Nice work. Nice to see actual numbers verifying what David Andersen knows in his bones. I'm particularly interested in the "clean G3-D5 twelfth". There's another concept floating around that if you tune perfect twelfths this will also accommodate inharmonicity as well as guaranteeing aural beauty. Several of us have pursued the P12 tuning as a slightly different flavor of holy grail. I wonder if you do this again, could you measure the other twelfths within your tuned double octave? Thanks Jason | || ||| || ||| || ||| || ||| || ||| || ||| || ||| jason's cell 425 830 1561 http://www.linkedin.com/in/jasonkanter | || ||| || ||| || ||| || ||| || ||| || ||| || ||| On Tue, Feb 17, 2009 at 2:07 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? 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