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<html>
<head>
<title>Triangle / Lambdoma</title>
<meta charset="utf-8" />
<meta
name="viewport"
content="width=device-width, initial-scale=1.0, maximum-scale=1.0, user-scalable=no"
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color: white;
text-shadow: 0 0 1px #000;
transition: background-color 100ms;
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grid {
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grid * {
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font-size: 18px;
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</head>
<body>
<div id="message"></div>
<div id="help">
<div class="content">
<h1>Triangle / Lambdoma</h1>
<p>
This instrument uses 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 ratios, and demonstrates how
one can use these intervals to find specific ratios. I had never seen
just intonation demonstrated so elegantly, so I made this page to
explore the concept interactively.
</p>
<p>
I later learned that I had rediscovered the
<a href="https://www.lambdoma.com/">Lambdoma</a>, so called by the
ancient Greeks for its resemblance to the letter Lambda. The synergy
of color and sound in the Lambdoma, linking the octave to the color
wheel, had been studied in depth by artist and sound practitioner
<a href="https://www.lambdoma.com/barbara-hero.html">Barbara Hero</a>.
Hero made the Lambdoma her life's work, and built an 8x8 electronic
Lambdoma instrument for sound healing purposes. Hero herself 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
>, which traces the Lambdoma back to Pythagoras (ca. 500 BCE) via the
<i>Introduction to Arithmetic</i> by
<a
href="https://archive.org/details/nicomachus-introduction-to-arithmetic/page/191/mode/1up"
>Nichomachus of Gerasa</a
>
(ca. 100 BCE). They suggest it has been rediscovered several times,
including in the 19th century by Albert von Thimus, who
<a
href="https://archive.org/details/bsb10527783/page/137/mode/1up"
target="_blank"
>depicts it</a
>
in
<i>Die harmonikale Symbolik des Alterthums</i> (1876). More 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>mathematics and perception</h2>
<p>
With the root, fifth, and fourth in the top-left corner, the Lambdoma
shows how the 3:2 proportion is essential to the perception of
consonance. The musical circle of fifths, derived from these simple
proportions, can be studied in more detail in this program's
Pythagorean scale mode. 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 is the "comma" which is
averaged out in various keyboard tuning systems.
</p>
<p>
Tuning systems must weigh the harmony of pure intervals against
musical versatility. A Pythagorean tuning system made from pure
fractions will include many different "fifths" and "thirds" between
the other notes, which makes each musical key sound highly
distinctive. Some intervals are extremely dissonant and harsh,
rendering certain keys unplayable.
</p>
<p>
12-tone equal temperament enables musicians to play in any key by
bending all of the notes slightly out of tune. This process invokes
irrational numbers, and creates in-between intervals which do not
exist anywhere in 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 whole-numbered fractions.
</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. One may easily grasp this countable infinity of rationals
by considering that, while there are infinitely many fractions, 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 Barbara Hero. 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>
<br />
</div>
<div id="help-button"><span>?</span></div>
</body>
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|
