Running Head: fingerboard intonation chart a fingerboard intonation chart for the violin and viola

A fingerboard intonation chart for the violin and viola

Guy Vandegrift and Michelle Smith

Wright State University

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

Abstract

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 18^th 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 (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
  

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:

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:

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.)

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 “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.