September 23, 2009 at 04:11 AM ·

I use a Korg chromatic tuner to tune my instrument.  I first tune the A string visually using the tuner, then I tune the other strings by ear, but check them with the tuner.  When finished, I try to get all 4 strings to move the needle of the tuner directly in the middle.  But after watching one of Todd Ehle's videos tonight, he seems to imply that a properly tuned violin does NOT move the needle exactly to the middle for all 4 strings. He seems to imply that if the A string moves the needle of the tuner exactly to the middle, then the D string should be slighly flat according to the tuner, the G string should be a bit MORE flat, and the E string should be slightly sharp.  Can someone shed some light on this?  Have I been tuning incorrectly all these years?

Here's the video.  His explanation starts at 0:28.

## Replies (101)

September 23, 2009 at 05:25 AM ·

It's true, only the A moves the needle to the middle.  Strange, but true.  Someone else can quote the physics;  I just tell the kids it's magic.

September 23, 2009 at 06:54 AM ·

Tis true, the tuner uses math--we use our ears.

September 23, 2009 at 07:12 AM ·

When we use our ears we are using the small integer ratio of 3:2 to tune a perfect fifth; it's explained here; http://en.wikipedia.org/wiki/Perfect_fifth It's easy to hear when it's perfectly in tune because the surrounding vibrations decrease the closer to "in-tune" it becomes. The auto-tuner is probably dividing the octave exactly by 12 so that each interval wont have an integer ratio.

September 23, 2009 at 07:56 AM ·

Nigel, I was trying to remember and explain to a student the role of the twelfth root of two in calculating frequencies of successive notes in a scale.  (I'm not sure what type of scale.)  I remember doing the calculations years ago, but I don't remember quite enough to explain it.  Can you help me?

September 23, 2009 at 09:33 AM ·

I should say at the outset that I'm not a mathematician! The twelfth root of two is used for the equal tempered scale and gives the ratio to divide the octave into twelve, so that the ratio between successive intervals (the smallest ones, ie. semitones) is always the same. So, the number is 1.05946309436. If you start with 440 and multiply it by that number, one gets the frequency of the next semitone up ie. B-flat. If that's done twelve times one arrives at 880, the frequency of the octave above A=440.

The frequency of the note that is "the perfect fifith" is not the same as what we'd get using the integer ratio of 3:2, which is why only the A will be in tune according to the tuner.

September 23, 2009 at 12:09 PM ·

I don't think there's a genuinely clear answer here if you think about it too much. After years of trying to out-guess customers, I've given up which method they will use: no matter what I do, a large percentage will retune to the other system. I know what sounds "best", but not everyone does that.

On issue is that It depends how often they practice with a piano. Players who play with piano a lot will tune close, to the "tuner" notes; players who play with other non-fixed instruments will tune to pure fifths. A friend who plays in a quartet added another twist to it, that the music being played influences the tuning, too, and the quartet has to come to agreement about what it's doing. I got to see another option recently when I commented to a cellist that he was tuning the low strings a few cents high; it turned out that he was preparing some modern piece that demanded that for some reason that I didn't understand, and had gotten into that habit.

Being aware of the problem seems only to be the first step. . .

September 23, 2009 at 12:24 PM ·

Wow!  I'm an engineer with a strong math background and am thoroughly confused by this concept.  Along the same lines, I sometimes check my intonation with my tuner.  I've noticed that when playing scales, the tuner needle stays right in the middle (and it sounds in tune) until I reach the E, one octave above open E.  At that point, if the notes are in tune by ear, then the needle is no longer in the middle; according to the tuner, each note is quite a bit sharp.

September 23, 2009 at 12:31 PM ·

Smiley - I was wondering the same thing. I had tuned with an electronic tuner right before my lesson and my teacher promptly tuned it for me, by ear. When I checked it later, it was out of tune....strangely, it sounded better. Guess I should have asked, but I didn't start wondering about it until a couple of days ago.

September 23, 2009 at 12:38 PM ·

The simple reason is that the correct mathematical proportions (3:2) to create a perfect fifth result in pitches which are slightly more distant from each other than will fit into octaves where the frequency doubles for each octave. You can sit down and do the math, with a starting pitch of a nominal 100. By the time you run upwards through an entire circle of fifths with the math (C>G>D>A>E>B, etc, back around to C) and supposedly end up on the note where you started (many octaves higher), you'll be about a half-tone sharp.

Bach's well-tempered clavier was his attempt to tune keyboard instruments so that with an instrument that couldn't adjust pitches depending on context and key, at least all keys could sound equally bad. Up to that point, keyboards were often tuned to favor common keys, and consequently, the unusual ones sounded horrible.

Instruments with somewhat fixed pitches can deal with these things in different ways--wind players customarily bend notes depending on their context, even though the keying remains the same. Guitar tuning structure is similar to a piano's and you will notice guitar players tuning by first tuning by ear, then playing a couple of chords and readjusting so that the commonly-used chords will sound sweet (they also have to deal with a separate issue with string stretch not related to temperament).

Other intervals have similar problems, but since you don't have to tune to them you're not aware of them.

http://en.wikipedia.org/wiki/Well_temperament

September 23, 2009 at 01:30 PM ·

By the way, Bach's method was essentially mathematical, not ear-based. There is still disagreements as to whether it's the "right" solution. Some people maintain that the original reason that composers selected certain keys was because of their individual characteristics, which are wiped out by even-temperament schemes, so that some attempt should be made to retain the individual characteristics of the different keys--not a bad argument, I don't think.

Piano tuners have a different discussion as to whether the compromises being dealt with mathematically result in the best possible compromise, or simply the easiest one for tuners to execute (most tuners tune by a system where certain note combinations beat in certain ways--they are "out of tune" but in a universally-accepted way) . If you want to read a fascinating non-fiction book where this issue turns out to be the hook, read "Grand Obsession".

September 23, 2009 at 02:59 PM ·

Smiley- Much of what has been explained is in a book called, "Temprament" and I recomend it for any music library.  As a matter of fact I think Michael D is who I learned about it from one of his other posts?

September 23, 2009 at 09:48 PM ·

But there is also the question of whether to tune with pure fifths or with narrow fifths, and the best answer seems to be that tuning the open strings in narrow fifths solves lots of problems most of the time.

To do this, having decided on the A tune the D ever-so-slightly sharp to the A, and then the G ever-so-slightly sharp to the D. Then there is a choice for the open E. Either you leave it pure, or you can tune it ever-so-slightly-flat.

Personally I don't think I do ever consider tuning the E string flat, but definitely tune the D and G strings up a fraction.

String quartets have to tune in narrow fifths, otherwise the C on the viola and cello end up way too flat.

This is nothing new. I can't remember who, but someone a very long time ago  - it might have been Leopold Auer - talking about playing with the piano, pointed out that if you tune the A perfectly to the A on the piano, and then tune your D string perfectly to the A, and the G perfectly to the D, and the E perfectly to the A, you will end up with only one of the four strings the same as those notes on the piano - the A. In other words, not such a good idea.

According to Dorothy DeLay, Zuckerman particularly recommends tuning in narrow fifths when playing solo Bach.

Anyhow, the zero on the chromatic tuner then becomes something you are more interested in.

September 23, 2009 at 10:30 PM ·

September 24, 2009 at 12:03 PM ·

Thanks to Bill Platt for sending me the following explanation:

-----------------------------

Hi Smiley:

It is surprising yet true that in European music, we don't really

learn--are not taught--the realities of what our "notes" are. I'm not sure why this is.

The problem of the "tuner is not in tune" stems from the fact that there

isn't one single correct relationship between pitches. In fact there are

really three different extreme camps if you will--all three of which are

part of violin technique:

1. Harmonic series, also called "Just"

2. Pythagorean, which is also harmonic, but based on the cycle of 5ths

only;

3. Equal Temperament, which is based on 12 equally spaced tones.

The latter, item 3, only works out because it is remarkably close to items (1) and (2) by mere coincidence. By tuning in equal spacings, all keys are equally fuzzy, equally imperfect, yet close to perfect.

So, the "perfect 5th" is a very special thing. Remember the Harmonic

Series, where frequency is F, F/2, F/3, F/4, F/5 etc?  Well, (F/3)*2 = the perfect 5th up from the tonic. This is A above D, or E above A etc. When you tune by ear, this is what you get.

The chromatic tuner gives you a 5th which is slightly flat: take

(2^(1/12))7 and you'll see that it is slightly less than 3/2.

That's pretty much it for the perfect 5th. It turns out that the 12-tone

system also lands remarkably close to all the other intervals, too! Just

dumb luck!

September 24, 2009 at 12:27 PM ·

I just did the following calculation, took 2 raised to (1/12) power, multiplied by itself 7 times and it equals 1.498307.  If you start at open A string (440 Hz), and multiply by this number, you get 659.25 Hz.  This is the tone you would get if you tune the E string with the tuner and get the needle in the middle.  The perfect fifth (by ear) is 440 Hz times 3/2, which is 660 Hz, a difference of 0.75 Hz -- a small, but perceptable difference.  So, I guess that explains it.

But I would have to conclude that it is not necessarily wrong to tune all 4 strings with the tuner; particularly if you are playing with a piano.  I have noticed that when playing with my piano trio, my high notes sound sharp relative to the piano, but they are in tune when I play alone (e.g., when practicing).  This sharpness on the high notes is a different phenomenon than the "imperfect fifths" explained above (I think).  And it is quite a bit more pronounced.  Can someone shed some light on this and offer practice tips to play in tune relative to a piano.  Does one practice with a tuner and keep the needle in the middle?  I am still quite baffled by this.

September 24, 2009 at 08:12 PM ·

Smiley you may find your answer with regard to higher tones if you look up 'octave stretch' on the net.

September 24, 2009 at 08:14 PM ·

TOO MUCH MATH! But I think I understand... a little bit.

September 24, 2009 at 08:29 PM ·

A faint urge to throw a fire-cracker into the fire makes me want to ask this question: how do you think of tuning first-finger B, first position on the A string, played on its own as part of a phrase or melody, and not as part of a double-stop?

September 24, 2009 at 09:17 PM ·

Smiley, thanks so much for starting this discussion.  It is very interesting, especially the parts I understand.

Nigel, thanks for answering my question so clearly.  It almost makes everything simple.

Smiley, I did the same calculations you did using the 12th root of 2, years ago.  I remember doing them for fun (!) with my father using a log table.  That was a very long time ago.  I don't know whether anyone here has ever used a log table.

I have one more question.  I vaguely remember that there is something important about two notes whose frequencies are in the ratio of the square root of two to 1.  I think that ratio is also important in Chinese classical music.  Can anyone help me with this?

September 24, 2009 at 10:51 PM ·

I agree with Michael...too much math. I need this in English.

September 24, 2009 at 10:55 PM ·

Now I'm dying to know the answer to Mr. Fischer's question.

September 25, 2009 at 12:44 AM ·

Cellist David Finckel recommends tuning all all 4 strings to an electronic tuner, and also all instruments in string quartets, etc.(I think he is the cellist in the Alexander Qt). It eliminates one sorce of disagreement.

Andy

September 25, 2009 at 01:31 AM ·

Greetings,

I wouldn`t use an electronic tuner as a rule.  I would rather take an a from a piano in which case I @play the d -major- chord with the sustaining pedal down.

The problem in a piano trio is that you often have to be in tune with a piano that is out of tune anyway so it makes tremendous demands on the ear and reflexes.  It also means that part of the preparation for any piano trio cocnert is gettuing used to the vagaries of the instrumetn supplied.  A well tunes piano is very rare.

Cheers,

Buri

September 25, 2009 at 06:47 AM ·

My answer to Simon Fischer about how I think of the b on the A string: it scares me. It has always been the 1st position 1st finger note that I cannot reliably fix to what I'm playing until I have completely, I don't know, absorbed the whole phrase it sits in. I often can't clearly HEAR what the note is, it disappears. The overtones or something overwhelm it, and the truth of the pitch doesn't make it into my auditory processing bits.

So, you got any great ideas for fixing that problem? My solution has been to do it by kinaesthetics, and have faith that it will be alright.

My brain hurts when I try to think of music as maths.

September 25, 2009 at 12:11 PM ·

After my lesson, I took my tuner and checked to see exactly how my teacher had tuned it (assuming it had remained in tune - it holds well) and it was exactly as Smiley's OP described from the video he'd watched.

September 25, 2009 at 12:21 PM ·

OP?  Original Post?

Sorry, I'm not up on the latest  web abbreviations.

September 25, 2009 at 12:47 PM ·

Sorry, Smiley. Yes, OP means original post.

September 25, 2009 at 03:59 PM ·

Finckel is the Emerson Qt's cellist.

September 26, 2009 at 07:34 AM ·

 Just Just in cents Pyth Pyth in cents Equal Equal in cents I 1,000 0,0 1,000 0,0 1,000 0 II 1,125 203,9 1,125 203,9 1,122 200 III 1,250 386,3 1,266 407,8 1,260 400 IV 1,333 498,0 1,333 498,0 1,335 500 V 1,500 702,0 1,500 702,0 1,498 700 VI 1,667 884,4 1,688 905,9 1,682 900 VII 1,875 1088,3 1,898 1109,8 1,888 1100 IIX 2,000 1200,0 2,000 1200,0 2,000 1200

.. but I still haven't got a clear concept of an a flat! Not one that would be any help in hearing whether I played it in tune, without any accompaniment. Such a difficult question to put to a little girl, Mr. Fischer.

As to the b': that has two "sweet spots" One has a second harmonic that resonates with a G string harmonic, and the other is in tune with the E string. Which I use depends on the key.Three sharps or more: the higher one; less than three sharps: the lower one. If I'm awake enough to make the difference at all.

Now play a g - e' - b' - e' chord. I cheat, if I have time, by going from lower b to higher b while playing the chord. That problem would be neatly solved by tuning narrower fifths.

Such a can of worms. And still people manage to play beautiful music.

Bart

September 26, 2009 at 12:52 PM ·

September 26, 2009 at 02:28 PM ·

Violin players have it easy. You want to tickle your math anxiety, try this:

September 26, 2009 at 03:22 PM ·

What's wrong with a fork? No batteries to run down. I was taught to tune the A and then tune the other strings to it so there are no beats. Simple and guaranteed accurate. After all, that's the only option in an orchestra, where you tune to the oboe/piano/leader depending on the instruments involved.

September 26, 2009 at 07:08 PM ·

Simon Fischer, I tune my first finger B natural to the E string, in accordance with pythagorean tuning.  I'm assuming this is also the note you would choose when playing a B major scale.  Is it also the note you would choose for a D major scale?  Only when playing it melodically?(I suppose I should refer to that chart that Bart so kindly posted.)

The thing that boggles me (as I wrote in a similar thread I started on playing with imaginary people), is that C natural is lower when using melodic (pythagorean) tuning, even if C natural is the tonic.  But when you add a harmonic E (i.e. when you add your duet partner, who is playing an E at the same time), it lifts the C up.

Is this right?  Can a C major scale begin on two different notes, depending on just or pythagorean tuning?  That's so trippy...  No wonder I always feel like I'm having to adjust my fingers.

September 26, 2009 at 07:14 PM ·

"The situation is hopeless, but not serious."

September 26, 2009 at 09:42 PM ·

I never mean to frighten the horses...

To my way of thinking, what confuses things is the notion that major, minor, augmented and diminished intervals can, to a certain extent, be adjusted wider or narrower according to taste, but that perfect intervals must be perfect, i.e. they are either in tune or not in tune, and there can be no disagreement.

The confusion I am thinking of is that surely this applies only to double-stops. After all, if you play an out-of-tune octave, or out-of-tune perfect fourth, you can't say 'I like it like that'. Well, I suppose you can, but it won't help matters.

But when it comes to single notes that are part of a phrase or melody, there is leeway. For instance, this first-finger B in first position on the A string. Much of the time it needs to be in a middle place where it is slightly sharp to the open D and slightly flat to the open E. When that is so, then the perfect fourth from B to E is slightly wide. But in a phrase or melody, and when tuning according to the harmony of the moment, the B is simply not noticeable as being 'flat', nor the E 'sharp'.

Other times the B needs to be slightly lower or higher depending on the context.

I always remember playing fourth violin in a student performance of the Vivaldi B minor 4 violin concerto. The fellow playing first violin could not be dissuaded from tuning his first finger B exactly to the open E, and the first-finger F# on the E string exactly a perfect fifth above the B. The result was horrendous sharpness that spoiled what would otherwise have been a concert to look forward to. I still haven't got over it, as you can see.

What I also have never forgotten, in a good sense, is a conversation with a certain very talented musician when I was about 25. I was explaining to her about 'fixed' and 'movable' notes in double stops, i.e. playing third-finger D on the A string, and first-finger F# on the E string, if you want the third tone to be in tune you have to play the F# flat. It is 'movable' while the third finger is 'fixed', since it must be in tune with the open D.

Or if you play second-finger F on the D string with third-finger D on the A string, if you want the third-tone to be in tune you have to play the F slightly sharp. In the first case, the movable note is above the fixed note, and in the second case it is below the fixed note. So the rule is: pull the movable note in the direction of the fixed note.

So anyway, my talented musician friend, who knew nothing of all this (but she was a terrific player - still is) listened patiently and politely until I had finished, and then said, 'But why do you need to know all this stuff?! Why can't you just listen?!'

Of course she was entirely right. First come your ears, afterwards any theory you may wish to examine or apply.  The fact is, surely, you really don't need to know much at all, intellectually, to be the most fabulous player and musician on the planet.

But not everyone is on the same level, and lots of players do not know exactly what 'in tune' is; so simply listening just isn't the point, and a little explanation, or some reference-points, can be a great help. But still, I've never forgotten her tart reply. It is worth always remembering: never put your thinking ahead of your ears.

September 26, 2009 at 10:49 PM ·

@ Mr. Fischer- In your earlier post, are you saying that the violin be tuned with the G & D strings slightly sharp and the E string slightly flat?

September 26, 2009 at 11:38 PM ·

Some people would say that. Others would say to make the E bang in tune with the A. I once tested this by asking a few people sitting around me in an orchestra, about how to tune in narrow fifths. Some said to tune the E flat, some said not to.

I think the interesting way to look at it is to consider what would happen if you had a fifth string a perfect fifth above the E string - a 'B string'. If, when you tune down from the A in exact perfect fifths, you end up with the G too flat, the same 'stretching' must apply when tuning up from the A in exact perfect fifths.

If the E were perfectly in tune with the A, and then the 'B string' perfectly in tune with the E, the 'B string' would be too sharp. Tuning in narrow fifths, around the A, logically means that the two strings either side of it (keeping our 'B string' for a moment), are pulled slightly in the direction of the A, the lower strings up and the upper strings down.

As a young teacher I didn't know much of this, and when I was teaching the Mendelssohn Concerto I ran into the following sequence more than a couple of times, before I knew not to:

The student would play the opening B too sharp (according to me, at least). I would point it out, but it would keep happening. So then I would say, look, find the same B on the A string (a tone above the middle of the string) and tune it as a perfect-fifth double stop with the open E. But when the student did that, the B that was in tune with the open E would be exactly the same B that they had been playing all along.

So then I would simply say - well, I don't know why that is, there must be something wrong with your strings or something, but all I can say is it's too sharp!

Then, if you do take a very slightly flatter B, a perfect fifth above the open E, this is exactly the same as the first-finger in first-position on the A string when the first finger is 'midway', i.e. slightly sharp to the open D and slightly flat to the open E, as mentioned previously.

I can't resist passing on the immortal advice - or immortalising the advice - of the British violinist Roy Gillard, who sadly passed away a few years ago while not yet at an age when you might expect him to. He would tell you: Intonation is simply a question of mind over matter. If the audience don't mind, it doesn't matter!

I always laughed when I heard Roy say this, but unfortunately it doesn't work if you don't care about the audience - if it is yourself you don't want to upset by playing out of tune! (Sorry to spoil a good joke by getting serious again.)

September 27, 2009 at 01:28 AM ·

I have one of these Korg tuners, I thought it would be handy when tuning up at noisy venues, so I have not used it much at all. When I do use it the strings are close enough for a little tweek. Just now I discovered the calibrator. If G is set at 438, D at 439, A at 440, and E at 441, the violin is even closer to being in tune, but still it requires a little tweek. I have always preferred the tuning fork, it’s much quicker by ear than watching a needle and waiting for the green light.

How do we tolerate the out-of-tune strings when we are so diligent to play all our notes in tune according to their position/function in the key. I have only seen the violinist tune to the ’piano-A’ then proceed to tune the other strings from the A-string. Perhaps they did tune to just intonation and then tweek it a little to reach equal temperament.

September 28, 2009 at 08:20 AM ·

The key of e minor presents the problems in a nutshell. One wants the e to resonate with the E string, and the g with the G string. Together, the minor third they form is too narrow for comfort. Perhaps that is why only 4% of v.commers wish to build their home in this key. Now how do they solve this?

September 28, 2009 at 02:59 PM ·

September 28, 2009 at 03:43 PM ·

Math is not my friend. Now I feel like I am doomed to play out of tune for the rest of my life.

September 28, 2009 at 04:11 PM ·

September 28, 2009 at 04:25 PM ·

Bill, yes, I can do those things. Perhaps I'll have to give up my math phobia.

September 28, 2009 at 07:02 PM ·

Bill,

Exactly!

To satisfy my own curiosity I made a few more tables, mostly about quarter-comma meantone temperament.

 two octaves and a third four fifths four meantone fifths one meantone fifth ratio 5 5,0625 5 1,4953 cents 2786,31 2807,82 2786,31 696,58

Comparing the three kinds of fifth, pure, equally-tempered and meantone, gives

 fifth ratio cents G D A E pure 1,5000 701,96 195,56 293,33 440,00 660,00 equal 1,4983 700,00 196,00 293,66 440,00 659,26 meantone 1,4953 696,58 196,77 294,25 440,00 657,95

Some string quartets tune to equal temperament -- and use electronic tuners to achieve it (and to end all discussion, as a professional quartet player told me).

The chief virtue of meantone temperament seems to be that it makes Simon Fischer's b' problem disappear. It is easy to recognize because the thirds sound so sweet, and many early music ensembles use it.

For the math geeks among us: the continued fraction expansion for the meantone fifth gives rise to the 31-tone temperament. (see here for English, but without the math). In 1978 or 1979 I heard a lecture about this by professor Otto Frisch, one of the discoverers of nuclear fission. You can play nearly consonant sevenths in it as well: sounds strange!

September 28, 2009 at 07:41 PM ·

I toyed around last night tuning my violin to various widths, and then tuning to my piano, which was tuned just last week.  I burned about an hour finding exactly what it was that my ear was telling me and why.  Then, when I went to practice, my fingers were so confused I played everything horribly.  Or at least it seemed.  Maybe I'm just too aware of it now.  Or maybe I just don't know what I want anymore--the voices in my head are contradicting each other...

September 28, 2009 at 07:51 PM ·

@ Emily-  When the voices in my head are like that I tell them, "Don't Make Me Come In There!!!" and they'll settle down!  I hope this helps. ;)

September 28, 2009 at 08:26 PM ·

I put my voices to work.....I'm a writer.

September 28, 2009 at 09:34 PM ·

I wonder, do the same physics apply to a digital tuner? or does the "needle" on the screen simply go to the middle when the string is perfectly in tune?

September 29, 2009 at 02:52 AM ·

Scott, both the digital tuner and the analog tuner (the one with the needle) follow the same laws of physics.

September 29, 2009 at 03:10 AM ·

Some people practice with the tuner on. I wonder if that is more a hindrance, a crutch, or an aid?

I just find it distracting with the needle dancing around.

September 29, 2009 at 05:55 AM ·

I don't recommend it.  First of all, most consumer or amateur electronic tuners (the Cdn\$30-40 tuners) don't respond accurately right away.  It takes too long for the needle or light to settle.  So, two problems with that - 1) by the time you see the needle or light settle, your sense of timing is totally off, and 2) if you don't wait long enough, you may get a false sense of intonation - or worse, you could actually be playing in tune to start with, but you believe the first indication from the tuner which says you're out of tune (before it settles), and you go and adjust your finger!

Having said that, I'm not totally against practising your scales to a tuner, but only if you're playing the scale VERY SLOWLY, AND if you have a tuner which responds very quickly and accurately.

September 29, 2009 at 07:39 AM ·

...it's like sticking your ear in a wheelchair and rolling it right to the note.

Go ahead and play with a tuner.  It's great at outputting unarguable answers, like a calculator.  Just promise me you'll hum the tune in your head when you go away, and when you sit back down, you'll try to remember to tap into the sound you've burned into your memory.  Better yet, hum along to every tune you hear throughout the day, and try to match the pitches you hear.  Maybe you'll find you don't need a tuner after all.  After all, that's what we're trying to achieve, isn't it?  A well trained ear is the best calculator of all.

September 29, 2009 at 12:26 PM ·

If you want to play out of tune, all the time, like a piano, using a tuner is definitely the way to do it. Unfortunately, the people who will be listening to you will be using their ears, and will know the difference.

September 29, 2009 at 03:14 PM ·

I have a neighbor who's car alarm repeats honking the car's horn and it went off when I had my 440-A electric tuner playing.  The car horn is... 440-A.  If the battery of my tuner/metronome runs out and I need to tune up I'll just kick his car!

September 29, 2009 at 05:13 PM ·

September 29, 2009 at 10:25 PM ·

There seems to be a consensus that practicing with a tuner is a bad idea, but I don't agree entirely.  Most music is intuitive; that is, you can hear when the notes are right, but sometimes when you have weird accidentals with awkward fingerings or positions, like 2nd or 4th position, I find it very helpful to turn on the tuner and play very slowly, just to make sure you are hitting the right notes.

September 29, 2009 at 10:49 PM ·

September 29, 2009 at 11:24 PM ·

The previous responses were good, but allow me to answer your question more precisely. I tune pianos, by the by.

Your chromatic tuner represents the notes found on a keyboard instrument like a piano. A piano is tuned (nowadays, and most often) to stretched equal temperament. The piano's A is tuned exactly to 440, and is usually the first note tuned before "setting the temperament." Setting the temperament involves tuning the fifths slightly flat (and thus making the fourths slightly sharp). Go ahead and try it if you have access to a tuned piano. Play some fifths and some fourths in the middle register. You will hear "beats" - those little wobbly noises that you have been using to tune your violin's perfect fifths. You'll hear them in just about every interval that you can play on a piano besides the octaves.

Right, so, this means that a chromatic tuner expects slightly flatted fifths and slightly sharped fourths. But - wait a minute... us violinists don't use slightly flatted fifths OR slightly sharped fourths. That would sound absolutely terrible! No, no, we tune our violins with perfect fifths, and when playing fourths, we make them as perfect as we can. This means that our notes do not always match up perfectly with the notes found on a piano or on your tuner.

When we say flatted fifths, it stands to reason that, if you start on middle C, and play to the fifth degree above it (G), that G will be slightly "flat" with regards to a perfect interval. It's the same idea if you go from D to A, and everything else.

So, this last paragraph indicates that if you start on your A string, and play the D below it (and it is tuned to a perfect fifth), then your chromatic tuner will indicate that the D is flat. This is because, in order to create a flatted fifth, the lower D must be slightly sharp in comparison to how you usually tune your violin. The G will likewise seem to be even more flat, according to the tuner. Your E will register as being a bit sharp, since it is tuned to a perfect fifth with your A string. Does this make sense?

Don't be fooled by ignorant comments about the violin or the piano playing "out of tune" because of this discrepancy. This is a very incorrect way of thinking. There are many, many tuning systems out there, and all of them are viable. For Western musicians, though, an approximation of equal temperament is always the most appropriate nowadays. If Mozart had heard a modern piano playing his music, he certainly would have mentioned that the music sounds a bit out of tune.

So, keep tuning to your perfect fifths, and try to pay a little less attention to your chromatic tuner. I find that a good way to practice is with Midi. Play with a pre-recorded rendition of the lines that your violin is playing, and thus you will shift your fingers into more of an equal temperament-like mode. Practicing with the chromatic tuner will just frustrate you. I don't recommended them to my students.

---

If you're auditioning with an orchestra that demands A=442 or something, then you might need a special tuner or fork. For the most part, I tune up with my Quartz metronome. It has the A=440. That's all you really need.

September 30, 2009 at 04:42 AM ·

After reading the above detailed comment, and other informative posts, I can finally understand why I don't enjoy playing the piano as much as I do on the violin. Everytime I strike a chord on the piano I feel the sound slightly clashed and not as harmonized as the violin. That weird "beat" or whatever you want to call it kills my ear, it just don't have that smooth feeling that I can get from the violin.

And yet, everytime I play with the piano (I practice alone often) I always thought "what's wrong with my intonation? They're all off!". Yikes!

September 30, 2009 at 09:04 AM ·

I am strongly opposed to playing with a tuner.  A tuner teaches you intonation by seeing rather than by hearing.

September 30, 2009 at 02:03 PM ·

I'm amused to note I find this all fascinating. Sharelle- I read that Casals said he spent the first hour finding E on the D string, and after that his 6-hour practice day was easy. I so get that. This discussion also may answer a number of things I notice in my teaching lately. 1)The adult student whose A on the E sounds flat to me, but who hears what I want as sharp, has to be operating in a different system. 2) Why I change or take away finger markers a lot, why I try to only mark specific spots for specific pieces, and why I'm regularly saying to edge that one up or down. 3) Why my advanced viola girl has herself written on her key-of-F solo "E's a fingernail high". 4) Why I say to my jam that I like my E string a little high. 5) Why I don't exactly agree that F# & Gb or C# & Db are the same key, but go w/the flow since I don't like fingering in Gb or Db a bit. (Just avoid all 4 unless absolutely required)   :)) Sue

September 30, 2009 at 10:51 PM ·

I generally have a chromatic tuner (Korg OT-12) permanently on when I'm teaching. Nearly all the time it is there only to provide an A to tune to. The student could walk six paces to the piano and play an A, or I could go to the immense trouble of picking up a tuning fork, or leaning forward and sounding the A on the metronome, but the tuner just sits in the corner of the room and they can see the needle and the lights from where they are, and nobody has to do anything, so it's nice and convenient.

For weeks on end, that is the only time it is ever used. But occasionally I might find that someone is playing, say, a C# consistently too flat. Having pointed it out enough times, and they still seem to be satisfied with the note when I am not, I might suggest we look at what the chromatic tuner thinks. Often it turns out that it isn't that they are playing a tempered C# while I am wanting a sharper, 'expressive' C# - but they are in fact playing slightly flat even to the tuner.

The machine is useful for that. Saves time, and stops matters being confused with the whole business of the exact tuning of accidentals being partly a matter of taste.

Another use I have sometimes found for the chromatic tuner is in checking or measuring sharpened sharps and flattened flats. The danger is that in one's musical zeal one ends up almost playing quarter-tones in the name of expressive intonation. So the thing is to see what the tuner thinks is a bang-on C#, and then move the needle just slightly to the right of the zero.

This isn't a matter of tuning by looking instead of by listening. First you hear what the tuner says is an in-tune C#, then you hear what the C# is like when you move the needle fractionally to the right. Or without looking at the tuner you play a passage and stop on a C#, and then see what the tuner thinks of it. Sometimes you may realise that the note could be flatter and still be a couple of degrees to the left of zero.

But personally I have always had a very short interest-span when it comes to chromatic tuners.

October 2, 2009 at 08:44 PM ·

I use a chromatic tuner and fight myself over whether to tune just the A or all 4.

I have to say that *if* I tune all four, the result is almost always unsatisfactory. Most of my teachers believed that a pure 5th (whether sharp or flat) reveals itself eventually and playing music that is reliant on good 5ths (esp Bach and Classical) outweighs any benefit from being exactly "in tune" with the tuner.

I advise my students to use a tuner just for the A and then tune the old way!!

IF you are interested, I will tune myself and then take a reading to see how the other strings match up.

October 3, 2009 at 01:15 AM ·

Simon Fischer:  "I can't resist passing on the immortal advice - or immortalising the advice - of the British violinist Roy Gillard, who sadly passed away a few years ago while not yet at an age when you might expect him to. He would tell you: Intonation is simply a question of mind over matter. If the audience don't mind, it doesn't matter!"

Aaron:  I have found this thread very interesting and liberating.  For me. considering issues from all sides helps me become comfortable with the fact that sometimes there is no one simple true answer and that I can be flexible in my approach.  I tune a guitar and steel guitar using a tuner for a "close approximation", then I play some chords and adjust to where the chords sound sweet. I will probably continue to do the same with the violin, but thinking about the various aspects of this issue is certainly useful.  I love the quotation above.  My intonation does vary with my audience: just me, me and a teacher, me and bandmates.

October 3, 2009 at 02:36 AM ·

October 3, 2009 at 03:52 AM ·

Well, this discussion sure has propelled me into a strange direction....
I did not do the math and define the exact ratio to compress the range when tuning, but I did follow the math in the discussion...
So, I decided to 'analog it a bit', and reset the tuning on my fiddle. I followed the advice proposed by Simon to make 'Narrow fifths', and my fiddle came alive in a manner it never had before.

When playing, I could feel harmonics that were never there, resonant sounds that I did not know I could generate, and it sounded wonderful. It was not only fine to my ear, but away from the violin too! My only audience was my wife, doing something else in another room, and she could recognize the difference.

I think I'll play that way a while, and then try other options... but I doubt I will generate anything like the result I got with this change.

October 3, 2009 at 04:28 AM ·

I got a better conception of all these things after watching Prof. Sassmannshauss elucidating  videos. It is worth having a look:

violinmasterclass.com/intonation.php

He concludes that practically we use three different intonation systems while playing violin, and that the choice between them and the place and time to use it is a matter of wise discretion.

I don't mean to start a parallel thread, but this also makes me curious to know in which system those with perfect pitch have their ears 'tuned'. If it is in equal temperament, I guess that the Pythagorean system with sound unbearable to them, and vice-verse.

October 3, 2009 at 05:55 AM ·

I think part of it may also be the violin; when they are created the tones are a major consideration of the final product (or so I understand; I have never made an instrument).

October 3, 2009 at 01:03 PM ·

@Roland, now curosity is killing me. I think I'm going to give Simon's advice a spin, too and see what happens with narrow fifths.

October 3, 2009 at 01:30 PM ·

My kids' teacher taught them to tune with the tuner for A only, and then D, G, and then E according to fifths by ear. After that, check (and adjust) each string with the tuner.

The problem is that they always find the strings slightly off (D, G slight flat, E slight sharp) when checked with the tuner. Then they would readjust everything (hence not trusting their ears) according to the needle. I am thinking - doesnt that defeats the purpose of tuning by ear?

What should I tell my kids to do?

October 3, 2009 at 02:19 PM ·

Tell your kids they have very good ears and let them in on the compromise that is piano tuning.

October 3, 2009 at 03:48 PM ·

"but this also makes me curious to know in which system those with perfect pitch have their ears 'tuned'."

I know only one person with perfect pitch, and she has told me that when she played in a different orchestra with a different, higher, "A" in the summer it took her about two weeks to adjust so that the new situation sounded good to her. . . and another two to adjust back when she got home. From that, I'd guess that perfect pitch isn't a fixed thing, but slowly slides around to the situation at hand. I would expect that the temperament involved would work the same--whatever you spent the majority of your time doing would rule.

October 3, 2009 at 04:58 PM ·

Michael...

The idea of perfect pitch is adjustable....in my opinion. If I play with a group that is actually not tuned to 440 then I am at sea for a while. However, it is not possible to get too far off before things get really difficult. I once played for a conductor that asked the orchestra to play something written in Eb in E. 20 minutes into the rehearsal, some of us (mostly those with perfect pitch) had rolled over and given up. If a player has perfect pitch he or she will have learned to monitor every aspect of their playing using the perfect pitch. We can sing intervals, of course and we can compare one note to another, but essentially, every note on the score is already pre-conceived in the brain/ear before playing. Of course, all players do this, but the method of learning this skill is established from either end of the spectrum. I can play with a group a tiny bit flat or sharp but no more. I cannot turn my hand to baroque pitch no matter how hard I try......any note I see WILL be played spot on (unless I mess up the fingering or shift) so even if my A is tuned way out, I'll still play stopped notes in tune.

PS - Nice to see you on here, we used to exchange messages on M-net.

October 3, 2009 at 05:58 PM ·

Thank god I don't have perfect pitch. That story has convinced me to detune and randomly change the a-strings around the house regularly.

I will do anything I can do to prevent this curse called "perfect pitch" from infecting my household.

Maybe some Penna Dutch Hex Panels would be helpful, too...

October 3, 2009 at 06:10 PM ·

Yeah, David, she also told me that hearing transcriptions of things she knows to other keys for other instruments is intolerable to her.

October 3, 2009 at 07:07 PM ·

Bill and Michael. Sometimes it's like a curse.....no, really.

As a child, I would go to concerts and my brain would try to process every note.....for example, I'd hear a tune on an oboe or clarinet (or violin!) and rather than hear music, I'd almost "see" the notes and their names.

The plus side, as a teacher, is that often I'll hear two-three-four? measures and then stop the pupil and say.....your D# in the middle is not good. They will ask me which bar (measure) and which note and I don't know!!! Anyone can do this (I presume) but it's the fact that I will "call" the note rather than say 4th 16th note of third group....

I often got criticised at the Conservatory for not playing in tune well enough. Most people couldn't understand this from a perfect pitch-er but the reality was the an out of tune note was far more annoying to me than others......sometimes even a split second ability to name notes and feel the tuning isn't fast enough to adjust....

October 4, 2009 at 01:13 PM ·

"I often got criticised at the Conservatory for not playing in tune well enough. Most people couldn't understand this from a perfect pitch-er but the reality was the an out of tune note was far more annoying to me than others......"

Yeah, I think the problem here is that violinists can vary their pitches according to just intonation. I imagine that people with perfect pitch have a very difficult time differentiating between equal temperament pitches and the necessary adjustments required by just intonation. With just intonation, all pitches will vary slightly depending on the musical framework. This must cause a lot of problems for those with perfect pitch, because they are used to a non-varying approach to intonation.

Anyway, this above statement is tricky. The professor at the conservatory tells you that you are playing out of tune, and you respond by thinking "I can't play out of tune, I have perfect pitch." A bit odd, to say the least. The reality of the situation is that your professor and you are working in two different systems. In his system (probably closer to just intonation than equal temperament), you are playing horridly out of tune. In your system (probably a very specific approximation of ET on the violin), you are playing in tune very nicely.

Personally, I am glad that I don't have this so-called "perfect pitch." I would much rather be able to hear and play with tone coloring than to be set on playing middle C at exactly 256 Hz every time.

October 4, 2009 at 02:37 PM ·

October 4, 2009 at 02:37 PM ·

Jerry Agin's program Intonia gives one the opportunity to compare equal, just, and Pythagorean intonation visually. It has a separate setting for tuning, which is more sensitive than the setting used for playing. I found it a great tool for exploring tuning and intonation.

October 4, 2009 at 02:38 PM ·

"The professor at the conservatory tells you that you are playing out of tune, and you respond by thinking "I can't play out of tune, I have perfect pitch."

Truth is a perfect pitch player can and will somethimes play out of tune. You cannot use the perfect pitch like an unfailing technique.....you have to practice the tuning "in" to your pieces.

I practice the notes to my system of hearing, which is certainly NOT the same as ET. However, in my experience, I cannot recall a time where I have met up with a pianist or another instrumentalist and I have not fitted 99% in to ET.

October 4, 2009 at 06:10 PM ·

I tried tuning the "narrow 5ths" as suggested, it was pretty hard to make that narrow 5ths to get in tune without the help of tuner. There's no references how much flatter for the E and how much shaper for the D and G, except that I play a B on A string, which should match E and D, likewise, E on D string which match both A and G. But that'll be much too troublesome and looks embarrasing to do that much work on tuning on stage.

I find I'm so used to the usual 5th tunings and I'm not experiencing any improvement, except some difficulties on intonation and strange sounding fingered 5ths (e.g. 1st finger on A and E on both G/D strings). Going back to the regular tuning and I find myself playing everything "right" again.

Am I missing something?

October 4, 2009 at 09:34 PM ·

I believe the narrow 5ths can be tuned without a tuner by first tuning up to the pure 5ths thus eliminating the acoustic beats then raising the lower strings until there is one perceptible beat per second, lowering the note in the case of the E string, appearently this is acceptable.

This tuning is to allow keyboards to be played in all keys, without extreme dissonance,

Bach may have tuned the keyboard differently per occasion, or per composition, throughout his career.

October 5, 2009 at 01:59 AM ·

This is certainly a thread to put away in a file!

On another note, it has been a while since we have had a thread reach that magic 100 posts!  Will this one do it????

October 5, 2009 at 02:13 AM ·

Royce,

I didn't expect this topic to spark so much interest and controversy, but it certainly has been educational.  In particular, the link posted by Rick Benford really helps to clarify the different types of intonation.  Unfortunately, I do not subscribe to any of the "standard" intonation types.  The fingers of my left hand are just not that accurate and repeatable to say that I intentionally flatten or sharpen a note here and there.  It takes all the concentration I can muster just to play something not too out of tune.  The more I learn, the more I realize how impossibly difficult it is to play violin.  But, therein lies the enjoyment.  It's the challenge that makes violin so much fun and so rewarding.

October 5, 2009 at 04:38 AM ·

Smiley- I agree with you.  However, my ears are good enough that I do make intonation adjustments and have been experamenting with the intonations & tempraments that have been brought up.  Facinating!  This thread has realy helped me to understand what is written in the book, "Temprament".  I highly recomend it, but it is heady stuff!  Wont it be cool if this hits the 100 mark?!?!?!

October 5, 2009 at 11:32 AM ·

Henry pointed out a very important practice of  the old times that seems forgotten nowadays. Especially Paganini adapted tuning to pieces by mean of the scordatura ( as in Carnaval of Venice ) or  by tuning the G string a quarter of tone higher for the purity of thirds in some caprices (That was one of this fourth secrets which I forgot))

October 5, 2009 at 07:35 PM ·

A quarter tone? That's fifty cents!

Kidding aside, was it really that much?

October 6, 2009 at 11:56 AM ·

I would'nt  assert the accuracy of  the quarter of a tone . I confirm Paganini  tuned  the G string a bit higher .He might have  tuned some  minor thirds  by choicing  the 7/6  ratio resulting in a decreasing of 33 cents from the temperated minor third (not very  far from the quarter of atone)

October 6, 2009 at 02:35 PM ·

Found something interesting: two well-known Dutch violinists, the late Jeanne Vos and Bouw Lemkes, have made special study of microtonality, and used the results in practice. Compositions have been written for them that make use of 31 tone equal temperament. Bouw Lemkes gives an explanation here.

I believe it is very interesting to read, even in the mangled version that automatic translators produce  from the carefully written Dutch in the article.

October 6, 2009 at 02:37 PM ·

@ Bart- Nice site!!!!! Thanks for sharing!

October 6, 2009 at 09:28 PM ·

This has been a fascinating thread.

I play old time fiddle music and have noticed that some of the older musicians that play in a more archaic style (sadly, mostly heard only on recordings now) are very consistent in playing some notes very different from equal temperament. I hear at least two distinctly different notes between C and C# in the key of A, and these notes sound very much right to my ear. A lot of folks who learn those old tunes force the notes up or down and it really changes the feel of the music.

I believe most of the tunes I hear this in are in the Mixolydian and Dorian modes (the scales with flatted seventh, and flatted third and seventh in relation to a major scale-if I've got it right).

October 9, 2009 at 07:41 PM ·

I recently downloaded an app for my iPhone called Cleartune.  It was \$3.99 and turns your iphone into a chromatic tuner.  I mention it for two reasons.  First, it is a handy app if you happen to own an iPhone -- avoids having to carry around a tuner.

Secondly, and more relevant to this discussion, it offers a setting for the type of temperament.  I thought I was already confused with the information in this thread.  The Cleartune app offers settings for the following temperaments:

- Equal Temperament

- Powertonal Guitar

- Pythagorean

- Pythagorean just

- Standard Just intonation

- 1/3 SC Minor Thirds

- 1/6 SC Attenuated

- 2/7 SC Extended or Zarlino

- Almost-equal

- Aron-Neidhardt

- Barnes' Bach

- Kellner's Bach

- Kimberger III

- Shifted Valiotti/Young

- Valiotti

- Werckmeister I/III

- Early French

- Homogeneous French 1/5 SC

- Rameau

- Roussear II

- Rousseau III

- Roussear IV

I'm not even going to ask what all these mean, but I guess it is safe to say that there are many different concepts of intonation.

October 9, 2009 at 09:30 PM ·

@! Smiley!!!!!!

DUDE!!!!!!  I'd be afraid of my iPhone giving me some sort of cancer!!!!!!

October 10, 2009 at 04:23 AM ·

Just looked at the scale on my Korg tuner.  Using Smiley's calculation that the difference would be .75 hertz and a ballpark 4 cents/hertz conversion, we are talking about three cents which is just bigger than half the space between dead center and the first tick mark.   Maybe I am an idiot but I would be very happy to be that in tune at all times.....actually I would be very happy to be able to tune my D string that accurately all the time.  Tom

October 10, 2009 at 10:34 AM ·

Greetings,

you guys do realize that once you have selected the tuning system of the day you have to match it with an appropriate vibrato or the whole point would be occluded.  Such systems are strapped on the back of the hand  (or botty) and administer a mild electric pulse.

Cheers,

Buri

October 10, 2009 at 02:18 PM ·

Tom, If you divide 440 by 1.5 twice you'll see that based on perfect fifths, the cumulative error to reach a perfect G would give a pitch 44cents lower than the ET pitch of 196hz.. How many marks is that on your Korg?

October 10, 2009 at 02:41 PM ·

Michael,

That's not the result I got. See tables, above. According to my computations the difference between a G tuned to perfect fifths or to 12ET fifths is 3.92 cents, or 0.44 Hz.

Yours, puzzled,

Bart

October 11, 2009 at 01:21 AM ·

Once I had the idea of becoming a ’Piano Tuner’ and I imagine that that list of tunning systems would be part of the curriculum. But, in regards to violin playing I don’t think we use any ‘single’ one of them, if we did the vln might sound out-of-tune. Maybe we use a combination of ‘all’ of them, because the violin is a ‘pitch-flexible’ instrument so we can play quarter tones and micro tones any where at any time we like. That’s what made this instrument appealing to me.

Pitch can be varied with no restraints and adjusted in the midst of performance, eg;

when playing a long note with the piano or tunning-in our double stops and then readjusting these notes to sound in-tune when playing melodic passages.

In Blues and Rock music there are the ’blues notes’ which are the 3rd, 5th, 7th notes of the

diatonic major scale, they are flattened by an ‘inexact’ amount, generally less than a semitone, and the performer has complete control according to ‘their’ taste.

If we consider that any given note could belong to many keys/scales/modes, ( hehe, I tried to count them ) including exotic ones, their position/function in each is different and so on the ‘pitch-flexible’ instruments we are able adjust them accordingly.

That list of tunning systems maybe daunting but all we need to rely on is our faculty to listen.

This is my ‘modus operandi’ and I hope that it is of help to others.

October 12, 2009 at 12:43 AM ·

With the tunings that Smiley bought, how would someone use them?  Just tune to them and play any Piece?

October 12, 2009 at 06:19 AM ·

I was wondering which one of those 'Bach' tunnng systems would be better suited  for the keyboard to play the 'six S&P for vln solo'. And then if the violin tunes-up the strings to the 5ths on the keyboard and both play in unison, note-for-note, how different would it sound to the violinist after they had played these pieces solo for so many years..? And how would the music sound if they used a Microtonal Keyboard...?

October 12, 2009 at 06:27 AM ·

Royce,

Opinions vary. On the one hand there is Luke, who says that being used to 12 tone equal temperament (12TET) we cannot expect ourselves to adapt quickly to other systems of tuning. On the other hand, playing with squeezed fifths immediately produces a different set of resonances which can be used straightaway. I believe it is the ear that guides the fingers. If that is correct, practice enables the fingers to follow the ear, rather than ingraining microscopically precise movements that are the same every time. As an example of this: it took a lot of time for me to learn to play octaves (Kreutzer 25) in tune. When they finally were to my teacher's satisfaction, he proceeded to detune my violin and asked me to play the octaves again. To my surprise, it wasn't all that difficult to compensate on the fly.

Here is another table, this time of several meantone temperaments. 1/3 cm means 1/3 comma meantone, produced by subtracting a third of the syntonic comma from a pure fifth. It is interesting to note that subtracting 1/11 of the comma nearly yields the equally tempered fifth: 12TET is 1/11 comma meantone.

 1/3 cm 1/4 cm 1/6 cm 12ET Pythagoras just cents ratios cents ratios cents ratios cents ratios cents ratios cents ratios fifth 694,79 1,494 696,58 1,495 698,37 1,497 700 1,498 701,96 1,500 1,500 do 0 1,000 0 1,000 0 1,000 0 1,000 0 1,000 0 1,000 re 189,57 1,116 193,16 1,118 196,74 1,120 200 1,122 203,91 1,125 203,91 1,125 mi 379,14 1,245 386,31 1,250 393,48 1,255 400 1,260 407,82 1,266 386,31 1,250 fa 505,21 1,339 503,42 1,337 501,63 1,336 500 1,335 498,04 1,333 498,04 1,333 so 694,79 1,494 696,58 1,495 698,37 1,497 700 1,498 701,96 1,500 701,96 1,500 la 884,36 1,667 889,74 1,672 895,11 1,677 900 1,682 905,87 1,688 884,36 1,667 ti 1073,93 1,860 1082,89 1,869 1091,85 1,879 1100 1,888 1109,78 1,898 1088,27 1,875 do 1200 2,000 1200 2,000 1200 2,000 1200 2,000 1200 2,000 1200 2,000 Hz Hz Hz Hz Hz Hz g 197,18 196,77 196,37 196,00 195,56 195,56 d 294,55 294,25 293,94 293,66 293,33 293,33 a 440,00 440,00 440,00 440,00 440,00 440,00 e 657,27 657,95 658,63 659,26 660,00 660,00

Have fun!

Bart

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