Are/were the tuning schemes really practical?

February 14, 2014 at 02:27 AM · I am confused about tuning schemes/scales and I do not find any handy answers to my particular questions. Maybe someone can offer an explanation or suggest where I could find some useful resources. I have already read some of the theoretical papers but my questions are not addressed.

First, I need/made a data base:

Equitemp. Just Int.

modern calculated calculated

293.66 293.55 294.33

329.63 329.65 327.03

349.23 349.27 348.7

392.00 391.92 392.44

440.00 440.06 436.13

Using "C4" as 261.63.

Notes in order D,E,F,G,A.

Assuming that my interpretation and arithmetic are correct, my questions are:

So what? Could musicians over the years consistently hear these slight differences and how about an untrained audience?

How did past musicians know how to tune for these subtle differences in frequency without benefit of measuring devices?

Have other instruments gone through similar evolution regarding tuning schemes?

Replies (30)

February 14, 2014 at 04:05 AM · I believe the tuning schemes come from the ears and not vice versa.

MMy facts may be inexact and somebody on here I'm sure will know enough to correct them, but:

Perfect-sounding i.e. just intonation happens when the frequencies line up perfectly so that the sounds 'lock' together. BBut we don't usually figure that out by measuring the frequency; you can tell when it just has that resonant, perfect blend.

PProblem is, mathematically, the perfect blend does not line up exactly from what note to another. For example, I can tune an F to sounds perfect with a C and an A to sound perfect to the F but the A may not be perfect to the C. That's just a clumsy example, I don't have the real lineups off the top of my head, but you get the idea...

TThis is where somebody should chime in about pythagorean tuning but i will make a total mess of the sscience if i try from the top of my head!

aanyway, if it's just little bit off no big deal, it won't be super-noticeable to most people and it will bother even fewer :) But, since most instruments are tuned to a given key with that emphasizes certain frequencies more that others, if you tried to play in a different key the out-of-tune notes become much more pronounced even to the casual listener.

And again, on a lot of instruments it's not a huge deal, since string and wind instruments can make adjustments so that things 'line up' again. Most of us do this by ear without really thinking about what system of tuning we are in. And no, most of us don't really go around thinking 'Oh, he's playing in equal temperament' etc.etc. :)

HHowever, it IS a big deal on the keyboard instruments because they have no way to adjust pitch as they play. So if the instrument is tuned for the notes to all line up 'justly' with C, and they try to play a piece in Db it is not so good! And yes, you would probably notice :). HHence equal temperament was invented-on an equal-tempered keyboard, all notes are tuned equally to each other, so none of the intervals may be 'perfect' but everything is balanced out to be close enoug for comfort.

TThe big reason this affects violinists is that we like to play with pianists, but for ourselves we also like to play with as close to pure intonation as we can, since it just sounds that much better! So sometimes we have to do a little compromising, but again it's usually minimal and we can usually do it by ear.

So why do we bother measuring and categorizing it? WWell, I think it can be helpful to be aware of as we train our intonation, to be aware of the variables and differences that we are working with. Also some of us are just fascinated with the measurements and knowking the science behind everything. I don't know that most of us spend too much time worrying over hundredths of cents..musical ones at least :)...unless it's at the very highest levels. But most of us-yourself probably included!-can hear when it's really off, and once you become aware of it you would probably start to recognize that extra beauty of the truly pure blend.

Does that clarify at all? The schemes are not the point; the intonation is. The schemes just developed to accommodate the limitations of the instruments and the inherent scientific limitations of intonation itself.

February 14, 2014 at 04:17 AM · There has been a great deal written about tuning and playing in different temperaments, here and elsewhere. I suggest you search the archive.

The subtle differences become most evident when playing fixed intervals or with other instruments that may be tuned differently. The violin is usually tuned to "beatless" fifths but the piano is not. Thus even though both A's may be 440, the note on the piano corresponding to the open G string will seem slightly sharp.

If you are just going to play some piano trio music you don't need to get uptight about it though. In my experience (and I don't claim to have a fantastic ear), it's hard to detect unless (1) The violinist has professional intonation -- the cello is less critical because the frequencies are lower so frequency differences become more difficult to detect, (2) The piano is freshly tuned so that the temperament is accurately set and the unisons are perfect, and (3) Nobody is using any vibrato.

Pianos are tuned systematically and you can definitely hear the difference between different tunings (equal vs. other temperaments), especially in pieces that are in certain keys. You might not be able to identify what's "wrong" with a piano that is not tuned to modern equal temperament, but you'll hear a difference in the sound and in the "activity" of certain intervals and chords.

I was stunned when my piano tuner proved to me that my *digital* piano is actually not accurately tuned. That is rather a different matter though.

And by the way, the reason a good piano tuner is so highly prized is because really it's not easy to get it just right.

February 14, 2014 at 01:24 PM ·

Luckily we are able to follow different tuning systems/temperaments, but we are not able to play them with out a reference. For instance, if you were recording violin and piano music, and you recorded the violin first, then piano. The violin will be noticeably out of tune.

frequency chart

February 14, 2014 at 03:36 PM · I remember my music teacher in school playing a record where the Bach prelude in C major was played on a two-row harpsicord. One row was tuned in equal temperament and the other in just temperament in the key of C. The pianist (or harpsicordist) would play the first half of every bar on the first row and the second half on the second row. It was a very clear illustration of the virtues of equal temperament - on keyboard instruments.

When playing - or listening to - piano concertos it is sometimes very obvious that the orchestra is adjusting to the equal temperament when the piano starts to play. The orchestra will play the introduction in just temperament of whatever key and then when the soloist starts playing they are reminded and adjust. A friend of mine describes it as the temperament blanket being put over the orchestra.

February 14, 2014 at 03:49 PM · Nice spread sheet CC.

Please note that my curiosity is not a challenge to whatever tuning schemes are out there or the volumes of available history.

I wonder what would happen if someone handed me a violin with some loose strings and required me to tune it to a certain scheme but without aid of meters, tuning forks or any other device?

@BP Now that is "wholesale" intonation! And I wonder if some music doesn't simply border between rigorous tuning(s) and "colorization"? I am amused that my key of Ab is a "happy" sound.

February 14, 2014 at 09:31 PM · The whole point of equal temperament ect. is for instruments that cannot intonate. For a violin ju just tune it in perfect fifths and adapt from there. If you play with a piano you adapt to equal temperament, if you play in g flat minor you adapt to that. If you play a flute quartet and the flute is rising with temperature, you adapt to that ;-)

February 14, 2014 at 10:04 PM · Over the years I've noticed that the better Irish fiddlers tend to play the 4th finger B on the E-string a shade flat when playing solo. Why is this? I think it is because the fiddler is subconsciously getting that B into resonance with a harmonic of the open G, bearing in mind that a lot of the music they play is either in G or a one-sharp modal scale. In these circumstances they probably feel that a B that is a perfect fifth above the E just doesn't have the right resonance.

February 14, 2014 at 11:06 PM · "The orchestra will play the introduction in just temperament of whatever key and then when the soloist starts playing they are reminded and adjust"

Yes Bo, but the adjustment is subconscious, and we have no real control over it.

February 14, 2014 at 11:40 PM · Listen to a recording made by Ysaye here:

http://www.youtube.com/watch?v=mO9zzok_uEA

Early cylinder and acoustical 78 recordings re-mastered into digital format are a good place to study intonation. Every so often, you read a comment about Ysaye playing out-of-tune... but if you listen carefully, his intonation is extraordinary-- its just tuned differently. He does not mimic the piano's pitch, but rather uses subtle pitch ranges as a means of expression. This is especially audible at about 2:19 in the recording. This is what often seperates the great masters-- the ability to bend pitches and control each note with phenomenal precision. Fritz Kreisler did this too, but he started to adapt to a more modern concept of intonation. To my ears, freedom to select micro-intonation for expressive purposes is one of the great "virtues" a violinist can have.

February 14, 2014 at 11:44 PM · " Could musicians over the years consistently hear these slight differences and how about an untrained audience?

How did past musicians know how to tune for these subtle differences in frequency without benefit of measuring devices? "

Do not let yourself get confused by absolute frequencies, a little music theory background, especially concerning intervals will help you to understand it all.

Your best tuner is your ears. Your violin itself has to be internally in tune, the strings are tuned in pure fifths. If you play alone, you even can choose your own A tone (though these days is better to stick to some standard around 440Hz).

If you want to play with other violinists, tuning fork as a reference tone for A string then becomes handy because all of the players in the group are tuned to the same reference tone.

If the strings are not tuned in pure fifth, it can be heard during open string double stop as a wa wa frequency that disappears when the strings are in tune.

Each string is one pure fifth apart from adjacent string, when A is reference tone E = (3/2)*A, D = (2/3)*A, G=(4/9)*A. In numbers A=440, E=660, D=293.333, G=195.555

A comparison of all the intervals in the western scale below.

Comparison of Pythagorean Tuning, Just Tuning and Equal Temperament

Interval NameInterval abbr.PythagoreanCentsJustCentsEqualCents
*M2 (alternate 10:9)
*M6 (alternate 12:7)
*m3 (alternate 19:16)
*m2 (alternate 10:9)
Unison/PrimeP11:101:101:10
Minor Secondm2256:24390.2216:15*111.732^1/12 : 1100
Major SecondM29:8203.919:8*203.912^2/12 : 1200
Minor Thirdm332:27294.136:5*315.642^3/12 : 1300
Major ThirdM381:64407.825:4386.312^4/12 : 1400
Perfect FourthP44:3498.044:3498.042^5/12 : 1500
TritoneTT729:512611.7364:45609.782^6/12 : 1600
Perfect FifthP53:2701.963:2701.962^7/12 : 1700
Minor Sixthm6128:81792.188:5813.692^8/12 : 1800
Major SixthM627:16905.875:3*884.362^9/12 : 1900
Minor Seventhm716:9996.099.51017.602^10/12 : 11000
Major SeventhM7243:1281109.7815:81088.272^11/12 : 11100
OctaveP82:112002:112002:11200

Only Pythagorean Tuning is important when you play melody, Just tuning only when you play chords and double stops on the violin, you can ignore Equal Temperament (piano) tuning.

Wikipedia articles on Pythagorean Tuning, Just Intonation and Equal temperament should give you a starting point. violinmasterclass.com videos on Intonation also provide good explanations.

February 14, 2014 at 11:58 PM · As Paul said, "And by the way, the reason a good piano tuner is so highly prized is because really it's not easy to get it just right."

We needed our piano tuned and I called the tuners in our town, and asked for a really good technician. They sent out someone named Oppornockity, and he did a fantastic job. A few years later we needed it tuned again and I called up and asked them to send out Mr. Oppornockity. "No, sir, we'll send someone else."

I insisted that we wanted the same person, and demanded to know why they wouldn't send him. The woman on the phone replied, "Well, you see sir, Oppornockity only tunes once."

February 15, 2014 at 01:33 AM · Thanks to all for the many informative replies.

Now I have to wonder if all this is influencing my opinion that the church piano is out of tune although the keyboardist is a first class musician?

And I believe that good Irish music contains a generous dose of appropriate "accent". However, I understand Ysaye's style but it's not for me. (I will be curious to find him playing a more lively tune.)

I am beginning to think that individual playing styles are more important than any underlying arithmetic.

@CC That is an exceptional frequency chart!

February 15, 2014 at 01:52 AM · Correction The super frequency chart was posted

by Pavel Spacek.

February 16, 2014 at 02:30 PM · I'm able to deal with the notion of different keyboard tuning.

However, I have spent a lot of effort in trying to convert printed notes into music and I can't understand (for violin) why subtle scale modifications even come into play except for open strings. The bottom line is the ear, not the arithmetic (?)

February 16, 2014 at 03:14 PM · The ear naturally hears in perfect intervals more similar to pythagorean tuning, equal temperment is completely arbitrary, I doubt anyone exists who hears perfectly in equal temperment, in equal temperment every note is slightly out of tune to what the ear would want to hear, singers do the same thing as violinists, they don't sing scales in equal temperment but rather hear and sing in perfect intervals for each individual key.

February 16, 2014 at 10:13 PM · Let's stay with the violin.

What you call arithmetic is derived from the characteristics of human hearing and physics. When you play open string, you hear a tone. When you stop that string in half length, ratio 2:1, you perceive the same tone, just an octave higher. Similarly, you can stop the string in various other ratios and derive further intervals and construct the whole scale. If you play natural harmonics on the violin, you are using those exact ratios. Intonation is then quite objective, either you play in tune or not. You can always check that with open strings. But there is a snag.

When you play the scale - melody, everything is fine. But once you start playing double stops or chords, the deficiencies of our universe show.

The perfect intervals like fifth - open strings or octaves 0-3 are fine. But try to play in the first position B on A string - 1 and open E string - pure fourth, everything is fine, sounds in tune.

Then play open D string and B on A string - 1 - major sixth and surprise, to hear it in tune, you have to adjust the B. How on earth, it is the same B? Well, in the first case of B-E you play the pythagorean interval which is identical with just interval. In the second case of D-B you play the just interval which is different from the pythagorean interval.

February 17, 2014 at 03:40 AM · It is a little too late for violin tests right now but I will try the double stop example. I just want confirmation, in the double stop example, how is/was the violin tuned?

February 17, 2014 at 02:52 PM · @ Pavel S.

I tuned with my standard Korg. I think B/E sounded "OK" but I had to play the B just a whisper flat for the D/B to sound right.

However, I do now understand the principles of the tuning controversies and it seems that there is no advantage to be gained beyond ET.

I am still in awe of Pythagaras (sp?) who was able to deal with this subject long before all the modern tuning tools and knowledge were readily available.

February 17, 2014 at 07:28 PM · @ Darlene, yes, for the double stop example the violin is supposed to be tuned the standard way, the perfect fifths. Glad the experiment worked for you and showed you the differences in violin play.

This page and the five videos violinmasterclass.com/en/masterclasses/intonation give the clearest explanation of different kinds of intonation used in violin play.

February 18, 2014 at 02:40 AM · Ok, I will try that out. I have a good high resolution computer program to handle exotic tuning.

But you can also help me out with a somewhat related question. Do you have any info about the "width" of notes? I mean, how wide (frequency) can a note be such that it would be recognized as some specific pitch in a musical composition?

February 18, 2014 at 11:24 AM · yet if you play an interval of a fifth you can easily hear a 2c difference, in fact thats the difference between a perfect fifth and a equal tempered one. so between intervals even a 1c difference is noticeable.

February 18, 2014 at 01:02 PM ·

"you can easily hear a 2c difference"

I don't think so. Maybe you think you can.

If we were able to hear a 2-4 cent difference in pitch, listening to music wouldn't be enjoyable.

February 18, 2014 at 01:13 PM · having tuned many a clavichord I assure you its quite easy.

the not being able to hear 5c differences, is referencing only one note sounding and your ability to notice if its sharp or flat.

when two notes are played together, you get beats the instant they are not perfectly in tune, very easy to hear

February 18, 2014 at 03:09 PM · When I started this thread I had hopes that I would discover some new esoteric information that would take me to the next level. There certainly is enough rigorous mathematical literature to suggest that mathematics are of major importance for playing the violin.

Well, now I have changed my mind.

I alone create the "personality" of my music which was not really my first choice. I would welcome the safety of comfortable boundaries, even numerical, but numbers alone are not enough.

February 18, 2014 at 05:07 PM · Basically, we "construct" our musical universe with perfect fifths, then "indulge" in those sensuous true thirds when we can.

Constructing with thirds leads to pretty awfull results (as in unaccompanied amateur choirs here in France, who start flat and then sink..

Fifths are the Harmony of the Spheres, thirds are the Pitfalls of the Heart?

Fiths are aerobics, thirds are a cuddle.

Fifths are salt, thirds are sugar.

Ying &yang, etc. etc.

February 19, 2014 at 01:03 AM · You made a significant distinction .... personal use vs. theory. I think that the theory is only important if (personal) performance is improved as a result of the theory. (Perhaps I might call theory "second fiddle"!)

February 19, 2014 at 02:42 PM · The question is whether the study of tuning schemes leads to improved performance RELATIVE to other available training? I don't find that to be the case but I recognize that knowledge of theory does serve to enrich the playing experience.

Who is not looking for new ways or understandings to improve performance (see SHAR catalogue!)?

Maybe, in a way, I am looking for some vulnerability in my violins "armor" but there are no openings so far!

February 20, 2014 at 02:07 AM · WOW!

That Borup paper is impressive! (and just happens to agree with my opinions! :)

February 23, 2014 at 04:24 AM ·

I find the 3rd and 7th notes of a scale have a lot of leeway(+\- 10cents), and I sometimes call them floating notes. I also believe that these notes need to bend in a piece, not a scale, to achieve excellent intonation. If the 3rd and 7th are not floating, the melody can sound a bit bland.

February 23, 2014 at 03:19 PM · @Charles C.

If it were not for leeway I would have no intonation at all !

Just kidding. I totally agree with you about "best" notes and context.

@John C.

I have a Yamaha keyboard (not professional version but a "good" one). Is there a chance I can learn something from that ?

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