From e46ca2bff72fed5c309ef7c5d04e97ae2eb7f2b6 Mon Sep 17 00:00:00 2001
From: julian laplace <julescarbon@gmail.com>
Date: Tue, 8 Jul 2025 13:28:31 +0200
Subject: docs

---
 index.html | 57 ++++++++++++++++++++++++++++++++++++++++-----------------
 1 file changed, 40 insertions(+), 17 deletions(-)

(limited to 'index.html')

diff --git a/index.html b/index.html
index ec5a70e..97bc708 100644
--- a/index.html
+++ b/index.html
@@ -220,8 +220,9 @@
           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.
+          <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>
         <p></p>
         <ul>
@@ -342,6 +343,7 @@
         </tableContainer>
 
         <h2>about this page</h2>
+
         <p>
           This webpage was inspired by
           <a href="https://www.youtube.com/watch?v=4pdSYkI86go"
@@ -354,36 +356,57 @@
           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.
+          explore the concept interactively.
         </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
+          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 CE), and suggests it was rediscovered by Albert von Thimus
-          who
+          (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
-          read in Hero's
+          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"
             >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).
+          >, <i>The Lambdoma Matrix and Harmonic Intervals</i> (1999).
+        </p>
+
+        <p>
+          In the Lambdoma, Hero also sees the image of Georg Cantor's
+          transfinite set of rational numbers ℚ, which Cantor proved countably
+          infinite by arranging fractions in the form of a matrix. 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. For example, the
+          common tuning system of 12-tone equal temparament is based on an
+          irrational interval of the 12th root of 2 (<super>12</super>√2).
+          Equal-tempered intervals like this do not exist in the Lambdoma - they
+          can only be approximated by rational numbers.
         </p>
 
         <p>
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