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Thread: Guitar tuning and temperament.

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    Guitar tuning and temperament.

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    Guitar tuning and temperament primer.

    Much has been said on this board and others about the reality behind getting your guitar in tune and what can and canít be achieved. Recently the subject came up and I was asked if any of the threads on the subject were worthy of being stickied. My sig a few links to some useful info on the subject but often people donít have the time or inclination to wade through them.

    This thread is an attempt to present that information and try to put it into terms that the average guitar player can understand and relate to. This content is taken both from my notes and handouts which are aimed at degree level students and those links I referred to.

    Iíll start simple and elaborate if anyone wants a more in depth understanding.


    So whatís the deal with guitar tuning?


    A common problem many guitar players encounter as they improve and start to listen carefully to what they are playing is that many of the chords they play just donít sound properly in tune. They will adjust one of the notes in that chord to compensate only to find that other chords are now out of wack. A frustrating problem and the first thing many will do is look to the intonation on their guitar as the source of the problem. Although the intonation needs to be right and should be set correctly it isnít the root cause of the problem and will not cure it.

    Why is this?

    So what is the problem? Basically in a nutshell the musical system of dividing the octave into 12 equal divisions that we use today is not perfect. The only intervals that will be perfect are the octave and unisons. Letís take a quick look at some of the science behind tuning. If we take a vibrating string it has a frequency that is dependent on the mass per unit length of the string, the tension in the string and the string length. To keep things simple for now we shall concentrate on the string length as this is what is important to us.
    We shall give the string length a value of 1. It has a fixed frequency at that length. Now we shall divide the string exactly in half and now the frequency is exactly an octave higher in pitch and where the 12th fret is on your guitar. For now assume we havenít changed the tension in the string. The octave can be described as 2:1 as a ratio.

    To bypass some of the more mundane maths for the moment accept that the ratio above is true. Pythagoras, way back when, discovered that the other notes that form perfect intervals can also be expressed as ratios of divisions of a string in the same way.

    They are the octave is 2:1, the fifth 3:2, the fourth 4:3, the major third 5:4, the minor third 6:5, the major sixth 5:3, the minor sixth 8:5, the whole tone 9:8. Dividing the string according to these ratios produces pure intervals without beats.

    In other words you need to divide the string using the ratios above to sound as perfect intervals. As an example to sound a perfect fifth we would need to fret the string exactly one third of the string length. We divide the string into three and stop it one third of the way along. A ratio of 3:2
    The other perfect intervals follow the same rules. For the fourth divide in to four and stop one quarter of the way along or a ratio of 4:3.

    Why donít we position frets to give perfect intervals?

    OK thatís the theory. So why doesnít it apply to the guitar? Simply if we did that then you would be able to sound a perfect fifth in one position and in relation to just one key note. Try and play the same interval starting on another note and it would be way off. We get round this by tinkering with the intervals by sharpening or flattening them by small amounts to bring things back into line

    The problem lies in the fact that by dividing the string into 12 equal intervals as we do today the ratios and as a result the intervals are not perfect. Each one is compromised to fit as closely as possible to give a system that can play ďalmostĒ in tune in every key. This tuning system is known as 12 note equal temperament or 12ET. Essentially to avoid having some perfect intervals and some imperfect and to facilitate playing in more than one key signature the notes of the scale are adjusted or ďtemperedĒ by varying degrees to even out the inaccuracies.

    12ET is a good system and it works but it isnít perfect. With the exception of the octave and the unison all the intervals on your guitar are off perfect by different amounts. It is unavoidable.

    So what can we do about it?

    In short there isnít much you can do to change the tuning system we use but you can do two things to help the situation. First, you have to understand and accept that 12ET is a compromise designed to allow you to play equally ďout of tuneĒ in all keys. Secondly you can tune you guitar with reference to a single note to avoid further compounding these compromises by transferring them from string to string.

    A method of tuning the guitar to play well in 12ET,

    I make no apologies for cribbing this instruction directly from an article in the GAL Journal published some years ago. There are quite a few methods of tempering your guitar when tuning but this one is about the best I have come across. Essentially you are trying to tune all strings with reference to a single note at either the unison or octave. Try it

    Tune the 1st and 6th strings: The E, open 1st string, should be in pure unison with the harmonic of the E, 6th string at the fifth fret.

    Tune the 4th string: Play a harmonic on the (in tune) 6th string at the 12th fret, adjust the 4th string until the E on the second fret is in pure unison.

    Tune the 2nd string: Play a harmonic on the (in tune) 4th string at twelve. As this sounds, adjust the 2nd string until D at the 3rd fret is in pure unison.

    Tune the 3rd string: Play a harmonic on the (in tune) 4th string at twelve and as this sounds, adjust the 3rd string until D at the 7th fret is in pure unison. Your done now check it.

    Double check: Play a harmonic on the (now tuned) G string at twelve, and as this tone sounds, play G on the 1st string at three. The two tones should be in pure unison. If they are not, you've done it wrong or the instrument doesn't fret in tune at seven. Go back to the beginning and carefully check each step up to this point. If the tones are still faulty, then readjust the 3rd string until the harmonic at twelve is in unison with the 1st at three. Do not tamper with the 1st and 4th strings because it is the 3rd string you are trying to bring in tune. When you have the 1st, 6th, 4th, 2nd and 3rd strings in tune, in that order, continue with the remaining 5th string.

    Tune the 5th string: Play the A on the (in tune) 3rd string, at the 2nd fret. Listen to this pitch carefully and now adjust the 5th string until the harmonic at twelve is in pure unison. When the foregoing steps are followed correctly, the strings will be tuned perfectly to equal temperament. No further tuning adjustments will help you now..

    Thatís all for now! Any questions?

    Thatís a very basic nuts and bolts introduction to temperament and tuning when applied to the guitar. As I started by saying, I hope to expand on any and all of this if you folks want me to. In the meantime if you have any questions or if any of it is unclear please ask away and Iíll do my best to fill in the gaps or explain it better.

    The links I have had in my sig for ever are listed below for any that may want to read up on them. They all relate to the subject and are good sources of information on the subject.

    => => Equal Temperament & Guitar Tuning.<= <=
    => => How to tune your guitar - Correctly...<= <=
    => => An excellent in depth look at musical temperaments.<= <=

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    Good job!

    But it won't stop all the folklore and urban legends circulating about "perfect tuning." For those who want to read an actual **book** on the subject, there's Ross W Duffin's How Equal Temperament Ruined Harmony (and Why You Should Care) (W W Norton & Co, New York, 2007), an entertaining look at the history of equal temperament from a violinist's point of view (hey, violins don't have frets -- you can just put your finger wherever your ear tells you to).
    "Digo: 'paciencia, y barajar.'" -- Don Quijote de la Mancha, Part II, Chapter 23

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    Quote Originally Posted by lpdeluxe View Post
    Good job!

    But it won't stop all the folklore and urban legends circulating about "perfect tuning." For those who want to read an actual **book** on the subject, there's Ross W Duffin's How Equal Temperament Ruined Harmony (and Why You Should Care) (W W Norton & Co, New York, 2007), an entertaining look at the history of equal temperament from a violinist's point of view (hey, violins don't have frets -- you can just put your finger wherever your ear tells you to).
    A better, more impartial and informative read is.Temperament: How Music Became a Battleground for the Great Minds of Western Civilization.

    The violin is indeed able to play in just about any temperament. That doesn't exclude the fact that when playing with modern instruments of fixed pitch it has to play in equal temperament.

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    Quote Originally Posted by muttley600 View Post
    The violin is indeed able to play in just about any temperament. That doesn't exclude the fact that when playing with modern instruments of fixed pitch it has to play in equal temperament.
    Blame the piano. If the pianists didn't insist on playing in multiple keys, with modulations and extensions and god knows what all, we wouldn't have this problem. Back when I was tying the gut frets onto my lute, I could use any interval I wanted. And don't forget those blasted horns with valves.
    "Digo: 'paciencia, y barajar.'" -- Don Quijote de la Mancha, Part II, Chapter 23

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    Thanks Mutt!!! This thread has been stickied!!! Hope people find it useful. I do!!

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    Thumbs up Equal Temperment forever!

    It is frustrating to have just set up and perfectly tune and intonate a guitar to have the owner immediately start messing with the machine heads. It is equally frustrating to try to explain Equal Temperment to those people who are so stubbornly proud of their "Golden Ears". In my recording studio I insist everyone use my Peterson V-Sam Tuner, there is always someone who tunes by ear or will start messing with the tuning after we start the session. I hope Equal Temperment will become more of a topic in the future.
    20 years ago I was confused by the science of tuning and stummbled on Equal Temperment in an Encyclopedia (remember those?) I didnt want to accept that music was not so exact and pure, but I got the message. I attempted to see what would happen if a plotted the "Circle Of Fifths" in a chart. I took an arbitrary value of 100 to start. multipied that by 3. I then divided by 2 to get 150 and plotted that value in its proper place in the 12 note interval I was plotting. I then multiplied 150 by 3 and then divided by 2 and plotted that value. (I would have to divide by 2 more than once to ensure my final value would end up between 100 and 200) I did this until all 12 spaces where full. I then compared the ratio of any 2 intervals. There were 2 ratios that repeated. I forgot exactly what they were, I have the whole experiment written down someplace. The most interesting part of all this was that I recognised a pattern of the 2 different ratios, It was the same pattern as the black and white keys of the piano, which is of course the Major Scale. I felt relieved that there seemed to be some order to this after all. This system is called "Just Intonation". The problem is you are stuck to play in the one key. Equal Temperment takes an "average" of these 2 ratios and makes one repeated interval. The ratio of Equal Temperment is 2 to the exponent 1/12 or 1.059463094. If you are curious you can take any number and multiply it by this ratio 12 times and you will end up with exactly an octave of your original number. A guitar tuned to 12ET with an accurate tuner is pure bliss!
    VP

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    Quote Originally Posted by lpdeluxe View Post
    Blame the piano. If the pianists didn't insist on playing in multiple keys, with modulations and extensions and god knows what all, we wouldn't have this problem. Back when I was tying the gut frets onto my lute, I could use any interval I wanted. And don't forget those blasted horns with valves.
    The blame if thats the right word is more down to the history of composition than any instrument or musician. The desire to modulate or shift through complex harmonic changes drove the need for tempered tuning systems.

    I trained making lutes and viols so tying the frets on is the easy bit trust me. And don't start me on organ pipes, horns, closed and open pipes, fipple reeds and the demands they all put on temperament.

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    Quote Originally Posted by muttley600 View Post
    I trained making lutes and viols so tying the frets on is the easy bit trust me. And don't start me on organ pipes, horns, closed and open pipes, fipple reeds and the demands they all put on temperament.
    Wish I had something that would improve your temperament.
    "Digo: 'paciencia, y barajar.'" -- Don Quijote de la Mancha, Part II, Chapter 23

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    Quote Originally Posted by lpdeluxe View Post
    Wish I had something that would improve your temperament.
    Beer will normally do that.

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    Quote Originally Posted by muttley600 View Post
    Beer will normally do that.
    Muttley beat me to it.
    "They can kill you, but the legalities of eating you are a little dicier." - David Foster Wallace (1962-2008)

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