Equal Temperament for the Violin

October 17, 2019, 7:57 AM · Hello everyone,

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.

Replies (102)

Edited: October 17, 2019, 8:09 AM · Concert pianos are almost never tuned in perfect equal temperament. For large concerti, the piano is always tuned specifically for the work being played.

In any case, you idea that you need such a high level perfect pitch to play the violin is ridiculous. There isn't a list of a million different versions of each note that violinists have to drill. We just play the note so that it creates a phre harmony with whoever else is playing.

October 17, 2019, 11:12 AM · 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.

This is why musicians (instrumentalists and singers) who practice matching pitch in equal temperament with a keyboard instrument all the time and don't listen to the quality of the intervals they create with the keyboard or other instruments/voices tend to generate major thirds too high, perfect fifths too low, and leading tones that "don't lead."

Also, one has to adjust for the environment. Temperature and humidity affect the pitch of instruments and the spaces they sound in...even trying to play equal temperament is a skill in itself!

Edited: October 17, 2019, 1:29 PM · 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.
When to strive for neutral/tempered tuning ? --When working with a pianist, the two symmetrical chords; diminished and augmented, the whole tone scale, and the fast chromatic scale. When you are practicing a solo part alone you will naturally prefer the Pythagorian/Melodic/Leading-tone intonation. If you are playing second violin or viola you will adjust to Chordal/Just tuning. Good luck with all that.
October 17, 2019, 2:25 PM · AND---What have you observed to be the effect of vibrato and the vibratos of different players?
October 17, 2019, 2:48 PM · 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.
October 17, 2019, 4:17 PM · 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.

"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."

Why doesn't this matter or why does it matter less for keyboard instruments? A piano and a violin both have strings, except the piano strings are struck with a hammer while the violin strings are vibrated with a bow.

Joel, I'm going to use your point about not bending in the wrong direction, thanks.
Two questions.
1. Like you said, the difference between these tunings is small, and I've been advised against using an electronic tuner, so how should I practice? Am I supposed to be able to hear the tiny differences? Because I can't.
2. Why do we prefer Pythagorean tuning for single line melodies and scales? What makes a tuning system based on perfect fifths superior to equal temperament? Does it sound "nicer"?

October 17, 2019, 5:33 PM · 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.

As a general rule, "just" intervals, which are intervals where the ratio of the frequencies of notes are ratios of simple integers, work very well when sounding chords. So think of "just" intonation systems for harmonies and chords.

For most notes of a melody, equal-tempered intervals work well with the occasional expressive intonation used where artistically appropriate.

It is the rare person indeed who can discern a difference of 4 cents or less. And for the violin the difference in tuning of adjacent open strings between equal-tempered and perfect fifth is a mere 2 cents.

You can find lots of master class videos that talk about the challenges of intonation systems from violin/piano pieces to quartets. There is no simple answer that applies to all situations.

Edited: October 17, 2019, 6:46 PM · > Why doesn't this matter or why does
> it matter less for keyboard instruments?

Well, for string instruments, it matters hugely if one wants to project their sound over an orchestra as a soloist. Perceived sonic strength of a player isn't just a function of amplitude, it's also the overtones present in their sound as well. Achieving those whole-number ratios for intervals (which Suzuki Method folks call "ringing tones") is an essential factor in good tone production.

On a modern concert grand, the cast-iron plate supporting up to 20-ton tension of the high-carbon steel strings over a gigantic tapered spruce soundboard make up for the slight loss of sympathetic vibration with all the fifths on the instrument being compressed.

> the difference in tuning of adjacent
> open strings between equal-tempered and
> perfect fifth is a mere 2 cents.

But this gets to be an issue as we get further away from the tuning A, so if a string quartet tunes to "pure" fifths (instead of compressing them slightly), the cello and viola C's are horrendously out of tune with the violin E's. :P

October 17, 2019, 6:55 PM · 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?
October 18, 2019, 1:04 PM · 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.
Why does piano tuning sound so good when all of the intervals are a little bit off ? Because it is close enough, and because it is a percussive instrument with lots of internal resonance.
Why do we naturally prefer the Pythagorian/melodic/leading-tone/"expressive" tuning when practicing a scale without a drone or accompaniament or electronic tuner ? I don't know why, but I suspect that our brains prefer the large (9/8) whole step. Two of those long whole steps added together gives us the long major third (81/64) which sounds out of tune when played as a double stop. To get the more pleasant Just major third of (5/4) we add the short whole-step (10/9) to the (9/8) whole step. [10/9 x 9/8 = 10/8 = 5/4]. We struggle with these problems when we work on the Bach Unaccompanied Sonatas. Und so weiter...
Or: just trust your ear and use vibrato.
Edited: October 18, 2019, 4:41 PM · John,
String players don't tune to equal temperament because they don't HAVE to. If a better way to tune pianos had been discovered, no one would use EQ because it's a compromise. The reason it is used is that it was discovered centuries ago that you can't tune every note to every other note on a instrument with a fixed pitch like a piano or organ or harpsichord (or guitar, which is also tuned to EQ because the frets are fixed).
Violinists don't have to play 10-pitch chords. We seldom even play 3-note pitches.

Even if you wanted to, you may not like the results. For example, on the piano, A4-C# is a wide interval, beating about 9 times per second. It, like the other beating on the piano, sounds acceptable, like a fast but pleasant vibrato. On the violin, it sounds buzzy and obnoxious. Tuning on the violin family is timbre-based: move your finger a zillionth of an inch and you go from clean and open to closed. That's why you can't fit, for example, D on the A string (first position) into some other tuning schema--there is only one place for it and if you move it it will sound dead, and simply out of tune.

If you listen to a young orchestra, you can predictably hear the tendency to play certain pitches too flat, like F# an C#. And you don't even have to hear any other pitch, like the G/D above for comparison. It just sounds sour.

An experienced teacher doesn't have to hear an interval to tell the student they're out of tune, especially in the high positions. You listen to the timbre, and you just know the pitch is not in its "slot." This is NOTHING to do with relative or perfect pitch. Just timbre. The violin is open, closed, or somewhere on the spectrum between.

There is also the added issue of inharmonicity, which causes the actual intervals of EQ on the piano to be (usually) more widely spaced. Inharmonicity varies with the piano, depending on the string tension and length. Steinway recommends that their tuners stretch the bass and treble due to the peculiarities of the human ear. Absolutely perfect octaves in the high treble tend to sound too narrow to most people.
I participated in an aural experiment once at the U. of Washington's psych lab, just sitting and listening to electronic intervals. It's true--mathematically perfect octaves sound dreadfully narrow.

Inharmonicity is not an issue on the violin.

"For large concerti, the piano is always tuned specifically for the work being played."

Re Cotton's assertion above: Although I'm a piano technician, I don't do much high-level concert tuning right now. However, I think if you asked concert tuners, they might say pianos are sometimes tuned for a specific work. "Always" is a stretch. I would be surprised if concert tuners always know what's on the program. They are more likely to simply get the best possible tuning for a particular piano. Don't forget that "program" implies a wide range of keys, even within just one work. You'd have to tune the piano before every movement or even modulation. Besides, what is more worrisome to tuners is unison stability, not tiny variations in tuning for this or that. Most audience members don't care about the specific tuning, as long as it's consistent and acceptable. But they can easily hear out-of-tune unisons.

Edited: October 18, 2019, 7:15 PM · 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.

"The limit of pitch discernment among trained, adult musicians has been reported to be about 5 cents."
I was actually going to ask about this but you beat me to it. Thanks for the clear explanation.

"It, like the other beating on the piano, sounds acceptable, like a fast but pleasant vibrato. On the violin, it sounds buzzy and obnoxious."
Very interesting, thanks for the insight. When you say tuning on the violin is timbre based, you're referring to sympathetic vibrations right?

Edited: October 18, 2019, 7:42 PM · 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.

Octaves and fifths resonate the best on the violin. As you play up a G major scale you should hear the pitches that "ring" with the other (at that moment unused) strings. Violinists learn to appreciate intervals that are "in the slot" (Scott's phrase) or "locked in." It's precisely because there are so many intervals that you can learn to play in tune that you don't need perfect pitch to play the violin.

October 18, 2019, 8:30 PM · 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.
October 18, 2019, 9:49 PM · Paul,
How do I go about learning intervals? With ringing notes it's easy, just adjust your finger till the ringing is loudest, that's the right position. Should I just listen and try to memorize the sound of thirds and fourths and the rest, or is there some easier way like with the ringing notes.
October 18, 2019, 10:33 PM · John,
Yes, sympathetic vibrations. The violin must be in tune with itself, and tells us when it isn't. It could care less about temperaments.

I have noticed, however, that some violins are better in this regard than others. A certain richness of overtones is necessary. Violins with overly bright, colorless tone can actually be difficult to tune well. When harmonic information is missing, the ear is disoriented, and things don't snap into or out of tune. Cheap student instruments can be like this. Fine instruments are like good telephoto lenses that snap into and out of focus clearly. I think that immediate color change with small finger movements is why the best violins are so easy to achieve a great vibrato on.

"Keyboards weren't tuned in equal temperament till the 20th century, originally they were tuned to favour simple keys.."

Lyndon, Google is your friend. Try it.

October 19, 2019, 12:57 AM · 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.
Edited: October 19, 2019, 3:23 AM · 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?

John, I wouldn't worry about it. You don't learn 4 notes for every note on the violin, you learn to hear when it sounds good. The concepts are hard to wrap your head around. I've asked the same questions as you many times.
Most recently I did some work on Pythagorean tuning, and came to the conclusion that JI, like any other mode, doesn't, or isn't intended to, work beyond the range of one octave. The Greeks constructed an octave from two tetrachords using Pythagorean principles, but if you add even a third tetrachord (you don't even need to add a second octave) the internal relationships fall apart because of the lack of symmetry in the first two tetrachords.
It can twist your melon.
Use your ears not your brain.

October 19, 2019, 4:23 AM · 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
Edited: October 19, 2019, 5:45 AM · 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.

Pythagorean tuning doesn't ring well in double stops, and pure tuning doesn't lead well in melody. Equal tuning is a pretty nice compromise, when you factor in that echoed note from hall reverb that will form double stops with your next note.

The trick here, really, is how you bow. Bow it right, equal tuning isn't really as bad sounding as you would have imagined.

October 19, 2019, 6:05 AM · Equal temperament has every single interval out of tune, some slightly, the thirds, quite a bit.
Edited: October 19, 2019, 7:16 AM · "Equal temperament has every single interval out of tune, some slightly, the thirds, quite a bit."

Out of tune compared with JI, but to assume that JI is "in tune" is begging the question. In fact JI is crap beyond the octave, as I said.

Beating and lack of beating are a quirk of physics. To use them as a definition of being aurally in tune is potentially arbitrary. The Greeks thought a JI major third was discordant.

They didn't all approve of Pythagoras. He wanted to amend a pre-existing system that had been designed for the ear by enforcing mathematical rules upon it. He wasn't the inventor of music. JI was the new-fangled system, and it caught on because mediaeval Christians thought that Pythagoras' theology complied with theirs. (and it occurs to me to suppose that, when the organ is your primary instrument, using maths to cut the pipes to length is the easiest way to build one)

Edited: October 19, 2019, 7:47 AM · John asked, "How do I go about learning intervals?"

You learn by practicing and playing and listening to yourself as you go. The most important intervals on the violin are the half step and the whole step because they are the most common in classical music. These are grooved by practicing scales and studies that have a lot of scale-like passages in them like the Wohlfahrt etudes or Dont Op. 20 (depending on your level, which you didn't say, so I'm assuming beginner to intermediate). Usually beginners' half-steps are too wide and their whole-steps are too narrow. When you are playing half-steps in first position your fingers should be touching. The "ring tones" are calibration points. You'll learn that there are more of them on your violin than just octaves and fifths. It's important to understand that your ability to hear/listen will advance in parallel with your skill at hitting the targets reliably.

I suggest you visit violinmasterclass.com and watch the video tutorials created for that site by Kurt Sassmannshaus. They are masterful. If you have a teacher then your teacher should be helping guide your intonation systematically. Listen to violin music with the score in your lap, especially the slow movements from the solo Bach Sonatas and Partitas. Did the performer play an interval (especially a double-stop) with intonation that surprised you? Does this happen when you listen to string quartets? If you are not very familiar with SQ literature I suggest you start with Haydn Op 20 and Beethoven Op 18 -- these are very good for listening to intonation. When I first started really listening keenly to professional recitals and recordings (well into adulthood, unfortunately), I had these "aha" experiences and I realized (very quickly) that it's not them. It's my sense of intonation that needed to be recalibrated. I grew up listening mostly to piano music, which, in the modern age, has been recorded almost entirely on instruments tuned to "equal temperament" which is not what you are learning on the violin. Don't get bogged down in the warfare over temperaments. You can see from this thread the amount of needless bellyaching it causes. Just learn to play your violin properly "by the book" and you'll be fine.

October 19, 2019, 8:23 AM · " Unless you have some semantics at work, Lyndon, you seem to be saying that Chopin, Brahms, Schumann, List wrote for JI keyboards."

If I recall correctly, pianos in the late 1800s were mostly in some sort of mean-tone tuning - everyone was aiming for equal temperament, but a systematic, reproducible method for tuning in ET wasn't published until something like 1912.

October 19, 2019, 8:43 AM · 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
Edited: October 19, 2019, 2:57 PM · 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.

One last question, does a whole step (or any other interval) have a characteristic sound one can learn? For example, if I learn to recognize the wide Pythagorean whole step between B and C#, will I be able to recognize other whole steps too?

October 19, 2019, 3:11 PM · 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.
October 19, 2019, 3:13 PM · Thanks a lot Joel.
October 19, 2019, 3:20 PM · 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.
October 19, 2019, 3:30 PM · Thanks Paul, the responses here have been enlightening.
October 19, 2019, 5:56 PM · "Beating and lack of beating are a quirk of physics"

Beating just means the coincident partials of two strings aren't in tune. When your violin isn't in tune, it beats. The faster the beat, the more out of tune. Not a quirk...a feature.

Edited: October 19, 2019, 7:13 PM · Actually, the faster the beat, the closer the vibrations and the closer to in-tune.

A string quartet I was playing with about 10 years ago hired a professional coach for 4 sessions (via the AMCP) and he worked on us (really WORKED on us) to get the Mozart Dissonance quartet in tune by ignoring where our fingers were and LISTENING. Those "damned" thirds! In the end we gave a nice "soiree performance" with that Mozart and a Haydn quartet - I don't recall which. It wasn't bad (I have the recording) and we did get those "damned thirds!" Clearly that is why
attending so many amateur quartet performancesa is unbearable.

ALSO - JOEL, thank you for your answer about vibrato. I was pretty sure of what you said from observing some violinists with really great vibrato. The former concertmaster of the chamber orchestra I play with had a magnificent vibrato. A small woman (about 5 feet tall) with hands big enough and arms of appropriate length to tame any violin. I heard her vibrato commented on by local violinists who knew of it only from hearsay. I have attended concerts by some of the great ones (Heifetz, Elman, Stern, Hahn, Friedman, Chang, Meyers etc.) and have never heard a more impressive vibrato. As a physicist with hearing poor enough that I have really studied human hearing and am able to measure my own with and without hearing aids - I am certain that she is able to add at least 20 DB to her apparent sound level with that vibrato. I used to think it was her violin that was responsible, but then I heard her testing new bows for a friend on the friend's violin - IT WAS DEFINITELY HER VIBRATO - and bowing. Fast and wide it always sounded in tune.

I recall Nathan Cole commenting that he had believed that vibrato should be below and up to the note, but when he saw an actual visual O-graph display of his vibrato frequency it was clear the pitch went over the note about 40% of the time. I too had always argued that we vibrato up to the note and I was put down by professional cellists who argued that they vibrate around the note. Just because we think that is what we are doing doesn't make it so. Lesson learned!

October 19, 2019, 7:20 PM · "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.

Scott is right about unisons. They are the bane of the tuner. Once I went to hear Emmanuel Ax perform a recital. As I was taking my seat I saw my piano tuner nearby and greeted him. At intermission I turned around to chat with him, but he was gone. A moment later he appeared on stage with a few of his tuning tools. He played a couple of intervals and then left without doing anything. When he returned to the audience I asked him what happened and he said "They asked me to check a unison but I decided it was okay."

October 19, 2019, 10:31 PM · Thanks Paul, Yeah, apparently I got that backwards.
Edited: October 20, 2019, 7:41 AM · 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.
October 20, 2019, 9:50 AM · 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."
October 20, 2019, 10:43 AM · Add Temperament by Stuart Isacoff to that list.
October 20, 2019, 2:13 PM · The sum of cos(x) + cos(x) is not cos(2x).

There is a nice discussion here:


October 20, 2019, 2:26 PM · Cos(x) + cos(x) = 2cos(x), in the same way that apple + apple = 2 apples.
Who expects otherwise?
Edited: October 20, 2019, 3:59 PM · "Hello everyone, I learnt about the different tuning systems"
Yeah, sorry, but reading that and then reading your question makes me think you really didn't learn anything about tuning systems.

Any "tuning system" that you apply to the violin is not gonna work, because violins don't "use" tuning systems. Tuning systems are used in instruments that work with specific fixed string lengths or notes that can't be modified once tuned: piano, guitar, flute...

Nevertheless, in the violin, there are only 4 fixed "unmovable" string lengths/notes: the 4 open strings. Actually 3, because we consider the whole orchestra is gonna have the same A440/A442. Any other note you play on the violin can be played perfectly in tune with whatever is going on in the background. That's the beauty of fretless fingerboards, that it's so hard to control them but you have all the freedom in the world to adjust every note.

So, to sum up, violins can't be tuned to equal temperament at all, violins are system-free instruments.

"It would simplify things so much if you only had to learn a single pitch for each note"
Unless you are some kind of super maestro playing the violin, I don't think you can really control and practice specific notes for different scales. In other words, I don't believe you can practice solidly and with precision a C note in a C major scale, the same C in a A minor scale, a C in a G major scale... because, in reality, all those "C's" are different, slightly different. But, what you will play is the same C all over again, which is hard enough to tune perfectly. If you are really good, then of course you can adjust the C so fast that is imperceptible for the scale you are playing on.

October 20, 2019, 4:07 PM · There are a large number of ear training lessons on youtube. Several teach intervals by referring to well-known songs.

For example:


October 20, 2019, 4:40 PM · 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...

> Why doesn't this matter or why does
it matter less for keyboard instruments? <

The answer is pretty simple, at least for pianos (not so much for harpsichord): most of the time, the strings are dampened by dampeners,and there will not happen any sympathetic vibration. Only if using the pedal or hold down a "silent" key, there are free strings available which can vibrate. While both are important components in expressive and colorful piano playing, the piano has to be constructed in a way that every tone will be able to stand for itself, as Gene explained.

Edited: October 20, 2019, 5:30 PM · "Cos(x) + cos(x) = 2cos(x), in the same way that apple + apple = 2 apples.
Who expects otherwise?"

That changes only the amplitude, not the frequency. Two tuning forks at 440 Hz should sound like a loud tuning fork. Not like a tuning fork an octave up (that is, at 880 Hz).

That brings me back to Gordon's statement, "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." I guess I'm just not understanding this statement.

@Nuuska, the better digital stage pianos (Kurzweil Forte, Dexibell Vivo S9) now model that resonance. It's pretty incredible what the digitals have been able to do.

October 20, 2019, 6:05 PM · 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: -
So cos(a+b)+cos(a-b)=2cos(a)cos(b)
So cos(a)cos(b) = half cos(a+b)+cos(a-b)
In other words carrier b, amplitude modulated by a (or vice versa), gives sum of frequencies a+b plus difference a-b.
October 21, 2019, 12:11 AM · Paul N.
"In other words, I don't believe you can practice solidly and with precision a C note in a C major scale, the same C in a A minor scale, a C in a G major scale..."

It's entirely possible, and not difficult to do so. It's like saying you shift to 3rd position on A string, 1sr finger, which is a D note, and it sound different all the time. It's not hard to imagine that it will create a lot of problem in any circumstances.

What really makes it difficult to make all C sound the same, is that your ear will tell you it sounds "wrong". Especially when playing alone, in a small space or dry room with no reverb. Anyone who has decent ear will be able to tell immediately something is wrong and adjust accordingly, and usually, end up in Pythagorean tuning.

Edited: October 22, 2019, 2:28 AM · Gordon, you are "mixing" up effects.

Your final equation,
  cos(a)cos(b) = half cos(a+b)+cos(a-b)
can be interpreted in two ways.

1. the sum of e.g. 440.5 and 439.5 on the right (a=440, b=0.5) amd indeed a 440 Hz tone amplitue-modulated at 1 Hz. This is what musicians and physicist recognize as 'beating'.

2. Nonlinear mixing (Tartini tones), where the ear registers a product of two signals. (EDIT: e.g. a=439.5, b=440.5) This effect is much weaker and only noticeable at high sound levels, like in your left ear while playing. The audience won't hear it.
In that case, indeed a difference frequency at 1 Hz is produced (but human ears can't register such deep infrasonics) and a sum at 880 Hz. All the other harmonics combine as well, so if you combine an equal-tempered D (harmonics 294, 587, 881, 1175 Hz) with an A (440, 880, 1320) you get sum and difference tones for all possible combinations, for example 587/1320 generate 733/1907 Hz. Hence the cacophony of Tartini tones when you're tuning a fifth.

Edited: October 21, 2019, 3:43 AM · I could have written cos(a)cos(b) = half (cos(a+b)+cos(a-b))
but either way you are mistakenly describing cos(a+b/2) and cos(a-b/2)

Normally you'd have A x cos(b) where A is the amplitude and is constant. When A isn't constant, it amplitude modulates b (and vice versa).

When you amplitude modulate a carrier you always get a sum and difference signal (sidebands if the modulating signal is complex). 440 amplitude modulated by 1Hz will give 439 and 441.

"product" is a mathematical expression meaning multiplication (to distinguish it from sum). "The product of 2 and 3 is 6". Confusing here. But with sine waves the product of two happens to be the sum of two others, lol.

"much weaker" - that's why I used the word 'subliminal'.
"All the other harmonics combine as well"
Of course, but I was only talking of tuning forks, which for practical purposes generate only sine waves.

October 21, 2019, 5:43 AM · Casey, may be you misunderstood or I couldn't explain it properly. My points are:
1. In violins, there are no systems, the idea of a tuning system in a violin is absurd.
2. You can't practice, control or remember the C note position for each specific scale, which has its own specific C note, differencing from each other C in terms of very few Hz.

What you can do is use your ears to adjust your C to an open string, for example, or tune your C when playing a chord. Specially in the first position, you can learn to play a B in the A string differently depending on the chord you want to play. 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. You have to match the proportions.

Edited: October 21, 2019, 7:58 AM · 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.
October 21, 2019, 7:51 AM · 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!
Edited: October 21, 2019, 8:59 AM · 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.
Edited: October 21, 2019, 9:19 AM · 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.

From Wikipedia:
"A tuning system is the system used to define which tones, or pitches, to use when playing music. In other words, it is the choice of number and spacing of frequency values used."

Being able to use multiple tuning systems doesn't mean the violin doesn't use any. I quote Kurt Sassmannshaus, "Most of the time we use the Pythagorean system".

Edited: October 21, 2019, 10:08 AM · "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"

Indeed. If, for example, you had a three-note chord with pitches open D, B on the A string, and then open E, you can't play the B in the exact position to make a non-beating 6th below or a perfect fourth above. You have to move your first finger as you roll the bow towards the top. Eventually, we do this unconsciously.

There is a position for the B that would be a compromise, making the 6th and 4th a little out of tune.
That, in essence, would be a tiny sample of equal temperament, or some kind of temperament. (Since the word temperament means to tweak or modify, I'm not sure that playing clean Pythagorean intervals qualifies as a "temperament." But that's just a rhetorical exercise.)

But again, for what ever reason, we can stretch the 4ths on the piano and get away with it, while on the violin they sound terrible. Perfect 4th are almost as difficult to achieve in chords as octaves, maybe more so. I'm not sure why playing intervals tempered by the same amount are acceptable on the piano but lousy-sounding on the violin. Maybe it has something to do with the much longer strings and greater resonance on the piano that makes them tolerable.

Edited: October 21, 2019, 11:21 AM · Scott,

"I'm not sure why playing intervals tempered by the same amount are acceptable on the piano but lousy-sounding on the violin."

Because there are two separated yet closely related problems here. One, is to bow with a pure tone that the harmonic contents are not disturbed and ring smoothly. Another, is that we perceive the temperament differently between what we hear under the ear and what people really hear (and what recording really sounds like).

Take the first movement of Tchaikovsky violin concerto, the first triple stop for example (D A F#), try to find a sweet spot for that F#. ;)

October 21, 2019, 11:04 AM · I agree with Scott that fourths on the violin are hard. I find them to be the hardest interval. I hate them.

Scott, since you know a lot about piano tuning, let me ask this: Is it possible that we get away with imperfect fourths on the piano because the unisons are never quite perfect and that masks the issue with the fourths? Or because the means of generating tone (striking the string rather than bowing it) causes a favorable combination of overtones and anharmonicities? Just guessing here.

Back to the violin, I wonder if fourths are more sensitive in finger position with respect to "beats" or "wobble" than other intervals. That would require math that I don't have time for because it involves beating among overtones.

October 21, 2019, 11:41 PM · Paul,
4ths are wide simply because they have to be for equal temperament to work. They're about 1 beat per second, and 5ths about 1/2 a beat. 5ths are so close to perfect that it's hard to tell.

If 4ths weren't that wide, it would screw up something else. It's nothing to do with unisons--we set them wide by tuning single strings anyway. The other two on each pitch are muted out.

I guess the fact is that we just simply learn to live with the imperfections of the piano.

October 22, 2019, 1:43 AM · "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?
October 22, 2019, 2:43 AM · Gordon, "either way you are mistakenly describing cos(a+b/2) and cos(a-b/2)"

i'm not sure what you mean exactly and whether we agree in the end. In case 1, a=440, b=0.5. (I mistakenly wrote b=1, now corrected). In case 2: a=439.5, b=440.5.

cos(440*t*2pi)*cos(0.5*t*2pi) = (cos(439.5*...)+cos(440.5*...))/2 sounds like a 440 Hz tone beating at frequency 1 Hz.

cos(439.5*...)*cos(440.5*...) = (cos(880*...)+cos(1*...))/2 sounds like an 880 Hz tone plus sime inaudible infrasonic.

October 22, 2019, 1:45 PM · 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
October 22, 2019, 2:43 PM · My 2 centimes d'Euro.

If we play with a piano, we have to play in equal temperament by imitation, while adjusting double stops as we go.
If we play with a harpsichord, we will be blending with some form of mean-tone temperament, again by imitation.
If we play unaccompanied, we have to invent our own tuning, often as we go.

Frequent listening, both attentive and casual, is vital to stock our aural memory.

Strict "Just Intonation" does not work for very long on a keyboard:
Lets say C-F and C-G are pure fifths;
Make pure major triads on C,G & F: we will have fifths G-D and A-E pure, but D-A will be far too small to use in a D minor triad.

Edited: October 22, 2019, 3:02 PM · 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. :)

Scott, yeah I guess we're just used to it. My comment about the unisons was that if the unisons are not perfect, the sin of widening the fourths could be somehow covered, in the same way that vibrato might cover some errors in violin intonation. But I think the unisons would have to be really bad before that happened.

October 22, 2019, 3:01 PM · What about a fifth?
I find it quite difficult to correctly match a perfect fifth because you use one finger only.
October 22, 2019, 3:13 PM · Paul,
The goal (my goal, anyway) is not to get a perfect tuning, but rather to be consistent. If all the 4ths sound about the same, that's much better than this one wide, then that one perfect.

"Do you mean after we have been told they exist or before we have been told they exist?"
Gordon, customer education, especially concerning expectations, is important. A technically well-tuned piano can still sound out of tune for many reasons, but most pianists don't know why. Also, people vary greatly in how they listen and what they find objectionable.

October 23, 2019, 6:33 AM · 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.
Edited: October 23, 2019, 11:08 AM · @Scott "Absolutely perfect octaves in the high treble tend to sound too narrow to most people.
I participated in an aural experiment once at the U. of Washington's psych lab, just sitting and listening to electronic intervals. It's true--mathematically perfect octaves sound dreadfully narrow."
Yes, I've noticed this. I assume my electronic keyboard has perfect 2:1 octaves, but they do sound narrow.
October 23, 2019, 7:12 AM · Equal tempered fourths and fifths are only 2 cents off perfect, equal tempered thirds are 14 cents off perfect, or is it 16 cents.
Edited: October 23, 2019, 7:56 PM · 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!
Edited: October 23, 2019, 8:01 PM · 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.
October 23, 2019, 11:42 PM · 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.
October 24, 2019, 10:21 AM · Joel -- great trick!
October 25, 2019, 6:45 PM · "Even though it is theoretically perfect, it sounds flat compared to the woodwind chords behind you. What do you do?"

Tune the woodwinds?

There's a tendency to go sharp, especially on high notes, so it's not surprising when you have to go sharp to match. I don't think you've conclusively shown here that the ear hears it incorrectly.

October 25, 2019, 9:07 PM · But the woodwinds aren't going sharp, and the harmonic should be a perfect octave.

Maybe the reason violinists tend to drift sharp is because they're overcompensating for the tendency for perfect octaves to sound flat.

October 25, 2019, 9:09 PM · 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.
October 25, 2019, 10:54 PM · "Equal tempered fourths and fifths are only 2 cents off perfect, equal tempered thirds are 14 cents off perfect, or is it 16 cents."

Fourths and fifths are not tempered equally. Fourths are wider than the amount by which fifths are narrowed. As one ascends into the high treble, fifths will sound perfect.

October 25, 2019, 11:06 PM · "the woodwinds aren't going sharp"

How do you know that? An open E sounding flat is my argument that they (and perhaps everyone else) have.

October 26, 2019, 12:12 AM · Scott, my statement is fact, your statement is totally confused.
Edited: October 26, 2019, 8:11 AM · Tuning sharp.
My woodwind colleagues find we string players tune sharp. It maybe to hear ourselves better, but it is also because we know that winds go sharp as their air columns warmsup.

Another point.
We don't always agree on intonation in the stratosphere. I fact the cochlea has fewer hair-cells per semitone as we go higher, and our poor brains have to interpolate more.

October 26, 2019, 7:31 AM · It's not the open E that sounds sharp, is it? Only the high harmonic?

"Equal tempered fourths and fifths are only 2 cents off perfect."

"Fourths are wider than the amount by which fifths are narrowed."

I'm kind of ignorant about this stuff but those two statements sound compatible to me.

October 26, 2019, 8:39 AM · that part of Scott's statement is correct, its the other part that makes no sense.
October 26, 2019, 8:42 AM · Aural uncertainty.
As well as the hair-cell distribution I mentioned above, there is the fact that we only hear really equal octaves in the middle range: the high end of the keyboard sounds flat, and the lower end sharp, to a degree that depends on the individual.
Pitch perception also depends to some extent on volume: in my case incresed intensity makes pith seem lower.
October 26, 2019, 8:47 AM · "It's not the open E that sounds sharp, is it? Only the high harmonic?"

That argument is strained. I'm assuming that the harmonic is actually in tune with the open E, then the tuning of the harmonic is equivalent to the tuning of the open E. If the harmonic is not in tune, then the argument is already invalid. And I think that the harmonic should be in tune and for us to be able to hear that.

There is a reference to inharmonicity, the basis for stretched tuning on pianos, but bowed strings are said to not exhibit inharmonicity. Ref: https://en.wikipedia.org/wiki/Inharmonicity

Moreover, that the winds are necessarily in tune does not have any basis. Unless they have a fixed reference, it's likely that they're out of tune, and the general push towards sharpness takes them there.

October 26, 2019, 4:25 PM · "Scott, my statement is fact, your statement is totally confused."

"that part of Scott's statement is correct, its the other part that makes no sense."

Lyndon, you seem like the confused person here. If you want to be the expert here on inexpensive old German violins that you sell, fine. However, until you pass all the tests of the Piano Technicians Guild and become a Registered Piano Technician, and have attended many years of classes with the best technicians and tuners in the country, then please don't call me the confused one.

If something I say makes no sense, then ask me and I'll explain it.

October 26, 2019, 4:38 PM · You idiot, I'm as clavichord maker and well informed about all manner of keyboard tuning.
October 26, 2019, 5:45 PM · Well that escalated quickly...
October 26, 2019, 7:31 PM · Temperament, temperament people.
October 26, 2019, 10:16 PM · "well informed about all manner of keyboard tuning."

Except the basics of equal temperament piano tuning.

Edited: October 27, 2019, 12:56 AM · 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.
October 27, 2019, 7:12 AM · 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..

High harmonics sound flat for the reasons I have outlined already.

October 27, 2019, 9:25 AM · 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.

"High harmonics sound flat"

This is consistent with my claim that we tend to play sharp, especially on high notes, which is also consistent with physical and experiential limitations of humans at higher reaches.

"I think we should defer to the winds and brass on intonation. It is so much harder for them to adjust."

Conversely, it's much easier for them to get it roughly right, so they should be in awe of string players making it far enough to even get close :), or something like that - also give strings credit for how much harder it is for them and the learning required to get there.

It'd be a mistake to assume that winds are right because their instruments have valves or keys instead of unfretted strings. Winds can do vibrato and therefore adjust the pitch, pretty much all the time, whether or not they're consciously doing so. Given that they have the same human physical limitations and tendencies as string players, they are also likely to be playing sharp when playing high notes - perhaps not as much as string players due to the valves or keys, but certainly within the limits of their discretionary pitch adjustment. The range of pitch differences could also be greater at higher notes for reasons similar to string effects at higher positions.

And with perhaps the strings leading the way, and winds attempting to play in tune consciously or otherwise, the entire orchestra could be sharper at the end of a performance, as another member has mentioned in this forum from experience - so much so that the strings need to be tuned higher to have the open strings match. (Alternatively the winds may have led by shifting sharper due to temperatures rising.)

But unless we have actual measurements of the pitches, we don't know who's right or wrong. I measured my E and harmonic, and they're right - so it's not the instrument which is wrong here; it's actually right when the fundamental is right. If that is perceived to be out of tune relative to another set of notes, then I'd think that they are actually out of relative harmony, so therefore the others are wrong. Comparisons to pianos don't make this easier to understand, because we've learned that pianos also stretch tuning and therefore are not valid as an absolute reference. Again, without actual measurements of the winds and violin in this situation, this is just conjecture.

If the argument was just that higher notes are perceived to be flat, I'd agree - as I see it, that's explainable simply because the "ear" doesn't really know and tends to overshoot.

October 28, 2019, 2:40 AM · Wind vs strings?
Many experienced amateur wind players have never been taught to play really in tune. The various instruments have different structures and are basically not in tune with each other, even when the A is assured.
For example the C# of a flute is way too high.. And it is easier for a string player to shift a finger a millimetre or two than for a wind player to adjust his/her embouchure.

Now retired, I play in an ambitious amateur orchestra, and playing in tune with the wind sections requires frequent tweaking of my four built-in adjusters...

October 28, 2019, 6:39 AM · 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?
Edited: October 28, 2019, 7:43 AM · Very roughly speaking, a vibrating string sounds "true" if the amplitude of each harmonic is proportional to 1/frequency.

Another way of saying this is the first harmonic's amplitude is 1/1, the second harmonic's amplitude is 1/2, etc. So if n the number of the overtone, its amplitude is proportional to 1/n.

In practice, the harmonic amplitudes do not decrease strictly as 1/n. The variation in the decrease results in the sound quality commonly called "timbre".

A simple test one can perform is to install a frequency spectrum app on one's phone and record each open string. 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. You can do those for various notes that are a problem, like B or C# on the G string, a common location of wolf and flat sounding notes, to get a visual picture of the bad note.

Aurally, this can be noticed by playing a cadence that attempts to land solidly on the tonic: like IV-V-I. For example, on the D string, the IV is the fingered G, a particularly resonant ring tone. The V is the open A, also very resonant. If you then land on the I, the open D, and it suffers from weakness in harmonics 1 or 2, the note will sound out of tune.

Edited: October 28, 2019, 11:01 AM · 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..

And the fundamental is not an overtone, by definition..
While the first "overtone" on a violin is the second harmonic, on a clarinet it is the third harmonic (closed cylindrical tube). On a bell, the overtones are not harmonic at all!

October 28, 2019, 11:55 AM · If you record a note and then perform something called a fourier analysis on it, you get a plot of frequency versus amplitude.

For the violin, a series of very distinct peaks occur at frequencies that are integer multiples of the base frequency of the note. Each of these peaks is called a harmonic.

You are correct in that the term overtone is technically used to describe harmonics other than the first one, but I am sure you get what I was trying to describe.

The amplitudes of such a plot are related to the loudness of each harmonic. Not the same as pitch or frequency.

I know nothing of bells, except the time I got my bell rung by a hulking linebacker when attempting an end-around in a high school football game. I can confirm the effect was very inharmonic!

October 28, 2019, 12:23 PM · 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"?
I took false to mean "out of tune", hence my confusion.

It's true that terms used by my high-school physics teacher did not always coincide with those from my acoustician father.

October 29, 2019, 9:12 AM · 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.
October 29, 2019, 10:26 AM · There is a difference between the THEORY of fourier analysis, and the SOLUTION of the theory for specific problems.

If one is perusing the literature for signal processing, then look for "discrete" fourier analysis, i.e., fourier analysis of a digitally sampled signal.

The phone apps do this in real time by a solution method call Fast Fourier Transform (FFT).

The popular freeware recording program for computers, Audacity, also uses one of these FFT techniques (there are several). So you can capture your playing with Audacity, then use the spectrum feature to observe the power of each harmonic relative to each other.

From a practical perspective, one usually does not need to worry about the details of how it is done. We are just looking for general trends.

Whether a tone sounds harsh as compared to false is a matter of personal opinion I guess. On violins where I noticed a weakness in the first couple of harmonics, it certainly sounded like something I did not want to hear in a performance. >grin<

Edited: October 29, 2019, 2:30 PM · To come back to th OQ (original question?) :
- Equal Temperament is an improvement on Pythagorean tuning, most of the time;
- On our string instruments, we should probably build our intonation with fifths (pure or tempered) and then indulge in pure thirds and sixths when we can.

For example, a G-major triad (G-B-D) can have a high (leading-note) B if it is a dominant chord in the key of C major, but a nice mushy low B if it is a final chord in the key of G major.

October 29, 2019, 3:21 PM · Why would the third of a chord be "mushy and low?" Why wouldn't you try to make a pure 3rd?
Edited: October 30, 2019, 6:56 AM · I meant a pure third.
I was just trying to be witty..
October 31, 2019, 2:52 PM · One leader (concertmaster) I played with seemed to play in piano (equal) temperament. And was sure he was right.
A little challenging!
Edited: October 31, 2019, 5:10 PM · Joel mentions Scherezade.
For exactly that reason - strings tend to go flat in the heat of the hall, and woodwind tend to go sharp, when I was sitting no. 2 to the best leader I've played for, he played the solos at the end of the 4th movement stopped, not as the (printed) harmonics.
So I had to join him on the E' ' as a stopped note. Luckily, I matched him.

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