From cd250e8ac29148941b33c61728d50d746288f680 Mon Sep 17 00:00:00 2001 From: julian laplace Date: Wed, 23 Jul 2025 14:04:38 +0200 Subject: copy --- index.html | 83 +++++++++++++++++++++++++++++++++++++++++++++----------------- 1 file changed, 60 insertions(+), 23 deletions(-) diff --git a/index.html b/index.html index 6837018..6212d61 100644 --- a/index.html +++ b/index.html @@ -623,39 +623,76 @@ >, The Lambdoma Matrix and Harmonic Intervals (1999).

-

the mathematics of perception

+

the music of whole numbers

- 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.

+

+ 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. +

+ +

+ 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. +

+ +

tuning systems

+

The musical Circle of Fifths, created by repeatedly multiplying a frequency by 3:2, can be studied in more detail in this program's - Pythagorean 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 syntonic comma, and tuning systems - try to correct for it in various ways. + Pythagorean 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.

- 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 - (1:12√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. +

+ +

+ The interval between this pseudo-unison is known as the + ditonic comma, and it is one of various "commas" that tuning + systems adjust for. Another important comma is the + syntonic comma, which is the difference between a just major + third (5:4) and the closest equivalent achieved from stacking fifths. +

+ +

+ 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. +

+ +

tone and transfinite sets

+ +

+ 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 (1:12√2), which can only be approximated by + whole-numbered fractions.

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