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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;
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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 simple 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 ...), while
the <b>undertone series</b> (1/1, 1/2, 1/3, ...) is shown by the
columns. The resulting intervals are instantly musically meaningful,
though they arise from simple ratios.
</p>
<p>
<b>Brightness</b> indicates octave. <b>Color</b> indicates position in
the octave, with red being the root or unison interval 1/1.
</p>
<p></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>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
understand the concept more deeply.
</p>
<p>
I later learned that I had recreated the
<a href="https://www.lambdoma.com/">Lambdoma</a> as described by
<a href="https://www.lambdoma.com/">Barbara Hero</a>. Hero made an
electronic lambdoma instrument for sound healing purposes. She traces
the Lambdoma back to 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 CE), and suggests it was rediscovered 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
read in Hero's
<a
href="https://lambdoma.com/pdfs/the-lambdoma-matrix-and-harmonic-intervals.pdf"
target="_blank"
>article</a
>, <i>The Lambdoma Matrix and Harmonic Intervals</i> (1999). In the
Lambdoma, Hero also sees the image of Georg Cantor's transfinite set
of rational numbers ℚ, which Cantor showed to be countably infinite
through use of a Cartesian plot (versus the uncountable continuity of
real numbers ℝ).
</p>
<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 "Pythagorean" mode.
Similar notes can be found by color and compared, and one can easily
hear how stacked fifths overshoot the octave by finding the next red
note.
</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>
<script src="bundle.js"></script>
</html>
|
