38 votes
Accepted

If equal temperament divides an octave into 12 equal parts, why are the hertz differences not the same but 12ths of two?

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 ...
Sagebrush Gardener's user avatar
31 votes

From the piano tuner's viewpoint, what needs to be done in order to achieve "equal temperament"?

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 "...
trw's user avatar
  • 4,833
30 votes
Accepted

Why is B♯ higher than C♭ in 31-ET?

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 ...
dan04's user avatar
  • 837
29 votes

If equal temperament divides an octave into 12 equal parts, why are the hertz differences not the same but 12ths of two?

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 ...
hotpaw2's user avatar
  • 1,802
28 votes
Accepted

Do the violins imitate equal temperament when accompanying the piano?

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 ...
MattPutnam's user avatar
  • 22.4k
26 votes
Accepted

Why did equal temperament become the standard tuning system for keyboard instruments?

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 ...
Laurence's user avatar
  • 92.2k
24 votes

With modern electronic technology is temperament unnecessary?

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 ...
Dave's user avatar
  • 17.8k
24 votes
Accepted

Do classical pieces sound different today than the originals due to temperament?

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 ...
Todd Wilcox's user avatar
  • 56.6k
23 votes

Is it mathematically possible to create an equal temperament which matches just interval ratios?

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 ...
phoog's user avatar
  • 22.3k
21 votes

Do the violins imitate equal temperament when accompanying the piano?

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 ...
PiedPiper's user avatar
  • 20.9k
20 votes

If equal temperament divides an octave into 12 equal parts, why are the hertz differences not the same but 12ths of two?

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,...
piiperi Reinstate Monica's user avatar
19 votes

What would've given more accurate consonant ratios than 12?

Before there were 12 notes one had 8 different notes (modulo octaves), formed by three different hexachords. Since ancient times musicology had the concept of tones and semitones, and these hexachords ...
Lazy's user avatar
  • 19.9k
18 votes

Why did equal temperament become the standard tuning system for keyboard instruments?

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 ...
leftaroundabout's user avatar
17 votes

With modern electronic technology is temperament unnecessary?

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 ...
user31386's user avatar
  • 171
17 votes

Why is B♯ higher than C♭ in 31-ET?

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 ...
phoog's user avatar
  • 22.3k
16 votes
Accepted

Whereabouts is G?

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....
Richard's user avatar
  • 84.3k
16 votes

Why did equal temperament become the standard tuning system for keyboard instruments?

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 ...
Tim's user avatar
  • 192k
15 votes

Is it mathematically possible to create an equal temperament which matches just interval ratios?

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 ...
guidot's user avatar
  • 11k
15 votes

When you specify an EDO (Equal Divisions per Octave) value, is it always a whole number?

Wendy Carlos used her Alpha scale at about 15.385 steps per octave and Beta scale, approximately 18.8 steps per octave on "Beauty in the Beast". They give up octaves altogether in favor of ...
Theodore's user avatar
  • 2,815
14 votes
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Why does this tonal scale from 1737 have both F and E♯?

Wow! What you've stumbled upon is not just an impressively early use of a tone-cluster; it's also an impressively late survival of a system of chromatic notation, universal in the Middle Ages and ...
Mirlan's user avatar
  • 1,786
13 votes
Accepted

"∞-TET": Is there music that doesn't use discretely pitched notes?

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 ...
Нет войне's user avatar
13 votes

With modern electronic technology is temperament unnecessary?

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 ...
ttw's user avatar
  • 25.2k
13 votes

Was equal temperament caused by the invention of the piano, or was it inevitable?

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 ...
guest's user avatar
  • 211
13 votes

Outside of Equal Temperament, what decides the spelling of notes in a major scale?

I think the matter wasn't about intonation, but is historically related to the fact that the flat sign was developed first and Medieval music didn't have a sense of fix pitches for the staff. Long ...
Michael Curtis's user avatar
12 votes
Accepted

When you specify an EDO (Equal Divisions per Octave) value, is it always a whole number?

A fractional value would represent an equal division of multiple octaves rather than a single octave. 12.5 EDO, for example, would be equivalent to 25 ED2O (equal division of two octaves). As an ...
Aaron's user avatar
  • 86.9k
11 votes

With modern electronic technology is temperament unnecessary?

"just intonation better than equal temperament" Judgement call there. When instruments are slightly off perfect ratios, there can be very appealing beating and chorus effects. Piano strings are ...
Phil Freihofner's user avatar
11 votes
Accepted

Cello tuning by harmonics: Just intonation vs ET

It does apply to any instrument, but for cello it's perhaps most notable because we so often play with instruments that have E-strings: violins, or else guitars. If you tune a cello in Pythagorean ...
leftaroundabout's user avatar

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