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|
<!DOCTYPE html>
<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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grid * {
position: absolute;
cursor: pointer;
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grid > div {
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grid > div > div {
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grid > div > div:nth-child(1) {
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grid > div > div:nth-child(2) {
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super {
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#modus {
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#modus .bandpass,
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.mode {
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/**
* Root select
*/
.root-select {
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.root-select .row {
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margin-right: 0.25rem;
margin-left: 0.25rem;
margin-bottom: 0.5rem;
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.root-select .row.klavier {
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Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji",
"Segoe UI Symbol";
color: white;
font-size: 18px;
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border: 0;
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height: 50px;
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transition: opacity 0.1s;
opacity: 0.9;
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box-shadow: 0 1px 2px rgba(0, 0, 0, 0.75);
}
.root-select button:hover {
opacity: 1;
}
.root-select button.ok {
width: 78px;
}
.klavier button {
text-shadow: 0 0 1px #000, 0 0 1px #000;
}
.fraction {
display: inline-flex;
flex-direction: column;
font-size: 12px;
margin-left: 0.25rem;
}
.fraction span:first-child {
border-bottom: 1px solid;
}
@media (max-width: 800px) {
.root-select {
align-items: flex-start;
background: rgba(0, 0, 0, 0.5);
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.root-select > div {
background: rgba(0, 0, 0, 0);
margin: 0;
width: 100%;
align-items: center;
justify-content: center;
border-radius: 0;
}
.root-select .row.form-row {
flex-direction: column;
}
}
</style>
</head>
<body>
<div id="message"></div>
<div id="help">
<div class="content">
<h1>Triangle / Lambdoma</h1>
<p>
This instrument uses simple ratios 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>
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 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 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, 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> 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 stacking fifths does not bring you
back to the starting note. Find two far-apart red notes and play both
at once. These two frequencies are not quite the same, and they will
audibly vibrate or "beat" against each other. The interval between
these notes is known as the "syntonic comma", and tuning systems try
to correct for it in various ways.
</p>
<p>
Within a Pythagorean tuning system, each scale can sound highly
distinctive, since the notes will not be distributed evenly within the
octave. In a sense, 12-tone equal temperament "bends" all of the notes
so they are evenly spaced. Fifths in equal temperament are not quite
pure but close enough, making it easy to modulate between keys. By
comparison, thirds are quite out of tune compared to a pure ratio
(5:4). Equal temperament invokes irrational numbers, creating
in-between intervals which do not exist on the Lambdoma. (Notes in
equal temperament are separated by an irrational ratio of 1 to the
12th root of 2 (1:<super>12</super>√2), which can only be approximated
by pure 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
<i>countably</i>
infinite. Consider that while there are infinitely many natural
numbers, we may count our way up to each one, starting from 1.
Similarly, we can count the cells in a Lambdoma in a snake-like
pattern starting from 1:1, and thus map all of 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 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>
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