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<html>
<head>
<title>Triangle / Lambdoma</title>
<meta charset="utf-8" />
<meta
name="viewport"
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</head>
<body>
<div id="message"></div>
<div id="help">
<div class="content">
<h1>Triangle / Lambdoma</h1>
<p>
This instrument uses basic fractions to make a wide palette of just
intonation intervals available all at once.
</p>
<p>
The rows display the <b>overtone series</b> (1/1, 2/1, 3/1 ...). The
columns multiply out the <b>undertone series</b> (1/1, 1/2, 1/3, ...).
The resulting intervals are instantly musically meaningful, though
they arise from simple 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>
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 this help</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>natural</td>
<td>Natural numbers: 1, 2, 3 ...</td>
</tr>
<tr>
<td>undertone</td>
<td>Subharmonic intervals under the line 1/1</td>
</tr>
<tr>
<td>overtone</td>
<td>Harmonic intervals above the line 1/1</td>
</tr>
<tr>
<td>primes</td>
<td>Prime numbers only (most dissonant)</td>
</tr>
<tr>
<td>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>Collatz</td>
<td>
<a
href="https://en.wikipedia.org/wiki/Collatz_conjecture"
target="_blank"
>Hailstone numbers</a
>
of Lothar Collatz
</td>
</tr>
<tr>
<td>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 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 fractions, 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 "lamboid diagram" or
Lambdoma, named for its resemblance to the Greek letter Lambda Λ. The
synergy of color and tone, linking the octave to the color wheel,
seemed intuitive, and revealed a beautiful pattern. 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 mathematics of perception</h2>
<p>
With the root, fifth, fourth, and octave in the top-left corner, the
Lambdoma shows how the 3:2 proportion is basic to musical perception.
</p>
<p>
The musical Circle of Fifths, derived from repeatedly stacking the 3:2
proportion, can be studied in more detail in this program's
<u id="pythagorean">Pythagorean</u> scale mode, where each ratio is a
power of 2 or 3. Similar notes can be found by color and compared. One
can easily hear how stacked fifths overshoot the octave by finding two
far-apart red notes and playing both at once, which makes them beat
against each other. This interval is the "syntonic comma" which is
averaged out in various keyboard tuning systems.
</p>
<p>
Tuning systems must weigh the purity of thirds and fifths. A just
tuning system made from pure fractions might use as its basis the
difference between 3:2 and 4:3, which constitutes a semitone. Such a
scale implies different ratios between "fifths" and "thirds" at
different points in the scale. Each mode can sound highly distinctive,
and some intervals may be considered dissonant or harsh.
</p>
<p>
In a sense, 12-tone equal temperament "bends" all of the notes such
that the fifths are more consisent, making it easier to modulate
between keys, though other intervals (like thirds) are quite different
from a interval. This process invokes irrational numbers, and creates
in-between intervals which do not exist anywhere on the Lambdoma, no
matter how far out you go. Equal-tempered semitones are separated by a
ratio of the 12th root of 2 (1:<super>12</super>√2). Irrationals are
real numbers, and these can only be approximated by fractions made up
of natural numbers.
</p>
<p>
In the Lambdoma, Barbara Hero also sees the image of Georg Cantor's
transfinite set of rational numbers ℚ, which Cantor proved countably
infinite by arranging fractions along two axes by numerator and
denominator, similar to the Lambdoma. One may easily grasp this
countable infinity of rationals by considering that, though there are
infinitely many rational numbers, in between any two there lies an
uncountable continuity of 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
>. 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>.
Gradient algorithm via
<a href="https://iquilezles.org/articles/palettes/">Inigo Quizeles</a
>. Thanks to <a href="https://hems.io/" target="_blank">Hems</a> for
the support!
</p>
<p>Jules LaPlace / <a href="/">asdf.us</a> / 2018-2025</p>
<div class="close">✗</div>
</div>
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