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Becky Paul

UTSA Horn Studio

Spring 2013

The Application of Pitch-Bending in French Horn Performance

This paper presents a calculation and analysis of the frequencies in the harmonic series of

the French horn, which can be used to mathematically and theoretically explain the methodology

of bending pitch.

The comprehension and ability to bend pitch assists the horn player to:

1. Understand the unique properties of certain given notes within chordal context

2. Manipulate pitch with respect to just intonation during performance by:

a) Audiating the pitch

b) Comparing the audiated pitch to the understood contextual pitch

properties

c) Adjusting note fingerings to alter tuning

The ability to bend pitch is important to the performer and the ensemble as a whole

because this results in a musical presentation that is blended, resonant, and optimally pleasing to

the audience. A theoretical approach to the harmonic series will be discussed, to include a brief

foray into the physics of sound and the roles of equal temperament and just intonation in

ensemble tuning and performance. Application of the pitch-bending method will be discussed in

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detail, to include examples of French horn literature to which this method can be effectively

applied.

Physics of Sound and the Harmonic Series

Sound is perceived by the human ear in a number of ways. For example, the intensity of a

sound (measured in decibels) is perceived differently from the frequency of a sound, which is

measured in hertz. Frequency is pitch-based in that it affects how in tune or how out of tune a

note is in comparison to the other notes around it, while intensity is observed as a quantity of the

strength of a sound. When the frequencies of two pitches match, they are considered to be in

tune, which creates resonance.

When sound is perceived as resonant, it tends to be richer, bolder, and louder than

dissonant sound. One manner in which dissonance occurs is when two similar sound waves

vibrate at dissimilar but close frequencies. The notes are then said to be “out of tune”. They

produce a sound that listeners often wish to resolve to a more stable and resonant one. Tones that

display this quality are frequently described as ‘clashing’ in that an observable beat-like

oscillation can be heard when they are played in unison. This is often regarded as unpleasant to

the listener, who expects a smooth, rich, and aurally pleasing musical presentation. Playing out

of tune not only causes strain to the ear but also negatively impacts the amplitude of perceived

sound waves, making the tone sound unsteady and harder to distinguish.

Amplitude is the measurement of energy that is carried by a wave of sound. Waves that

carry more energy sound louder than waves that carry little energy. When the frequency of an

active force (air speed, in this case) matches the natural frequency of the object being acted on

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(the horn), it causes an increase in the amplitude of the produced sound. This is called resonance,

and when this resonance is achieved while playing an instrument, it produces a set of overtones.

Each fundamental frequency (which is the lowest sounding natural frequency) produces a set of

overtones, which means that when one hears a particular note, such as a ‘C’, one might also

notice the quiet presence of several other notes which make up the harmonic series of the

fundamental pitch. For example, while hearing the ‘C’, one might also faintly hear the higher

pitches of a ‘C’, a ‘G’, and an ‘E’. The harmonic series establishes an instrument’s unique timbre

due to a pronounced presence of certain harmonics. This causes variation in the waveform,

which distinguishes a certain instrument from the many other instruments within the ensemble.

The combination of multiple distinct timbres causes music to sound different depending

on what instruments are playing and directly affects tone color and tuning. When the overtones

within the harmonic series of different instruments match up (meaning, theoretically, that the

entire ensemble plays a note that is well in tune, matching not only the fundamental frequency of

the pitch with one another but also each overtone), the resulting sound is resonant, bold,

harmonious, and stable with no need for resolution of dissonance. When this occurs, the

ensemble is said to be well in tune. There are many ways to tune instruments, but the two

methods which are most commonly used among ensembles are equal temperament and just

intonation.

Common Application of Equal Temperament and Just Intonation

The equal-tempered scale was created with keyboard instruments in mind1. One

advantage of using the equal-tempered system of tuning is its proportionality between the

1 Campbell, Jim. "The Equal Tempered Scale and Peculiarities of Piano Tuning." The Equal Tempered Scale and Peculiarities of

Piano Tuning. N.p., n.d. Web. 01 Feb. 2013

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intervals in every key, which allows music to be transposed to any key and sound equally in tune

because the octave is divided into twelve equal semitones (or half-steps). This method of tuning

is, in some respects, mathematically precise but does not sound perfect to human ears due to

alteration of the fundamental frequencies of certain pitches.

Equal temperament and other tuning methods do not necessarily line up precisely with

the notes in the scale. For example, the frequency of an ‘A’ in a D major chord (the ‘A’ being the

dominant of the chord) would deviate quite a few cents from the frequency of an ‘A’ within an F

major chord (the ‘A’ now functions as the mediant), even though they are the same exact pitch.

Our ears want to hear harmonics’ frequencies in chords rather than scale frequencies. This

deviation can be easily remedied in real time by adjusting each of these pitches to fit within the

given chord, but it is not possible for a pianist to react in this manner.

In just intonation, the semitones within the octave are not equally-spaced and it is up to

the individual musicians within the ensemble to adjust their tuning based on the scale or mode

that they are playing in. Within this overtone-based system of tuning, performers must

understand where each pitch fits within the context of a chord in order to apply the pitch-bending

method. In this way, the individuals involved in the music-making are responsible for how ‘in-

tune’ or for how ‘out-of-tune’ the ensemble sounds.

Many professional ensembles consistently practice just intonation for tuning purposes

because wind and string instruments can adjust pitch to varying degrees in real time. In

performance, an instrumentalist might be perfectly in tune while playing a note with a tuner

which is calibrated to A=440 Hz, but they will be slightly out of tune if the orchestra around

them is justly tuning by ear to A=443 Hz. Musicians are able to play in tune with one another by

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employing pitch bending while overcoming challenges such as key modulations and temperature

changes.

Application of Just Intonation Within an Ensemble

The use of equal-temperament to tune an instrument well to a specifically-calibrated,

mathematically proportional frequency works particularly well with instruments which are only

played by one person and are not susceptible to pitch alteration based on external, environmental

factors. Instruments like the piano, organ, vibraphone, and chimes are tuned in equal

temperament and cannot bend pitch. Musicians who play stringed instruments and wind

instruments are constantly adjusting based on temperature, humidity, individual technique, chord

structure, and other outside factors. Thankfully, these instrumentalists are able to easily

manipulate pitch by changing something about the way that they play. For example, a brass

player might use an alternate fingering and a violinist could place their ring finger in a different

spot, thereby matching the perceived resonant pitch that is in tune. This is one example of

ensemble tuning based on just intonation. The method for tuning based on just intonation is

entirely centered on the human ear and the knowledge of individual pitch properties within

chordal context.

Chordal Tuning and Pitch Properties

Chordal tuning based on just intonation requires extensive knowledge of the harmonic

series. To apply the method of audiation, comparison, and adjustment, the individual performers

must understand the pitch-properties of specific tones within the harmonies of the piece that they

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are performing and ‘bend’ the tuning of the pitch to fit within the chord. Performers employ their

understanding of these tendencies in a number of ways. In a sense, each musician plays their role

within a piece of music by listening to one another and understanding where they fit within a

chord’s harmonic structure. Intermediate knowledge of music theory and the physics of musical

sound is required in order to fully understand pitch-properties.

Data Analysis and Relation to Pitch-Bending

The first attached chart displays the proportional frequencies of the overtones within the

harmonic series, assuming that A=440 Hz. Additionally, it displays the pitch-properties of each

harmonic in comparison to equal temperament. The ‘cent differential’ is in reference to the

amount of cents, sharp or flat, a pitch is in relation to equal temperament. The human ear is

particularly skilled at noticing small differences in pitch. A cent differential between 3 and 10

would be slightly noticeable. Anything beyond 10, and especially differentials of 25 or more,

would be immediately evident as being out of tune.

Note that the cells which are highlighted in purple are examples of frequency variations

of the same pitch, which is a concert ‘A’. Equal-tempered tuning is based on the foundation that

a concert ‘A’ frequency would be exactly 440 Hz, and yet concert ‘A’ is repeated multiple times

throughout the harmonic series with significant deviation from 440Hz. This is a visual and

mathematical representation of the problem with equal temperament; it is not a perfectly tuned

system. In fact, it is purposefully out of tune so that pianists, organists, and a multitude of

percussionists do not have to constantly re-tune their extensively complicated instruments every

time that they encounter a key change. As previously stated, the equal tempered system of tuning

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works effectively on solo instruments like the piano and the organ, which have no other

instruments to compare pitch with, but is not as practical when used exclusively within an

ensemble setting.

The second piece of attached data is a horn fingering chart which displays the most

commonly-used fingerings for each chromatic note within a logical range. In addition to the

common fingerings, several alternate fingerings are presented as additional options for tuning

purposes. Both F-side and B-flat-side fingerings are represented in the chart. A majority of these

fingerings can be applied effectively with accurate knowledge of which harmonic a horn player

is playing within and the pitch properties of that harmonic.

Pitch properties are displayed visually within the frequency chart as cent differentials.

For example, the first harmonic is the fundamental frequency of the sounding pitch. It is not an

overtone, so it cannot be compared with other tones and therefore cannot be in or out of tune.

The second harmonic is where the first overtone occurs, sounding an octave higher than the

fundamental frequency. There is still no cent differential because an octave is simply a repeated

tonic pitch, which is the pitch that just intonation is based on. The third partial contains the first

audible overtone that is actually perceived as a component within the chord. This overtone

sounds as the dominant of the chord, though it tends to tune two cents sharp compared to equal

temperament, making it the performer’s responsibility to choose an alternate fingering if the

player chooses to place the note into equal temperament. This is an example of the application of

the pitch-bending method.

Another common situation during which multiple performers should use pitch-bending

for tuning purposes is when three parts of a chord are split between three or more people. Two

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adjustments to tuning need to be applied in this scenario. The person playing the dominant pitch

would need to compare his tuning to the person playing the tonic pitch. The harmonic is higher

than equal temperament. The person playing the mediant pitch may need to choose an alternate

fingering. His choice may need to be more radical because the mediant, being placed on the fifth

partial, tends to tune 14 cents lower than equal temperament, which is very obviously deviant

from just tuning. The result which is attained from applying the pitch-bending method in this

way includes achievement of resonance frequency, warmer tone color, and elimination of

dissonance.

Application to French Horn Literature

Applying pitch-bending to music that moves quickly is more challenging. There are,

however, instances during fast-moving pieces of music where pitch-bending is not only

appropriate, but is essential. Upon examination of Reicha’s Minuetto trio for three horns, it

would appear that the notes change too quickly for anyone to notice a subtle difference in tuning,

yet in the case of repeated notes that outline a chord, pitch-bending is strongly encouraged.

(Reicha Horn Trios op. 82- Partitur in F)

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In the first two measures, the key of C major is clearly outlined by the repeated tonic in

the first horn part, the mediant in the second horn part, and by the eighth notes which highlight

the dominant in the third horn part. Though brief, the repetition of a tonic chord is enough to

strongly emphasize the established key. The same thing happens in m. 3, but this time the

emphasis rests on the dominant chord, leading to a half cadence in m. 4. In both of these places,

adjustments to tuning need to be made. The second horn part often encounters the mediant of the

chord, which tends to tune very sharp in equal temperament, so the person playing second horn

would need to choose an alternate fingering for the ‘E’ in mm. 1-2 and for the ‘D’ in mm. 3-4.

Additionally, the person playing third horn would need to use alternate fingerings for the ‘G’s, as

the dominant of the chord tunes slightly flat. These modifications to tuning, while they may seem

trivial, often make the difference between a professional quality performance of literature and an

amateur performance.

Closing Thoughts

There are multiple factors which come together to create a memorable musical

experience. One of the most basic and simultaneously complex elements of a professional

performance is the individual application of just intonation within an ensemble setting. This

requires performers to understand what a chord should sound like, and where they each fit within

that chord. They need to know what to adjust in order to achieve the best tuning possible and

how to make those adjustments based on their understanding of the harmonic series. Pitch-

bending is a process that every professional should be familiar with, but even more important

than the ability to change one’s tuning is the ability to understand the context within which one

should apply that process. It is not simply the comprehension of the application of a tuning

method that makes a musician wise and experienced; it is the understanding of where that

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method fits into the musician’s daily life as a member of a living, breathing, and consistently

changing ensemble. An in-depth knowledge of the harmonic series and its tendencies is essential

to any performer who seeks not just to know how to tune, but also to understand why the

ensemble tunes the way it does, and what that means for musicians, conductors, and audience

members alike.

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Works Cited

Bain, Reginald. "BAIN: The Harmonic Series (Overtone Series)." BAIN: The Harmonic Series

(Overtone Series). University of South Carolina School of Music, 1997. Web. 01 Feb. 2013.

Boerger, Ron. "French Horn Fingerings." French Horn Fingerings. N.p., 1999. Web. 01 Feb. 2013.

Campbell, Jim. "The Equal Tempered Scale and Peculiarities of Piano Tuning." The Equal Tempered

Scale and Peculiarities of Piano Tuning. N.p., n.d. Web. 01 Feb. 2013.

Farkas, Philip. The Art of French Horn Playing. Evanston, IL: Summy-Birchard, 1956. Print.

Hass, Jeffrey. "What Is Amplitude?" What Is Amplitude? Center for Electronic and Computer Music,

School of Music, Indiana University, 2003. Web. 01 Feb. 2013.

Hulen, Peter L. "A Musical Scale in Simple Ratios of the Harmonic

Series..." Http://persweb.wabash.edu. Wabash College Department of Music, n.d. Web.

Millican, Si. "Turn Off the Tuner for Better Ensemble Intonation." SBO Magazine RSS. N.p., 2 Aug.

2011. Web. 01 Feb. 2013.

Nave, R. "Overtones and Harmonics." Overtones and Harmonics. Hyperphysics, n.d. Web. 01 Feb.

2013.

Rothstein, Edward. Emblems of the Mind: The Inner Life of Music and Mathematics. New York: Avon,

1995. Print.

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Suits, B. H. "Frequencies of Musical Notes." Frequencies of Musical Notes. Physics Department,

Michigan Technological University, 1998. Web. 01 Feb. 2013.

Suits, B. H. "Scales: Just vs Equal Temperament." Scales: Just vs Equal Temperament. Physics

Department, Michigan Technological University, 1998. Web. 01 Feb. 2013.

Definitions

Some key terminology used in this paper is defined as follows:

Amplitude- The measurement of energy carried by a wave of sound; the higher the amplitude,

the more energy that is carried with the wave, which causes it to sound louder.

Audiation- The act of hearing music in one's mind

Cent- A unit of pitch based on equal temperament, used to describe small differences in

perceived tuning. (R. Nave, J. Campbell)

Dissonance- When two waves vibrate at dissimilar frequencies, they produce an unstable sound,

which feels the need to resolve to a sound that is more ‘stable’ and resonant. Dissonant tones are

sometimes described as ‘clashing’ and an observable wave-like oscillation can be heard between

the two contrasting sounds.

Equal temperament- A method of tuning which divides an octave into twelve equal semitones.

This method of tuning is proportional in any key and allows music to be transposed without any

change in musical intervals. (R. Nave)

Frequency- The number of sound wave cycles that occur over a period of time

Fundamental Frequency- The lowest natural frequency that exists to create a pitch

Harmonic Series- A series of tones consisting of a fundamental frequency and the harmonics

related to it by an exact fraction. (Merriam Webster)

Harmonic- An overtone accompanying a fundamental tone at a fixed interval, produced by

vibration of a string, column of air, etc…in an exact fraction of its length

Hertz- A unit of measurement used for values of frequency. One hertz is equal to one wave

cycle per second.

Just Intonation- A method of tuning which varies based on the overtone series of a particular

scale. This method includes matching pitch, based on pitch properties of each of the overtones in

the scales harmonic series.

Overtones- A musical tone that is a part of the harmonic series above a fundamental note and

may be heard with it (Merriam Webster)

Pitch-bending- A method used to alter a sounding pitch in order to match the understood

properties of said pitch and its contextual relationship within a chord.

Pitch properties- The tuning tendencies of fundamental frequencies and their overtones

Resonance- A frequency at which an object vibrates naturally, at specific amplitudes. Resonant

sounds tend to be more powerful, richer, and louder than dissonant sounds.

Resonance Frequency- A frequency at which a driving force (in this case, air speed) matches

the natural frequency of an object, causing an increase of amplitude in a sound. When resonance

frequency is achieved while playing an instrument, it produces overtones.

Semitone- Also referred to as a 'half-step', an equal-tempered semitone contains 100 cents. (R.

Nave)

Sonority- Often in reference to non-traditional harmony, sonority refers to a chord’s

individuality, which identifies it as unique in tone color. Not to be confused with quality, though

inclusive of quality (major/minor, etc…).

Sonorous- Deep, rich, and capable of producing a resonant sound

Timbre- The unique sound of an instrument, which distinguishes it from other instruments

Reference

440

466.16 116.54

0 cent differential 2 cent differential (sharp) 0 cent differential

Fundamental Octave Fifth Octave

Concert Pitch Horn Pitch H1 H2 H3 H4

A#/Bb1 F 58.27 116.54 174.81 233.08

B1 F#/Gb 61.74 123.47 185.21 246.94

C2 G 65.41 130.81 196.22 261.63

C#/Db2 G#/Ab 69.30 138.59 207.89 277.18

D2 A 73.42 146.83 220.25 293.66

D#/Eb2 A#/Bb 77.78 155.56 233.35 311.13

E2 B 82.41 164.81 247.22 329.63

E#/F2 B#/C 87.31 174.61 261.92 349.23

F#/Gb2 C#/Db 92.50 185.00 277.50 369.99

G2 D 98.00 196.00 294.00 392.00

G#/Ab2 D#/Eb 103.83 207.65 311.48 415.30

A3 E 110.00 220.00 330.00 440.00

A#/Bb3 E#/F 116.54 233.08 349.62 466.16

14 cent differential (flat) 2 cent differential (sharp) 31 cent differential (flat) 0 cent differential

Third Fifth Seventh Octave

H5 H6 H7 H8

291.35 349.62 407.89 466.16

308.68 370.41 432.15 493.88

327.03 392.44 457.84 523.25

346.48 415.77 485.07 554.37

367.08 440.50 513.91 587.33

388.91 466.69 544.47 622.25

412.03 494.44 576.85 659.26

436.54 523.84 611.15 698.46

462.49 554.99 647.49 739.99

489.99 587.99 685.99 783.99

519.13 622.96 726.78 830.61

550.00 660.00 770.00 880.00

582.70 699.25 815.79 932.33

4 cent differential 14 cent differential (flat) 49 cent differential (flat) 2 cent differential (sharp)

Second Third Tritone Fifth

H9 H10 H11 H12

524.43 582.70 640.98 699.25

555.62 617.35 679.09 740.82

588.66 654.06 719.47 784.88

623.66 692.96 762.25 831.55

660.75 734.16 807.58 880.99

700.04 777.82 855.60 933.38

741.66 824.07 906.48 988.88

785.76 873.07 960.38 1047.68

832.49 924.99 1017.48 1109.98

881.99 979.99 1077.99 1175.99

934.44 1038.26 1142.09 1245.91

990.00 1100.00 1210.00 1320.00

1048.87 1165.41 1281.95 1398.49

41 cent differential (sharp) 31 cent differential (flat) 12 cent differential (flat) 0 cent differential

Raised Fifth Raised Sixth Seventh Octave

H13 H14 H15 H16

757.52 815.79 874.06 932.33

802.56 864.30 926.03 987.77

850.28 915.69 981.10 1046.50

900.84 970.14 1039.43 1108.73

954.41 1027.83 1101.24 1174.66

1011.16 1088.94 1166.73 1244.51

1071.29 1153.70 1236.10 1318.51

1134.99 1222.30 1309.61 1396.91

1202.48 1294.98 1387.48 1479.98

1273.99 1371.98 1469.98 1567.98

1349.74 1453.57 1557.39 1661.22

1430.00 1540.00 1650.00 1760.00

1515.03 1631.57 1748.11 1864.66

Concert Pitch Horn Pitch Standard Fingerings Alternate Fingerings

A#/Bb3 F3 1/T-Open n/a *Blue indicates the Bb side of the horn,

B3 F#/Gb3 2/T1+2+3 n/a which is to be played with the trigger

C4 G3 Open/T1+3 1+3

C#/Db4 G#/Ab3 2+3/T2+3 n/a *Orange indicates the F side of the horn

D4 A3 1+2/T1+2 3, T3

D#/Eb4 A#/Bb3 1/T1 n/a

E4 B3 2/T1+2+3 T2

E#/F4 B#/C4 Open/T1+3 2+3, T-Open

F#/Gb4 C#/Db4 1+2/T2+3 1+2+3

G4 D4 1/T1+2 1+3, T3

G#/Ab4 D#/Eb4 2/T1 2+3, T1+2+3

A4 E4 Open/T2 1+2, 1+3

A#/Bb4 E#/F4 1/T-Open T2+3

B4 F#/Gb4 2/T1+2 T3

C5 G4 Open/T1 1+3, T1+3

C#/Db5 G#/Ab4 2+3/T2+3 1, T2

D5 A4 1+2/T1+2 3, 2, T3, T-Open

D#/Eb5 A#/Bb4 1/T1 Open, 2+3, T1+3

E5 B4 2/T2 1+2, 1+3, T1+2+3

E#/F5 C5 Open/T-Open 1, 2+3, T1+3

F#/Gb5 C#/Db5 1+2/T2+3 2, 3

G5 D5 1/T1+2 1, T3

G#/Ab5 D#/Eb5 2/T1 1+2, T2+3

A5 E5 Open/T2 1+2+3, T1+2, T3

A#/Bb5 E#/F5 1/T-Open 1+3, T1, T2+3

B5 F#/Gb5 2/T2 2+3, T1+2, T1+2+3

C6 G5 Open/T-Open 1+3, 1+2+3, T1, T1+3

C#/Db6 G#/Ab5 2/T2+3 2+3, 1+3, T2

D6 A5 Open/T1+2 3, 2+3, 1+2, T1+2, T-Open, T3

D#/Eb6 A#/Bb5 1/T1 1+2, 3, T1+2, T3

E6 B5 2/T2 1, T2+3, T1

E#/F6 C6 (Siegfried) Open/T-Open 1+2, 3, 2, T2, T1, T1+2