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@@ -378,7 +378,7 @@ Lambdoma, named for its resemblance to the Greek letter Lambda Λ. The synergy of color and tone, linking the octave to the color wheel, seemed intuitive, and revealed a beautiful pattern. This pattern had - earlier been uncovered by artist and sound practitioner + previously been uncovered by artist and sound practitioner <a href="https://www.lambdoma.com/">Barbara Hero</a>, who built an 8x8 electronic Lambdoma instrument for sound healing purposes, using the same pattern of colors. @@ -430,23 +430,11 @@ <p> With the root, fifth, fourth, and octave in the top-left corner, the - Lambdoma shows how the 3:2 proportion is essential to human - perception. + Lambdoma shows how the 3:2 proportion is basic to musical perception. </p> <p> - Mapping colors logarithmically to this wheel of fractions between 1 - and 2, with red at the octave (1:1), it naturally follows that the - fourth (4:3) is green, and the fifth (3:2) is blue. These ratios seem - to correspond to the photoreceptors in the human retina, which are - sensitive to wavelengths of light in three different ranges: long, - medium, and short. These ranges are perceived as "red, green, and - blue" in the brain, yet our mind's eye sees a continuous cycle of - color that loops back on itself. - </p> - - <p> - The musical circle of fifths, derived from repeatedly stacking the 3:2 + The musical Circle of Fifths, derived from repeatedly stacking the 3:2 proportion, can be studied in more detail in this program's <u id="pythagorean">Pythagorean</u> scale mode, where each ratio is a power of 2 or 3. Similar notes can be found by color and compared. One @@ -457,20 +445,20 @@ </p> <p> - Tuning systems must weigh the purity of thirds and fifths. A - Pythagorean tuning system made from pure fractions will include many - different "fifths" and "thirds" at different points in the scale, - especially when using basic ratios to determine a 12-tone scale. Each - musical key sounds highly distinctive, and some intervals may be - considered dissonant or harsh. + Tuning systems must weigh the purity of thirds and fifths. A just + tuning system made from pure fractions might use as its basis the + difference between 3:2 and 4:3, which constitutes a semitone. Such a + scale implies different ratios between "fifths" and "thirds" at + different points in the scale. Each mode can sound highly distinctive, + and some intervals may be considered dissonant or harsh. </p> <p> In a sense, 12-tone equal temperament "bends" all of the notes such that the fifths are more consisent, making it easier to modulate - between keys, though other intervals (like thirds) are far from a pure - interval. This process invokes irrational numbers, and creates - in-between intervals which do not exist anywhere in the Lambdoma, no + between keys, though other intervals (like thirds) are quite different + from a interval. This process invokes irrational numbers, and creates + in-between intervals which do not exist anywhere on 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 fractions made up @@ -482,8 +470,8 @@ transfinite set of rational numbers ℚ, which Cantor proved countably infinite by arranging fractions along two axes by numerator and 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 + countable infinity of rationals by considering that, though there are + infinitely many rational numbers, in between any two there lies an uncountable continuity of real numbers in ℝ. </p> @@ -499,8 +487,12 @@ <a href="https://www.nyz.recycled-plastics.net/" target="_blank" >Dave Noyze</a > - for telling me about Barbara Hero. Thanks to - <a href="https://hems.io/" target="_blank">Hems</a> for the support! + for telling me about + <a href="https://www.lambdoma.com/" target="_blank">Barbara Hero</a>. + Gradient algorithm via + <a href="https://iquilezles.org/articles/palettes/">Inigo Quizeles</a + >. Thanks to <a href="https://hems.io/" target="_blank">Hems</a> for + the support! </p> <p>Jules LaPlace / <a href="/">asdf.us</a> / 2018-2025</p> |