>> My piano is in standard meantone temperament, which has two 4:5:6:7 (German >> augmented sixth) chords: Bb D F G# and Eb G Bb C#. These chords sound >> absolutely convincing and beautifully resonant to me. The "subharmonic" >> equivalents, Bb C# E G# and Eb F# A C#, are also beautiful in their own way. >> The tuning I described has three 4:5:6:7 chords, and three more in the >> "subharmonic" configuration. That's the point of it. >At the moment my piano happens also to be in meantone but I would call it "Historic >Meantone", specifically what is called "quarter comma Meantone" where all the fifths >are flattened by 1/4 comma (21.5cents/4) execpt G# and Eb. >When you say 4:5:6:7 do you mean that Bb--D is pure, and D--F is pure and F--G# is >pure? No, I too tuned my piano by ear to an approximation of 1/4-comma meantone, so Bb--D is pure but D--F is slightly flat. Theoretically correct 1/4-comma meantone would have a maximum error of 6 cents in the 7-limit tetrads, while the second tuning I gave for the Keenan/Fokker/Lumma scale reduces that to 2 cents. >And if two >supersonic tones yield an audible resultant then Helmholtz et al are up to date. Richard, difference tones and beat frequencies, although they often have the same CPS numbers attached to them, are two completely different phenomena. Difference tones arise from nonlinearities in the auditory system, and when the real toned increase (or decrease) in volume, the difference tone increases (or decreases) in volume much more rapidly. For quiet tones, you may have beating but no difference tone. Even a 33Hz beat can be perceived without the slightest hint of a 33Hz tone. One additional phenomenon, the virtual pitch, is often close to the difference tone, and can beat against it. The understanding of these phenomena has increased tremendously since the 60's -- I suggest you go to the library and look up "psychoacoustics". >In other words the beat frequency, 44, is a lower harmonic of both 220 and >264. Or 220 is the fifth harmonic of 44 and 264 is the 6th harmonic of 44. So when >you say the 5:4 is louder and beating rapidly but does not distrub the consonance >of the 6:5 this may be the reason why. If you replace "beat frequency" with "difference tone" I would be inclined to agree with you. The same thing would work with 7:6 and 6:5. >I am wondering if you have explored >musically the series of what is called superparticular ratios...3:2, 4:3, 5:4, 6:5, >7:6, 8:7, 9:8, 10:9 Like Harry Partch, I find all the small-number ratios significant, not just superparticulars. >But, ancient Greek physists (Pythagorus, Didymus, Aristoxenes) have given us what we >are talking about right now. Where else does 4:5:6:7 come from? Actually, the ancient Greeks and the Europeans until the 1600's would have called this chord 105:84:70:60, since string lengths were the only known physical parameter to exactly quantify chords. In the 1600s or 1700s the vibrational nature of sound, and the fact that frequency was inversely proportional to string length, was discovered, and the more relevant 4:5:6:7, a comparison of frequencies, would have had the meaning it does today (although string lengths remained the convention for some time).
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