Download Comparison of Interval Size in Equal Temperament and Just Tuning

Comparison of Interval Size in Equal Temperament and Just Tuning
Interval
Half step
Whole step
Minor third
Major third
Fourth
Augmented fourth
Diminished fifth
Fifth
Minor sixth
Major sixth
Minor seventh
Major seventh
Octave
Equal
temperament
100 cents
200
300
400
500
600
600
700
800
900
1000
1100
1200
Just tuning
Difference
112 cents
204
316
386
498
590
610
702
814
884
996
1088
1200
+ 12 cents
+4
+ 16
- 14
- 2
- 10
+ 10
+2
+ 14
- 16
- 4
- 12
0
To play intervals in Just tuning (with no beats):
1. One player (or other source of the reference note) sounds the lower note of the
interval desired. Use a tuner to insure that the lower note is accurately placed in the
equally tempered scale.
2. A second player uses a tuner to adjust the upper note of the Equal Tempered
interval (half step, whole step, etc.) up or down by the number of cents indicated in
the “Difference” column. A pure beatless Just interval will result.
For example: to play a Just tuned major third above C, look in the Difference column
to see that the (E) of this major third needs to be played 14 cents lower than the equal
tempered interval indicated when the tuner indicator is centered.
Key Symbolism (Affektenlehre)
(from Key Symbolism by Charles McDermott)
“The concept of key symbolism was not only rooted in the modal distinctions of ancient theorists* … but received no wider
discussion than in Bach’s own lifetime.”
*Pythagoras, Aristotle, et alia; also, theorists of the Mediaevum and the (especially Italian) Renaissance. Originally associated
with planets in Greek Classical philosophy (the planets & their gods being, in turn, associated with states of mind and
morality), the Greek Toyol were evolved into the ecclesiastical eight modes which eventually expanded to twelve (see Grout,
HWM, Chapters 1-5). [source: J. Mattheson; Das Neu-Eroffnete Orchestre (1913)]
1.
D minor​ – submissive and quiet, but also grand, pleasant and satisfying: it can promote devotion in churchly subjects,
but calmness in secular. Generally flowing when pleasant.
2.
D major​ – by nature rather penetrating and obstinate. Best suited to joyous, warlike, enlivening material, but may be
used on a delicate subject for agreeable and unusual effect, especially in flute and violin music.
3.
Eb major​ – lofty and earnest, often plaintive (but, to Bach, a grand key symbolic of the Holy Trinity).
4.
E minor​ – pensive and profound, often gloomy and sad. Yet, the listener still may hope for consolation. Swift music may
be in this key, but merriment is not therefore implied.
5.
E major​ – despair or mortifying sadness; most comfortable (sic) with subjects of helplessness (and helplessness from an
excess of being in love), and sometimes so cutting severe, sorrowful and mordant that one could compare it to a fatal
separation of body and soul.
6.
F minor​ – gentle calm, although dark and heavy emotional pain associated with some despair; and immoderately
moving. Expresses beautifully a black, hopeless melancholy and may cause fear and dread in the listener.
7.
F major​ – capable of expressing the most beautiful and virtuous sentiments with a perfect grace, whether
magnanimosity, perseverance, or whatever, with a natural moderation and incomparable facility.
8.
F# minor​ – leads directly to great affliction (especially in mean-tone tuning!). It is more languishing and lovelorn than
fatal. Has an abandoned and singularly misanthropic aspect about it.
9.
G minor​ – almost the most beautiful key. Combines earnestness, amiability, grace and complaisance. In this it is, above
all, flexible to delicate and refreshing qualities as well as to longing or amusing, or to temperate lamenting or
cheerfulness.
10. G major​ – has much of insinuating and expressive in it. For all that, it is no less brilliant and is as well adapted to serious
as well as gay subjects.
11.
A minor​ – plaintive, decorous, composed and somewhat soporific (but not therefore displeasing).
12. A major​ – extremely exhausting in spite of some brilliancy. More suited to sorrowful and sad passions than to
divertissements. Very effective for violin compositions.
13. Bb major​ – very entertaining and sumptuous. Something modest is contained in it, enabling it to pass simultaneously as
magnificent and dainty.
14. B minor​ – bizarre, listless, melancholic and therefore rather seldom encountered.
15. B major​ – vexatious, hard, somewhat desparate, and quite uncomfortable.
16. C minor​ – exceedingly lovely but also sorrowful. Since the first quality wants far too much to prevail, and one can
easily be satiated thereby, it is not inappropriate to enliven it through a livelier movement, that one might not become
sleepy with the gentleness. “But if it is rather a piece which should promote sleep, then one can leave this remark aside
and naturally attain this end soon.”
17. C major​ – somewhat coarse and impudent, but not unsuitable to rejoicing and to where one otherwise lets joy run its
course. In despite of this, a clever composer can rechristen it to something quite charming, and apply it to tender
situations.
1
SELECTED TUNING BIBLIOGRAPHY Compiled by Peter Middleton August 2016 Books & Articles Averitt, Frances Lapp. “Tuning with Difference Tones.” ​F​ lute Talk, April 1986, 25­27. Barbour, J. Murray. ​Tuning and Temperament: A Historical Survey, ​
2​nd​ ed. East Lansing: Michigan State College Press, 1953. Benade, Arthur H. ​Fundamentals of Musical Acoustics
​
. Revised edition. New York: Dover Publications, 1990. Includes good basic information about woodwind acoustics. Benson, David J. ​Music: A Mathematical Offering. ​
Cambridge: Cambridge University Press, 2007. Thorough coverage of scales and temperaments. ​
Blackwood, Easley. ​The Structure of Recognizable Diatonic Tunings
​
. Princeton: Princeton University Press, 1985. Boehm, Theobald. ​The Flute and Flute Playing in Acoustical, Technical, and Artistic Aspects
​
. New York: Dover Publications, 1964. Bohn, Dennis A. “Environmental Effects on the Speed of Sound.” ​Journal Acoustical Society of America, Vol. 36, No. 42, April 1988. Detailed analysis of the environmental effects of temperature and humidity on the speed of sound. Bosanquet, R. H. M. ​An Elementary Treatise on Musical Intervals and Temperament
​
(London 1876). Edited by Rudolf Rasch. Utrecht: The Diapason Press, 1987. Brown, Judith C. “Frequency Ratios of Spectral Components of Musical Sounds.” ​Journal Acoustical Society of America
​
, Vol. 99, No. 2, February 1996, 1210 –1218. Confirms that the harmonics of wind instruments, voice, and bowed string instruments are in exact integer ratios with their fundamentals. Inharmonicity in piano sounds is in agreement with earlier studies by Fletcher (1964). Slight inharmonicity was measured for violin and viola pizzicati. Brown, Rebekah Ann. “Dynamics of Intonation in Performances by Artist Violinists.” Ph.D. dissertation, Indiana University, 1996 Coltman, John. “Flute Intonation in Performance,” ​
​
The Flutist Quarterly (Fall 1998): 42­47. Demany, Laurent and Catherine Semal. “Pitch versus Brightness of Timbre: Detecting Combined Shifts in Fundamental and Formant Frequency.” ​
​
Music Perception, Fall 1993, 1­14. Study of whether tone color influences perceived pitch. Concludes that pitch and brightness of timbre are independent perceptual dimensions, but that this relationship may depend on context of some tasks and not others. 2
​
​
Denis, Jean. ​Treatise on Harpsichord Tuning. (Translated and edited by Vincent J. Panetta, Jr.) Cambridge: Cambridge University Press, 1987. One of the first (1650) detailed descriptions clearly directed to practicing musicians of the application of one­quarter comma meantone temperament applied to the harpsichord. Di Veroli, Claudio. ​Unequal Temperaments. Buenos Aires: Artes Graficas Farro, 1978. Doty, David B. ​The Just Intonation Primer: An introduction to the theory and practice of Just Intonation. San Francisco: The Just Intonation Network, 1993. Duffin, Ross W. ​How Equal Temperament Ruined Harmony (and Why You Should Care). ​
New York: W. W. Norton & Company, 2007. Maintains that the exclusive use of equal temperament in modern performance has compromised harmony. ​
Ellis, Alexander J. and Arthur Mendel. ​Musical Pitch. Amsterdam: Frits Knuf, 1968. Articles covering the history of musical pitch, with Mendel clarifying and extending the work of Ellis. ​
Folkers, Catherine. “Playing in Tune on a Baroque Flute.” ​
​
Traverso, January 1998, 1­4. Fransson, Frans. “The Scale in Played Music.” ​
​
Swedish Journal of Musicology, 56, 49­54. Scales played alone by flute, oboe, and violin have intervals stretched upwards as scale ascends. Octaves were stretched about 15 cents. Goode, Elizabeth. “Fundamentals of Intonation.” ​Flute Talk, December 2005, 14­20. Goodway, Martha and Jay Scott Odell. ​
​ ​
The Historical Harpsichord: The Metalurgy of 17th and 18th Century Music Wire. Stuyvesant, New York: Pendragon Press, 1987. Includes discussion of the inharmonicity of strings used on the harpsichord, and of stretched tuning on the piano. ​
Haynes, Bruce. ​A History of Performing Pitch – The Story of “A.” Lanham, Maryland: Scarecrow Press, 2002. An extensive survey of European pitch levels from the 16​th​ century to today. Heller, Eric J. ​Why You Hear What You Hear, Princeton University Press, 2013. Excellent and comprehensive guide to sound waves and how they behave. Many examples that describe exactly how instruments work, and there’s a thorough discussion of psychoacoustics. Superb graphics illustrate nearly every topic. Helmholtz, Hermann L. F. ​On the Sensations of Tone. New York: Dover Publications, 1954. A classic early book on physiological and musical acoustics, and probably the greatest single contribution to the psychology of hearing. Helmholtz was a strong advocate for Just Intonation. Isacoff, Stuart. ​Temperament: How Music Became a Battleground for the Great Minds of Western Civilization. New York: Vintage Books, 2001. The paperback edition includes an Afterword that “expands the technical backdrop on which the original story was placed.” (Isacoff 2001) ​
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Jorgensen, Owen. ​The Equal Beating Temperaments: A Handbook for Tuning Harpsichords and Forte­Pianos, with Tuning Techniques and Tables of Fifteen Historical Temperaments. Raleigh: The Sunbury Press, 1981. Jorgensen, Owen. ​Tuning the Historical Temperaments by Ear. Marquette: Northern Michigan University Press, 1977. Keislar, Douglas Fleming. “Psychoacoustic Factors in Musical Intonation: Beats, Interval Tuning, and Inharmonicity.” Ph.D. dissertation, Stanford University, 1991. A detailed examination of whether the perception of intonation in music is dependent on the actual interval tuning rather than the beat rate among partials. Excellent bibliography. Kellner, Herbert Anton. ​The Tuning of My Harpsichord, 2​nd​ ed. Frankfurt: Verlag Das Musikinstrument, 1980. Klop, G. C. ​Harpsichord Tuning. Garderen, Holland: Werkplaats voor Clavecimbelbouw (distributed in the U.S. by The Sunbury), 1974. Simple descriptions of various temperaments and how to tune them on the harpsichord. Koenig, David M. ​Spectral Analysis of Music Sounds with Emphasis on the Piano, Oxford University Press, 2015. Emphasizes visualizations of musical sounds with many examples that include flute and other wind instruments. Introduces the idea of using autocorrelation to accurately measure the pitch of a wide range of sources. Kohut, Daniel L. ​Instrumental Music Pedagogy: Teaching Techniques for School Band and Orchestra Directors. Champaign: Stipes Publishing, 1996. Includes considerable practical information on tuning and basic musical acoustics. Kopiez, Reinhard. “Intonation of Harmonic Intervals: Adaptability of Expert Musicians to Equal Temperament and Just Intonation.” ​Music Perception, Summer 2003, 383­410. Concludes that “even in expert musicians, intonation is not determined by abstract tuning systems but is the result of interaction among compositional features, the acoustics of the particular musical instrument, and deviation patterns in specific intervals.” Krell, John. ​Kincaidiana – A Flute Player's Notebook. Culver City, California: Trio Associates, 1973. Includes Kincaid’s suggestions about tuning, as well as some difference tone trios. Leukel, Winfried and Stoffer, Thomas. “The Influence of Harmonic Context on the Tuning of Thirds Played by Professional Flautists.” Psychology of Music, 32(1), 75­89. Levitin, Daniel J. ​This Is Your Brain on Music. New York: Penguin Books, 2006. Provides a scientific understanding of how humans experience music. Lindley, Mark. ​Lutes, Viols & Temperaments. Cambridge: Cambridge University Press, 1984. Discussion of issues involved when applying Pythagorean, Equal, Meantone, and Just systems to fretted instruments such as viols and lutes. 4
Lloyd, L. S. and Hugh Boyle. ​Intervals, Scales, and Temperaments, 2​nd​ ed. New York: St Martin’s Press, 1979. Loosen, Franz. “The Effect of Musical Experience on the Conception of Accurate Tuning.” ​Music Perception, Spring 1995, 291­306. Violinists, pianists, and non­musicians evaluated scales in various tuning systems. Results showed that violinists preferred Pythagorean scales to equal­tempered scales; pianists preferred equal­tempered to Pythagorean scales; both violinists and pianists judged just­tuned scales to be less accurately tuned than either Pythagorean or equal­tempered scales; non­musicians did not show any preference. This study confirms that conception of accurate tuning is determined by musical experience rather than by characteristics of the auditory system. Loy, Gareth. ​Musicmathics: the mathematical foundations of music. Cambridge; MIT Press, 2006. Thorough treatment of musical scales, tuning, and intonation. Also excellent chapters on the physical, geometrical, and psychological basis of sound. Mathieu, W. A. ​Harmonic Experience: Tonal Harmony from Its Natural Origins to Its Modern Expression. Rochester, Vermont: Inner Traditions, 1997. Mathieu draws from many musical traditions, and encourages singing and playing various combinations of notes in tune as a way to understand the ambiguities and possibilities of equal temperament. McLeod, Philip. “Fast, Accurate Pitch Detection Tools for Music Analysis.” Ph.D. dissertation, University of Otago, Dunedin, New Zealand, 2008. Describes development of the “Tartini Tuner” real­time pitch measurement software and its applications for musicians. Neuwirth, Erich. ​Musical Temperaments. Vienna/New York: Springer­Verlag, 1997. Explains the principles behind various musical temperaments. Includes many illustrations and a CD­ROM with audio examples demonstrating each temperament. Obgushi, Kengo. “The Octave Enlargement Phenomenon and Its Origin.” ​Electronics and Communication in Japan, 62A (6), 11­18. Describes a theory to predict the psychological octave and why stretched octaves are preferred. Partch, Harry. ​Genesis of a Music. New York: Da Capo Press, 1974. Partch was a visionary composer, theorist, and creator of musical instruments, and a strong advocate of various forms of Just Intonation. Chapter 15 is an excellent overview of the history of tuning and intonation. Podnos, Theodor. ​Intonation for Strings, Winds, and Singers: A six month course. Meutchen, New Jersey: The Scarecrow Press, Inc., 1981. Discusses Tartini’s approach to tuning, and Campagnoli’s applications of Pythagorean tuning. More than half the book is devoted to the placement of pitches within various chordal structures. Interesting perspective from a string player’s point of view. Includes pitch analyses of performances by several well­known violinists. Potter, Chris. ​Seven Steps to Better Intonation. Nashua, New Hampshire: Falls House Press, 2000. Exercises to improve intonation, and charts to fill in to keep track of progress. Reynolds, Verne. ​Intonation Exercises for Two Horns. Century City, California: Wimbledon Music, Inc., 1980. 5
Includes exercises for each interval from unison to the octave. Can be played by any two instruments and transposed as needed. Schellenberg, E. Glenn. “Asymmetries in the Discrimination of Musical Intervals: Going Out­of­Tune Is More Noticeable Than Going In­Tune.” ​Music Perception, Winter 2002, 223­248. The title says it all! Smith, Fenwick. “Keeping Your Temper: A Flutist’s Guide to Intonation,” ​The Flutist’s Handbook: A Pedagogy Anthology. Santa Clarita, California: The National Flute Association, 1998. One of many publications that describes the piano (incorrectly) as being equally tempered. Snow, Donald Bradley. “A Conductor’s Guide to Wind Instrument Deficiencies: A Practical Addendum to the Undergraduate Conducting Text.” D.M.A. dissertation, University of Southern Mississippi, 2006. Spell, Eldred. “Scales: An Incomplete Look at What Every Flutist Should Know,” ​The Flutist Quarterly (Spring 2012): 20­23. Excellent overview by a real expert on efforts to improve the modern flute scale. Steblin, Rita. ​A History of Key Characteristics in the Eighteenth and Early Nineteenth Centuries. 2nd ed. Rochester, NY: University of Rochester Press, 2002. Sullivan, Anita T. ​The Seventh Dragon: The Riddle of Equal Temperament. Lake Oswego, Oregon: Metamorphous Press, 1985. Takes a new look at equal temperament through the eyes of a piano tuner and presents some of the alternatives. Tipton, Albert. “An Approach to Just Intonation by Employment of Difference Tones,” ​The Flutist’s Handbook: A Pedagogy Anthology. Santa Clarita, California: The National Flute Association, 1998. Vos, Joos. “Purity Ratings of Tempered Fifths and Major Thirds.” ​Music Perception, 1986, 3:251­257. Study showing beats to be important for perceived intonation of both intervals and musical passages. Weisberg, Arthur. ​The Art of Wind Playing. Galesville, Maryland: Meredith Publications, 2007. Detailed and highly practical information about the woodwind instruments and their use in ensembles. Werckmeister, Andreas. ​Musicalische Temperatur (Quedlingurg 1691). Edited by Rudolf Rasch. Utrecht: The Diapason Press, 1983. White, Joanna. “Intonation—Just and Tempered.” ​Flute Talk, April 2002, 8­13. Wilkinson, Scott R. ​Tuning In: A Basic Guide to Alternate Scales, Temperaments and Microtuning Using Synthesizers. Milwaukee: Hal Leonard Books, 1988. Good overview of the basic principles and history of tuning and temperament, and includes a section on how we perceive and distinguish musical sounds (psychoacoustics). Wye, Trevor. ​Practice Book for the Flute, Volume 4: Intonation. London: Novello, 1983. 6
Includes discussion of the harmonic series and its application to the flute, just and equal tempered scales, application of difference tones to tuning, and numerous excellent exercises to help develop tuning awareness. Wye, Trevor. ​Proper Flute Playing: A Companion to the Practice Books. London: Novello, 1988. Discusses general intonation issues as well as specific suggestions for note endings, and playing loudly and softly in tune. Wye, Trevor, William Bennett, and Eldred Spell. “​Cooper’s Scale Revisited.” Bennett, Spell, Wye, March 2011. Young, Robert W. ​Making Sense Out of Cents. Oak Brook, Illinois: C. G. Conn, 1976. Includes tables of frequency ratios and corresponding cents in equally tempered and just scales. RESOURCES Tuner apps: Peterson iStroboSoft Tuner: https://www.petersontuners.com/products/istrobosoft/ Tunable: https://play.google.com/store/apps/details?id=com.affinityblue.tunable&hl=en Clear tune: https://play.google.com/store/apps/details?id=com.bitcount.cleartune&hl=en TonalEnergy Chromatic Tuner and Metronome:
https://itunes.apple.com/us/app/tonalenergy­chromatic­tuner/id497716362?mt=8 Soundcorset Tuner and Metronome: https://play.google.com/store/apps/details?id=com.soundcorset.client.android&hl=en General information on tuning: https://en.wikipedia.org/wiki/Musical_tuning http://www.earmaster.com/music­theory­online/ch06/chapter­6­2.html Charts showing the measured amount of deviation from Equal Temperament for pianos of various sizes. An excellent graphic to show that pianos are clearly not equally tempered. http://www.pianosupply.com/tuners/490st.html#appendixc Excellent cents calculation tools: http://www.sengpielaudio.com/calculator­centsratio.htm Calculators for change of pitch with change of temperature: http://www.sengpielaudio.com/calculator­pitchchange.htm Flute Acoustics​: www.phys.unsw.edu.au/music/flute 7
An excellent and comprehensive site devoted to flute and other woodwind acoustics. Smart Music​: www.smartmusic.com Accompaniment software that includes a versatile tuner and can record practice sessions. Tuning CD: Schwartz, Richard A. ​The Tuning CD. Cherry Hill: Richard A. Schwartz, 1997. On iTunes: ​https://www.amazon.com/Tuning­C­D­440­Richard­Schwartz/dp/B002COP51Q​) Excellent details about using a tuning CD, and basic instrument acoustics. A few typos and unclear use of the terms “harmonic” and “overtone” but otherwise very useful. http://www.dwerden.com/soundfiles/intonationhelper/the_tuning_cd_booklet_free_version.pdf Anyone with questions related to the presentation is welcome to contact me at: [email protected] 1
TUNING GLOSSARY
Compiled by Peter Middleton
August 2016
Beats
A periodic rise and fall in strength (amplitude) heard as a change in
loudness that occurs when two pitches of slightly different
frequency are played together. If, for example, two instruments
play A = 440 Hertz (Hz) and A = 442 Hz together, the listener will
hear a tone of 441 Hz (the average) whose loudness fluctuates at 2
beats per second with a “wah-wah-wah” sound. (442 Hz – 440 Hz =
2 Hz or beats per second)
Cent
One one-hundredth of a semitone in equal temperament. For
example, the equally tempered half-step interval from C to C# is
100 cents. An octave contains 1200 cents (12 semitones x 100 cents
each = 1200). Electronic tuners are almost always calibrated in
cents.
Comma
The discrepancy in cents between pitches arrived at by two
different successions of perfect fifths, the “leftovers.” For example,
when a succession of perfect beatless fifths starting on C (C-G-D-A,
etc.) is continued so as to produce every pitch, the resulting final B#
will not be the same pitch as C, but will be 23.5 cents higher. This is
called the Pythagorean comma. There are several different types of
commas.
Difference tone
Tones that occur w
​ ithin the brain​ and whose frequency is usually
equal to the difference in frequency of the two primary tones. For
example, if 1400 Hz and 1000 Hz are sounded together, a
difference tone of 400 Hz will result. Sometimes called resultant
or subjective tones. Very useful for tuning just intervals.
Frequency
The number of repetitions per unit of time, usually seconds.
“A 440” has a frequency of 440 repetitions, or complete cycles, per
second. Musical frequencies are measured in Hertz (Hz).
Fundamental
The lowest frequency component of a complex tone. A complex
tone will contain harmonics or partials in addition to the
fundamental. Notes are “named “ based on their fundamental
frequency; when we play A in the treble staff, the fundamental has
a frequency of 440 Hertz (repetitions per second).
Harmonic
A component of a complex tone that is a whole number multiple of
the fundamental frequency. The fundamental is called the first
harmonic. For example, the third harmonic of A = 440 is 1320 Hz (E
above the staff in just intonation), since 440 x 3 = 1320. A common
harmonic exercise involves progressively overblowing middle C to
produce C in the staff, then G on top of the staff, C above the staff,
2
E, G, Bb, C. These are all harmonics, and members of what is called
the harmonic series or harmonic spectrum.
Hertz
A unit used to describe frequency, specifically repetitions per
second. Abbreviated “Hz” as in A = 440 Hz.
Interval
The distance separating two pitches that corresponds to a unique
frequency ratio. The interval of an octave has 2:1 frequency ratio.
Intonation
The degree of adherence to correct pitch within a specific
harmonic context; more generally refers to the variable and
dynamic inflections of pitch that form an inherent part of
performance.
Just (pure) intervals Intervals derived from the natural harmonic series that have simple
whole number ratios: 2:1 (octave), 3:2 (fifth), 5:4 (major third), etc.
Just intervals have no beats.
Overtone
A component of a complex tone (See F
​ undamental​) that has a pitch
higher than the fundamental. Partials above the fundamental are
sometimes called overtones. The first overtone is the same as the
second partial, since the first partial is the same as the fundamental.
The term is not in as wide usage as it once was, due to the different
numbering system used to identify overtones as compared with
harmonics or partials.
Partial
One of a group of components of a complex tone (See
Fundamental​) that are n
​ ot necessarily​ harmonically related to the
fundamental, and may or may not be in exact whole number
multiples of the fundamental. Bells, xylophones, chimes and many
other percussion instruments produce harmonically u
​ n​related
partials, as does the piano.
Pitch
The name of a note on a scale from low to high; also the subjective
psychological response to a particular frequency. Usually the same
as frequency, but very high and very low pure pitches that are loud
can sound respectively higher and lower than they actually are
when measured.
Resonance
Consider a child in a swing. The swing has a natural frequency that
is determined by the length of its ropes. If the swing is given a small
push at the right time in each cycle, it gradually goes higher and
higher. This is an example of resonance—the swing receives only a
small amount of energy during each push, but continues to go
higher. A flute is resonant at a frequency determined by the length
of the tube (which keys are closed), and the way it is blown.
3
Temperament
Timbre
Literally means “adjustment” or “modification” and is used to
describe intervals that have in some way been altered from their
pure or just whole number ratios to facilitate modulation. Used
particularly to help make the intonation of instruments with fixed
pitch like the piano, harpsichord, and organ sound acceptable in
various keys.
Tone color. A characteristic of a tone that distinguishes it from
others of the same frequency and loudness. Describes why a
trumpet and a cello playing the same pitch sound different. Tone
color is determined primarily by three factors: 1) which harmonics
are sounding, 2) the strength of the individual harmonics (or
partials in some cases) that are sounding, and 3) the time or phase
relationship among the harmonics sounding, which is related to
the attack or onset.
Tuning
The process of adjusting​ ​an instrument to a particular pitch.
Tuning system
A scale of tones created by use of a succession of pure or just
intervals that have no beats. Applies to systems such as Just and
Pythagorean in which all intervals are expressed as ratios of two
whole numbers.
(3:2, 4:3, etc.)
Well temperament Devised in 17th
​ ​ & 18th
​ ​ century Europe to permit performance in
many keys without retuning, yet retain the particular sound of each
key depending on the closeness to or remoteness from the key on
which the tuning was centered. This characteristic was considered
desirable by many composers who believed that each key was
associated with particular emotional affects and colors.
Wolf tone
An interval that is (usually) so wide or narrow that the beating is
severe and considered not acceptable or “agreeable.” The name
came from similarity to the sound of wolves howling discordantly.
Comments and Suggestions on Ways to Improve Tuning Compiled by Peter Middleton August 2016 A. Tuning ­­ general information 1. Be sure the cork in the headjoint is set properly. Generally, the line on the cleaning rod should be centered in the embouchure hole. 2. Set up the flute the same way every day. As a starting point, align the center of the embouchure hole with the center of the majority of the keys on the body of the flute. 3. For most flutes of recent design, keep the headjoint pulled out about 1/8 to – 1/4 inch. Rarely will the headjoint need to be pulled out less than this, and may be need to pulled out even more. 4. The primary controls we have for pitch and timbre (tone color) are embouchure hole coverage, air speed, and the length, size, and angle of the air jet. 5. Change one thing at a time so you can observe how the variables interact. 6. A flute ­­ or other wind instrument ­­ can be optimized to play only a single pitch precisely. Every other pitch except the design pitch requires some sort of compromise in the design. 7. Don’t worry about actually tuning the harmonics you produce when the flute is blown: the fundamental and harmonics created by the headjoint for a particular pitch are always in tune with each other. 8. Many different colors are possible as the flute is blown in different ways, but the actual harmonic components of the sound are still in their whole number ratios, only their proportions have changed. Their tuning does not change. 9. Shorter tube length yields greater pitch flexibility. So C# in the staff is the easiest note on which to alter the pitch. 10. The lowest notes are typically the most stable and therefore the most difficult to tune. 11. As the flute descends from open C# toward the foot joint notes, each lower note is affected less and less by a change in headjoint position because the proportion of the total length of the flute represented by a change in the headjoint position becomes less and less as we go down the scale. 12. Intonation problems are almost always caused by tone production problems. 13. Breath control has a profound effect on intonation, and the speed of the air stream is a critical factor. Practice playing harmonics to gain control of the air speed. Try to minimize embouchure adjustment and work with primarily the air speed.​ ​(see Floyd/Gilbert 1990) 14. Air jet speed can vary from about 40 to 130 MPH depending on register and dynamic. 15. Humans generally prefer stretched intervals. Remember when approaching notes from below that the higher note tends to sound flat, and the bigger the interval the greater the effect. 16. The most important basic tuning skill is to learn how to hear beats in unisons­­beats are periodic changes in volume­­then learn how to eliminate them by slowly moving the pitch up or down as needed. Work with another player if possible, or use a tuner that can sound a steady pitch, or another source of stable, accurate pitch like a tuning CD. 17. The object is always to eliminate unwanted beats, or adjust any difference tones so they are “in tune” with what you’re playing. See section C for information on difference tones. 18. The faster the beats, the farther out of tune for at least one of the notes. There will be a point at which the ear cannot distinguish the individual beats as they get faster and faster. At this point a low pitched difference tone will emerge. 19. One way to visualize this transition from distinct beats to a continuous difference tone is to think of a child’s bicycle with a playing card in the spokes. At slow speeds the individual “snap” of the card is clearly audible, but as the speed gets faster, there is point where there is only a continuous “buzz” as the spokes move past the card at high speed. 20. Well­tuned chords sound fuller because of the overlapping reinforcing effect of harmonics of the same frequency generated by different pitches in a chord. It would be impossible to tune chords played by acoustic instruments if their individual harmonics could be out of tune! 21. Almost all instruments begin notes (the attack or onset) sharp and flatten immediately. To experience this, play tongued notes against a unison drone and practice bringing the pitch down to match the drone. 22. Remember that the harp, guitar, and keyboard percussion are always tuned in Equal Temperament. 23. When playing with a piano, remember that the piano gets sharper the higher it is above middle C. Adjust accordingly, especially when playing softly. 24. The smaller the piano, the sharper it will play above its middle octave. 25. Equal temperament is an excellent point of departure. B. Embouchure 1. Check embouchure hole coverage by the bottom lip. The amount of embouchure hole covered generally increases as we go higher and/or get louder. A good starting point is about one­third covered in the lower register, moving to maybe two­thirds covered in the top register. 2. The left hand notes G up to C# are progressively sharpened by a more open (uncovered) embouchure hole. Adjust accordingly. 3. Try pitch bending exercises to enhance flexibility. Start on C in the staff and bend it down to B, then Bb, then A if possible. Then bend C in the staff up to C#. It’s difficult to bend C higher than C#. Try to do this without rolling the flute much, but by lowering/raising the jaw and air stream. It’s challenging, and is closely related to the way we play most of the time, but here with more exaggerated motions. 4. Try not to roll the flute in or out or move your head up or down excessively. It takes energy and changes pitch, timbre, and dynamics at the same time. 5. Excess embouchure tension has the effect of pulling back the corners making lower lip effectively thinner, which uncovers the embouchure hole and can raise the pitch and diffuse the sound. C. Difference Tones (see Glossary for definition) 1. Difference tones can train the ear to hear subtle pitch changes, and are useful for developing a feel for Just tuned (pure) intervals. 2. Difference tones sound something like a mosquito buzzing nearby and are easiest to hear when loud and high­pitched. Two piccolos are perfect to create some examples: one holds a C above the staff, the other starts on the same C and moves slowly up a chromatic scale. You will hear a series of difference tones starting with a very low pitch as C and C# are sounded together. 3. To hear a difference tone, the original pitches must be at least 20­30 Hz apart. Much below this, and we hear a frequency difference as beats. 4. When the interval between the original pitches is less than an octave, the first order difference tone (higher note minus lower note) is lower than either of the original pitches. 5. When a difference tone is precisely in tune relative to the notes that are generating it, the original interval is a pure just interval (no beats). 6. See the Bibliography for resources about tuning with difference tones. D.
Tuning Devices & other technical resources 1. The calibration accuracy of tuning devices is sometimes questionable. Be sure yours is set where you want it. (A440, etc.) 2. When using a meter, LED, or strobe­wheel type tuner, it’s often helpful to have someone else look at the indicator for you so you can concentrate on ​listening. Play with your best sound and have a partner tell you how close you are, or whether flat or sharp. 3. Some tuners will sound pitches: set the tuner to the tonic or dominant of the key your practicing and play slow scales listening for beats and difference tones. Try to anticipate the note you will be playing. 4. Tuners that play pitches rich in harmonics create a sound that makes tuning easier. Listen before you buy if you can. An external output is convenient for connecting to a sound system. 5. Practice with a commercial Tuning CD (see Resources in the Bibliography). 6. Most tuners are built to measure or sound pitches in Equal Temperament, though there are models that will measure some of the historical tunings we have discussed, and still others that will sound pitches in various historical tunings. 7. Record yourself with a computer running suitable software (like Audacity—a free download) or a portable digital recorder like the Zoom H1, H2n, H4n, etc. Play tuning duets with yourself. 8. Try Smart Music software (www.smartmusic.com). The software can accompany much of the flute repertoire, includes a metronome, a tuner with useful features, and can also record your practice sessions for later study. 9. Use a tuner as point of departure. If you can play every note in tune with a tuner, you’re off to a great start. The types of adjustments needed to refine your tuning even more to fit particular ensemble situations will become very manageable. Making the adjustments required to tune beatless intervals is not much of a problem compared with “simply” playing every note in tune with a tuner! E. Tuning “TO DO” list 1. Check some harmonics and compare with the normal fingering. For example, overblow low C an octave and compare its pitch with first finger C in the staff. Then follow the same procedure on C#. A close match provides a general indication of whether the headjoint is adjusted appropriately, and provides feedback on how your flute’s scale was designed. Move the headjoint a bit to make the harmonics match the normal fingering while trying not to make any embouchure adjustments. You may need to split the difference between the two settings depending on your flute’s scale. 2. Practice tone color change exercises to both improve the richness of your sound and make tuning easier. Remember, the more harmonics present in the sound, the easier it is to tune. (See Wye Vols. 1 and 4, for example). 3. Observe posture and position. Bring the flute to you. Moving the head forward to “meet” the flute uncovers the hole, raises the pitch, and makes the sound more diffuse. 4. Practice tuning unisons and octaves with a tuner or another player. Unisons and octaves share all harmonics and are the easiest to tune. 5. Use the chromatic tuning chart in the packet on the NFA website. Make several copies first. Work with a partner and record each other’s “pitch profile.” Look for tendencies. If you are confident that your embouchure is working well and placement of flute on bottom lip is appropriate, then when you see a consistent pattern after several days, make an adjustment to the headjoint to adjust the basic pitch level of the instrument. Also make charts while playing at different dynamics. 6.
Know your flute’s pitch tendencies on every note. (see item 5) 7. When changing dynamics, some kind of corresponding pitch correction is ​always required. To experience this, play crescendos and diminuendos while watching a tuner. Keep track of the timbre (color) too as the dynamic changes. 8. Use a pitch reference some of the time: an electronic tuner, tuning CD, keyboard, but depend ultimately on your ears. 9. Before tuning, warm up the flute up by gently blowing warm air through it while fingering low C/B. Don’t do this with wood flutes or piccolos. 10. When tuning to a piano, for variety try playing ​before the piano sounds an “A” to see where your pitch is prior to any adjustments you may make. Be sure to play with your best sound. Consider also tuning to a second note as well that uses a greater length of the flute, such as D or E. 11. When tuning to a piano, hold the sustain pedal down and play into the piano strings. Then gently play the same note on the keyboard and compare. Repeat. 12. Usually tune without vibrato (which adds changes in pitch, timbre, and loudness), but with a full sound. Or start with vibrato if you’re more comfortable, then remove it so any beats can be heard. It is impossible to hear beats while playing with vibrato unless what is being tuned to also has the same vibrato! 13. Take plenty of time to tune as you develop your “tuning technique.” Play at the same dynamic level you will practice or perform. Use your best sound. Play long notes­­it’s almost impossible to tune using notes shorter than a 16​th​ at quarter = 72. 14. Think of notes growing out of bottom octave of the instrumentall upper notes are really harmonics anyway. 15. Experiment with alternate fingerings to adjust pitch as appropriate. 16. Play tuning duets such as those by Trevor Wye, Harriet LeJeune, and Verne Reynolds (originally for French horn). 17. Practice singing intervals, and then play them. Try singing and playing at the same time. The book Rhythmical Articulation by Pasquale Bona (intended for singers, but which works perfectly for flute players) will not only improve rhythmic accuracy, but is written to be sung. 18. Anticipate the placement, tuning, and sound of the next note you will be playing. “Think in tune,” and visualize intervals that are perfectly in tune. 19. When playing with a piano or ensemble, and the tuning doesn’t sound right, first try to play with a better, more resonant sound. Many of the issues involved with good tone production are also critical for good intonation. 20. Listen to as much live and recorded music as you can to develop sensitivity to good intonation. Include string quartets and Renaissance and Medieval vocal ensembles. 21. While practicing, strive to think in terms of performing music, not just in terms of fixing tuning problems. The goal is to control all aspects of our playing simultaneously in support of the music. 22. To help control flat playing: • Use faster air (make the lip aperture smaller), but maintain a constant quantity of air so the dynamic doesn’t change • Move lips/jaw to aim air higher and uncover embouchure hole somewhat • Keep some air going past the end of the note you’re finishing • Keep the flute warm during rests, especially if there’s a soft entrance coming up • Avoid pushing the headjoint in all the way to compensate for a lack of air speed or breath control. Perhaps the headjoint may be rolled in too far, or the lip plate too low on the bottom lip resulting in too much embouchure hole coverage. 23. To help control sharp playing: • Decrease airspeed and/or cover more of the embouchure hole while lowering the airstream (try separating your teeth more) • If the pitch is too high, the flute may be too high on the lip • Lips may be too tight, and/or too much tension in the body generally • Be sure not to pull corners of the mouth back. This uncovers the embouchure hole and raises pitch • To crescendo without going sharp, increase the air speed, lower the air stream by opening your mouth a bit, and cover more of the embouchure hole.