Precise Temperament Tuning: A5 = 432.08121Hz
In July 2020, I developed a new, purely Geometric Tuning approach called 'PRECISE TEMPERAMENT TUNING' (A5 = 432.081216hz). This tuning was achieved using Just Tuning (Pythagorean) as the base (Perfect Fifth 3/2) and only ONE (+0.01) modification to the Major Third interval (1.25 (5/4)), adjusting it to 1.26 (2^1/3). The new scale achieves mathematical/geometric symmetry in the intervals of the Major 2nd (9/8), the 4th (4/3) and the Perfect 5th (3/2). Additionally, all hertz values' digital roots sum to 9 (matching the phenomenon inherent to all Regular Geometric Polytopes). This is also true when considering the decimal extension 'tails' ('mantissas') that form blockchain like recognizable relationships to Pythagorean Just Tuning notes as well as the overtone series.
Despite its direct correspondence with Pythagorean Just Tuning intervals which normally don't sound very good because the Major Third is incapable of doubling an octave (eg starting with for example 9hz using the 5/4 Major Third relationship will yield 9hz x 1.25^3 = 17.58hz NOT the full 18hz required to double the octave from 9hz. When we replace 1.25 with 1.26 this gap is closed (9hz x 1.26^3 = 18hz) to create the geometric symmetry with the Perfect Fifth (which can), due to this small 0.01 ratio addition this music has a beautiful quality, texture and meaningful contrast, particularly in melancholy (emotive) pieces like Moonlight Sonata.
All this is achieved without the need for the Pythagorean Comma or other adjustments for versatility required when venturing into various key changes/transitions.
Background music: Moonlight Sonata by Beethoven tuned to Precise Temperament Tuning (432.081216hz) played by Jason Martineau
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Robert Edward Grant
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