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@@ -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> |