Fifth Beats

This is PIANO CONCERTO......NOT ORGAN CONCERTO.....maybe I'm wrong..

But precisely because it is not my instrument, I am more interested in the subject; Mr. Saverio, I am counting on more stimulating replies, please.

I don't know why you were answered in the terms mentioned above by Saverio, I am usually used to asking many questions and receiving answers as well as, in general, being very polite.

Yes sir, I believe that a comparison between enthusiasts of different instruments is important, as different musical experiences can be a useful stimulus. But also be careful to remain within music and not privilege too much certain points of view which, although extremely relevant, are "sector-specific".

I should preface that, for an exhaustive discussion of the subject, too much time would be required and it would be necessary to resort to mathematics that is not necessarily intuitive to everyone. For this reason, my answer will be quite simplified and concise.

In practice, there are several factors to consider.

On one hand, ancient musical scales, built with a cyclic approach (use of beats) or with a divisive approach (division of the fourth), have the well-known problems of transposition and lack of octave alignment (Pythagorean comma), in addition to the fact that chromatic scales built with such approaches have semitones of unequal ratios (therefore not equal to the ear).

On the other hand, equal temperament formalized by the mathematician Stevin, due to certain properties of natural numbers, cannot be formulated as a cyclic algorithm; this entails the practical impossibility of obtaining such a scale as "temperament" in a cyclic tuning, due to the lack of correct reference intervals.

It follows - at a theoretical level - the impossibility of suppressing beats in equal temperament and, conversely, if one wants to approximate such temperament in tuning, the need for some "deceptive" trick based on the physiology and psychology of the perceiving subject.

For example, beats are hardly distinguishable in struck string instruments - such as, for example, the Piano - due to the short persistence and decay of sounds, whereas they are easier to encounter in wind instruments or those with air reserve - such as, for example, the organ - due to the longer duration of the sound.

To this is added the fact that, when two "pure" notes are played together, the ear perceives new additional notes at frequencies equal to sums or differences of multiples of the frequencies of the emitted notes, a phenomenon described as combination tones (remember the third tone of Tartini? ) or, for example in electronics, as intermodulation (are you familiar with distortions?). This is due to the fact that the human ear is not a linear transducer (amplifier).

Well, the things we have mentioned can make perception difficult and human intelligence is designed to disregard things that are difficult to process if they are not useful (for example, if a car is coming towards me and also a fly comes towards me, I don't waste time analyzing the Doppler effect produced, but I hurry to dodge THE FLY? nohhhhh, THE AUTOMOBILE!); it follows that - according to the point of view where what is not perceived does not exist - the statement of the person who asserted "as they are performed, going up, they also become pure", an assertion that turns out to be pragmatically true, even if mathematically "not exact"(among other things, I too saw the videos some time ago and I didn't notice it and, in my opinion, I wouldn't even notice it now, because I am not going to see if they know Lebesgue's integral theory well Lebesgue from those who, like PianoExpert, talk to me about music!).

I am a physicist, mathematician and engineer by training, but I come here on this site because I love music and I leave aside exact numbers and Fourier Fourieranalysis.

Finally, but I do not want to be mistaken, it seems to me that both phenomena we have discussed (beats and intermodulation) are also used in organs (otherwise, how would I create almost human timbres with tremolo or what enormous pipes would I have to build to achieve the extremely deep basses of some organs?).

Kind regards

Hello, Gennarino's explanation is excellent.

I believe Saverio was simply joking. I have read some of his posts, always very friendly and self-ironic...

I also think he was joking, but it's likely not so much! (don't touch Pianoexpert!)

You are absolutely right Gennarino.......mine only wanted to be a "bad joke"..I ask forgiveness from the Lords...yes it is true anyway PIANOEXPERT (to me) is a beacon, a point of reference for our beloved instrument which is the piano, I apologize again, but as a good Vesuvian I "went to small schools" I am not very erudite in music......

Regards

saverio

No one knows everything. Much less someone like me who is neither a Physicist, nor a computer scientist, nor a mathematician, nor an engineer. I believe that one must follow the beats, count, divide, debate, contest, measure, theorize etc...., but as a crude experimenter, I believe that the ear is the "final, inexorable judge".

For example, both in the extreme lows and the extreme highs, the ear wants a deception. The same choice of making major thirds increasing and fifths decreasing, in the central range of the tempered scale, respects the "pleasantness" that the ear desires. Numbers, exercises, formulas can be found in any book dealing with tuning via the tempered scale.

Besides, Pythagoras had done the exact calculations... so exact... that they were wrong!!!! And even his successors.....with the advent of polyphony, leaving monodic music behind, the ratios did not work. Hence the natural scale which respects the subtle differences in the tones of the diatonic scale. With a "ten-finger" keyboard, a new problem is created. An octave with 12 keys!!!!! And then it would be enough to calculate the twelfth root and apply the value to each semitone!!!! Not a chance!!!!! And then one piano sounds like this, another sounds like that.......Our Physicist friends also tell us that these blessed multiples of the fundamental ....here they do not correspond!!!!!! I'm going crazy!!!!!!!! Calm down. AAAA H!!!! If only I could have a theoretical string!!!!!! One of infinitesimal thickness and infinite length!!!!!!!! Instead, I must settle for these steel "wires", with more or less carbon... of different thicknesses and even wrapped in copper... to shorten the lengths. Such strings behave like many infinitesimal strings welded together!!!!!! In short, the "harmonics" are "disharmonics". And so it is. Otherwise, it would be too easy. And then this ear of ours, so demanding!!!!!! If we question it about absolute sounds... it might err... but on relative sounds, on differences... on sounds that do not please it......it doesn't care about anyone!!!!!!

And so let us arm ourselves with theory, with numbers, with electronic tuners....but let us put in our pocket a large dose of good taste....we cannot do without that...it is indispensable to us in making a deliberate "error" pleasant.

....When it takes me like this........stop me.......I beg your pardon!!!!

GREAT PAOLO !!!!!! Besides knowing a lot (in my opinion... you know Everything!!), you are also humble. Great person and professional.

I SUBSCRIVE YOUR HOLY WORDS 1000 TIMES !!

Look closely at my photograph......it is redder!!!

Dear Paolo,

it is truly true that "as a very fine experimenter" the ear is the "final, inexorable judge."

In fact, mathematics is beautiful but, one must understand, it is also abstract!

For example, the fact that the ear is non-linear makes mathematical treatment almost inaccessible.

And, if we ever wanted to be "mathematicians to the core," we would have to give up tuning any instrument.

In fact, if we tuned by unison, we would have to wait an "INFINITE" amount of time to be sure that we have tuned correctly, and if we tuned by exact semitones, we would still have to wait an "INFINITE" amount of time to be sure that the beat frequency is the correct one (in Naples they say: Nemmeno na ntecchia e meno o e cchiù!).

But then, how the hell do you do it, when you tuned my Schimmel, that you managed to pull it off perfectly even though you didn't have tuning forks or frequency meters in your hands?

Could it be that, as You say, the ear wants its illusions, i.e., "Pleasantness"?

Could it be that notes are not "pure" but are "timbral"?

Could it be that the strings are not "ideal"?

Could it be that no one beats a lucky ear at perceiving the relative relationships between notes?

Could it be that knowing is not doing, and is not even understanding?

There, it is precisely this that convinces me that a theory without practice makes no sense at all, once again.

Warm regards

I confirm that beats at the same ratio (i.e., interval between two notes) increase/tempo as the frequencies of the two notes generating them increase.

E.g.: if a fifth "shortened" or "tempered" by one twelfth of a comma in the first octave beats 2 times per second, ta...ta ...ta... in the last octave (the same fifth) beats much more rapidly ta.ta.ta.ta.ta. The same applies to thirds, etc.. trust me, an organ builder says so, someone who tunes whistles with few harmonics!!! (they can certainly be heard)

Paolo,

so true!! mathematics and the purity of music!! nothing adds up!! Pythagorean comma, natural harmonics that do not correspond, it is all a compromise, and rightly so. I remember perception and music psychology courses; the brain does not follow mathematics. But the organ is more difficult, requiring more rigor for two fundamental reasons: first, because each pipe produces few harmonics (it is purer than a string); second, the stops are transposed in octaves up and down, coupled with others. The piano (and I am making good Saverio happy) is more "human".

Beating stops:

the human voice, unda maris, viola concerto (terrible to tune because the two pipes in unison often mock you and pretend to be in tune, but when you go to lower the third, you realize they are not!!!)

The same as in any interval, these two notes (one from the principal and one from the beating stop) slightly out of tune with each other create about 3 beats per second in the middle octave, and if I maintain the same "detuning" ratio throughout the keyboard, it will result in being slow in the bass and very fast in the treble. But I (cleverly) correct it to keep the beats constant as a singer would do with their voice; let's just say that in this case we organ builders also "stretch".

by the way, when we are at the register and we spend hours there, there is no point in listening to the beats..... they bark in your brain like a chihuahua, they pierce your eardrums, like a chihuahua barking in your ear; often, when they are in the last rows of the filling, they are pulsations of physical pain, an English colleague of mine lost his hearing at 40.

The beat frequency between two (pure) frequencies ω1 and ω2 is determined using prosthaphaeresis formulas (I apologize to those who hate mathematics and/or did not encounter Trigonometry in their course of study!).

If, for mere simplicity, we use as sounds two sine waves of constant amplitude 1, frequencies ω1 and ω2, and they start in phase, i.e., S1 = Sin (ω1 . t) and S2 = Sin (ω2 . t), the resulting sound is:

S= Sin (ω1 . t) + Sin (ω2 . t) = 2 Cos((ω1-ω2) . t/ 2) Sin((ω1+ω2) . t/ 2) = 2 Cos (ωt) . Sin (ω.t), where I have set:

ω = (ω1 + ω2) / 2 and ω = (ω1 - ω2) / 2. You see that ω is the frequency of the resulting sound, while ω is the frequency at which its amplitude varies (Beat); it is also seen how ω is intermediate between ω1 and ω2 and how ω is lower than both ω1 and ω2.

It can then be seen that if I double ω1 and ω2 (I move up an octave and keep the ratio ω1/ω2 unchanged), the two values ω and ω also double.

To be clear, if ω'1 = 2 . ω1 and ω'2 = 2 . ω2, then ω' = (ω'1 + ω'2) / 2 = (2 . ω1 + 2 . ω2) / 2 = 2 . ( ω1 + ω2) / 2 = ω1 + ω2

and the same story for ω.

The problem, as I believe I have already said, lies in the ear's ability to perceive beats in a real situation.

It must be remembered here again that, as already emphasized, the sound of struck strings does not have the persistence of the sound of an organ pipe, which has the fortune of having a reserve of energy (in the form of air that continues to be pushed into the pipe), which compensates for losses due to friction (of the air, of the bridges, and so on!) that are present in piano strings (also in pipes, it's just that the air reserve is the trick you don't see!). To this is added, as PianoExpert has already pointed out, the fact that piano strings (and, I believe, also organ pipes, although perhaps to a lesser extent) are not ideal strings at all.

Ultimately, if we then finally focus locally on the world as it is perceived by others, we can also understand that, in the presence of an amplitude that damps rapidly and a non-linear ear (with distorting phenomena), beats are perceived with greater difficulty, even becoming imperceptible, unless one has a trained ear, quite far above the human average.

I conclude with a "long live beats" (at least those audible and even with the vibrating body itself), otherwise bells would sound damnably cold to me!!

And now, since I have lost a great deal of time writing the blessed formulas above, I would be pleased if we returned to talking about more exciting things and, if you allow me, I will be grateful.

Kind regards.

Gennarino's explanation was beautiful; I would like to add a consideration if I may. It would be much easier for the human ear to distinguish beats in a pure sound, that is, a sound where partials are not superimposed. If we had an oscilloscope at our disposal, we would discover that the sound wave is approximated to a sinusoid for calculation purposes, but in reality, the matter is much more complex; we could think of a carrier (the sinusoid used to perform calculations) upon which other sound waves are loaded. The result varies from instrument to instrument, and this determines its timbre. Obviously, with a superposition of so many waves, in the high frequencies where Gennaro rightly reminded us, we have a very rapid decay of sound that makes it impossible for the tuner to maintain unchanged the beat ratios, and the ear approximates many beats that decay very quickly into staticity. Unfortunately, we humans are very limited, even if there are people who believe they are God on earth. Just think about when, with moving machines, we cannot understand if a wheel is solid or spoked, due to our visual limitation of about 24fps; this is also why, under acceleration, it seems to us that the wheels suddenly stop. To our eye, they are stationary when we reach the stroboscopic limit of vision, which is when the wheel completes about 24 revolutions per second, if I remember correctly. In light of this, why should the ear be unlimited?

Very well done, Simone,

this was exactly the additional difficulty I was referring to when, starting the previous post, I said "If, for mere simplicity, we use two sine waves as sounds...", because true sound (that which is perceived) is anything but a sine wave and, as many of us know, it isn't even a perfectly periodic one, being subject to decay, environmental noise, friction, and transmission delays of the ear's mechanics, etc., etc..

The reference to similar limitations of our beautiful and sacred eyes and to stroboscopic phenomena is also wonderful! It is no coincidence that in workshops where machines with exposed rotating elements are present (e.g., saws), neon lamps are prohibited!

🙁 I want to acquire this knowledge too, ugh. I would like you to recommend texts that explain well how the equal temperament system was reached. Is it possible??😃

The sources I have studied are not very exhaustive and I would like some help in finding the right path.

thanks 😉

I have already recommended the books by my Teacher Pietro Righini: L'acustica per il musicista and Lessico di acustica Ed. Zanibon. I find them fundamental

Hi everyone! People always talk about fifths and fourths, but how should major and minor thirds and sixths be descending or ascending?

If one wants to tune by thirds (creating the temperament), how does one proceed and what speed should they have?

Thank you very much

Lorenzo

ascending and progressive major thirds and sixths. The progression in the central section of the thirds F3-A3, A3-C#4, C#4-F4, F4-A4, at approximately 5-7-9-11 beats per second respectively. One can start already with the thirds. They are heard better. And then, perhaps, verify fourths and fifths.