Friday, 31 July 2009

Just intonation vs Equal temperament

This isn't really quite about horn playing, but it quite clearly and obviously is about music - and if you want to be a good horn player then you must also be a good musician as well. And if you want to be a good musician, you have to have some kind of understanding of pitch and tuning.

So here's another way of understanding how scales work, and why equal temperament is a compromise that makes notes sound equally "nearly" right in all keys.

First of all, understand that for two notes an octave apart, the higher note is twice the frequency of the lower. For a true perfect 5th, the frequency ratio is 3/2. A perfect 4th is 4/3. All the intervals of the major scale can be expressed in terms of these whole number ratios (i.e. positions in the harmonic sequence relative to the key note). The example of C major scale is as follows Each note is described in terms of its frequency ratio relative to the C at the bottom end of the scale.
  • C 1/1
  • D 9/8
  • E 5/4
  • F 4/3
  • G 3/2
  • A 5/3
  • B 15/8
  • C 2/1
Now, if you look at a different key, G major for instance. All these frequency ratios above apply, but relative to the G at the bottom of the scale. But the G itself has a frequency ratio to the original C. So let's see what these ratios work out as when you multiply up.
  • G 3/2
  • A 3/2 * 9/8 = 27/16
  • B 3/2 * 5/4 = 15/8
  • C 3/2 * 4/3 = 2/1
  • D 3/2 * 3/2 = 9/4 (halve the frequency to go down an octave = 9/8)
  • E 3/2 * 5/3 = 5/2 (down an octave = 5/4)
  • F# 3/2 * 15/8 = 45/16 (down an octave = 45/32)
  • G 3/2 * 2/1
Now, compare these with the notes that are common to the C major scale. G, B, C, D and E have precisely the same tuning. But A does not. In C major with just intonation, A has a ratio of 5/3, or 1.6667. In G major, the frequency ratio is 27/16, or 1.6875. So in G major, A is slightly sharper than in C major (assuming that G is tuned to be 3/2 relative to C).

As you work your way through the various scales, you find that for every single scale, just intonation gives you different frequencies for some notes as compared to other scales which have those notes in common.

Now, if you are singing, or playing a stringed instrument without frets, then this is not a problem, you can sing or play in just intonation for whichever key you happen to be in and you can make the necessary adjustments as you change key. But it is a bit of a problem for a keyboard instrument. You can't instantaneously change the tuning of the a proportion of the strings whenever a piece modulates into another key! This mean that if a keyboard was tuned to just intonation in one key, it would sound distinctly odd if you play a piece in a key that is distant from it.

Enter equal temperament. I'm not sure anybody knows who invented it, but Vincenzo Galilei (father of the astronomer Galileo Galilei) was one of the first recorded advocates of it. It took a while to catch on, but by the time of Mozart, it was universally used for the tuning of keyboard instruments. Bach wrote the Well-Tempered Clavier in order to demonstrate the possibilities of "well tempering" which was a form of nearly-equal temperament, showing that a single keyboard instrument could play reasonably in tune in all 12 major and minor keys.

True 12-tone equal temperament, which is what we generally mean by the phrase these days, works on the principle that an octave is divided into 12 exactly equal semitones. By equal, that means equal in frequency ratio. But if you divide a 2:1 ratio into 12 equal ratios, you don't get integer ratios. The frequency ratio is 21/12 or about 1.059. This is not an integer ratio - you can't get a pair of integers where you divide one into the other to get exactly 21/12.

Now, when you compare the integer ratios with the frequencies obtained by equal temperament, you find that there are some differences. The following list gives you the difference in cents (100ths of a semitone) between equal temperament and just temperament for the notes of a major scale. Negative numbers indicate that just intonation is flat relative to equal temperament, positive numbers indicate that just intonation is sharp.
  • C 0
  • D 3.91
  • E −13.69
  • F −1.96
  • G 1.96
  • A −15.64
  • B −11.73
  • C 0
There are some quite substantial numbers there. The difference between just and equal temperament for an A is almost a sixth of a semitone. That is easily discernible.

The Wikipedia entry on Just intonation includes some sound samples which enable you to compare chords using just intonation with equal temperament. (You will need to have a player that can play OGG files to listen to the samples.) If you compare the sound sample that plays a scale and then various triads in just intonation, and then the sound sample that plays the same scale and triads in equal temperament, you will probably be able to hear some "beats" in the equal temperament version that aren't present in just intonation. So the advantage of equal temperament is that everything sounds about as good in all keys, and the disadvantage is that in all keys, you lose a little bit of harmonic purity through the frequency ratios of chords not being true harmonic (i.e. integer) ratios.

Now all that horrid maths is out of the way, you still are left with the question of how do you tune your horn?

And the answer has to be that because tuning varies, if you want to eliminate beats, especially when the horn section is playing as a quartet, you need to listen to and adjust if necessary every note you play. What is more, you can't assume that a particular adjustment (e.g. of hand position) will work the same way in two different pieces, especially if they happen to be in different keys. And then again, if you are playing with a piano, you are going to have to adapt to its equal temperament, whereas if you are playing the Beethoven Sextet (for 2 horns and string quartet), making the tuning sound "right" will involve something very close to just intonation.

Tuning is a dynamic thing - you never achieve a perfectly tuned instrument because the tuning varies according to circumstance, from piece to piece and even within a piece when the key modulates. You have to stay on your toes all the time.


  1. Yes. As a viola player, it's sometimes difficult to know what to do when playing with the piano. Some passages never quite come out right.

  2. So what can a humble music student do to develop this level of flexibility and prepare her/him-self to make these adjustments as well as possible? It seems there is both the mental aspect of recognizing the discrepencies in the various situations, and the physical ability to adjust in time. Do you know of a good comprehensive method, or at least exercises that target the most important issues for which to be ready, that a student can add to her/his daily practice?

  3. That's a very good question. I'll see if I can expand on that in a future article, but here's something to be going on with.

    You can develop basic pitch bending skills just by playing notes with a tuner in front of you. Play a note, see where it is on the tuning indicator, then see how quickly you can get the note steady one or two notches sharper or flatter. Then try playing notes and aiming to get them almost instantaneously out of tune by a controlled number of notches, without making any adjustment to your tuning slides. The aim of this is to gain fine control over the pitch of the note. Whether you use the hand or the lips to adjust the pitch, or some combination of both, is entirely up to you. Do whatever works for you and doesn't compromise tone quality.

    This gives you control over the pitch, but the other half of the skill, the listening, is not something you can do much of on your own. To develop the skills of listening and working out how best to fit your pitch into a chord, you need to take every possible opportunity to play in groups. Get four players together and play some quartets. Start out by just playing simple chords and hold them until you are really happy with the tuning. Gradually you will find that the tuning settles more quickly as you get used to each other. Make sure you swap around and take turns as to whether you are in the top, middle or bottom of the chord.

    Then progress to things like hymn tunes or Bach Chorales. Record yourselves and listen back to see whether the tuning is really as good as you thought it was.

    If you spend a reasonable amount of time and effort really concentrating on accurate tuning, it will become second nature, and you'll find that you can't not do it, whatever group you are playing in.

  4. I would suggest practicing with a drone rather than a tuner. If you want to learn how to play with pure intervals that are spot on, there is truly no better way. Whereas equal temperament is an arbitrary system, not really suited for instruments like viola or trombone (or the voice for that matter!) just intonation is very easy to hear if you have a steady pitch to reference against. Simply move toward the place of most consonance (no audible beating). New worlds open up. You have more choices for color and expression. for example: more than one pure minor third each with its own character, very consonant sounding tritone (7/5) and dominant seventh (7/4).
    Unfortunately there aren't alot of great resources for the study of just intonation, but a good place to start is Partch's book "Genisis of a Music"

  5. When one says that a trumpet is tuned in Bb, does that somehow relate to this? Does it mean that the trumpet will sound best when played in the key Bb, since Bb has been used as base tone when computing the correct frequencies for the other notes?

  6. It is indirectly related. What it means is that the lowest note a trumpet in Bb can play is a Bb. Without pressing any of the valves, the next note is the Bb an octave higher - twice the frequency. The next note is the F above that - three times the frequency of the bottom Bb. And so on.

    These notes are what is known as a harmonic series - notes whose frequencies are integer multiples of the fundamental.

    Pressing the valves down on the trumpet or the horn has the effect of changing the length of the tubing by switching in an additional length, and so changing the fundamental note of the instrument.

  7. Wow! Finally an answer to this question :) I've been looking around a lot on the Internet for an answer what it really means.

    "The lowest note it can play", do you mean except pedal tones?

    Or do you mean a Bb pedal tone and the normal Bb (that trumpeters think of as "low C") is an octave above? Then trumpeter's G in the middle of the staff i 3/2 of the pedal Bb, for example.

    As far as I have understood, just intonation corresponds exactly to harmoic series. So the conclusion is that when pressing no valve the most natural key for the trumpet would be Bb, but that would change as soon as one presses a valve. Pressing the second valve make the trumpet optimized for playing a just intonation scale based on A (trumpeter's B).

    Or did I miss something? This kind of fundamental understanding of the instrument is really great.

  8. Sorry for the bombardment of basic questions about the horn but can one analyze why the low F# sounds so fake on the trumpet in terms of just intonation?

    I mean, is it not an integer multiple of a fundamental tone? Or what? Why does it not sound as vibrant as other notes?

  9. On the trumpet, the harmonic sequence goes low C, middle C then successively higher G, C, E, Bb, C and so on.

    The trumpet is in Bb because each C sounds as a Bb, and before valves were invented it was convenient for the music for both trumpets and horns to be transposed into the key of the harmonic sequence of the instrument. Eventually with the invention of valves trumpets were standardised in Bb and horns in F.

    The low F# in a trumpet sounds dodgy for a variety of reasons, all of which I suspect make some contribution to the overall effect.

    1. The note is in the very low register, not natural territory for the trumpet.

    2. With all three valves pressed, the air is passing through nearly 50% more tubing than when no valves are down. This will make a noticeable change to the tone color.

    3. With all 3 valves down, the adjustment in the overall tubing length is a bit less than is necessary to get the F# in tune. It will tend to sound sharp. That is why many trumpets have an additional tuning slide on the 3rd valve which can be accessed using the player's fifth finger. The idea is that at need you can push the slide out a bit and so flatten the note.

  10. Thanks a lot Jonathan for your comments, much appreciated :)

  11. Equi-tempered western musical instruments, especially the keyed ones, have really diluted many a musical systems including the ancient Carnatic Music system of India. This Indian musical system has always endeavored to use the beauty of infinite number of assonant and consonant notes. For more on this pl read my blogs under:
    Ragas in Indian Music – FAQ2
    Swaras, Swarasthanas and Sruthis

    I value your comments on them.