56

There are physical and psychoacoustics reasons behind it. A vibrating string held by its two extremities can only vibrate at certain frequencies (cycles per second, expressed in Hertz, i.e. 440 Hz = 440 cycles/second), which relates to the characteristics of the string (e.g. its weight per unit of length, its flexibility) and how it is used (e.g. the ...


47

By definition this is not possible. Just intonation ratios are rational numbers, N/M where N, M are integers. Equal temperament is based on defining the smallest ratio as the n-th root of 2, 2^(1/n). For 12TET n = 12. What you are basically asking is if an irrational number can be made to exactly match a ratio of integers. This will never be possible....


38

The intervals between notes are "equal" not in the sense that the difference in Hz between them is the same, but the ratio a between them is the same. Let's say g is one semitone higher than f, then g = a f. Note Hz Ratio a to previous note, rounded to 3 decimal places A4 440.00 A#4 466.16 1.059 (466.16 / 440.0 = 1.059, and so on down the column)...


30

Let's get some terminology straight. In equal temperament, octaves aren't merely perfect; they are "just" or "pure". "Just" and "pure" are synonyms while "perfect" has a different technical meaning in music. Nevertheless, "just", "pure", and "perfect" happen to be the same when ...


28

This question seems to arise from a “linear” mental model of notes. C♭ C C♯ D♭ D D♯ E♭ E E♯ F♭ F F♯ G♭ G G♯ A♭ A A♯ B♭ B B♯ C♭ C C♯ Like a piano keyboard, but somehow with 31 notes per octave instead of 12. (Building or playing such an instrument is left as an exercise for the reader.) But instead, look at the notes in Circle of Fifths ...


28

The division of notes has to do with human perception and psychoacoustics. One description of human perception is the Weber-Fechner law, where a human will perceive equal changes in some sensory input, such as sound level or sound pitch, not by absolute level or value difference, but by the ratio of the change. e.g. larger values need a proportionately ...


28

Tuning in an ensemble is a skill in relative pitch, not absolute pitch. Players will hear what others are doing, and the group will come to a consensus organically. With instruments that are capable of microtuning adjustments, this will also lead to more just-tuned intervals. But what's important to note is that these tuning decisions happen on the fly, and ...


27

Note: For the sake of discussion, I'm limiting myself here to equal temperaments, which is the most common way of tuning keyboards. Other systems exist, of course, but would probably only confuse the matter. Why do B and C and E and F not have a sharp note between them? Simply because, acoustically speaking, there is no room in our current system for ...


27

Partly to allow the same, diatonic, piece to be played at different pitches as @Tim suggests. But also, I think, because music started getting more tonally adventurous within the SAME piece. When you start wanting to visit (say) the mediant key as well as just the dominant and subdominant, equal temperament is a must.


24

In principle, the answer is yes, with software instruments it is feasible to (re-)set the tuning so that you can realize music with modulation that stays in just intonation across these changes. The frequencies are directly accessible in sound synthesis environments like PureData or Overtone, and even just by setting the tuning information in a set of MIDI ...


24

Yes, but also due to the changes in piano construction. In some ways, a classical piece played on a modern piano might sound more true to the composer's original intent than the piano it was originally played on. Modern pianos are generally louder and brighter than the ones in the late 1700s and early 1800s. So loud passages, such as might be found in some ...


21

The other answers approach this from dividing the octave and showing that equal divisions must be irrational. Another way of looking at this is to consider whether we can compose an octave by successive multiplications with a rational number. The result is of course the same: we can't. Start with the Fundamental Theorem of Arithmetic: every integer ...


21

The temperament in a professional orchestra tends to vary between just intonation and equal temperament. If they are playing with a piano the strings will tend to adjust to the piano temperament where the difference would be audible and tend towards just intonation when they are playing alone. Woodwind instruments are built with the aim of playing in tune ...


20

What happens if you go down by the same steps: 440Hz 1 step down : 403.33Hz 2 steps down : 366.67Hz 3 steps down : 330.Hz ... 11 steps down : 36.67Hz 12 steps down : 0Hz 13 steps down : -36.67Hz So, using your "equally divided" logic, we are at zero Hz after 12 steps, and the next step beyond that is minus 37 Hz! What does that even mean? But ok, let's ...


19

Every note has a pitch, determined by the fundamental frequency of the sound wave that produces it. When you have two different notes, you have two different pitches, caused by two different frequencies. The distance between those pitches is called an interval, and corresponds to the ratio of the note's frequencies. For example, if one note is an octave ...


18

Some people seem to make the case that having some keys beat more than others (as is in the case in the older well-tempered tuning systems) is a feature not a bug. Yes, but I don't think that was ever a major consideration. Originally, all tuning systems just tried to give good approximation to just intonation (JI). At first just for a few neighbouring ...


17

You cannot even realize "just temperament" reliably when you are working with continuous-tone instruments like singers and trombones. Take a look at even something as old as J.S. Bach's mass in B minor, like the "Confiteor" which goes off-tonality somewhere after 2:30 (in this recording) and loses tonal center rather thoroughly between 3:00 and 4:00. The ...


16

You are exactly correct that it is the logarithmic nature of pitch that causes this effect. In cases like this, I find that a picture is helpful. Here I've labeled equally spaced octaves (1200 cents) along the x-axis (representing pitch). I've then labeled the corresponding frequencies on the y-axis as multiples of some arbitrary base frequency f. Note that ...


16

Yes, if not far more than 7 when you consider pitches outside of the diatonic scale and variations on A440! By "stray slightly to make a note sound more in tune," you're talking about just intonation. In 12-tone equal temperament, in C major, let's say C is our "zero-point." For the ensuing discussion, all pitches are based off of this 12TET where C is "0 ...


16

It is because B and C are closer together than the difference between B and B♯ and the difference between C and C♭. That is, they are all some sort of semitone apart. Alternatively, note that B♯ is also higher than C♭ in every 12-tone temperament, because in the 12-tone system B♯ is the same pitch as C, while C♭ is the same pitch as B. but this doesn't ...


16

Simply so that any music could be played in any key and it would sound the same. Problem with tuning to another temperament means that pieces sounded particularly good in some keys, and particularly bad in others. And re-tuning often isn't a quick answer - especially on instruments such as piano! Non-fretted stringed instruments, such as violins, trombones ...


15

As I understand the question, this is pure mathematics: No it is impossible. No matter, how many divisions you have, say n, the step width will always be nth root of two and therefore an irrational number. The just relations are rational numbers, so there will always be approximations, but the more you choose, i. e. the higher n is, the closer you will be ...


13

It is significant when you are trying to tune an instrument by ear, using the purity of intervals as your guide. You (and the pages you link) refer to jumping up 7 octaves vs. 12 fifths, but don't forget that any notes you reach that way can also be brought down by one or more octaves as well. To illustrate this, let's bring all the notes down into the same ...


13

It's a bit more complicated than may appear at first glance. Within a single key, if Just Intonation makes the I,IV, and V chords all (4,5,6) ratios, the ii chord will be off. The other question is what note to play as a melody note. Often, melodies are somewhat independent of the underlying chords (at least in CPP if not in Jazz and other Pop theories). I ...


13

The earliest use of equal temperament was on fretted instruments with fixed frets. The ratio of 17:18 for the string length for successive frets is a good approximation to equal temperament. The errors were well within the tolerance of other intonation issues such as non-uniform gut strings, and the different amounts of string bending on different frets ...


12

Standard tuning for solo violin in classical music is just intonation. Tune the A string and, from there, tune the other strings with just-intonated perfect fifths. Some times, as a compromise you may need to tune the violin temperate, for example when you need to play many open strings in duo/ensemble with a instrument not capable of just-intonation. ...


12

A trivial answer : yes. When I was quite young I wrote a computer program to spit out a succession of 'beeps' at random frequencies not related to any musical scale; I suspect many people who have a computer and a bit of an interest in music have done the same. In practice how close you could get to infinity (!) would be limited by the resolution at which ...


11

My answer is no, it isn't really possible to use a fingering technique to play an A440 recorder at A415. No professional would even try; they would instead, as Wheat noted, have a real A415 instrument or, if extremely confident and well rehearsed, transpose on-the-fly down a half-step. One can indeed bend most notes up or down quite a bit using half-holes, ...


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