Update on Linus Tuning - the "QQS tuning" (Quantum-Pure, Quasi-ET, Stretched-Octave)

All,

I have replaced my last video. The last one had two problems, 1. I made a mistake (sorry, but I always do), 2. The violin strings
drifted, cannot maintain the tuning through the duration of the piece. I have to neglect the upper two
strings and just mind arriving at the interval in the end. The interval in the double stop is as I first
worked out to be, 9/8 * 9/8 * 81/40 (x 293.66 Hz = 752.63 Hz. The strings may have drifted, though.
https://youtu.be/4G5WNdXMOvM https://youtu.be/4G5WNdXMOvM

I will later make another video to demo how "Pure" intervals are not more pure than the "Quantise" ones

Mendelssohn violin piece played with violin tuned (Hz):
E : 669.00; A : 446.00; D : 293.66; G : 195.778
While some violinists tune up their G-strings in a concert, I am doing the exact opposite. The "standard" violin tuning (exact 3/2 P5's) is E 660, A 440, D 293.33, G 195.56. Each string down is 1.96 cent against the "ET" scale. Violinists do this to compensate for the 1.96 x 2 cent error against a supposedly 2:1 ocatve, ET piano.
In the 80's I proposed the 81:40, 1,221.5 cent octave which I named "Stretch Octave". It has subsequently been proven scientifically (1) that the music octave as perceived by the listener is about 1220 cent.
Here I postulate the reason why the octave is 1,221.5 cent : consider the whole tone as 9:8 (as the "Pythagoras Scale"). Stack six of these together, and it makes 1,223 cents - complies with the finds of the scientists (about 20 cents bigger than the 2:1 octave). Apparently, the ear likes to construct the octave according to the size of this 9:8 interval.
I therefore call this tuning system "Quasi-Equal" (or "Pseudo-Equal").
Re "Quantum" intonation" - in contrast to "Pure" intonation. I noticed that the ear is actually also sensitive to complication ratios, contrast to the old belief that ratios need to be "simple". Therefore ratios which are "relevant" in music are just numerous. This recording is a example to illustrate this "fact".
"Stretched" music "needs" an "ET" accompaniment. Instead of contradiction, the "Stretch" scale consists the 4/3 tetrachord, which agrees with ET (with small error). Because of the small size of the "half-tone", the lower note in the "half-tone" step sounds a tiny "sharp", the upper one a little "flat". This is how the "sensation" in music comes about, or how the "leading note" gets "resolved".
In the very last double stop of this piece are the two notes, F# on high position, against an open D (presently, 293.66), a horrifying endeavour, theoretically impossible to be in tune. The old school says the "Pure" F# is 734.16 Hz (under a 446A, 293.66 x 5 / 2), otherwise, 740 Hz in ET. Recording here plays F# at a not-so-simple ratio of 9/8 * 9/8 * 81/40 (x 293.66 Hz = 752.63 Hz). Notice in particular, the double stop immediately prior plays this same F# with the D a "Stretched Octave" higher than the open D (interval = 9/8 * 9/8), Both these intervals sound "pure". The postulate is, any interval any multiples of the syntonic comma, 81/80, sounds pure or significant. In short, any interval which sounds in tune are, by default, significant. Hence this name "Quantum".
The note D tuned at 193.66 is an attempt to render this last note same as the D on the ET,440 piano. A at 446 adapts the tuning to the "Stretched" environment.
(1) OCTAVE DISCRIMINATION: TEMPORAL AND CONTEXTUAL EFFECTS Lola L. Cuddy and Peter A. Dobbins, 1988

Linus Liu
2016-2