Please excuse me if I was unable to respond further this afternoon, but I had some commitments; I am trying to reply to you now that I have a few moments of peace.
The subject is colossal in scale, and even by condensing it as much as possible, it remains very difficult and long to describe, so I will have to put my abilities of synthesis and exemplification to the test. Therefore, I apologize if my intervention may seem approximate to the experts; in fact, I invite them to take part in the discussion by helping me integrate where I might, most likely, have omissions or excessive approximations.
The key point is choosing where to start. I would start from the conversion of a stimulus into a sensory response.
We can divide the environment, which belongs to the domain of physical reality, from man, who belongs to the domain of perceptive reality.
In the environment, we have sound waves that propagate through a medium (which can be air or a solid). In the air, they move at a speed of approximately 331 meters per second at a temperature of 0°C. As the temperature increases, the ability of sound to propagate in the air also increases, and thus its speed increases according to the formula 331.6 + 0.6*T. It propagates as a longitudinal compression wave where the particles transmitting this energy through space move back and forth relative to their position of rest. These mechanical waves, through the air, as we were saying, arrive in our ear. At this point, we should already make an important consideration. Each of us is anatomically different from one another; therefore, our auditory canals, although differing only slightly from one another (I am not referring to the two ears, but from person to person). We can view the auditory canal as a tube having one closed end. The resonance frequency for an open-closed tube is therefore c/4L, where c is the speed of sound in meters per second in the air at a certain temperature (this is why I previously made this digression on the speed of sound), divided by 4 times the length in meters of the tube. The length of the auditory canal ranges from 15 millimeters (in children) up to a maximum of 30 millimeters in adults. I report below some values in relation to different lengths of the auditory canal (you can perform the same calculations yourselves using the above formula and verify; I might have made a mistake in some calculation): 15 millimeters = resonance at 5733 Hz; 20 millimeters = resonance at 4300 Hz; 25 millimeters = resonance at 3500 Hz; 30 millimeters = resonance at 2867 Hz.
But what is resonance? It is the phenomenon for which some frequencies, bouncing off a surface, return in phase with those coming towards them, resulting in a sum of amplitudes and thus a reinforcement of the mechanical wave, which can be calculated with the definite integral of the two positive semi-waves or, in an entirely analogous way, negative ones, of the direct signal and the refracted signal; but let's keep things simple and intuitively accessible to everyone. It is enough for us to know that in the case of the resonance frequency of our auditory canal, we are inclined to hear those particular frequencies better.
Returning to the resonance frequencies we calculated for smaller measurements of the auditory canal, this should make us reflect on why, when we speak to children, we tend to use that silly little voice. It is done precisely because one seeks to work within a range of frequencies (around their resonance frequency or as close to it as possible) so that they can hear better. However, we have seen that even changes of a few millimeters (from 25 to 30, for example) can determine a fairly large delta in frequency (about 700 Hz), which in the range we are analyzing is not at all small. We can calculate this amplitude in cents for the interval (3500-2867) Hz so as to have a linear magnitude instead of logarithmic magnitudes with the formula: cents = 3986 log(3500/2867). We get approximately 345 cents, which corresponds to three and a half semitones (an interval wider than a minor third!).
Having made this premise, we can return to our macro areas (man and physical reality) because we have already seen how in physical reality there can be physical variables that can modify (given the same human perception) some important parameters. Speaking of frequency, I would attribute to it the most immediate magnitude perceivable by everyone; not without reason, when speaking of music, even those least interested immediately think of musical notes and thus of what musicians call "pitches" and what physicists, engineers, and mathematicians call frequencies.
Human perception, however, is by no means an equal or identical parameter, to use the same term as before, for everyone; in fact, the human response to a physical stimulus is anything but a common parameter. Everyone has the
hers. However, what determines awareness? From a set of parameters, one important one being memory and experience. Listening to something beautiful could give us ideas; re-listening to that same thing could provide points of comparison and other ideas. We are then considering only acoustic stimuli when we should also consider other types of stimuli, such as visual ones (the gestures and movements of a pianist in a concert that could emphasize certain moments), or olfactory or gustatory ones. In essence, given the enormity of parameters at play, each of us responds to the same stimuli differently. None of us, in light of the considerations we have made (at least in the psychoacoustic field), can say who is right or who is wrong because these are reactions of our bodies to external stimuli which, as we have seen, vary according to different physical and psychophysical parameters from person to person.
For the moment, we have limited ourselves to analyzing only one anatomical parameter of our auditory system; anyone among you who wishes to study or delve deeper into the auditory system (I recommend it because it is truly fascinating) will discover that reasoning analogous to ours regarding the auditory canal can be extended to every single component of our ear.
We can, however, use so-called isophonic curves (Iso-Phonos = Same sound, precisely because they represent the ear's ability to perceive the same sound at different sound pressure values), to evaluate how the ear typically behaves at different frequency and pressure values.
Look at how from 3000 to 5000 Hz the ear performs better, with a maximum peak at 3500 Hz. This is because the resonance of the auditory canal helps us, as we have demonstrated.
The low frequencies are very difficult to listen to because the wavelength of a 20 Hz sound is around 17 meters. It becomes difficult to hear it at low volume and, above all, it also becomes difficult to locate the source in space. It is commonly said that one can hear sounds from 20 to 20,000 Hz; in reality, this view is very optimistic. Typically, a good ear "hears" well up to 15,000 Hz, then can only perceive a few changes (for example, by turning the sound source on and off), and subsequently hears nothing. It is nonetheless possible to hear some variations up to 20,000 if compared with other sounds. In the case of the piano, the last note (C8 in American notation - C7 in European notation) is 4186 Hz; considering the second harmonic (C), we will find it at 8372 Hz, the third (G) at 12543, the fourth (C) at 16744, and it is already out of reach. Therefore, regarding extreme notes on the piano, we can barely hear the first three harmonics (N.B. Three harmonics include the fundamental sound, which is the first harmonic!). I note that even at 12543 Hz, according to the isophonic curves, our ear begins to drop significantly; therefore, given also the very rapid extinction of piano sounds at those pitches, we can quite comfortably consider those sounds as having two audible harmonics (fundamental and first harmonic; these things also depend on factors such as age and other factors that we have seen to be of an anatomical type).
A to these considerations, another fact is added, namely what I would call, quite unofficially, the "degree of ear training," precisely because this graph depends heavily on the ability acquired over time to distinguish pitches with more or less precision.
I find it necessary to attach an image from the slides of the Acoustics and Psychoacoustics course taught by Prof. Maurizio Massarelli, whom I hope will not mind; in fact, I am sure he will appreciate the dissemination.
Here in this slide—excuse me if my notes are also present—we observe that after a certain pitch (circled as frequency 523.2, which would be C one octave above middle C), the ear begins to drop. In particular, we notice that it does not drop in the same way for every category of people. We notice that in curve C, in the case of non-experts, it is possible to drop by even more than an octave. (Naturally, we are not making references to those undergoing the test; we send frequencies and ask when, according to them, we reach double the previous frequency). In the case of musicians, the ear drops significantly less; in the case of tuners, even less. However, we see that even tuners (the most trained) cannot achieve perfect linearity. For all the reasons stated above, therefore, even in the case of tuners, there will be those who drop more and those who drop less; this graph is based on statistical tests, obviously, because it is not possible to measure sensations without an empirical approach.
Adding to the tangle of things that
we said that even the intensity at those frequencies plays an important role. Increasing the intensity (while keeping the frequency constant) leads one to listen to a higher pitch sound, even though in reality the sound is always the same. The same goes for very low sounds which, by increasing the intensity at the same frequency, still seem lower.
We conclude with a shock definition... Perfect tuning does not exist, because it is affected by so many subjective variables that change from person to person. In the same way, we could speak of physical variables that change from object to object. If it's of interest, we will also make a digression on this.
But this statement (that perfect tuning does not exist) must not mislead us; there are still adjustments that suit our ear and therefore, even if it is not perfect, it still sounds good. Electronic tuners do not work the way the ear works while tuning. This is where my idea begins: to create a tuner that works like the ear and requires us to listen for harmonic ratios. After all, all the technological wizardry that works properly is created by emulating things in nature. We shall see...
If there are any doubts, of course, I remain at your disposal...
