Running Head: FINGERBOARD INTONATION CHART
A fingerboard intonation chart for the violin and viola
Guy Vandegrift and Michelle Smith
Wright State University
Author Note
Guy Vandegrift and Michelle Smith
Wright State University, Lake Campus
Correspondence concerning this article should be addressed to Guy Vandegrift, Wright State
University Lake Campus, Celina, OH 45822. Email: guy.vandegrift@wright.edu
1
FINGERBOARD INTONATION CHART
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Abstract
Intonation tests using double-stops played to just consonance can be used to help violin and viola
students check intonation. The discrepancy between just and equal tempered tuning can be quite
noticeable to students, especially when just thirds and sixths are involved. Fingerboard charts
have been developed that display these errors for almost every note played by on the bottom half
of each string. These visual cues offer students the opportunity to simultaneously hear, see, and
feel the distinction between just and equal temperament. Just intervals that are more difficult for
students to hear can be constructed systematically from simpler intervals, and confidencebuilding exercises have been devised for students who are practicing without supervision.
FINGERBOARD INTONATION CHART
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A Fingerboard Intonation Chart for the Violin and Viola
Testing one’s own intonation requires both a reference pitch, as well as a means for
gauging one’s ability to match that pitch. While an electronic tuner provides the necessary
feedback for the student working alone, such devices can reduce the joy of practicing. Burns
(1999) explains how just intervals correspond to the frequency ratios {3/2, 4/3, 5/3, 5/4, 6/5,
8/5}. Some of the proposed intonation tests use compound intervals associated with the
frequency ratios {5/2, 10/3}. Intonation tests are no better than the student’s ability to hear and
play the interval to just consonance. Students who lack this skill have little choice but to practice
with incorrect intonation until the teacher intervenes at the next lesson. This paper introduces a
procedure for constructing the more difficult intervals (e.g. thirds) from simpler ones (e.g. the
major sixth). The fingerboard charts of Figures 1 and 2 have been posted on the internet for
violin and viola students to use as reference. The charts differ from those the 18th century
(Barbieri 2008) in that the spacings have been carefully calculated for both equal tempered
tuning of the open strings. While many students play on open strings with just tuning, a chart for
such tuning would would be hopelessly complicated. Most of the exercises described in this
paper could be performed with either just or equal tempered open strings, especially at the level
of less advanced students.
Two groups of students might benefit from these exercises: Those with a keen interest in
mathematics are apt to find these exercises fascinating. Students who spend long hours
practicing might also benefit from these “mechanical” exercises, which can be performed even
after artistic alertness has been dulled by too much practice. Music is ultimately an art, not a
science, and pitch must be judged subjectively, not by calculation or measurement. What Ross
FINGERBOARD INTONATION CHART
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(2004) calls “melodic tuning” is important, but Ross also proposes mechanical tests such as
using a finger to briefly stop a resonating but un-played open string.
All activates described here have been tested on a viola and a 3/4 sized violin. Passages
that include a cello part were tested on a full sized cello.
“Large” and “Small” Errors
In order to remove unwanted clutter from charts and passages, we adopt the “%” symbol
as shorthand for “cent” (1 semitone = 100 cents = 100%). Loosen (1993) pointed out that
accomplished violinists routinely deviate by 8% from the tempered scale, and we shall refer to
any deviations from equal temperament as an “error”, with the full understanding that
accomplished violinists deliberately deviate from equal temperament. As discussed in the
(optional) appendix, each factor of {3} in the frequency ratio adds a “small” error of
approximately 2%. With the exception of one 6% error, all “small” errors in the fingerboard
charts are 4% or less. The fingerboard chart also depicts “large” errors of either 14% or 16%.
The “large” errors contain a factor of {5} in the frequency ratio. If open strings are tuned to just
fifths, most “large” errors are equal to 21.5% (the syntonic comma).
We suggest that students should first become aware of the “large” error, and then learn to
play to within this tolerance. It is unlikely that the methods of this paper can bring a student to a
precision much better than 4%. If we accept this tolerance of 4%, all “large” errors can be
treated as roughly equal, and to this extent, the situation regarding “large” errors can be
summarized as follows:

Just major thirds and sixths are “narrow”

Just minor thirds and sixths are “wide
FINGERBOARD INTONATION CHART
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“The just minors are too wide” makes for a good mnemonic if one imagines overweight coal
miners who have just cause for refusing to work because they cannot fit through the escape
tunnel.
Testing Notes with Just Intervals
Figure 3 can serve to introduce the conventions used in both fingerboard charts. Observe
the cluster of three circles representing E♮ on the violin’s open “D”. Finger locations for equal
tempered notes are denoted by solid circles centered on the string (with no left or right
displacement). White circles denote natural (tempered) notes and dark grey circles denote the
(tempered) accidentals. Therefore a white circle represents the tempered E♮.. The red circle is
displaced to the left, towards a lower string, because it represents a test against the open “G”.
The blue circle is displaced to the right because it references the open “A” (to the right of the
“D” string.). Sample passages can be found on the fingerboard chart. In our example, we have:
(1)
The quarter and half notes in these passages are not intended to denote time or tempo. Instead,
the quarter notes are to be held (or repeated) until intonation is achieved; the whole note denotes
the final result. The simplified fingerboard chart of Figure 2 displays the following pair of tests:
(2)
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The first bar depicts a “multi-step” test described a website by Sassmannshaus (2012). Such
tests are difficult because it is nearly impossible to adjust one finger while the other remains
motionless. Multi-step tests must be practiced slowly and carefully, training both fingers to land
lands properly, which after all, is how the instrument should be played.
Testing Notes with Harmonics
The fingerboard charts show five harmonics for each string, but only a few are useful as
intonation tests. Each harmonic must be identified with two locations on the fingerboard chart:
A diamond ( ) indicates where the finger lightly touches the string, while a cross ( ) identifies
the pitch actually sounded by the harmonic. This pitch actually sounded by a harmonic is
identified by situating the cross ( ) where one would press the finger (hard against fingerboard)
in order to play the note. The vertical location of a cross indicates whether the harmonic sounds
sharp or flat.
The simplified fingerboard chart of Figure 3 can be used to illustrate where to touch a
string in order to achieve a desired pitch: A cross labeled “-14% h5” is situated where one would
play a flattened version of B♮ on the violin’s “E” string. Here, “h5” stipulates that the string is
touched at the h5 harmonic, and this harmonic is associated with the label
“b-f#-c#-g# (-14%) h5”. The lower case letters denote the pitch sounded by this harmonic, with
the convention that the leftmost letter denotes the pitch as played on the leftmost (i.e. lowest
string). Thus the pitches B♮-F#-C#-G# are sounded on the G-D-A-E strings, respectively. To
test this B♮, play the h5 harmonic on the “G” string. Continuing with h5 on the other strings, the
chart informs us that it sounds as F# on the “D” string, C# on the “A” string, and G# on the E
string (the latter being well outside the normal range of the instrument.)
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The h4 harmonic can serve as an intonation test when played on the instrument’s lowest
string to test G♮ on the violin’s “E” string (C♮ on the viola’s “A”). Harmonics can also be
played as shifting exercises to keep the left hand relaxed and in proper position. An excellent
warm-up exercise involves playing h2 on the instruments lowest string, and shifting down to first
position to play h3 through h6, using the second finger to play both h5 and h6 (which teaches
finger independence). Students might enjoy playing the bugle tunes Taps and Reveille, as well
as the theme song from Spielberg’s movie, Close Encounters of the Third Kind. The harmonic
shifting exercise also motivates students to hold the instrument properly. Folk fiddle players
know that it doesn’t matter how the violin is held if one stays in first position. The truth is that
proper hand position is not needed for the repertoire used by beginning students, leading some to
wonder why they can’t hold the instrument any way they wish. Young people often exhibit two
personality traits that adults find both refreshing and frustrating: They like to (1) think for
themselves, and (2) set lofty goals. While first trait makes them apt to invent their own way of
holding the instrument, the second trait can be used to keep the first at bay: The student whose
aim is to shift like a virtuoso will want to practice the harmonic shifting exercises and therefore
be forced to hold the violin properly.
Another fun exercise is to play h2 on one string and h3 on the adjacent lower string, in
fourth position. The two harmonics should sound at nearly the same pitch.
Constructing and Testing Just Intervals
Certain basic intervals must be mastered a priori, before any construction of just intervals
can begin. These just intervals are the unison {1/1}, octave {2/1}, perfect fifth {3/2} and fourth
{4/3}. The most problematical might be the octave. Sundberg (1973) pointed out that the ear is
remarkably tolerant to octaves that are slightly out of tune, especially when widened (or
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“stretched”). Ability to play these basic intervals to just consonance should be verified with an
electronic tuner and/or qualified teacher before proceeding to construct the other intervals.
A string player typically “hunts” for consonance by gradually varying the pitch until
maximum consonance is achieved. Many of the exercises that follow are designed to narrow the
range of pitch over which this “hunt” takes place. This is useful because slowly if the search
range is too wide, students are apt to “hunt” either too quickly or too slowly. “Hunting” too
quickly causes the student to pass through consonance without noticing it. “Hunting” too slowly
leaves the student waiting in vain for an already consonant interval to suddenly improve, as the
dissonant sound becomes forgotten. By using easily recognized intervals to place the fingers in
approximately the right place, the “hunting” range for more difficult consonant intervals (e.g.
thirds) is reduced.
Constructing the Just Major Sixth
Since the goal is to construct just intervals, open string tuning to just fifths is
equally acceptable. Passage (1), above shows how to construct just versions of the perfect fourth
and major sixth. This is one of many passages and tests that the string player should also
practice either a fifth higher or lower (e.g. by testing on A♮ the viola or B♮ on the violin). To
enhance confidence in the just major sixth, the following passage should be played:
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(3)
Observe that the highest note is a bit flat, and listen for a beat pattern when it is played against
the highest open string. The tempo will depend on whether the open strings are just or equal
tempered fifths. With just-tuned open strings, the tempo will be 163 beats per minute for the
cello, with the viola and violin beating twice and thrice as fast, respectively. This passage was
played by an amateur violist on all three instruments. Clear and pleasant beats at approximately
the expected tempo were heard on the cello, but the two higher instruments made annoying
dissonant sounds that seemed to be beating at or near the correct tempo. Six college-age nonmusicians could recognize the presence of beats, but were unable to assign any tempo to it. A
biology professor with a good sense of rhythm, and who often conducts for local amateur and
high school musicals was able to hear a beat pattern of all three instruments that was roughly
consistent with the theoretical tempo.
The beat pattern for each instrument was measured electronically using readily available
technology: The output of a Radio Shack tie clip microphone was fed into the sound recorder
application found on a Windows PC at a sample rate 44100 Hz. When a passage by the violin
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was replayed at exactly half the sample rate (i.e. down one octave), it sounded like a cello and
made clearly identifiable beats. When the viola’s signal was played at half speed, and the beats
were marginally discernible. The recorded signals were also analyzed by Matlab to produce the
spectrograms shown in Figure 4. The four red horizontal patterns show the four harmonics for
each instrument, with the fundamental appearing at the bottom. Note that the second harmonic
beats at twice the fundamental’s beat frequency, with the third harmonic beating at thrice the
fundamental’s frequency, exactly as one would expect. . Also note the absence of beats in part
of the viola’s signal corresponds to a weak fundamental harmonic in the spectrograph. The
spectrograph allowed us to count the beats and establish the tempo on each instrument with the
following results:
Instrument
Tempo
(beats/min)
Tempo
Error
Fingered Pitch
Error (%)
“ideal”
163
0
-21.5%
violin
109
-54
-14.4%
viola
132
-31
-17.4%
cello
169
+6
-22.3%
In this table, the “ideal” pitch would be a final fingered note is flat by 21.5%, and the three
instruments did produce flattened notes. (Here the pitch deviation from equal temperament was
calculated from the beat frequency.) The largest error in the beat tempo occurred for the violin,
which beat triplets at a tempo of 109 beats per minute. An absence of beats is seen in the first
two seconds of the viola’s spectrogram, and the presence of beats at the second harmonic
suggests that the viola has a tendency to not sound at the fundamental harmonic at this pitch.
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This was confirmed by the viola’s weak strength in the fundamental as seen in a power spectrum
that was also calculated by Matlab.
Back in 1965, when music seemed to play a more prominent role in public education,
students in a middle school string orchestra played a game involving bow control. The goal was
to hold a note on one bow as long as possible, which challenged the students’ ability to control
the bow. An analogous (and more advanced game) might be to see who can make the best beat
pattern in passage (3).
Constructing the Just Major Third
Once the just major sixth has been mastered, the first bar of this passage can be used to
construct the just major third:
(4)
The second bar in (4) can serve as verification by challenging students to begin with the major
third and then release the third finger. If the resulting major sixth is at just consonance, the
student knows that the just major third was correct. An entirely different construction of the just
major third introduces the concept of “hunting”:
(5)
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In this passage, the violin’s C♮ must be adjusted upward to make the just third. (With tempered
open strings, Figures 1 and 2 show that the adjustment is 16%.) Passage (5) should be practiced
in three variations:
1. Do not sharpen the second finger until after the double stop with the open string has
begun. This allows the student to “hunt” upward for the consonant sound.
2. Sharpen the second finger before playing the double stop. This is more difficult and
forces students to “learn the comma”.
3. Play the passage in reverse order, with the strings tuned to equal tempered fifths. This
challenges the student to create a deliberately dissonant major third that is a mere 2%
wider than equal temperament.
Constructing the Just Minor Third
After the just major third is mastered (or sufficiently practiced), construct the just minor
third using the first bar of the following passage:
(6)
The second bar serves to verify, and both bars can be practiced in reverse order. The confidencebuilding exercise involves the need to sharpen the E♮ in the following passage:
(7)
Repeat this passage in the three variations described after passage (5).
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Constructing the Minor Sixth
Construct the minor sixth from an octave and minor third:
(8)
To build confidence, repeat the following in the three variations described after (5):
(9)
Constructing Just Intervals Larger than one Octave
Two intonation tests require compound intervals. To hear them, play the test note in
fourth position with the 2nd finger and use the 4th finger to play the h2 harmonic on the “G”
string. This converts the compound intervals into major and minor thirds:
(10)
The second bar of the following passage will be discussed in a later section. For now, play only
the first bar to build confidence in constructing the minor third:
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(11)
Repeat the first bar of (11) in the three variations discussed after (5).
Tempered Intervals
The following summarizes the construction of nearly equal tempered intervals using just
intervals on an instrument with equal tempered open strings. In this case it is important to tune
the instrument to equal tempered open fifths. All “quarter notes” are played to just consonance.
In each case the resulting “whole note” interval is with 4% of equal temperament. This is only a
theoretical precision, as so many other intervals must be also be correct to achieve it.
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(12)
The last bar of passage (11) represents a unique approach to equal temperament, but only
works for the minor sixth. If the open strings are tuned to equal temperament, and if both minor
sixths are made equally dissonant, then these minor sixths will be equal tempered, exactly. It is
difficult to assess the efficacy of this exercise, as one must use an electronic tuner to verify that
both sixths are tempered. And this renders it impossible to determine whether it was passage
(11) or the electric tuner that was responsible for learning this interval.
Issues in Cognitive Psychology
It is not uncommon for a student to spend many hours practicing something that is wrong.
The point of these activities is that they can be self-correcting by a student who practices alone.
But, since these exercises do not place fingers exactly where they belong in ordinary playing and
shifting, will they teach bad habits? A definitive answer would require careful experimental
studies, so we must be content with expressing the question in the language of cognitive
psychology. Behavioral psychologist Arthur Staats (1996) introduced the concept of Basic
Behavioral Repertoires (BBRs), which has been applied to sensory-motor skills by a number of
researchers (Riedel, Heiby, & Kopetskie, 2001). The idea is that one must first master basic
FINGERBOARD INTONATION CHART
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skills before progressing to more complex ones; much like learning how to print before
progressing to cursive writing.
Studies of how people acquire a second or even third language might also be relevant.
Perhaps basic skills such as shifting might be best learned as early as possible, just as language
skills are best acquired at an early age. Anyone who has attempted to learn a third language will
tell you that it was easier to become semi-fluent in the third language, than it was to achieve the
same level in the second language. (Bialystok & Hakuta, 1999; Perani, et al., 1998). This is
especially true when one acquires the first language early on in development (Levelt, 1989).
Two other issues are a bit more problematical: (1) If the tempered scale is analogous to
learning a second language, then perhaps it needs to be learned first. (2) These interval tests do
not work well in keys such as A or E major. These intonation tests based on just intervals should
not be viewed as a substitute for extensive exposure to music played at equal temperament, at the
earliest possible stage in a child’s life. And, at some point the student needs to move beyond
tests with just intervals.
Acknowledgements
We are grateful to Everet Kalcek for many helpful conversations about music instruction,
as well as to David Smiley, a teacher and member of the San Francisco Symphony who
introduced interval testing to young viola player back in the 1960s.
Appendix: Mathematica est diabolus in musica
All frequency ratios in the tempered scale are of the form, 2𝑛/12, where 𝑛 is an integer.
Just intonation occurs when the ratio of frequencies is 𝑝/𝑞, where p and q are reasonably small
positive integers. [Plomp 1965] These fingerboard charts are based on four equations:
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12 log 2 2 = 12.00 (exactly)
12 log 2 3 ≈ 19.02 ≈ 19 + 0.02
12 log 2 5 ≈ 27.863 ≈ 28 − 0.14
12 log 2 7 ≈ 33.688 ≈ 34 − 0.31
(For more precision, replace 0.02 by 0.0196, 0 .14 by 0.1369, and 0.31 by 0.3117)
The close proximity of logarithms involving {2, 3, 5} explains why ratios involving these
intervals are so suitable to the tempered scale. Logarithms are convenient because they allow us
to represent ratios of frequencies as subtraction between logarithms. In fact, music students
might find the logarithmic scale more familiar than the idea that just intervals involve ratios of
frequency. For example, consider the perfect fifth, or 7 halftones on the tempered scale. The
frequency ratio, 3/2, is sharp by 2%:
3
12 log 2 ( ) ≈ 19.02 − 12.00 ≈ 7.02
2
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Performing this calculation with all the just intervals, we have:
Interval
Ratio
Half-tones
Error
octave
2/1
12
fifth
3/2
7.02 = 7 + 0.02
+2%
fourth
4/3
4.98 = 5 − 0.02
−2%
major sixth
5/3
8.84 = 9 − 0.16
−16%
major third
5/4
3.86 = 4 − 0.14
−14%
minor third
6/5
3.16 = 3 + 0.16
+16%
augmented fourth
7/5
5.825 = 6 − .175
−17.5%
minor sixth
8/5
8.14 = 8 + 0.14
+14%
0%
The syntonic comma is 12 log 2 (81
) ≈ 21.5%. It is common for “large” errors when just
80
tuning is used. A different comma that arises in passage (11) is 12 log 2 (2048
) ≈ 19.6%.
2025
Since ratios involving smaller numbers tend to be more, these intervals have been ranked
according to the sum of the numerator plus denominator: For example, 2+1 = 3 is smaller than
3+2 =5, so we rank the octave as more consonant than the perfect fifth. This ranking roughly
matches the ranking of consonance most people perceive. Malmberg (1918) used a jury of
musicians and psychologists to rank the consonance in roughly the same order, except that the
minor sixth is regarded as being considerably more consonant: 2/1, 3/2, 5/3, 4/3, 5/4, 8/5, 6/5,
7/5. .
The Pythagorean intervals are approximations to the tempered scale that use higher order
fractions that do not contain the integer {5}
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Pythagorean
Ratio
Half-tones
Error
minor third
32/27
2.94 =3 -0.06
-6%
major third
81/64
4.08 =4+0.08
8%
minor sixth
128/81
7.92=8-0.08
-8%
major sixth
27/16
9.06=9+0.06
6%
Pythagorean intervals occur if multi-step intonations tests are used with an instrument tuned to
just open fifths. If one inspects passage (12) and assumes that the student can reliably create an
interval that is be between one-half and three-quarters a comma, the maximum error in the
resulting interval is 3%.
Intervals containing ratios with {7} have been called “the devil in music” (Barbieri
1991). The 31% error associated with this number makes it closer to the midpoint between two
halftones than it is to either 33 or 34. Nevertheless, something close to a tempered interval can
be achieved by arranging the error in {5} to partially offset the error in {7}:
(34 − .312) − (28 − .137) = 6 − .175
The adjacent string test associated with this interval is shown in the fingerboard charts, but not
highlighted because the test is neither close to an equal tempered interval, nor very consonant.
Though the 7/5 interval is not well suited as an intonation test, the attempt to hear this
resonance makes for yet another interesting game for those who love mathematics: Since the
tritone’s tempered frequency ratio, 26/12, is the square root of two, and since 49 almost equals
50, the following three numbers are nearly equal:
7
10
< √2 <
5
7
We see that the fingerboard charts require additional thin blue circles that are 17% flatter than
the “G” string’s tempered G#. The student’s attempt hear these just intervals as consonant might
prove futile, perhaps because there are other pairs of number that approximate the square root of
two, leading to a rather large cluster of just intervals: 25/18 < 7/5 < 45/32 < √2 < 64/45 <
10/7 < 36/25.
.
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Sassmannshaus, K. (n.d.). Intonation. The Sassmannshaus Tradition for Violin Playing.
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FINGERBOARD INTONATION CHART
Figure 1: Violin fingerboard chart
22
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Figure 2: Viola fingerboard chart
23
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Figure 3: Simplified chart for introducing conventions used in the fingerboard charts
24
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25
Figure 4: Spectrograms for violin, viola, and cello. A double stop is played between the highest
string and a fingered note on the second highest string. The fingered note is an octave and just
major sixth above the lowest string.
Remove from submitted document 266d, 266e, 266f. Alt-X: x♭x
x♮x
x♯x