From f68d55431f3c401523369ae504c51cb83446c179 Mon Sep 17 00:00:00 2001
From: julian laplace <julescarbon@gmail.com>
Date: Tue, 8 Jul 2025 13:47:42 +0200
Subject: docs

---
 index.html | 43 ++++++++++++++++++++++++++-----------------
 1 file changed, 26 insertions(+), 17 deletions(-)

(limited to 'index.html')

diff --git a/index.html b/index.html
index 97bc708..e61f53e 100644
--- a/index.html
+++ b/index.html
@@ -392,31 +392,40 @@
           <a
             href="https://lambdoma.com/pdfs/the-lambdoma-matrix-and-harmonic-intervals.pdf"
             target="_blank"
-            >article</a
+            >paper</a
           >, <i>The Lambdoma Matrix and Harmonic Intervals</i> (1999).
         </p>
 
+        <h2>mathematics and perception</h2>
+
         <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
+          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>
+          12-tone equal temperament is based not in harmonics, but in irrational
+          numbers. Equal-tempered semitones are separated by a ratio of the 12th
+          root of 2 (1:<super>12</super>√2). Equal-tempered intervals like this
+          do not exist in the Lambdoma - irrationals are real numbers, and these
           can only be approximated by rational numbers.
         </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.
+          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>
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