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 2^1/12 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 2^1/12.

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.

How flat is that open E on the F side?

Of course, it depends.

But if you make the assumption that the horn is in tune for its C (which is your responsibility to achieve) and that all the other harmonics are perfectly in tune to the harmonic sequence (which isn't true, but you get a decent approximation to it for the better brands of horn, at least if you have a good embouchure) then it is possible to calculate how far out the other harmonics are from equal temperament. So I've done that. Taking the various harmonics upwards from middle C, they are out from equal temperament as follows:

• C = in tune
• E = 13.7 cents flat
• G = 2 cents sharp
• Bb = 31.2 cents flat
• C = in tune
• D = 3.9 cents sharp
• E = 13.7 cents flat
• F = 51.3 cents sharp
• G = 2 cents sharp
• Ab = 40.5 cents sharp
• Bb = 31.2 cents flat
• B = 11.7 cents flat
• C = in tune

Move all those pitches up a perfect 4th to get the tuning of the equivalent harmonics on the Bb side.

From this you can see easily why some harmonics were for the most part avoided by the classical composers. from middle C upwards, the E, G, C, D, E, F (with some bending by use of the hand to flatten it), G and top C harmonics are adequately in tune.

Although the E is 13.7 cents flat (100 cents make a semitone), when playing hand horn, you would generally be crooked in the key of the piece, and so that note would expect to be flat relative to equal temperament - in fact making it so will make it sound in tune for the key you are in.

So if you are playing a transposed part, if you see a written E, you know it is probably the third relative to the key of the piece, and so a bit of flatness on that note is OK.

But if you are playing a romantic or modern piece for valve horns with the part written in F irrespective of the key of the piece, then you can't make that assumption. That means that you have to think about the key and listen out for the rest of the orchestra for your tuning.

Just because you have your horn in tune to the oboe's A doesn't mean that you don't have to adjust note by note during a session. String players of my acquaintance say that when they are playing in a key with sharps, they position their fingers to play slightly sharper for C# and G# than they do when playing Db and Ab, in other words they are using just temperament rather then equal temperament. You will have to match them - whatever key the piece is in.

I also recall a story of a pianist who was the rehearsal pianist for a big choir. When he played the rehearsal piano by itself (tuned of course using equal temperament) it sounded perfectly well in tune, but when he accompanied the choir it always seemed a bit off. The singers would have (without realising it) been singing to just intonation in order to get their chords perfectly in tune, and so the piano would have sounded a bit odd.

Tuning is a complex business. Having a double horn makes it easier to play in tune, but don't assume that simply because you have a good instrument and you have tuned it, that you need do nothing more. When playing in an ensemble, you need to listen to every note, and make instantaneous adjustments where necessary.

Playing chamber music

Playing chamber music is great fun. There's a far more intimate feel to it as compared to playing in an orchestra, and much more opportunity for you to express your individuality.

But there are certain other skills and talents that you need to hone if you are going to be a successful chamber player. They are somewhat different from those necessary to play well in an orchestra.

The first thing is that you play more in chamber music than in an orchestra, so your endurance needs to be better. A 2 hour rehearsal will tire you out far more than an equivalent orchestral session. So make sure you are up to scratch for your endurance.

There are a number of chamber groups which commonly include horns, some aspects of playing are common to all of them, and some are unique to each.

Wind quintet
This is one of the most common groupings. One thing to remember here is that if you play at full volume, you can easily drown a mere four woodwind players. So your playing needs to be gentle and toned down in terms of volume. Also, many woodwind quintet pieces (especially those by French composers) have textures that are gossamer-light, and your playing style needs to reflect this.

Brass quintet
This is the other most common combination. Your problem here is the precise opposite of your position in the woodwind quintet - your bell is the only one which faces away from the audience, so your sound will tend to appear muffled compared to the the other players, especially the trumpets. So you need to go in for sharper attacks and a solid bright tone.

One or two horns and strings
I'm thinking here of pieces like the Mozart Horn Quintet and the Beethoven Sextet. In essence the horns are solo instruments, though in a smaller context than that of an orchestral concerto. You have to think soloistically when playing such pieces.

Mixed wind and string ensembles
Pieces such as the Beethoven Septet and the Schubert Octet are what I think of for this ensemble. They have a bit more of an orchestral style to them - you are frequently providing notes in the middle of the harmony, but you have to keep the style nice and light to compensate for the fact that you have only a single string player to each part instead of the serried ranks of players in an orchestra.

Larger wind ensembles
Anything from a wind octet for pieces such as the Mozart C Minor Serenade, through the Gounod Petite Symphonie (9 players), the Mozart "Gran Partita" Serenade for 13 wind, to even the Richard Strauss "Happy Workshop" Sonatina for 16 wind instruments. The more players you add, the closer you become to working as a small orchestra, especially if (above 10 players or so) you have a conductor. With more players, you can afford to produce a slightly more solid tone than for a wind quintet.


But there are some things which you need to do in all of these ensembles.

First is the one single most important thing about playing chamber music. You cannot bury yourself in your part and plough on with playing the notes. Just as in an orchestra you always need to keep one eye on the conductor, so in chamber music it is vital that you exchange regular eye contact with the other players as you play. You rely on each other to stay together and to coordinate the beat. By means of gestures and eye contact as you play, you can achieve far more flexibility with regard to changes in tempo than can be managed in an orchestra.

In a wind quintet, the upper instruments - flute or oboe, sometimes the clarinet - will have the tune more of the time, and so have more of the responsibility to set the style and tempo. Mark in your part where necessary who you need to look at for leadership as regards the beat. You will as a group need to decide at the start of each movement who is going to give the upbeat.

Because you don't have a conductor, the responsibility for maintaining the beat and the responsibility for deciding on phrasing an interpretation is shared among the players. It has to be done by agreement and consensus, with grace and good humour. Each player is playing his or her own unique instrument, and so everyone brings their own unique contribution to the group. There almost always is some kind of "pecking order" but hopefully is isn't too restrictive, and ideas can come from anywhere.

If you make a suggestion in rehearsal, the aim should always be to improve the performance as a whole. Try to phrase it as such. Saying "could we try at bar x again, I don't think we were quite together" is much better than turning to a specific player and saying "you were late at bar x". If you want the group (and your participation in it) to last, you have to be diplomatic.

If there is a weak player in the group, your performance is only going to be as good as that player can be cajoled into producing. So you need to provide encouragement and confidence to the weakest members of the group in order to get them to play better than they realised was possible for them.

Horns in small wind groups, particularly for classical pieces, often find themselves playing repeated quavers, while the tune goes on above. Here is a wonderful opportunity to be of service to the group as a whole. Don't play all the quavers the same strength, even if they are all marked with a common dynamic. Stress the first one of each bar, and possibly the one halfway through the bar as well. The aim is to offer an emphasis to the beat and enhance the rhythm. If you do things like this, the other players will find themselves surprised at how easy and enjoyable it is to play with you.

In brass quintets, the leader is almost inevitably the first trumpet. The same principles apply in terms of looking at each other to coordinate phrasing and speed. The first trumpet will most often be the person gesturing the upbeats at the start of a piece, and will also usually be the person coordinating rubatos in the piece itself.

When you play something like the Mozart Horn Quintet, the primary responsibility for how it goes is shared between you and the first violin (yes, I know that the Mozart has only one violin and two violas. You know what I mean.)

For wind and string pieces such as the Beethoven Septet, the first violin is the de facto leader, and for small groups with piano, such as the Brahms Horn Trio, you are three entirely equal participants.

Even though groups of different types have a natural leader, that doesn't mean that you only look at the leader. For any particular passage, you look at whoever has the need at that specific moment.

In orchestras, you can get away with not knowing all that much about what is going on around you. So long as you are with the conductor things can't go too badly. But in chamber music, you have to have a far greater awareness of what your fellow players are doing. In fact, having this awareness is a thoroughly good idea for orchestral playing as well, and so you will find that playing in a chamber group will greatly improve your orchestral playing and your enjoyment of it, as you learn to listen better.

When to use the F side and when to use the Bb side?

Previously, I said that you should get familiar with both sides of the horn, so that you can choose which side to use on any occasion.

But how do you choose which side to use when playing in a performance?

There are a number of things to consider

• Tuning
• Security
• Tone
• Fingering

Start with tuning. As a general rule, you want to avoid using a 5th harmonic or octaves thereof since these are flat relative to equal temperament, and you want to avoid 13 and 123 valve combinations, as these will tend to be sharp.

So, on the Bb side, that means you avoid playing an open A, or playing Ab, G or F# with 2, 1 or 12 respectively. The A can be played 12 and the Ab 23 with no difficulty, but G will be flat if played 1, and sharp if played 13. So you play the G open on the F side if possible. The same consideration puts F# on to the F side using 2. Of course, in the octave above, G can be played open on the Bb side and F# with 2. Those fingerings are fine.

In the octave below, G and F# should also be on the F side.

First space F can be played either open on Bb side, or 1 on F side. Both should be fine from a tuning point of view. E played open on the F side will be a bit flat, but will be in tune played 2 on the Bb side. The same consideration applies to Eb, D and C#.

Security - avoiding cracks and clams - will tend to cause you to want to play predominantly on the Bb side particularly in the upper register, since the harmonics are further apart on the Bb side than on the F side.

There are those who suggest that you ought to play predominantly on the F side from 2nd-line G down, and predominantly on the Bb side from Ab upwards. Personally, I don't see the need for such a hard-and-fast rule, especially as just below that transition, you come to the range E to C# which will be flat if played on the F side. Also, if you make that a general principle, then you will find yourself with quite a few awkward fingerings and poor-sounding slurs if you have a passage which takes you over the break a lot.

Tone is a bit harder to decide on. In principle, you ought to be able to produce a tone on both sides of the horn such that the audience can't tell which side you are using. If you have achieved that, then for the most part tone colour doesn't need to be a consideration. In practice, the F side does tend to produce a slightly more veiled or velvety sort of tone, a bit less brash than the Bb side. If the piece you are playing requires that, then using the F side might be a good idea, if only to put you in to the right state of mind for the piece.

As for fingering, take a look at the following excerpt from the Scherzo of Mahler 6. (You can click on the graphic to see it larger.)



Look first at Horns 1 & 2 on the top line. Under normal circumstances, you would think that this is upper register, straightforward Bb side territory, especially as the notes are D# and E which would tend to be flat on the F side. But with the grace notes like this, going from 1 on the D# to 2 on the E won't give you the quickest or cleanest possible slur. Much easier is to go from 2 to 0 on the F side. But what about the tuning? Well this is a rather brash passage, we don't want all that much in terms of subtlety here, so you can open your right hand a bit, which will have the dual effect of raising the pitch and giving you a brighter and more raucous tone, which is precisely the effect required!

Horns 7 & 8 an octave lower should use the same fingering, for pretty much the same reason.

The middle two parts are more interesting - you have more different notes to consider. But almost all the note pairs are a semitone slur up from the grace note. The same consideration applies, you want if possible to avoid having two valves going in opposite directions in the slur. C#-D is OK 23-3 Bb side, or 2-0 on the F side. Either will work perfectly well. B#-C# can also be on either side, as can B-C. For A#-B you can take advantage of the slightly flat open Bb harmonic and play this note pair 0-2 on the F side. The A# is only a grace note, it will pass so fast that nobody will have time to notice that it is a trifle on the flat side. G#-A can be played 23-3 on either side, and so on. So in fact, the great majority of these slurs are either as easy on both sides, or significantly easier on the F side, albeit occasionally with an unconventional fingering. So this part should probably also be played on the F side, even though it is in a range more normally played on the Bb side.

I'll leave it to you to work out the rest of the horn 3-4 stave and the horn 5-6 stave to see what fingerings are appropriate and which side you should play.

The key point is that even if you make a decision predominantly to use one side, you have to know both sides sufficiently well so that when a passage comes up such as the one I have described, you can switch in order to make the passage easier to play effectively.

Right Brain, Left Brain and Making Music

Lyle Sanford, on his Music Therapy blog was kind enough to reference my post on practicing. He had been musing on right-brain and left-brain approaches to music. That seems to be a good topic to explore a bit.

The left side of the brain is associated with concentration and rational thinking. The right side of the brain is connected more with openness to experience and more intuitive ways of doing things. As Lyle correctly pointed out, the kind of practice technique I described in Practicing a difficult passage effectively is entirely left-brain in its approach. You have a task you have set yourself and you use your analytic skills to achieve it.

But there is a distinction to be made between playing your instrument and playing music. Unless and until you have sufficient control over the instrument that you can stop devoting all your effort and concentration to it, you can't play music - all you can play are the notes. It is instantly obvious to the listener when somebody is so consumed by the the technical issues of controlling the instrument that they have no thoughts for phrasing and musicality. Even somebody wholly uneducated in music will sense that there is "something missing" in such a performance even though they can't express what that something is. In contrast, even in a beginner, if they are playing a sufficiently easy piece, it is easy to tell whether they have "got it" and are thinking musically.

This isn't limited to classical music. I enjoyed watching X Factor last autumn on ITV. Both I (with an extensive musical education) and my partner (with no formal education in music beyond class music lessons at school) both picked out Alexandra Burke and JLS as the two front runners from early on in the series. They both had sufficient mastery of themselves that they could just get on with singing the music. This is something distinct from mere talent. Almost all the singers had lots of talent, but none of the others quite managed to achieve that presence that comes from being able to leave all your troubles behind and put everything into the performance. We wondered as the series went on whether anybody else would be able to make that step up, but nobody else did.

In my articles on playing the Eroica, I described some left-brain sorts of things that you do, to decide beforehand how you will approach a passage, in terms of dynamics, fingering or phrasing. But that is merely scratching the surface. Once you have sufficient control over the instrument, you can stop thinking in a left-brain sort of way and just leave yourself open to what is going on around you. So you instinctively know that a particular phrase has to be played just so, and that it is obvious for reasons you don't quite understand that the top of a crescendo must be exactly there.

So, how do you get into right-brain mode? Unfortunately, that is a very left-brain way of expressing the problem, which is a bit self-defeating!

First of all, you have to stop worrying. That means you need to know what you are doing well enough that you don't have cause to worry.

Then you have to listen. You need to do that anyway, so that you make sure you play together with the other players. But I am talking here of a deeper immersion in the music. You stop thinking about the particular way the first oboe has phrased something and you let the music wash around you. You become part of the music intimately attuned to everything else going on. Your left brain is still active - after all you do still have to blow down the instrument and make the necessary noises, but it is no longer in sole control. You have stopped merely playing the instrument and have started playing the music.

On a lesser level, this sort of thing is routine among most professional and the better amateur musicians. The most intense experiences however are rare, and give rise to a kind of euphoria which can leave you walking on air for days afterwards. In my own experience, a particular combination of circumstances is necessary. First of all the group I'm part of is of a quality such that I'm not distracted by other people's technical shortcomings. Second, the group as a whole needs to be sufficiently on top of the music that uncertainty doesn't get transmitted around. A top-notch and inspiring conductor is a big help, though not an absolute necessity. And finally everything just has to gell on the night, in a way which means that everyone gets a boost from everyone else's confidence, so that the whole group ends up playing a bit better than they realised they could. When this happens, there is nothing quite like being in the middle of an orchestra playing some thoroughly romantic piece of music and being completely surrounded by this great wash of sound, knowing you are contributing your own bit to it!

If you know your instrument sufficiently well, to some degree you can get on with playing the music in just about any concert, and the heightened concentration and awareness that comes from having to perform rather than merely rehearse brings enjoyment to music making. It is an act of sharing - with the audience and with your fellow players.

I've played hundreds of concerts in my time, at all levels of music below the professional, and I've had this kind of intense experience probably less than a dozen times. I hope that the audience shared it to some extent. Almost every concert brings a faint echo of the intense experience - enough for me to want to play the next concert and see if it will come next time!

Concerts on the Edinburgh Fringe

Anybody in Edinburgh on 12-14 August? I'm taking part in 4 concerts on the Fringe as part of St Clements Wind Ensemble, as follows:

Tickets £8, concessions £6, except for the lunchtime concert in Canongate Kirk, which is free.

12August 2009, 8pm
St Marks Unitarian Church (Venue 125)
7 Castle Terrace Edinburgh, EH1 2DP
www.edinburgh-unitarians.org.uk

• György Ligeti (1923 - 2006) Bagatelles (1953)
• Darius Milhaud (1892-1974) La Cheminee Du Roi Rene (The Chimney Of King Rene), suite for wind quintet, Op. 205, (1939)
• Jacques Ibert (1890-1962), Trois pièces brèves (3), for wind quintet 1930

-Short Interval-

• Sir Malcolm Arnold (1921 – 2006), Three Shanties, opus 4
• W A Mozart, Fantasy in f minor for a mechanical organ, KV 594, arr. for wind quintet by W S Meyer
• G Bizet, 'Jeux d'enfants', op.22, arr. for wind quintet by Gordon Davies
• and some other pieces to be announced in the concert. Duration ca. 70 min

13 August 2009, 8pm
St Marks Unitarian Church (Venue 125)
7 Castle Terrace Edinburgh, EH1 2DP
www.edinburgh-unitarians.org.uk

• W.A. Mozart: Fantasy in f minor for a mechanical organ, KV608. arr. for wind nonet by K H Pillney
• Darius Milhaud (1892-1974) La Cheminee Du Roi Rene (The Chimney Of King Rene), suite for wind quintet, Op. 205, (1939)/Jacques Ibert (1890-1962), Trois pièces brèves (3), for wind quintet 1930/Sir Malcolm Arnold (1921 – 2006), Three Shanties, opus 4
• Christopher Irvin: Sea Breeze, for 10 wind players, first performance

-Short Interval-

• György Ligeti (1923 - 2006) Six Bagatelles (1953)
• W.A. Mozart: Serenade No. 12 in C minor, K. 388.

Concert Duration: ca.80 min

FREE LUNCHTIME CONCERT 14 August 2009, 1pm at Canongate Kirk

Canongate Kirk (Venue 60)
153 Canongate, Royal Mile EH8 8BN
www.canongatekirk.org.uk

• Nicholas Sackman: 'Folio III' for flute and piano (M Heidemann)
• G F Handel: Sonata for Oboe and piano (R Kozam)
• Nino Rota: Sonata for Clarinet and piano (J Peacock)
• Tbc: Music for bassoon and piano (S Rennard)
• Nicholas Rimsky-Korsakoff: Quintet for wind quartet and piano (with J West, Horn)Michael Round, Piano

14 August 2009, 5pm Canongate Kirk (Venue 60)
153 Canongate, Royal Mile EH8 8BN
www.canongatekirk.org.uk

The programme will include elements from both of the St Marks programmes.

If you come to any of the concerts, do please come over and say hello afterwards.

It's a good group, you would enjoy hearing them. Last year we played the Brahms Serenade in D in an arrangement I made for 13 winds. BBC Radio 3 broadcast two movements of it in their Play to the Nation series on October 2nd last year.

Random thought

You don't ever master the french horn. The best you can achieve is to come to an arrangement with it.