Stopper tuning based on equal division of pure twelfths (octave and fifth) or Pure Twelfth Equal Temperament (EUROPIANO 3/1988)² has become the standard on concert stages all over the world in just 30 years.
The tempered intervals of this tuning create a strong beat masking effect, so that a very clear and resonant sound results.
The Stopper tuning is available for concert tuners and technicians with my tuning app Tunic OnlyPure, or by ear using the OnlyPure method.
I regularly give lectures and seminars for colleagues, e.g. at the PTG 2008 conference on the new OnlyPure tuning software: Stopper Temperament.
A new sequence was presented for tuning the Stopper tuning by ear with a new equal beating interval test, which considerably limits possible chain errors, at the PTG conference in Tucson, Arizona in 2019: Tuning Sequence Stopper Temperament
in Kansas City 2011, the following video was made with Grigory Sokolov:
Stopper tuning = transformation of the Pythagorean tuning by replacing natural octaves with acoustic octaves in every interval
It is philosophically interesting that the Pythagorean tuning (in which any interval is formed from power fractions of the numbers 2 and 3) transforms directly into the Stopper tuning by replacing the octaves with the natural size 2/1 with octaves Ω that are perceived as "more in-tune".
Ω = 2s/1 = 2.0014269…
p=pythagorean comma = 3^12/2^19
s= p^(1/19) = Stopper Tuning Comma
|Pythagorean Intervals||Stopper Tuning Intervals||Harmonic Intervals|
|4/3 = 2^2/3||Ω^2/3||4/3 (Fourth)|
|81/64 = 3^4/2^6||3^4/Ω^6||5/4 (M3)|
|27/16 = 3^3/2^4||3^3/Ω^4||5/3 (M6)|
|9/8 = 3^2/2^3||3^2/Ω^3||(maj. sec.)|
|32/27 = 2^5/3^3||Ω^5/3^3||6/5 (m3)|
|16/9 = 2^4/3^2||Ω^4/3^2||7/4 (m7)|
|243/128 = 3^5/2^7||3^5/Ω^7||8/15 (M7)|
|256/243 = 2^8/3^5||Ω^8/3^5||(minor sec.)|