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           >, <i>The Lambdoma Matrix and Harmonic Intervals</i> (1999).
         </p>
 
-        <h2>the mathematics of perception</h2>
+        <h2>the music of whole numbers</h2>
 
         <p>
-          With the root, fifth, fourth, and octave in the top-left corner, the
-          Lambdoma shows how the 3:2 proportion is essential to musical
-          perception.
+          The smallest whole numbers have the greatest significance to our
+          understanding of music. With the root, fifth, fourth, and octave in
+          the top-left corner, the Lambdoma shows how the 3:2 proportion is the
+          foundation of tonality. These frequencies sound quite similar to each
+          other, which as any musician knows, can sometimes fool the ear.
         </p>
 
         <p>
+          The next prime number interval, 5:4, is a just major third, and its
+          inverse, 4:5 is a minor third. Thus the overtone series sounds
+          "major", and the undertone series sounds "minor". The next prime
+          number out, 7, expresses a "Lydian" tonality.
+        </p>
+
+        <p>
+          At this point, we have just enough notes to form a fairly melodious
+          just-intonated scale, which you can play using your computer keyboard.
+          Within a just-intonated tuning system, each scale can sound highly
+          distinctive, since the notes are not distributed evenly across the
+          octave.
+        </p>
+
+        <h2>tuning systems</h2>
+
+        <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 <i>syntonic comma</i>, and tuning systems
-          try to correct for it in various ways.
+          <u class="mode" name="pythagorean">Pythagorean</u> setting, where each
+          ratio is a power of 2 or 3. Powers of 3 move by fifths, and powers of
+          2 by octaves. Using this principle, we can transpose any note back
+          down into the same octave and create a scale.
         </p>
 
         <p>
-          Within a Pythagorean tuning system, each scale can sound highly
-          distinctive, since the notes will not be distributed evenly across the
-          octave. In a sense, 12-tone equal temperament bends all of the notes
-          to make them evenly spaced. Fifths in equal temperament are all nearly
-          pure, 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.)
+          On this instrument, 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 of the same brightness,
+          and play both at once. These two frequencies are not quite the same,
+          and they will audibly vibrate or "beat" against each other.
+        </p>
+
+        <p>
+          The interval between this pseudo-unison is known as the
+          <i>ditonic comma</i>, and it is one of various "commas" that tuning
+          systems adjust for. Another important comma is the
+          <i>syntonic comma</i>, which is the difference between a just major
+          third (5:4) and the closest equivalent achieved from stacking fifths.
+        </p>
+
+        <p>
+          In a sense, 12-tone equal temperament "bends" all of the notes to make
+          the intervals evenly spaced. Fifths in equal temperament are all
+          nearly in tune, making it easy to modulate between keys. By
+          comparison, thirds are quite out of tune compared to a just major
+          third (5:4). An alternative is meantone temperament, which favors
+          harmonious major thirds over out-of-tune fifths, by distributing the
+          comma differently.
+        </p>
+
+        <h2>tone and transfinite sets</h2>
+
+        <p>
+          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
+          whole-numbered fractions.
         </p>
 
         <p>