Bob: I am very interested in this also. You are not alone. I do not understand all of this, but I am sure I will learn something. I agree with what you did by determining the twelfth root of the actual octave interval. I did the same thing in the calculations I already posted. What I also found was that when calculating a 4:2 octave for the next higher D, it was a different frequency than what is required by the twelfth root of the next A-A octave ratio. I am not sure what this means, but I think it may mean that it is necessary to continually stretch the octaves more and more to stay as close to ET as possible, which may or may not sound "best." In step 5 you tuned by ETD and tuned a narrow octave. I may be a bit of a dinosaur. I don't have an ETD and I don't know what settings you used to tune this narrower octave. Perhaps you continued with the twelfth root of the first octave interval? But since the octave was a narrower type than the one below it, I would expect that the fourths would not double each octave, but the fifths would at least double. In step 6 you "Convert cents into beats = Actual frequency at coincidence * (octave ratio ^ (cents/1200))." I take this to mean you used the measured octave ratio rather than the number 2. I expect this would cause an error, since I think your ETD would be measuring standard cents not ones corrected for the actual octave ratio. The error may not be significant though. Step 7 reinforces what I already believe: Any one interval can be tuned so that the beat rate is the same across the keyboard, but not two. On Tue, Feb 17, 2009 at 5:07 PM, <BobDavis88 at aol.com> wrote: > Okay, I did some actual measurements, as well as some better calculations. > 1) The speed of fourths does not double each octave, or anywhere close. > Demonstration below. > 2) The 12th root of 2 is indeed 1.059463, but is irrelevant to our needs, > even in equal temperament. > 3) Geometric progressions are harder to visualize than the simpler > arithmetic ones erroneously used in some textbooks. > 4) Tuning is complex, and an insoluble puzzle. Although the ear is always > the final arbiter, I care about this hair splitting, because facts and > figures always show me something else I should be listening to more > carefully, which will make my tuning sound better. I'm glad it came up. > > My own experience had shown that fourths don't speed up like I thought the > theory predicted, but I had long been curious why. After reading and > understanding why, in the math given in Dan Levitan's articles, I decided to > take some careful real-world measurements as a demonstration, and I see > David Andersen has offered to tune in person, which will show the same > thing. I consider myself an aural tuner, although I regularly use, and am > facile with, ETD-assisted tuning. Although I usually use Pocket Reyburn > Cyber Tuner, for this experiment I used my old AccuTuner II, for > repeatability, and because I'm faster at switching back and forth from > calculated tunings to direct interval measurement, and quicker at altering > the stretch to fit the piano (although PRCT will do this, too). > > To get to the meat first, here are the beat rates I measured, followed by > the methodology. The piano is my own Steinway A-3, so I could take as long > as I wanted, and it's not a bad piano. > > Fourth: Beats per second @ 4:3 > A1-D2 1.2 > ... > A3-D4 1.32 (#17 wire) > A#3-D#4 1.19 > B3-E4 1.26 > C4-F4 1.33 > C#4-F#4 1.28 > D4-G4 1.15 > D#4-G#4 1.22 > E4-A4 1.22 > F4-A#4 1.13 > F#4-B4 1.37 > G4-C5 1.45 > G#4-C#5 1.25 wire size changes to 16.5 @ G#4 > A4-D5 1.83 wire size changes to 16 @ D5 > ... > D5-G5 1.76 > E5-A5 0 (yes, 0. Some higher fourths are narrow.) > F5-A#5 0 > > These are not calculated, but actually measured. It is apparent that the > rate does not double every octave. In fact, it stays fairly constant, with a > couple of anomalies due to wire size, and perhaps very small measurement > errors in my interpretation of the movement of the lights. > > To anybody reading this far, here's the protocol: > 1) Tune A=440 Hz > > 2) Tune A4-A3 AURALLY so that it sounds cleanest. This was between 4:2 and > 6:3, slightly closer to 6:3. I lowered the stretch on the SAT a couple of > tenths, so that it also produced this octave. Interval width was then > measured directly. For instance, a "4:3" A3-A4 octave is measured by > listening where they are coincident (at A5). On the SAT, it is set to listen > at A5 (in Tune mode) and we then subtract the measurement of A3 (at A5, its > fourth partial) from that of A4 (also at A5, its second partial). It showed > about 1.1 cents wide at 2:1, 0.5 cents wide @4:2, and 0.3 cents narrow at > 6:3. I think this is representative of what most aural tuners do. It also > produced an A3-D4 fourth of 1.32 beats/sec, and a D4-A4 fifth of just under > 1/2 beat/sec. > > 3) Divide the octave into 12 equal pieces. This was done at the 4th partial > for accuracy, but I also checked at the fundamental. A word about that: > Although the twelfth root of 2 is 1.059463, that is irrelevant, except in > instruments without inharmonicity. The actual ratio of equally tempered > minor 2nds is the 12th root of the octave ratio. For instance, if A4=440, > and A5=881, the m2nd is the twelfth root of 881/440, or 2.002272^(1/12). > Cents would be 2.002272^(1/1200). This may not seem like much difference, > but higher up the piano it makes a greater difference. In the top 8ve it > might be the twelfth root of 2.0365. Math geeks please correct me if I'm > wrong. > > 4) Check contiguous thirds F3-A3-C#4-F4-A4 by measurement. I got 13.6 cents, > 13.8, 13.6, 13.7. Close enough for me to assume smoothly progressing thirds. > > 5) Tune notes of next octave up by ETD. This produced an A4-A5 between 2:1 > and 4:2, and an A3-A5 double 8ve about 1/2 beat wide at 4:1. It also > made D4-D5 just wider than 4:2, and a clean G3-D5 twelfth. A wider 8ve might > have kept the 4ths moving, but would have made a rough 8ve and double 8ve. > > 6) Start measuring 4ths. Again by actual measurement: set SAT in tune mode 2 > 8ves above lower note, read the difference between two notes of 4th @ > coincidence. Each 4th was retuned right before measurement. Convert cents > into beats = Actual frequency at coincidence * (octave ratio ^ > (cents/1200)). > > 7) I haven't made the same careful measurement of 5ths yet, but they > progress more normally with this stretch. > > 8) In the extremes of the scale, these measurements depend some on the rate > of change of inharmonicity (wire size, bridge progression), and the amount > of stretch chosen by the tuner, but there's really not much place to go in > the middle, so I think the principles hold, with most reasonable tuning > styles. Because inharmonicity is the cause, and varies from piano to piano, > progression of fourths will be different from piano to piano. Fourths can > even slow down. > > Any comments/corrections? > Bob Davis > > > > > > ________________________________ > Need a job? Find an employment agency near you. -- Regards, Jeff Deutschle Please address replies to the List. Do not E-mail me privately. Thank You.
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