Equal Temperament for the Violin
I learnt about the different tuning systems recently and a question occurred to me. Many many beautiful pieces of music have been written for the piano, which is tuned in equal temperament. If this tuning system is good enough for the piano, why isn't it good enough for the violin? It would simplify things so much if you only had to learn a single pitch for each note.
Concert pianos are almost never tuned in perfect equal temperament. For large concerti, the piano is always tuned specifically for the work being played.
Equal temperament is a tuning system where every single half step is the exact same size. While that works for a keyboard instrument so that it can approximate every single key, on a stringed instrument, everything just sounds out of tune because none of the intervals form the proper ratios for sympathetic vibrations to take place.
The topic of intonation comes up frequently. We don't think about all that math while playing. On a practical level I like to think of each note being a very close cluster of three pitches; low-neutral(tempered)-high. We can bend the notes high or low according to the harmonic or melodic context. The difference between tempered and "perfect" tuning is small, almost always close enough. That's why it works so well on the piano. For violin in first position the difference between a tempered pitch and a bent note is hard to hear and very hard to feel; ~ 10 cents, or 1/10 of a half-step, 1/10 of 11 mm, or ~1 mm (!). The difference between the high and low spots is the "comma", 22-24 cents, or about 2 mm, which can be heard and felt. In practice, just don't bend notes in the wrong direction and people will hear good to excellent intonation. The differences are more discernible on the Cello, with its much longer string length.
AND---What have you observed to be the effect of vibrato and the vibratos of different players?
Andrew, et al,-- Ah yes, vibrato. Vibrato is as wide or wider than the comma, so it fixes a lot of the problem. I suspect that when the vibrato is fast enough the mind of the listener will pick out the pitch that it Wants to hear. The majority opinion is that the vibrato should start on the flat side of a note. Some insist that all of the vibrato is on the flat side, but if you watch your vibrato in a mirror you will see that some of it moves sharp. If the vibrato mechanics are wrong, it can distort the posture of the hand and cause worse intonation instead of better.
Cotton, I didn't know concert pianos were tuned for specific pieces, interesting. As far as I understand, there are three (or four, just intonation?) different pitches to learn for each note, except for the notes of the open strings.
World class soloists make conscious decisions about playing notes "flat" or "sharp". It is one of the advantages of a musical instrument with a frequency range that can be easily varied with a simple shift of a finger.
> Why doesn't this matter or why does
You just need to listen to someone playing Bach in equal temperament on a violin and you'll understand. Any of y'all want to volunteer?
continued - I sometimes do this experiment in my classes: Tune my Viola to the piano A, then tune to perfect 5ths (3/2 frequency ratio), compare the viola pitches to the piano pitches and ask the students to raise their hands when they hear a difference. No one hears the D or G as being different, a few hear the difference on the C, which will be 6 cents (6/100) flatter than the piano C. The limit of pitch discernment among trained, adult musicians has been reported to be about 5 cents. Depending on the key, chamber music cellists and violists will sometimes tune the C up to match the piano C. The "error" between the cello C and the violin E will 8 cents, which is not bad. When we play two pitches as double stops our margin of error gets a lot better because of beats or difference tones, which is a different topic. That's why we can tune perfect 5ths, octaves, thirds, sixths, as double stops.
Julie, it'd be awesome if someone could record themselves playing a piece in equal temperament. I'd do it myself but I can only play in the intonation system I developed, it involves random placement of the fingers and lots of prayer.
Basically the violin needs to be in tune with itself. That's what Scott is driving at as well. When you play "G" on the E string (2nd finger), the G should "ring" by resonating with the two-octave harmonic of the G string. If you're off by half a millimeter the ringing is lost.
Keyboards weren't tuned in equal temperament till the 20th century, originally they were tuned to favour simple keys over complex keys with lots of sharps or flats, even Bach's Well Tempered Klavier was not intended for equal temperament.
equal temperament was proposed in the 1700s but it wasn't put into practice till much later, I don't need to google it, I'm very well informed on the topic, you don't seem to be.
Unless you have some semantics at work, Lyndon, you seem to be saying that Chopin, Brahms, Schumann, List wrote for JI keyboards. I suppose you are using ET in its strictest sense, where every semitone is exactly the twelfth root of 2?
well into the 20th century, piano tuners where still tuning to give some advantage to simpler keys, true equal temperament didn't really exist until the strobotuner
Violinists who play with Pythagorean intonations (or close to it, let's not play the word game) that came to my mind are David Garrett, Kristof Barati. Most soloists, to my ears, sound closer to equal tuning, including double stops.
Equal temperament has every single interval out of tune, some slightly, the thirds, quite a bit.
"Equal temperament has every single interval out of tune, some slightly, the thirds, quite a bit."
John asked, "How do I go about learning intervals?"
" Unless you have some semantics at work, Lyndon, you seem to be saying that Chopin, Brahms, Schumann, List wrote for JI keyboards."
meantone dies out in the early 1700s, what survived into the 1800s is well tempered tuning which make some fifths perfect so that thirds in simpler keys are not as far out of tune as equal tempered, the problem with these tuning is the thirds in complex keys are worse, not better, but this was considered an acceptable compromise since the complex keys were not used as much
Paul, I'm actually on Wohlfahrt book 1 right now. I've seen the video tutorials of Mr. Sassmannshaus, they're what made me want to learn more about intonation systems and music theory in general. My teacher never discussed these things with me when I was taking lessons for some reason (I'm picking up the violin again after a 6 year hiatus). I'll remember your point about listening to professional recordings.
Don't worry about it. I would call these fine intonation differences advanced class topics. Trust your ear. For a melody imagine how it would sound if you were a singer. Professional singers spend a lot of time on interval study. If you play tight half-steps, fingers touching, your whole steps will be OK. There are three aspects to developing good intonation; hearing, technical facility, and theory. A lot of really good violinists don't know about the theory, but it doesn't stop them from playing in tune.
Thanks a lot Joel.
I agree with Joel. The reason kids are taught to play familiar tunes is because you have some internal knowledge of how they should sound. In a major scale, the third should sound high and bright. It will be if the first two intervals (whole steps) are wide. We're talking about subtle differences here, so Joel's strategy of focusing on the half-steps (because then your fingers will touch) is very practical.
Thanks Paul, the responses here have been enlightening.
"Beating and lack of beating are a quirk of physics"
Actually, the faster the beat, the closer the vibrations and the closer to in-tune.
"Actually, the faster the beat, the closer the vibrations and the closer to in-tune." That doesn't jibe with what I learned in college physics. If you have two tuning forks at 440 and 441 Hz, you will hear "beats" at 1 Hz. If they are 2 Hz apart, you will hear beats at 2 Hz (twice per second, i.e., faster). You can prove this to yourself by setting up the respective cosine functions in Excel and just adding them together. You can hear it too when your piano tuner is working on your unisons.
Thanks Paul, Yeah, apparently I got that backwards.
If you have two tuning forks at 440 and 441 Hz, you will hear sum and difference frequencies at 881 Hz and 1 Hz. You can't hear 1Hz as a pitch: what you hear is traditionally known as a beat. Physics doesn't recognise things as being in tune. Physics doesn't wince if something is out of tune. We use lacking-a-beat-frequency as a definition of "in tune": we don't use in-tune as our definition of "lacking a beat frequency". If two tuning forks are both at 440Hz, you will hear their sum 880 as though it were a harmonic. But it would be subliminal.
Here are two books on this subject that I have enjoyed reading: Ross Duffin's "How Equal Temperament Ruined Harmony and Why You Should Care" is one, and the other is Alain Danielou's "Music and the Power of Sound."
Add Temperament by Stuart Isacoff to that list.
The sum of cos(x) + cos(x) is not cos(2x).
Cos(x) + cos(x) = 2cos(x), in the same way that apple + apple = 2 apples.
"Hello everyone, I learnt about the different tuning systems"
There are a large number of ear training lessons on youtube. Several teach intervals by referring to well-known songs.
I had to skip a few funny things about ET and pianos due to lack of time. So please forgive me if this should have been mentioned already by someone else, but since I've been a frequent user of multiple pianos in a former life I'd really like to come back to that question...
"Cos(x) + cos(x) = 2cos(x), in the same way that apple + apple = 2 apples.
Your ears (or your brain) are non-linear mixers. Non-linear mixers generate sum and difference frequencies. Off the top of my head it's about amplitude modulation. One frequency acts as a carrier wave and the other modulates its amplitude. The maths of that can be performed by an 18-year-old as follows: -
Gordon, you are "mixing" up effects.
I could have written cos(a)cos(b) = half (cos(a+b)+cos(a-b))
Casey, may be you misunderstood or I couldn't explain it properly. My points are:
Nobody sings in equal temperament, for a given key, singing is probably closer to pythagorean centered on that key. The violins should be much the same way, tune the notes as if you were singing the tune.
Nuuska: harpsichords have dampers on each jack. Thus unplayed strings are damped and are not free to vibrate sympathetically. I don't agree that "it" matters less for keyboard instruments; but there's not much control over pitch left the hands of the player. Now a clavichord is a whole 'nother matter!
Yes and no, Lyndon, violin and voice don't use any system, they are system-free instruments. Tuning systems were "created" for instruments with fixed unmovable pitches. Voice and violins don't have fixed notes or pitched (except open string), so it's absurd to try to tag one system to the violin. You simply can't. Gosh. The OP must understand that his question doesn't make any sense.
Paul, I recognize that the violin, being a fretless instrument, can play any pitch within its range. That doesn't mean it doesn't use tuning systems.
"I remember one piece I played where I actually had to play two different B's, because it sounded out of tune and bad if I played the same B"
I agree with Scott that fourths on the violin are hard. I find them to be the hardest interval. I hate them.
"I guess the fact is that we just simply learn to live with the imperfections of the piano." Do you mean after we have been told they exist or before we have been told they exist?
Gordon, "either way you are mistakenly describing cos(a+b/2) and cos(a-b/2)"
P4 ;-- The perfect fourth as a double stop is difficult to tune for a couple of reasons. The 3rd overtone of the lower note is the same as the 2nd overtone of the upper note, so it generates audible beats when even slightly off. The distance between the two fingers is a little bit wider than the equivalent whole-step on the same string. The notes immediately before the P4 double stop can throw you off. If the music allows it, release both fingers right before the P4.. For me, my intonation on all double stops and chords is more reliable if I hit them fresh, fingers off the string immediately before
My 2 centimes d'Euro.
Joel, that's the kind of explanation I was hoping for. I'm glad to have my distaste for playing fourths confirmed by cold science. :)
What about a fifth?
Paul N, fifths are hard too. But somehow they don't sound nearly as bad when they're a little off as fourths do. Maybe others don't agree. As Scott said, we're probably all sensitive to different things.
@Scott "Absolutely perfect octaves in the high treble tend to sound too narrow to most people.
Equal tempered fourths and fifths are only 2 cents off perfect, equal tempered thirds are 14 cents off perfect, or is it 16 cents.
I've read a few books on early temperaments but confess my playing itself still suffers crude intonation. After teaching myself violin over 4 years, for the past year I've realized I can go no further without a teacher...so I found one. My interest is in early music (I'm also taking viola da gamba lessons) so I found a teacher who specializes in early music. She made me start all over, in how to hold the bow, how to hold the violin, how to stand (never sit!).... I've never sounded so terrible, I used to at least hack through some basic repertoire but now I don't even know how to hold or finger...I've started over. So lately instead of whipping through the pieces I just slowly play them with a tuner reading me the pitch...how off I've been! My point? Instead of theory, perhaps set a tuner to whatever temperament interests you (mine is at 415A sixth comma meantone) and just listen carefully as you play through...is this the temperament you want for that piece? Try a few...the point isn't "temperament" but rather your ear and emotional satisfaction with how you play the piece. And if you're playing with others? Time to make practical agreement on the key notes (esp open strings) so you'll be in tune together!
Gordon wrote, "I assume my electronic keyboard has perfect 2:1 octaves." Depending on the keyboard, that could be a bad assumption. My piano tuner proved to me that my Yamaha stage piano is (slightly) out of tune. He showed me thirds on it that he said were incorrect, right in the middle of the keyboard. Was the piano used for sampling not in perfect tune? Is it deliberate on the part of Yamaha to make a piano that sounds more realistic? Who knows.
Octaves - in high registers. The ear, the mind, is not perfectly calibrated. At the very end of Scheherezadhe, the solo violin hangs out on the double harmonic E. Even though it is theoretically perfect, it sounds flat compared to the woodwind chords behind you. What do you do? Either push it sideways, or, what I did, right before the 4th movement, crank the open E a little sharp.
Joel -- great trick!
But the woodwinds aren't going sharp, and the harmonic should be a perfect octave.
Section First Violins can push the pitch sharp, deliberately or unconsciously, and it annoys the woodwinds. If you playing in tune, high on the E string, during a loud tutti, you can't hear yourself, and some will push the note higher to check it. An experiment anyone can do on a well-tuned piano; Tune the open E to the same piano note, then compare the double-harmonic E to the piano. I think the piano tuners call it stretched tuning. Scott will know more than I about this.
"Equal tempered fourths and fifths are only 2 cents off perfect, equal tempered thirds are 14 cents off perfect, or is it 16 cents."
Scott, my statement is fact, your statement is totally confused.
It's not the open E that sounds sharp, is it? Only the high harmonic?
that part of Scott's statement is correct, its the other part that makes no sense.
"Scott, my statement is fact, your statement is totally confused."
You idiot, I'm as clavichord maker and well informed about all manner of keyboard tuning.
Well that escalated quickly...
Temperament, temperament people.
"well informed about all manner of keyboard tuning."
I didn't expect to start another controversy. My best violin teacher told me that harmonics are flat. Actually, they SOUND flat, which is not quite the same thing. I'll leave it at that. As string players, I think we should defer to the winds and brass on intonation. It is so much harder for them to adjust. I am always impressed by woodwinds and brass with good intonation.
To be brief but factual, (from previous threads) the overtones of a bowed string are precise multiples of the fundamental frequency, but a plucked or struck string is another matter, where segments of a worn string can produce bell-like chaos..
I think it's good to dig into and learn more about intonation and perception, and if we happen to be wrong, that's great, because we've learned something in examining our beliefs.
Wind vs strings?
Interesting that you actually measured your harmonic to see if it's in tune. That got me thinking about whether testing a series of harmonics would help characterize a string as false (or not). Any of you physics braniacs thought about that?
Very roughly speaking, a vibrating string sounds "true" if the amplitude of each harmonic is proportional to 1/frequency.
Carmen, I find your use of "amplitude" confusing, as I usually link it to loudness. If I replace it with "frequency" or even "pitch" I make better sense of your very precise remarks..
If you record a note and then perform something called a fourier analysis on it, you get a plot of frequency versus amplitude.
Carmen, what bothered me was "If the amplitude of the first or second harmonic is noticeably lower than one or more of the higher harmonics, then the string will sound false". "Surely "harsh", rather than "false"?
Strictly speaking, Carmen should have said spectrum analysis (the machine is a called a spectrum analyser). Fourier analysis is the mathematical derivation of a spectrum from a mathematically defined waveform.
There is a difference between the THEORY of fourier analysis, and the SOLUTION of the theory for specific problems.
To come back to th OQ (original question?) :
Why would the third of a chord be "mushy and low?" Why wouldn't you try to make a pure 3rd?
I meant a pure third.
One leader (concertmaster) I played with seemed to play in piano (equal) temperament. And was sure he was right.
Joel mentions Scherezade.
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