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@@ -375,12 +375,15 @@ <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 + wheel, was first 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 + Lambdoma instrument for sound healing purposes. + </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 @@ -388,7 +391,7 @@ <i>Introduction to Arithmetic</i> by <a href="https://archive.org/details/nicomachus-introduction-to-arithmetic/page/191/mode/1up" - >Nichomachus of Gerasa</a + >Nicomachus of Gerasa</a > (ca. 100 BCE). They suggest it has been rediscovered several times, including in the 19th century by Albert von Thimus, who @@ -407,7 +410,7 @@ >, <i>The Lambdoma Matrix and Harmonic Intervals</i> (1999). </p> - <h2>mathematics and perception</h2> + <h2>the mathematics of perception</h2> <p> With the root, fifth, and fourth in the top-left corner, the Lambdoma @@ -417,16 +420,16 @@ 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. + makes them beat against each other. This interval is the "syntonic + comma" which is averaged out in various keyboard tuning systems. </p> <p> Tuning systems must weigh the harmony of pure intervals against musical versatility. A Pythagorean tuning system made from pure - fractions will include many different "fifths" and "thirds" between - the other notes, which makes each musical key sound highly - distinctive. Some intervals are extremely dissonant and harsh, + fractions will include many different "fifths" and "thirds" at + different points in the scale, which makes each musical key sound + highly distinctive. Some intervals are extremely dissonant and harsh, rendering certain keys unplayable. </p> @@ -437,17 +440,17 @@ exist anywhere in the Lambdoma, no matter how far out you go. Equal-tempered semitones are separated by a ratio of the 12th root of 2 (1:<super>12</super>√2). Irrationals are real numbers, and these can - only be approximated by whole-numbered fractions. + only be approximated by fractions made up of natural numbers. </p> <p> 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. + denominator, similar to the Lambdoma. 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> |