Bob: Thank you so much for giving your conclusions on Dan Levitan's articles. They are not available to me, but I was guessing at similar conclusions. All: I calculated the following beat rates, taking into account inharmonicity, of a piano with an iH constant of 0.1 at C3 that doubles every 8 semi- tones with the fundamental of A4 at 440hz. This is a sample piano's iH from the well known article "Inharmonicity of Plain Wire Piano Strings" by Robert W. Young. D4 and D5 were tempered within each of their A to A octaves. D3 and D6 were calculated directly from their respective D's rather than being tempered within their own A to A octaves. (There is a difference I hadn't realized before…) Intervals Octave Types 3:2 Fifths 2:1 4:2 8:4 D6-A6 -9.86 -2.41 +31.61 D5-A5 -2.88 -2.03 +3.15 D4-A4 -1.23 -1.00 -0.09 D3-A3 -0.58 -0.49 -0.11 4:3 Fourths A5-D6 -1.89 +2.41 +19.77 A4-D5 +1.11 +2.03 +5.73 A3-D4 +0.84 +1.00 +1.65 A2-D3 +0.51 +0.49 +0.39 You each may arrive at your own conclusions, here are mine: In the mid-range, 4:2 octaves will cause the beat rate of both 4ths and fifths to double each octave. If the octave is wider than 4:2, the fifths will beat slower and progress slower than double each octave and the fourths will beat faster and progress faster than double each octave. If the octave is narrower than 4:2 the opposite occurs. Higher in the treble, where iH increases sufficiently, the beat rates of both fourths and fifths progress slower than double each octave. Regardless of iH, widening a fourth narrows a fifth changes the beat rate and visa-versa. You can have all fifths beat at the same rate or all fourths beat at the same rate, but not both. A 8:4 (m6-M3 test) octave in the temperament is a good choice if the goal is beatless fifths. There is an added benefit to this because the beat rate of the highest m6 within an octave can be substituted for the beat rate the M3 above the octave. This virtually gives you 4 contiguous M3s within an octave rather than within a M10. On Sun, Feb 8, 2009 at 5:49 PM, <BobDavis88 at aol.com> wrote: > Do ascending fourths beat faster? The answer is "Sometimes." > > Ironic that this discussion on fourths and fifths is just about to produce > more heat than light, just at the time that something useful is beginning to > emerge.... Hang on, hang on.... > > Given a choice, I'll obviously rather have my tunings sound good than > understand them, but intellect and intuition are not mutually exclusive. > They are both human attempts at understanding, but they do not change the > nature of things; they are just approximations which we find helpful. I have > huge regard for the human wetware, so I like to think of intellect as > informing intuition, to make sure it has all the correct inputs it needs to > function well. If the result is good, it doesn't matter how we describe it, > but keeping both channels open surely makes the job more interesting. > Sometimes our doing is ahead of our understanding; sometimes the other way > around. > > That said, the math must itself be based on correct inputs. Calculations > based on frequency tables may be correct to six decimals, but are not the > information we need for tuning. (Because of inharmonicity, if A4 is 440.00 > Hz, D4 will not be 293.664776, but something closer to 293, and their 3:2 > coincidence will not be exactly 880 and 880.99xxxx, but something different > depending on the inharmonicity of each note). Inharmonicity is exactly what > this whole conversation hinges on, being the cause of fourths and fifths > staying close to the same up the scale. Fourths will typically, but not > necessarily, increase in speed very slightly as we ascend through the > temperament area, but in no case will their beat rates come close to > doubling each octave, and in most pianos, they stay close to the same above > (and below) the temperament. This is counter-intuitive only if we don't have > a full understanding of the effects of inharmonicity. > > I just re-read Dan Levitan's excellent articles which begin in the August > 1994 Journal. I'm no math whiz, so for me they're heavy going. It took him > several articles to explain this, so I can't summarize his reasoning in a > sentence or two, BUT, some of the conclusions are that > 1) Inharmonicity would cause ascending fourths to be wider (beat faster) and > fifths to be narrower (beat faster), but > 2) it also expands octaves even more, which more than counteracts this. > 3) as inharmonicity increases up a good scale, the ever-wider fourths, > ever-narrower fifths, and ever-wider octaves keep counteracting each other, > keeping the beat rate nearly the same, at least at the lowest coincidence, > depending on the tuner's choice of octave width. > 4) Although temperament beat rates are not the same from piano to piano, > most well-scaled pianos, even with different primary inharmonicities, > produce very similar beat speeds, and similar (but not necessarily identical > or linear) rate increases on the temperament thirds, because of the scaler's > attention to the rate of change of that inharmonicity. > 5) Some of our temperament tests which rely on equal beating, such as the > inside M3 - outside M6, are unreliable at the finest level, since the speed > of each of those intervals depends upon octave width, and they change either > at different rates, or, in the case of tests using a minor interval and a > major one, oppositely. > > Taking the D5-A5 fifth as an example: the inharmonicity of the third partial > of D4 (A6) is greater than that of the second partial of A4 (A6), which > would make it sharper, and therefore the fifth narrower EXCEPT that the > A5-A6 octave is wider than the difference. On a typical scale, if A4=440 Hz, > its 2nd partial (A5) will be about 2 cents sharp of 880, or about 881; its > 4th partial (A6) will be around 10 cents sharp of 1760, or ~ 1770Hz. This > will cause the 5th to be slower than if the octave were an exact 2:1 ratio. > The third partial of D5 (A6), will actually wind up very close to 1770. > > I hope Dan will forgive (or correct) my (very) condensed (incomplete) > version of his fine work. > > Bob Davis > ________________________________ > Who's never won? Biggest Grammy Award surprises of all time on AOL Music. -- Regards, Jeff Deutschle Please address replies to the List. Do not E-mail me privately. Thank You.
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