Equal Temperament

It is a type of convention, a system of pitches (frequencies)

tempering steel is a bit difficult O.o (at most it can temper a pencil). What you are referring to is called tempered steel.

Equal temperament would be the formation of the musical scale, i.e., the division of the monochord. If you are in front of a piano, any sound it can produce belongs to that scale.

I don't remember who created this division, but Bach in the early 1700s, through a work (The Well-Tempered Clavier), split into the 24 keys, testing the potential of the "new model" that previous ones did not have.

Giove wrote:

What is being referred to when one speaks of temperament? ... more equal?

It is a completely theoretical musical system through which the octave is divided into 12 semitones identical to each other, eliminating the so-called "just" or "pure" intervals. Before the adoption of this system, therefore until the mid-1700s, it was not possible, for example, to modulate within the same piece with the same ease as would be done later, simply because instruments (particularly keyboard instruments) had to be tuned taking into account the key of the piece being performed, with the consequence that other keys sounded decidedly out of tune.

JSB's Well-Tempered Clavichord was not written for an instrument tuned according to equal temperament, but rather for an instrument tuned according to one of the many possible unequal temperaments (there are many, but I do not know them...) which, however, allowed playing in all keys while appreciating their differences (something that we today on an instrument tuned with equal temperament can no longer listen to...).

Equal temperament or tempered system is the division of the octave into 12 equal semitones, thus identifying the chromatic semitone with the diatonic one.

The tempered scale is therefore a "compromise" that the tuner "embraces" during the tuning of the piano.

Pianoaccordatore wrote:

Equal temperament or the tempered system is the division of the octave into 12 equal semitones, thus identifying the chromatic semitone with the diatonic one.

The tempered scale is therefore a "compromise" that the tuner "embraces" during the tuning of the piano.

...and what about the octave? Since you take it as the starting point for your reasoning?

Giove wrote:

What is being referred to when one speaks of temperament (...to put it briefly, tempered steel comes to mind)? ... more equalized?

It is mathematical proof that music is mathematics, but musicians understand nothing about mathematics.

In fact, even today they cannot simply explain the difference between pre-equal temperament scales and equal temperament.

The former are based on rational numbers, while the latter on irrational numbers..

A musician's excuse could be that irrational numbers, already intuited in the time of Pythagoras, only began to be digested around the 1600s, and indeed shortly thereafter the equal system arrived.

An octave is that interval between two notes in which one note has a fundamental frequency (what our perception calls "pitch") exactly double that of the other. I recommend http://it.wikipedia.org/wiki/Temperamento_equabile

Just a terminological clarification...

"Temperament" is not the scale itself, nor the main tuning system. Temperament is the subsequent adjustment of certain dissonant and poorly consonant or unequal intervals. In other words... the division of the octave into 12 semitones is essentially made according to Pythagorean tuning, which is a tuning, a constant mathematical-acoustic criterion, let's say. When, after tuning, it is discovered that some of these intervals are not pleasing (for the myriad reasons that have existed throughout history), then certain intervals can be tempered. In the typical tuning system of European classical music, the equal temperament system, all intervals are tempered so as to be made, mind you, equal, identical to one another. There is not only equal temperament, to be clear; in the 1700s there were many different temperaments. And The Well-Tempered Clavier was most likely NOT written for an instrument tuned with equal temperament :-) I am not an expert, but I have read several times that this has been established for a long time; in Bach's time, there were many temperaments called "well temperaments" that adopted interval adjustment criteria different from those of equal temperament. In fact, equal temperament "detunes" all intervals, privileging their equality in order to allow, for example, greater ease of modulation. Other temperaments elevate more important and less important intervals, in a sense, privileging consonance or dissonance from time to time. But it is complex to explain...

The octave, as was being said, is a relationship. Ethnomusicologically, it is considered a universal, meaning a musical element present in almost all known musical cultures. It is not clear why this happens, but it probably has to do with the acoustic fact that the first harmonic of every sound is the upper octave. That is, if I play middle C, within that sound wave there is "in the background" the C an octave higher. This leads to the musical consequence where the most important consonance in music history after the unison is the octave :-) it may seem trivial, and it is, but this is probably the reason why it is very easy to tune a note to the upper or lower octave (that is, even people who are generally out of tune can do it). This also means that Pythagoras did not invent the octave, but it was likely a concept already present in musical practice. Probably...

Dilettante wrote:

...and the octave what is it? Since you take it as a starting point for your reasoning?

The octave, for example, is the interval between two consecutive Cs, or two consecutive Fs (or whatever you want).

ClaudioBernardino wrote:

An octave is that interval between two notes in which one note has a fundamental frequency (what our perception calls "pitch") exactly double that of the other. I recommend http://it.wikipedia....amento_equabile

Yes, but do you understand that if someone doesn't know what an octave is and you tell them like that, they will continue not to understand?

Marebu wrote:

It is the mathematical proof that music is mathematics, but musicians understand nothing about mathematics.

No, quite the opposite, since if you use a natural temperament, by the time you reach the end of the circle of fifths you have unusable intervals.

With equal temperament, you have neither pure fifths nor pure thirds (that is, with ratios like 3/2 for the fifth, for example).

Marebu wrote:

In fact, even today they cannot simply explain the difference between the scales of the pre-equal temperament era and equal temperament.

...huh?

Marebu wrote:

The former are based on rational numbers, while the latter on irrational numbers..
A musician's excuse could be that irrational numbers, already intuited in the time of Pythagoras, only began to be digested around the 1600s and indeed shortly after that the equal temperament system arrived.

Equal temperament comes much later

Dino wrote:

Equal temperament comes much later.

When?

Is the WTC not made with that?

Marebu wrote:

When?

Is the WTC not made with that?

Sorry, but the first volume is dated 1722 ... the second 1744... so even further ahead.

And then I am not so sure that Bach used our temperament ...

Marebu wrote:

Isn't the WTC made with that?

No.. well-tempered has nothing to do with equal temperament (it simply means that all keys were playable).

Bach often used a temperament close to Kirnberger and anyway it is still an open debate

http://en.wikipedia....empered_Clavier

Wikipedia would have been enough eh!... the part on "intended tuning" and "Title page tuning interpretations"

Fortunately musicians "don't understand anything" about mathematics.... who knows what nonsense they would write if they did...

Ettore wrote:

Fortunately musicians "don't understand anything" about mathematics.... who knows what nonsense they would write if they did...

For me, the length of time that passes from the discovery of irrational numbers to their acceptance and use for the solution of the age-old problem is significant!

Marebu wrote:

For me, the span of time that passes from the discovery of irrational numbers to their acceptance and use for the solution of the age-old problem is significant!

Which age-old problem? Which solution? Even temperament is not a solution, because there is no problem. Here we are talking about styles! You won't believe it, but Vivaldi played in natural intonation is more beautiful and, as they used to say, other temperaments like the good temperaments of Bach's era already resolve for themselves the possibility of writing in all major or minor ways.

Even temperament allows, rather, modulation to distant keys (modulation, not tonicization, eh) even within the same piece. And this is a Romantic "necessity"...

Marebù, even if there could have been suspicions about who I was... I was hoping for an unfortunate namesake... you are perfectly recognizable by what you write; in fact:

Marebu wrote:

For me, the span of time that passes from the discovery of irrational numbers to their acceptance and use for the solution of the long-standing problem is significant!

Wrong here too... the problem is a non-problem. In fact, it is dictated by the development of the only instrument that actually uses equal temperament, which is the piano. Before its appearance, there was no need for equal temperament, just as there is none now for anything that is not a piano. I don't know if you realized that a violin is tuned by PURE fifths (that is, NOT according to equal temperament). No orchestra plays in equal temperament.

Other trivialities to propose, or is this fine? No, because to err is human, to persevere is diabolical.

Carlos wrote:

Yes, but do you understand that if someone doesn't know what an octave is and you tell them like that, they will continue not to understand?

D'andiamo dicendo che l'ottava so cos'è in termini musicali, meno in termini acustici

Am I mistaken, or does even the piano not tune exactly equitably?

I seem to remember that strings are not perfectly elastic as a physical model would require, and by retaining a bit of stiffness they generate harmonics slightly sharper than theoretical ratios. Therefore, when tuning keys in octaves, if one wants to avoid beats, the octave must be just a tiny bit wider than the theoretical 2:1 ratio, so that the upper key is in unison with the harmonic generated by the lower key, which is indeed a little sharper than the pure octave.

?