Composing for the standard double horn
using extended just intonation
(and other non-standard tuning systems)
by Dr. Michael H Dixon
I will cover a spectrum of methods to interact with brass players when composing with extended just intonation and other microtonal systems. The performer needs to know WHAT pitches are required, HOW to implement them and know what they sound like. The composer can describe the tuning implementation to the performer and give no further assistance. At the other end of the spectrum, the composer can provide intricately detailed assistance. For clarity the amount of assistance can be expressed in 3 main areas:
1) The performer takes responsibility.
The minimum instruction from the composer is to show WHAT pitches are required.
This can be done in various ways in the notation: non-standard specific symbols (beyond standard tones and semitones); ratios; cent values; colours to show the place in a chord.
2) The composer suggests some brass technique.
The composer articulates some answers to the performer’s question, ‘HOW do I do this?’
An example would be to include fingering or slide position suggestions (tube length choices) and leave the performers to do much of the fine-tuning by lipping, right-hand technique or fine slide adjustments.
3) The Composer makes detailed instructions.
The composer articulates as many answers as possible to the performer’s question, ‘HOW do I do this?’
In addition to showing the precise tuning of each valve slide (pitch matching between harmonics of various tube lengths) and showing which fingering to use for every pitch, the cent value of each pitch can be shown. Cent values are useful to the performer due to the prevalence of tuning meters. Cent values also show how much different a pitch is to a neighbour, making it easier for the performer to choose how much to adjust by lip or hand, or find a useful alternative fingering (tube length combined with harmonic).
Providing a recording of the music is of great value to the performer.
Being present at rehearsals to sound the pitches in some way is equally useful.
Performers can then copy the sounds. Often performers will seek reassurance and questions such as: ‘Is this right’ or ‘Is this what you want?’ and be satisfied when hearing the pitch in context.
I have used all 3 methods, even when writing for myself to play the music.
Method 1) requires a lot of information about the pitches (the WHAT) and/or sound files.
Method 3) requires a lot of precise practical knowledge (the HOW) and for some situations does not need any WHAT. (However, many musicians are curious about what they are doing. One of my earlier works had a keyboard part with 24 pitches each octave. The keys were individually tuned. When the player first encountered the sounds, he commented that there was something wrong with the instrument due to the strange pitch movements. I had notated his part to be read in standard notation. Once he knew this he was much more comfortable. Similarly, in a horn quartet, I had the same written note played by two musicians, each with a different tube length, and the required sound was the ‘beating’ of two very close pitches. This knowledge is mandatory otherwise the musicians will naturally bring the two pitches to a perfect unison.
The more one is interested in the HOW aspect, the more detailed knowledge one must have regarding brass playing and the nature of the instruments.
I will spend most of our time on this.
Even if the detail is neither remembered nor implemented, a new feel for
composing for the horn (and other brass) may develop.
Harmonics of the Horn – 8 Tubes, B♭ Horn
(Notated in F)
Harmonics of The Horn – 8 Tubes, F horn and Lower
(Notated in F)
These 16 tube lengths are part of the standard double horn. The two primary tubes are the B♭ horn and the F horn. The player changes from one to the other with a thumb valve. The other 3 valves add length in the following way: middle valve (2nd valve) by a semitone; 1st valve by a tone, 3rd valve by a tone and a half. The combinations are successively short of the expected length: 1 & 2 is approximately 10 cents sharp to valve 3; combination 2 & 3 is approx. 15 cents sharp to 2 tones below the open tube; 1 & 3 is approx. 30 cents sharp to 2 and a half tones below the open tube; 1 & 2 & 3 is approx. 50 cents sharp to 3 tones below the open tube.
Longer tube length generally equates to more difficulty in pitching.
I capped the upper harmonic at 19 due to the extreme difficulty in pitching the actual harmonic. The tuning of an upper harmonic can be achieved lower in the range through the skill of the player. 17th and 19th harmonics, being very close to equal tempered tunings can be easily reproduced with alternate fingering (tube length) choices.
Many tube length fundamentals are very difficult to produce. These are shown with diamond note heads. some of them can be produced under some circumstances. For example, I successfully recorded the fundamental of the long C horn using a particular mouthpiece, but only produced the pitch once or twice in live performance.
A further note about the notation in the previous two pages: I use colours in my just intonation compositions primarily to show musicians how their note fits into a chord. This can assist in guiding the ear. The colour doesn’t show how many notes are in a chord, nor which other notes in a harmonic series are present, nor whether pitches from more than one harmonic series are present. The colour will assist musicians to find the best tuning as long as they have an idea what the ‘root’ of the chord is sounding like, whether it is present or not. The colours are aligned with prime-numbered harmonics and I’ve used basically the same colours that Robin Hayward uses in his Tuning Vine. I have added magenta for the 15th harmonic for my own convenience.
Choosing a scale – selection of pitches (choice of fundamentals)
A large number of pitch/tuning (ratio) choices in the mid/high range of the horn depend on the tuning of the fundamentals.
We can use some 1st or 2nd harmonics of tube lengths to construct a simple just intonation scale. For example, let’s call the low written F (B♭ concert) the scale tonic.
Descending to the 7th degree of this simple just intonation scale, we use the 2nd valve, which is easily tuned to the ratio 15:8 (1088 cents) by moving the valve slide a few millimetres further out from the usual setting.
We get the 6th degree of the scale using the 3rd valve or valves 1 & 2, easily tuned to ratio 5:3 (884 cents) by moving the 3rd valve slide a few millimetres further out than normal. Using the 3rd valve is simpler here than using valves 1 & 2 because the 2nd valve slide has already been repositioned.
The 5th degree of the scale is easily produced using the open F side. It should be almost perfectly in tune at ratio 3:2 (702 cents) if the F and B♭ are tuned normally.
The 4th degree of the scale is easily produced by using the F horn and valve 1, providing the ratio 4:3 at 498 cents.
The 3rd degree of the scale is easily produced by using the F horn and valve 3, providing the ratio 5:4 (384 cents). Valve 3 is slightly longer than valves 1 & 2 combined so the narrow major third (ratio 5:4) is simpler using valve 3.
The 2nd degree of the scale can be produced by using the F horn with valves 3 & 1 together.
Adjusting the tuning to ratio 9:8 (204 cents) is slightly problematic because slide 1 is set to -2 cents from the normal setting and slide 3 is set to -14 cents from the normal setting. This combination should provide a tuning of -16 cents yet the combination is in itself about 30 cents sharp, producing a result of +14 cents which is 10 cents sharp from the desired ratio. A solution is possible by producing scale degree 3 with valves 1 & 2, extending the 2nd valve slide for the tuning and freeing up slide 3 to extend for scale degree 2 tuning.
Many 7 note scales are easy to produce with any pitch as tonic.
[Ratios are an integral part of just
intonation. They were first used when dividing string lengths. They are also
the relations between pitches in a harmonic series. The latter can more easily
bring larger numbered ratios into focus rather than sorting out relations to
multiples of familiar harmonics high up in the series that are not playable in
their natural place on brass instruments.]
Choosing – The Tuning Reference Pitch
The Notation Reference Pitch
A Tonic (or No Tonic)
It seems useful when planning a group of ratios on the horn, to set the tuning reference to one of the two lengths that do not use valves: F horn or B♭ horn. All pitches that require the use of a valve to produce them also use either the F horn or B♭ horn. The air vibrates throughout the main tube as well as the smaller slide length. Therefore it seems easier to calculate ratios from these starting points. Once the possibilities have been discovered and checked, the central reference (ratio 1:1) can always be set to another pitch to suit the music.
This process is helpful when planning how far from standard pitch a standard brass instrument can play. There is not much upwards movement available. For instance, a scale tuned in Pythagorean ratios is possible. The sharpest note is the 7th degree: ratio 243:128 (1110 cents to 1:1) which is 10 cents higher than the equal tempered version.
When using scales with ‘sharper’ tunings such as the ratio 8:7 (231 cents), in the F scale, slides 1 & 2 could be pushed all the way in but slide 2 could not be pushed in far enough to produce 9:7 (435 cents).
A solution could be made by moving the main tuning slide of the brass instrument further out than usual. The disadvantage of that approach is that 1:1 may be lower than standard concert pitch relative to when A is 440Hz.
The extreme ‘lower’ tuning with valve 2 would be ratio 11:6 (1049 cents) which requires some lip bending. The issue with lip bending is that too much will lessen the focussed quality of the tone and work the lip muscles more than usual.
In a recent composition, Seven Portals, I successfully tuned the 3rd valve slide to ratio 13:8. The ‘D horn’ harmonic 16 matched the ‘F horn’ harmonic 13. The slide did fall out of the horn a few times in rehearsal.
This process of setting the tuning reference to F or B♭ horn is not important when the tuning of the horn part is not complicated.
Harmonics that are too high in the harmonic series to produce can be achieved using other tube lengths. For example, the 33rd harmonic of the B♭ series can be played next to the 8th harmonic of that series by using the F horn, harmonic 11 when the F horn is tuned perfectly tuned to B♭ horn harmonic 3. The 64th harmonic of the same series can be produced using the D horn, harmonic 13 when the D horn is tuned perfectly to B♭ horn harmonic 5.
*An earlier version of this paper was presented at EUROMicroFest 2017