Yahoo Tuning Groups Ultimate Backup tuning [tuning] Re: Complex number temperaments

Bill Alves appears to have confused tempo or metronome markings (beats per minute) with frequency, which as cycles per second (Hertz) is sixty times faster. I have a Franz electric metronome with a marked range of 40 to 208 beats per minute. Divided by 60 to convert to Hz, this a frequency range of 2/3 Hz to just under 3.5 Hz. So you can see that tempo is some "octaves" below the sensation of tone. Both are a result of volume (or amplitude) variations, however. Changes in what we perceive as loudness are merely normally slower than the frequencies we perceive as pitch or tones. What we perceive as pitch through our ears gradually shifts into a perception of touch or vibration (including through the chest) around the "octave" of 10 to 20 Hz. A few organs with "64 foot" pipes produce tones down to about 15 Hz. Emory Cook recorded one in a Mexican cathedral.
Most signal generators, test records and even CD's don't go below 20 Hz, nor do most analog tape recorders. And with LP's you start running into problems with arm mass versus cartridge (stylus assembly) compliance resonances, in some bad cases at frequencies well above 20 Hz. But with digital recording, more attention is being paid to this range, including the development of subwoofers.
Perhaps I should have mentioned something fairly obvious in replying to Sarn Ursell's inquiry. That is, whether you are looking at, for example, the square root of nine or minus nine, the numerical (number) part is three in either case, apart from the square root of minus one, or i.
Please don't misquote me. I didn't invoke Occam's razor and say the simplest concept with the fewest dimensions is best. Obviously, for the performing musician, the concept of everything being reduced to a series of air pressure changes is useless. But for the sound recording engineer, the separation of music into a multi-dimensional complex of melody, harmony, rhythm, tempo and loudness is useless. Only the limitations of analog equipment made loudness particularly important. With digital technology, loudness is scarcely a limiting factor, or is low (bass) frequency response.
Even how multiple dimensions are defined can make a difference. Until about twenty years ago, audio amplifier specifications were based on frequency response, power, phase, distortion and signal-to-noise ratios. It gradually became clear that frequency response at the treble end might better be expressed as the speed or quickness of certain parts of the circuit (the output transistors were usually the limiting factor) and so an alternate specification, rise time, became the buzz word. Actually, a rise time of a certain number of microseconds corresponds exactly to a flat frequency response to another certain number of Hz in the hundreds of thousands (far above the range of human hearing). The key here was to improve transient response.

🔗Bill Alves <ALVES@ORION.AC.HMC.EDU>

🔗Paul Erlich <PERLICH@ACADIAN-ASSET.COM>

🔗John F. Sprague <jsprague@dhcr.state.ny.us>

I realize that you wish to disillusion me of what you consider to be my misconception. I beg to disagree. You may have been referring to a book such as Bartholomew's "Acoustics of Music", which makes the distinction between frequency and loudness. From the musician's standpoint, this is perfectly reasonable and corresponds to our sense perceptions. I've been taking the viewpoint of the physicists, such as in Knudsen and Harris' "Acoustical Designing in Architecture".
Our situation reminds me somewhat of the plumber who wrote to the National Bureau of Standards about his discovery of what he thought was the perfect drain cleaner. After several letters back and forth, someone at the NBS realized that the plumber simply wasn't understanding their replies in the usual governmentalese, which went something like this: "While the application you have recommended may appear to prove initially efficacious in dissolving system blockages, its use is contraindicated." The assignment of making the next reply was given to another employee, who wrote: "Don't use muriatic acid. It eats the hell out of the pipes!"
Perhaps you are looking at loudness in terms such as VU (volume units)or as an RMS (root mean square) average. It is indeed commonly measured as such. However, that is essentially an alternating current phenomenon (after transduction by a microphone from sound waves to electricity) converted (rectified) into direct current. Now it is convenient to think of frequency (rate of vibration or pitch) as independent of loudness (amplitude of the vibrations).
My point, which you consider to be a misconception, is that within an individual vibration (the complete cycle of a sine wave, for simplicity), there is a variation in air pressure from average through compression, back through average through rarefaction and returning to average. (Either the compression or rarefaction may come first.) Through the range we consider as audible tones, it doesn't seem like huffing and puffing, but it is. Whether it appears soft or loud depends on the relative sensitivity of the ear to different frequencies as well as the actual sound pressure level at the ears. This sensitivity varies with the overall level as well as with frequency, and from one individual to another, partly as a function of age. Our hearing is not very linear.
Sound, as far as the microphone, loudspeaker or tape is concerned, is not made up of an FM component (frequency variation) and an AM component (amplitude variation). Within the limitations of the equipment, the changes in air pressure and electricity are merely matters of how fast and how much. (Now a microphone of rather less than the best quality may produce no output from tones of 20 HZ and 20 kHz. But a really good one will, and over a fairly wide range of loudness.) This is why sound can be considered one dimensional. (This is mono, not stereo or surround.) Think "wire recorder".
Now it is true that analog tape recorders have a bias frequency, although you can record without one. But LP records do not (except for CD-4 quadraphonic types). The signal in a mono LP groove is a stretched out transverse wave. Sound waves are longitudinal, but after amplification and conversion back into sound in headphones or loudspeakers, the sound output is once again longitudinal waves. The air pressure can vary rapidly or slowly, a little (softly) or a lot (loudly), but it is all just pressure changes, If the loudspeaker can't keep up, there may be a lot of distortion and/or reduced output. Too loud a high frequency may melt the tweeter voice coil. Too loud a low frequency may tear the cone suspension, tear the voice coil from the cone, slam it into the rear pole piece or blow it out the front of the magnetic gap.
In the case of wind instruments, the tones do modulate an air stream. That is usually slow enough to avoid any noticeable Doppler effect. You may hear the hiss of a pipe organ in quiet passages, or the breathy quality of a flute miked too closely. But you can also hear the breathing of a violinist miked too closely. The air stream may provide a small amount of acoustical amplification, but nothing like what you get from a fog horn (which is much louder).
Knudsen and Harris write of particle velocity as directly proportional to the product of the displacement amplitude and the frequency of the sound wave. They also state that, "The changes in pressure, density and temperature due to the passage of the sound wave through air are usually extremely small." An example is given of particle velocity of a few inches per second, but you need to realize that within a second, the particle's direction will have changed many times, so its actual displacement is very small.
Theoretically, microphones might be buiit which detected changes in the speed of air in a tube, or the changes in density or temperature (all versus time). With acoustical (non-electrical) recording, horns were used to concentrate or magnify these changes during recording and amplify them during playback. All the electrical types of microphones that come to mind (piezoelectric or crystal, dynamic and condenser) detect pressure changes, have very low output and need (or have built-in) a mike preamp. (Even dynamic loudspeakers can be wired to act as microphones - a spy technique, as is measuring the reflections from a flexing window.) Wireless mikes simply add a transmitter.
In the simple case of the sine wave, for example, the pressure extremes are greatest at its peak and trough, where the particle velocity, because it is changing direction, is least. Similarly, the pressure where the curve crosses the time axis (which represents normal pressure) is least, but the particle velocity is greatest.
Rhythm may match the tempo (beats per minute) or the time signature (such as 3/4, or waltz time) or neither. But it also depends on some fairly regular changes in loudness and/or frequency or tone color for its emphasis to be noticeable. But it doesn't have to be based on sine waves, and probably seldom is. A steady tone without variation has no rhythm but may still contribute to a musical effect. Think bagpipe drone.
This has all come very far afield from the question about complex numbers and temperament. Let me close by pointing out that the calculation of an equal temperament is simply a problem in compound interest. If based on "octaves", the present value is one, the future value is two, the number of periods is whatever number of tones you want and the rate is the unknown to be calculated. That number will let you calculate frequencies from whatever starting point upward. But it will not tell you what musical properties the scale may or may not have!

🔗Barbara Demetz <BARBARAD@CYBERHIGHWAY.NET>

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

Dear John F. Sprague --

It is odd that you claim to be taking the viewpoint of the physicists, since
my training is as a physicist. Anyway, I could respond in detail to your
long post (you seemed to sidestep my points) but it's time to take this
discussion off the list -- it's off-topic. Suffice it to say that _some_
vibrational phenomena will elicit a perceived rhythm in one frequency range
and a perceived pitch in another frequency range (and even both
simultaneously around 20Hz), while many phenomena are specific to one
sensation or the other and will not produce the other sensation even when
transposed to the appropriate frequency range, even if human sensitivity is
adjusted for, etc.

Whatever the nature of the misconception or lack thereof, it is unfortunate
when some composers use this logic to "tune" the tempo and the key of piece
to one another. There is absolutely no evidence, or theoretical reason to
believe, that such a "tuning" could have any audible effect.

-Paul

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