervals are only <i>approximated</i> by the intervals of the
Lambdoma.
</p>
<p>
In the Lambdoma, Barbara Hero also sees the image of Georg Cantor's
transfinite set of rational numbers ℚ, which Cantor proved
<i>countably</i>
infinite. Consider that although there are infinitely many natural
numbers, each of these numbers is by definition finite, and we may
count up to it by starting from 1 and adding 1 repeatedly. Similarly,
we can count the cells in a Lambdoma by starting from 1:1 and moving
outward diagonally in a snake-like pattern, thus mapping the rationals
to the natural numbers. Though there are infinitely many rational
numbers, by their nature they are discrete, countable, and not
completely dense. Between any two rational numbers, there lies an
uncountable continuity of irrational real numbers in ℝ.
</p>
<h2>thank you!</h2>
<p>
Sansula samples by Freesound user
<a
href="https://freesound.org/people/cabled_mess/packs/21410/"
target="_blank"
>cabled_mess</a
>. Gradient algorithm via
<a href="https://iquilezles.org/articles/palettes/">Inigo Quizeles</a
>. Thanks to
<a href="https://www.nyz.recycled-plastics.net/" target="_blank"
>Dave Noyze</a
>
for telling me about
<a href="https://www.lambdoma.com/" target="_blank">Barbara Hero</a>.
Thanks to <a href="https://hems.io/" target="_blank">Hems</a> for the
support!
</p>
<p>Jules LaPlace / <a href="/">asdf.us</a> / 2017-2025</p>
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