Equal temperament

[Jens]

So I have a question that is pretty esoteric, so I doubt I'll get any responses, though I know there are people here who seem to know about everything, so I'll give it a try. In equal temperament, a fifth interval is slightly (I think about 2 cents) lower than the natural fifth, which would be precisely 1.5 times the fundamental. Now the question is, when people sing, say a capella, do we tend to sing:

-In the Pythagorean scale, so with the fifth sung as 1.5 times, or
-In equal temperament, as a result of being used to it from pianos, etc. or
-The difference is not enough to be important, so it might be one or the other or somewhere in between.

[Jens]

Actually, I guess I'm not the only one to have wondered about this. I even found a study about it.

https://pubmed.ncbi.nlm.nih.gov/17613791/

Apparently, the first is correct, that they tend to use non-tempered tuning.

[profloater]

It should not be A surprise that a singer would choose natural intervals rather than well tempered ones, because they are already aware of the key in which they are singing. A piano keyboard is usually adjusted so that the intervals sound correct or approximately correct in every key. It has often been noticed that a piece changes its feeling subtly when placing a different key on the piano because equal tempering does not in fact make every key sounds exactly the same. There is more going on in a piano because the strings are struck usually at 1/12 of their length to make the best compromise between the string frequencies and the bar frequencies of the high tension strings. This compromise is what gives the piano it’s unique sound which can then be altered by different makers by tiny adjustment of these basic parameters. This is also why a piano tuned below concert pitch, in order to make it keep in tune for longer, has a different tone which can be recognised by musical experts. As you reduce the tension in the steel string the ratio between the string frequencies and the bar frequencies is changed and therefore the compromise set by the manufacturer for concert pitch is changed. This complicates the question of what a fifth, for example, sounds like in every key that you can play on the piano.

[Trebuchet]

Upon seeing the thread title, I was expecting the thread to be about psychology....

[Jens]

Upon seeing the thread title, I was expecting the thread to be about psychology....

Yeah, I can understand that. "Equal temperament" is the kind of word that people who are seriously interested in music know, but otherwise it is pretty hard to understand!

[Hornblower]

The Pythagorean intervals give the sweetest sounding harmony as long as you stay in one key, but they get gross when you try to play a piano in some other keys. The equal tempered system makes it only slightly out of tune, by the same amount in all keys.

There are additional complications from the fact that piano strings are not perfect harmonic oscillators, meaning the overtones are not perfect intervals above the fundamental tone. Is I am not mistaken, tuning the octaves to a strobe tuner does not make the best sound. When a skilled piano tuner tweaks them by ear, they end up slightly wider than an exact 2:1 interval.

[ngc3314]

I'e read some reviewers note that ensembles consisting only of string instruments or trombones tend to shift to the "untempered" intervals if not influenced by tempered instruments.

[Robert Tulip]

Yeah, I can understand that. "Equal temperament" is the kind of word that people who are seriously interested in music know, but otherwise it is pretty hard to understand!

The Well Tempered Clavier by JS Bach includes 48 preludes and fugues, composed in every key to illustrate the equal tempered tuning for the keyboard.

Choral music requires just intonation to produce accurate harmonic overtones. Looking for sources, here is a PhD thesis that recommends "the use of just intonation; an approach in which notes are tuned to overtones of the harmonic series. By tuning this way, beats are avoided, and the sound is perceived as more consonant, resonant, and in tune."

A friend with expertise in this topic recommended this excellent book Tuning Timbre Spectrum Scale, published by Springer-Verlag by Bill Sethares, Professor of Electrical and Computer Engineering in Wisconsin.

[Jens]

I'e read some reviewers note that ensembles consisting only of string instruments or trombones tend to shift to the "untempered" intervals if not influenced by tempered instruments.

That makes sense. It's kind of equivalent to the point about people singing a cappella, since our voices are like violins and trombones, so not tempered.

[Jens]

The Well Tempered Clavier by JS Bach includes 48 preludes and fugues, composed in every key to illustrate the equal tempered tuning for the keyboard.

Choral music requires just intonation to produce accurate harmonic overtones. Looking for sources, here is a PhD thesis that recommends "the use of just intonation; an approach in which notes are tuned to overtones of the harmonic series. By tuning this way, beats are avoided, and the sound is perceived as more consonant, resonant, and in tune."

I would intuitively guess that choruses, without piano accompaniment, would naturally go to the just intonation as they wouldn't have the slightly out-of-tune instrument influencing them.

[Robert Tulip]

I would intuitively guess that choruses, without piano accompaniment, would naturally go to the just intonation as they wouldn't have the slightly out-of-tune instrument influencing them.

Acapella singing such as motets and masses generates a depth of resonance through the overtones that come from just intonation due to the simple multiple frequencies.

The major scale in just intonation has notes in the ratio 24:27:30:32:36:40:45:48. Or 1 9/8 5/4 4/3 3/2 5/3 15/8 2. With equal temper these numbers are functions of the twelfth root of two.

I made the attached graph of the errors produced by equal temper against these harmonic ratios. Chords with the biggest variance from just intonation can sound out of tune, such as the chord DFA on a guitar tuned to the key of C.

[plant]

https://www.youtube.com/watch?v=6NlI4No3s0M

comparison of just intonation vs equal temperament

[Jens]

I made the attached graph of the errors produced by equal temper against these harmonic ratios. Chords with the biggest variance from just intonation can sound out of tune, such as the chord DFA on a guitar tuned to the key of C.

Though guitars are equal tempered in reality. It's actually a cause of problems, because guitarists (I have been guilty of this myself) try to tune their guitars by tuning (for example) that natural harmonic taken at the seventh fret of the fifth string with the (tempered) fifth fret of the sixth string, and then the tuning gets off.

[Jens]

Acapella singing such as motets and masses generates a depth of resonance through the overtones that come from just intonation due to the simple multiple frequencies.

The major scale in just intonation has notes in the ratio 24:27:30:32:36:40:45:48. Or 1 9/8 5/4 4/3 3/2 5/3 15/8 2. With equal temper these numbers are functions of the twelfth root of two.

Also, there's a chart in Wikipedia that does a similar thing (the numbers seem to be slightly different from yours, but I haven't tried to calculate why). And it also includes the five notes that you left out of your chart.

https://en.wikipedia.org/wiki/Equal_temperament

[Robert Tulip]

Also, there's a chart in Wikipedia that does a similar thing (the numbers seem to be slightly different from yours, but I haven't tried to calculate why). And it also includes the five notes that you left out of your chart.

https://en.wikipedia.org/wiki/Equal_temperament

Thanks, I found the chart at https://en.wikipedia.org/wiki/Equal_...ust_intonation and the numbers are the same as the ones I gave.

[21st Century Schizoid Man]

In equal temperament, a fifth interval is slightly (I think about 2 cents) lower than the natural fifth, which would be precisely 1.5 times the fundamental.

1.4983:1 vs 1.5:1

With instruments that are physically tuned (until a few decades ago, all of them), you have to choose one system or the other. Switching is costly.

However, with electronic instruments, you could switch the tuning with a push of a button. I wonder if any of them do that.

[Robert Tulip]

with electronic instruments, you could switch the tuning with a push of a button. I wonder if any of them do that.

Here are some references
https://www.midi.org/midi-articles/m...nation-systems
https://www.microsoft.com/en-au/p/ju...ot:overviewtab
https://patents.google.com/patent/WO1995022140A1/en
http://www.dbdoty.com/Words/What-is-...ntonation.html

[Jens]

1.4983:1 vs 1.5:1

With instruments that are physically tuned (until a few decades ago, all of them), you have to choose one system or the other. Switching is costly.

Well, not exactly true, because there are instruments where you can adjust the tuning on the fly, like trombones. With fretless stringed instruments it's a bit of a pain to adjust the string tunings, but you can easily change the finger location to adjust the tuning.

[21st Century Schizoid Man]

Well, not exactly true, because there are instruments where you can adjust the tuning on the fly, like trombones. With fretless stringed instruments it's a bit of a pain to adjust the string tunings, but you can easily change the finger location to adjust the tuning.

OK, trombone should work.

A fretless string instrument, though, you would need to use a finger even for a note that would normally be "fingerless", would you not? I mean, the strings would not be tuned in the right ratio to each other.

[Robert Tulip]

The violin family use just intonation - eg violin tuning is G D A E. Using A 440 Hz that means G 195.555 D 293.333 A 440 E 660

[Jens]

OK, trombone should work.

A fretless string instrument, though, you would need to use a finger even for a note that would normally be "fingerless", would you not? I mean, the strings would not be tuned in the right ratio to each other.

Yes, that's exactly what I meant to say when I wrote "With fretless stringed instruments it's a bit of a pain to adjust the string tunings, but you can easily change the finger location to adjust the tuning." I guess my explanation wasn't so clear. Yes, it would be a problem with open notes (you would have to adjust the tuning pegs).

[21st Century Schizoid Man]

Yes, that's exactly what I meant to say when I wrote "With fretless stringed instruments it's a bit of a pain to adjust the string tunings, but you can easily change the finger location to adjust the tuning." I guess my explanation wasn't so clear. Yes, it would be a problem with open notes (you would have to adjust the tuning pegs).

Come to think of it, is it impossible to change between the two systems on a fretted stringed instrument? You could change the tuning of each string, but the increments would be off.

[Jens]

The violin family use just intonation - eg violin tuning is G D A E. Using A 440 Hz that means G 195.555 D 293.333 A 440 E 660

You can tune a violin in different ways, and I think professionals would tune it differently, depending on whether they are playing in a string ensemble or accompanied by a piano, for example. If you are playing with a piano you would not want to use just tuning.

[Robert Tulip]

Here is an explanation of how violinists tune their instruments in an orchestra

"The A string is played by the concertmaster (lead violinist) to make sure all members and sections of the orchestra are tuned properly. As a violinist, you use the A that is 'passed' to you to tune your other strings. To do this, you first make sure the A string is in tune, then play the A and D strings together, listening for the perfect fifth interval to ring in tune. Then play the D and G strings together, followed finally by the A and E. In order to tune like this you must have a good knowledge or what the strings sound like, and what a perfect fifth sounds like too. Perfect fifths tend to sound very resonant, so you may be able to tune this way if you listen carefully to the ringing sound, or by looking at how the strings vibrate when you play them together. Start listening for these notes when you tune as a beginner as it is a great skill to be able to internalize a set note and understand the exact perfect frequency of the note by ear." (source)

The point here is the strong resonance of the just intonation interval that creates the perfect chord. And even if a violin is playing with a piano, adjustments to the pitch of each note are achieved by ear rather than by changing the pitch of the open string, which is always traditionally tuned to just intonation.

[7cscb]

I have a cousin who hears the tonic in most sounds. But he's been playing piano since the age of three. As with all our senses, precocity helps but honing through practise has a remarkable impact. I can identify the intervals but I recall being taught to recognize octaves, minor vs major and other basic sounds. They're so obvious now. Still, for me, well tempered or not is essentially academic.

[Jens]

The point here is the strong resonance of the just intonation interval that creates the perfect chord. And even if a violin is playing with a piano, adjustments to the pitch of each note are achieved by ear rather than by changing the pitch of the open string, which is always traditionally tuned to just intonation.

See, I'm under the impression that that's no always the case (that the open strings are always tuned to just intonation. It's true that if you use relative tuning by using the fifth, you will end up that way. But this article seems to be saying that there are different ways to tune the open strings depending on the situation. If you are playing with a piano, then some of the open strings will be out of tune with the piano if you use the relative method. In the article they describe several ways and recommend a general compromise tuning. I could be wrong about this--actually I'm a guitarist, and we have no choice but to use equal temperament because of the frets--but that's what the article seems to be saying.

[Jens]

Come to think of it, is it impossible to change between the two systems on a fretted stringed instrument? You could change the tuning of each string, but the increments would be off.

Yes, I think that is is impossible. I think you could tune the individual strings to be in just intonation, but then the same note played on different strings would be off. But I could be wrong.

[Jens]

I have a cousin who hears the tonic in most sounds. But he's been playing piano since the age of three. As with all our senses, precocity helps but honing through practice has a remarkable impact. I can identify the intervals but I recall being taught to recognize octaves, minor vs major and other basic sounds. They're so obvious now. Still, for me, well tempered or not is essentially academic.

I'm afraid to say I'm not exactly sure what you mean by "the tonic in most sounds." A tonic is the basic tone in a scale, so like the "do" in "Do, a deer." If you just hear a single C, you can't know if it's the tonic in a scale of C or if it's part of a G major scale, for example. You would have to hear a melody to be able to tell what the tonic is. Also, I don't think that octaves are generally things that we need to be trained to recognize. I think even children when they are singing a song will change octaves to suit their voices.

[7cscb]

I'm afraid to say I'm not exactly sure what you mean by "the tonic in most sounds." A tonic is the basic tone in a scale, so like the "do" in "Do, a deer." If you just hear a single C, you can't know if it's the tonic in a scale of C or if it's part of a G major scale, for example. You would have to hear a melody to be able to tell what the tonic is. Also, I don't think that octaves are generally things that we need to be trained to recognize. I think even children when they are singing a song will change octaves to suit their voices.

Hello Jens.

Big Doh! I suppose he hears the fundamental tone. Just like when I hear a string. But he has perfect pitch.

To my point, children sing off-key. I knew chords around the campfire. Then I played bass many years in working country rock bands and started learning intervals, scales and chords in my early 20s. Sure, I could (barely) sing Do-Ré-Mi but there was no comprehension. I remember repeatedly exercising recognition of octaves and intervals because it was the only way I could learn them. The notion we intuit octaves is, imo, wrong. I get that in some cases of tones and geometry, resonance causes octaves (and others) to become even more obvious. But distinguishing and recognizing and understanding those tones is different.

cheers,

[Hornblower]

If I am not mistaken, violinists avoid open strings when the fingered notes are tweaked to fit the music. The treble clef A can be played by fingering the D string and tweaking it as needed.

My horn is neither exactly just in any key nor exactly equal tempered. That is the nature of the beast with wind instruments. I just tune it slightly sharp and bend the notes down as needed. That is done with the lips along with the right hand in the bell, and becomes a reflex like singing in tune. Because of the acoustic characteristics it is much easier to bend the pitch down than up. A trombone must be tuned so none of its first position notes are flat.