understanding just temperament

April 22, 2009 at 10:56 PM ·

I have been studying temperament and I have a few questions about just temperament that are not quite answered by what I have been able to find on the wired.  I am hoping someone here could direct me to a bit more detailed (math oriented) discussion of the topic or maybe just answer these questions off the top of their head if they know.

Okay, first question. (nomenclature)

Suppose I start by tuning the open A-string to 440.  And then tune the E string to that A string by playing both strings together and adjusting the E string until the ratio of frequencies is exactly 1.5:1 (or 3:2) which is 660 Hz.   Is this 3:2 ratio on fifths all that is required to establish a "just" temperament?  I think so, therefor I am gonig to use that terminology in the next question, but if not, some of the questions below might server to elucidate where I have gone wrong in my understanding of just temperament.

Question 2  (playing octaves on justly tuned instruments)

First, suppose a break out a hypothetical perfectly tuned, equal temperamnet piano.  If we play that with our justly tuned violin, then that "E" at 660 Hz, will be slightly sharp (about 1.955 cents sharp) compared to the piano's 659.2255 Hz.  For that matter, every note we play on that E string will be likewise sharp (from the point of view of the piano) by exactly the same amount.  So far, so good.   From what I understand, that is just the consequence of playing a justly tuned instrument alongside an instrument tuned with temperament.  So put the piano away and stick with justly tuned instruments for now.  Suppose I crawl up the E string to the A above E.  That note should be exactly one octave above the open A-string.  But that E-string is sharp... and if you play it simultaneously with the open A there will be dissonance (about 2 cents worth).   Is it the responsibility of the violinist to play that note on the E-string just slightly flat so that dissonance goes away?  What if your are playing the higher A (on the E-string) by itself, without the open-A would it still need to be flattened? (if you don't you won't get the harmonic resonance with the A string.)

This brings us to question 3 (the notes before the 5th)

Suppose that one the A's on my hypothetical piano is tuned to the same 440 Hz as the open A on my justly tuned violin.  Now if the Piano were present I would want to climb the A string in such a way that the 2nd (B), the 3rd (C#), and 4th (D) were in tune with my equal temperament piano.  In that case I would have 100 cents between each equally spaced tone.  But now consider that I have only justly tuned instruments nearby.  Do I need to adjust my 2nd, 3rd and 4th so that when I get to the 5th (E) it is tune with my (sharp - 660Hz) open E?  That is to say, do I use evenly spaced, but wider, steps between each note on the A-string so that 100 "just-cents" are equal to about 100.2793 "equal-temperament-cents"? Or do I play the 2nd, 3rd, and 4th on the A string as if I were playing with the piano and then sharpen only the 5th to make it match my open E (to catch the harmonic resonance off the open E string)?  Perhaps I do neither of those things!  Perhaps in a "just temperament" I play 3rds and 4th slightly flat (relative to equally tempered piano) to achieve precise 4:3 and 5:4 frequency ratios. Then I could then play the 2nd (B) slightly sharp so that it is an exact octave below the (justly tuned) fifth over open E (also a B but one octave higher).  The point is, that the spacing between on a just instrument tones is not constant at 100 cents or some other value but dependent whatever you might have played recently or simultaneously

Replies (21)

April 23, 2009 at 12:52 AM ·

There's a handy book out there for you:

How Equal Temperament Ruined Harmony (And Why You Should Care):


April 23, 2009 at 11:43 AM ·

Haha I am laffing because I caused a whole lot of maths jokes doing this sort of thing in the past.

Forget equal temperament and stop talking cents. That's just confusing.

1. There is not one and only one "just" temperament.

2. An instrument tuned in perfect 5ths does not have a dissonance when you play octaves. You play octaves period. The E string is stopped at a' and besides, if you were tuned in perfect 5ths (1.5:1) there's no dissonance with the open strings either.

3. The 2nd 3rd and 4th are in tune with the fundamental or with each other. That is the nature of a just tuning, in that order. Constructed from the fundamental, all notes have a relationship of low-ratios viz. 9/8 6/5, 5/4, 4/3, 3/2 2/1 etc. the 6th and 7th are essentially reciprocals of the 2nd and 3rds though there are interesting alternatives too. Certain ratios are disfavored in western music, others are natural to us.

Why close ratios? They sound pure. They create smooth harmonies. It is really all about the sound and a good fiddle player finds these just ratios as a matter of course. The guitarists do to, by bending, pushing, flatting and sharping strings. (Easier to do in high positions). The mandolinists just play fast and live with shimmer.

4."The point is, that the spacing between on a just instrument tones is not constant at 100 cents or some other value but dependent whatever you might have played recently or simultaneously" Yes that is correct. The spacing falls out from the harmonic structure and isn't a fixed distance.

I could go on about this all day but the real fun is for you to explore. Don't take my word for it. Everything I just told you I figured out myself with a slide rule, a tape measure and a guitar.

There is one more beautifully wierd thing about temperament and why it is always imperfect but I won't tell you. You have to find it out for yourself--much more fun that way.

April 23, 2009 at 07:40 AM ·

The short answer is that you shouldn't play with a tuner, you should play in tune. What "in tune" means varies with context--even the same note differers within a piece, depending on how it's used in comparison with the other notes being played at that time. Give up the idea that each note has a pitch.

April 23, 2009 at 11:20 AM ·

Hi Derek,

The Wikipedia articles on just intonation, music and mathematics, meantone temperament, equal temperament and Pythagorean tuning are most informative and as mathematical as anyone would wish.

And if you like Abelian groups, try "isomorphic keyboard".

Good hunting,


April 24, 2009 at 04:15 PM ·

Thanks for the book and web references.


I am trying to finish reading it all.  In essence it seems like "just temerpament" means different things to different people.  And that ultimately, from a practical perspective, what really matters is "what sounds good".  Musicians just seem to "play it by ear".  And when it comes to tuning... maybe that really is the ultimate authority.

Thanks again everyone... I will continue to monitor this thread in case other posts filter in...

April 24, 2009 at 06:04 PM ·

Yes, what sounds right.

But that doesn't mean it is meaningless to analyze these things.

You may find that digging into the following will pique your interest:

1. Near-eastern music: especially that played by the nomadic North-African peoples, Syrian music, Persian music, Arab music. These wide ranging musical galaxies make use of what we term "quarter tones" but it isn't that simple. They purposely raise certain intervals to places that sound out of tune to our piano-assimilated ears, but which are expressive and natural. Even from tribe to tribe there are variations. Look into it, it is fascinating.

2. Historical keyboard temperaments. Before ca. 1900 or a bit before, keyboards were tuned to various compromises, and this varied by region and by instrument. Organs, harpsichords, pianos being the three main groups.  Farther back you go, the more "pure" but less flexible you get. Read about the various approaches to mean-tone temperaments. In times past, there were even keyboards with split sharps/flats--i.e. more than 12 keys per octave.

3. Melodic versus harmonic intonation.

4. 20th century experimenters such as Harry Partch and more recently Kyle Gann.

Fretless stringed instruments have always been free to play "just" in any key, which is something that a keyboard is incapable of. Therefore harmony, music, and tonal color were very different for keyboards especially in the past. Because our modern equal temperament is generally quite close to melodic intonation mathematically (but not sonically!), we don't discern this so well in our own playing with keyboards. Play with an early mean-tone instrument and you will notice a difference.

Fretted instruments are not necessarily equal temperament and in fact even guitars are only equal temperament in theory. In practice they tend to be tweaked to some favored key and they have non-linearites as well. Viols have tied-on frets and in some cases even "tastini" or "little frets" which give the different harmonic enharmonics...I don't know the proper term here.

April 24, 2009 at 08:27 PM ·


Stuart Isacoff has written a book called "Temperament" which is probably the last word on all this and may tell you more than you want to know.

April 24, 2009 at 10:42 PM ·

Derik, intonation is a stylistic and interpretive choice. A judgment call. It won't correlate precisely with musical math, particularly if you calculate based only on the fundamental frequencies. Perceived pitch doesn't involve only the fundamental, but includes brain processing of all the harmonic components, which aren't always whole number multiples of the fundamental on the bowed string.

Which components will you incorporate, and how will you "weight" the value of each? It varies from one person to the next. Electronic tuner manufacturers have developed varying strategies for determining "correct" pitch.

Since you're a physicist, I'll mention that an instrument string doesn't behave exactly like the hypothetical "ideal" string. It has bending stiffness, so it incorporates some characteristics and inharmonicity of a solid rod.

And since this is a violin site, I'll mention that a violin produces negrligeable sound below middle C (261 ish). Why do we hear notes below this pitch? These are tones which are "implied" from the harmonic structure above that pitch, where a violin body begins to become an efficient sound radiator. Brain processing again.

You already know a lot. Look into "phsychoacoustics", and it will keep you busy for years. :-)

April 25, 2009 at 01:23 AM ·

Hi David:  Isn't the Helmhotlz resonance at about 290? There isn't much below that , though, like you said:




It is interesting how there is a notch at about 190, and then a brief spike at 180. Depending on where you tune your A, that could either notch out the open G, or notch out the A on the G string--like with Baroque tuning...

This also seems to explain why 5 string violins are tricky, and why small violas have such insipid C strings.



April 25, 2009 at 07:52 PM ·


I'll just answer this one question of yours: 

"Is it the responsibility of the violinist to play that note on the E-string just slightly flat so that dissonance goes away?"

This would not be good. The E and all tones related to it--especially octaves and 5ths--must have an open, ringing sound. If it doesn't, it sounds choked off. If you're playing by yourself, it's not a problem. If you're playing with a piano, it probably doesn't matter, either. I think the human ear separates the violin and piano sounds (especially due to vibrato) and gives the violin some slack. It's all a big fat bag of compromises.


April 25, 2009 at 10:39 PM ·

Ah Bill, I see you've been doing your homework!

The Helmholz resonance is rarely as high as 290, which would be close to the open D. Strong resonances don't mix well with open strings, which are themselves strong resonances. Strong resonances in close proximity create "hot notes" and "wolfs".

It's hard to tell exactly what frequencies Schleske's graphs are showing. On most frequency analysis software (as opposed to a "screen shot"), one can position the cursor and get an exact frequency readout.

Scott, well said in that "It's all a big fat bag of compromises".

April 27, 2009 at 05:37 AM ·

I must say thanks again to all the people that have recently posted.


I really did go back to wiki and did a more thorough search and did indeed find about 5 hours worth of entertaining reading. I guess since someone else too the trouble to write all that, I can't be the only one who finds the theoretical stuff endlessly fascinating.


Perhaps some day, I can listen to a sololist and say "Hmm...an instersting choice... he chose to blend it this way instead of that..."


Until then, I have a lot more reading to do. Does anyone know, was this a big topic in the muscial community itself? Did famous instrument makers and instrument players and composers argue and write papers? Or was it more a matter of "what a teacher taught to his apprentices"?


Anyway thanks again for all the input.



P.S.  David Burgess said:  "I'll mention that an instrument string doesn't behave exactly like the hypothetical "ideal" string."

This reminded me of a something a one of my first professors told me.  He said "Now whenever a physicist says 'arbitrary system' he is secretly thinking of the simple harmonic oscillator."  I thought it was mildly amusing at the time but as studied more and more and saw how much truth there was in that, it just got funnier and funnier.   But maybe you have to be a physicist to see the humor.

April 27, 2009 at 02:21 PM ·

April 27, 2009 at 04:19 PM ·

For me, the most striking physicists' abstraction is dry water. It's surprising that they haven't come up with an ideal violin, yet.

April 28, 2009 at 02:27 AM ·


Is this what you speak of:


April 28, 2009 at 11:59 AM ·


No, the "dry water" I was referring to is water without viscosity. When modeling water waves it is easier to leave viscosity out and to correct for it later. The undergraduate text I used called this dry water. I remembered because it sounds contradictory.


April 28, 2009 at 01:01 PM ·

Ahh. I get it.  It is a language thing. Your textbood in Dutch?  What is the Dutch word?  (I like Dutch, especially that iceskating race: Elfstedentocht).


In English we call this weird absurdity an "ideal fluid."  Ideal my foot. Without viscosity you have no lift!

April 28, 2009 at 04:15 PM ·

Bill, the book was 100% American: Berkeley Physics Course, Volume 3: Waves.

Dry water must have been the invention of the author, whose name I don't remember. It can be bought in the same shop where they sell the massless spring and the dimensionless mass, so that we can make our arbitrary system.

April 28, 2009 at 04:50 PM ·

And smokeless tobacco, and cold fusion, and free money, and...:-)


April 28, 2009 at 08:36 PM ·

I'm also a physicist and I think there is something fundamental here about how people who spend so much time thinking like this approach the violin (btw, I do get the simple harmonic oscillator joke). When I started out, I spent a lot of time reading about things like temperment and the science of music and the violin in general. I found it helped me a lot when I started playing as well, since I better understood how the violin worked, and the basis for our modern music nomenclature. Of course, this only takes you to a point, and to develop any sort of technique, you have to practice, but it definitely gave me a leg up when starting out.

P.S. if you didn't get the simple harmonic oscillator joke, don't worry. A physics problem can't get much simpler than something that oscillates back and forth, it's one of the first problems you learn to solve. When you start to learn more advanced things with long, scary names like quantum field theory, second quantization, phonon/magnon/spinon/excition/-on theory and string theory, you are initially freaked out. Once you begin to understand the problem however, you begin to see that all that the people who discovered how to solve these problems did was stick a bunch of simple oscillators together in clever ways so that they can solve the problem. They then give their method an obtuse name so that reviewers are impressed and let them publish their work in high impact journals (OK, maybe that's a little harsh, that only happens in the biosciences :) ).

April 28, 2009 at 09:42 PM ·

I can't get my head round the Theremin.  No false harmonics (?), no wolf notes and absolutely no possibility of getting into an argument about whether or not to use a shoulder rest.

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