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Harmonics
BASIC INFORMATION
Harmonics are produced when the plucked/struck/bowed string is touched at particular points with light finger pressure. The string vibrates on both sides of the touching finger. The pitches of the harmonics correspond to each partial of the open string; the first harmonic is the string’s fundamental. The points on the open string at which harmonics are generated lie at the nodes of each ascending partial (i.e., where the string is divided by a whole number, e.g., 1/2; 1/3, 2/3; 1/4, 2/4, 3/4; 1/5, 2/5, 3/5, 4/5…). A harmonic can be produced at several of its nodal points, for example the third harmonic is obtainable by touching the string at 1/3 and 2/3 of its length. However a harmonic is not necessarily found at all of its associated nodes, for example the fourth harmonic is obtainable at 1/4 and 3/4 of the string length but not at 2/4. I will refer to the nodal points on the string at which a particular harmonic can be produced as ‘touch points’ for that harmonic, i.e., the set of ‘touch points’ for a harmonic is a subset of its nodal points. In other words a touch point is always a nodal point but the reverse is not true. Finding alternative touch points for a particular harmonic and knowing how many equivalent touch points are available can be useful to cellists in minimising shifts in the left hand, especially in fast passages.
Harmonics become quieter, more difficult to control (it is more difficult to find their position, they are more sensitive to changes in bow pressure and speed) and less overtonerich as they ascend. Harmonics are most overtonerich on the A string, and are increasingly less overtone rich on the thicker and stiffer D, G and Cstrings.
The sound quality of harmonics is characterised by their weak overtone content relative to equivalent pitches on the stopped string. The first harmonic, the open string, is the only exception to this, having a very rich overtone content. However, harmonics become overtoneweaker as they ascend. The reduction in higher partials is significant at first and gradually becomes marginal. This reduction can be expressed in the harmonic sequence: 1/1, 1/2, 1/3, 1/4,1/5…1/n, i.e., the fundamental (first harmonic) has the ‘maximum’ number of overtones, the second harmonic has half as many as the first, the third harmonic has a third of the amount of the first etc.
How harmonics work
Touching the string (without pressing it to the fingerboard) fixes the area under the finger as a static point during vibration. Partials with antinodes at this point are unable to vibrate under these conditions and partials with nodes at this point are free to vibrate.
The second harmonic is produced by touching the string at its mid point, an antinode for all of the odd partials. The odd partials are ‘blocked’ from the sound. The overtone content is halved since only the even partials (that is every other partial in the finite set of partials available in the open string) are able to vibrate. The new fundamental is the second partial of the open string (since the first partial has been excluded), the new second partial is the open string’s fourth partial, the new third partial is the open string’s sixth partial, etc.
Similarly, in touching the string at 1/3 or 2/3 of its length, the finger blocks partials with antinodes at, or close to, this point. All partials that have orders of a multiple of three are free to vibrate because they have a node at this point. Consequently, only every third partial sounds, the timbre contains a third of the number of overtones of the open string. The new tone’s fundamental is the open string’s third partial, the new second partial is the open string’s sixth partial, the new third partial is the open string’s ninth partial, etc.
So, the second harmonic contains half of the number of overtones of the open string, the third harmonic contains a third of the number of overtones of the open string, the fourth harmonic contains a quarter, the fifth a fifth, etc. This is true because there are a finite number of available partials on a cello string, limited by several factors including: string width relative to length, point of contact, exciter properties and cello body responses. These factors are flexible. Therefore, it is more accurate to say that the potential overtone content of a harmonic for various points of contact/excitation forces, etc., decreases as harmonic order increases.
IN CONTEXT
The reduced overtone content of harmonics in general means that they are particularly susceptible to damping; small increases (for example, using a slightly less dense plectrum) result in a relatively larger reduction in overtone content than for the stopped string. This is why cellists often use the fingernail for pizzicato harmonics. The exception to this rule is the first harmonic, the open string, which, is unrestricted by a touching or a stopping finger. The combined effect of this and the open string having the largest number of potential overtones means it can ring very ‘brightly’. The timbre can stand out as harsh compared to the stopped string. Cellists often avoid playing open strings because of this effect, preferring to stop a lower string at the same pitch.
Note
In reality, the pattern of overtone reduction in ascending harmonics might be more complicated than the above description suggests. Partials might only be subdued rather than completely excluded from the sound or extra partials might be included because of responses from the cello body/room or in combinations of various other partials. Moreover, upper partials of the higher harmonics are restricted even further because the effects of damping at the touching finger, which are relatively larger for higher harmonics. However, despite this, the broad pattern is generally true.
Loudness of Harmonics
For a fixed excitation force and contact point, ascending harmonics become quieter. This is clear since at a particular excitation force, the deviation of the string from its rest position is greatest for the first partial and reduces with ascending partials. Amplitude depends on this deviation. The figure below shows a hypothetical situation where a particular excitation force displaces an ideal string. The total amplitude of the first partial is A1, which is greater than that of the second partial A2, which is greater than that of the third partial A3 etc. More precisely, the amplitude of the second partial is half that of the first, the amplitude of the third partial is a third of that of the first, etc.
IN CONTEXT: Controlling overtone content and loudness of harmonics
The reduced overtone content and reduced loudness of ascending harmonics mean that, while they broadly adhere to the principles of the effect of changing contact point, excitation force and means of excitation described in the case of the stopped string, the limits of these inputs are different (for fixed conditions, there are more possibilities for overtoneweak, quieter sound and fewer possibilities for highovertone, loud sounds).
Moreover, since there can be multiple nodes on the string for the harmonic’s fundamental (this is a unique situation: the open/stopped string always has two nodal points at the nut/stopping finger/bridge), the ‘pattern’ of changing contact point is much more complicated in the case of higher harmonics than the stopped/open string. In reality, this means that a sul ponticello effect for harmonics is difficult to achieve: it will not contrast greatly with a sul tasto effect, and, since there are multiple nodal points on the string, there are also multiple points where sul ponticello and sul tasto might be found.
Note: ACOUSTICAL INFORMATION
The above figure only describes the situation for an ideal string. In reality, other factors influence the loudness of the partials. In fact, the struck string follows the above pattern most closely, with the amplitude of the partials decreasing by 1/n for ascending harmonic order. However, extra effects cause the equivalent equation for the plucked string to be (1/n)^{2 }– in other words, the amplitude of the higher partials decreases more than the example above and more than in the struck string. For the bowed string, the equivalent formula is 1/n multiplied by a numerical constant (these formulas can be found in Author Benade’s The Fundamentals of Musical Acoustics (2nd edn., new York: Dover Pubications, 1990)). Therefore, the amplitude of the higher partials is much more reduced for the plucked string than the bowed string, which is more reduced than for the struck string. In fact the ‘clavichordtype’ battuto tones can negate this trend because of a masking effect. Applied to real circumstances, amplified by a cello body in a particular room, these results can diverge significantly from the above formulae. However, the pattern of amplitude reduction for partials can be heard loosely to follow the above description, and it is clear that the reduction of amplitude in the plucked string is greater than that in the bowed string.
FINDING TOUCH POINTS
The set of touch points for a harmonic is a subset of the harmonic’s nodes. For some harmonics, every node is also a touch point, and for others there are fewer touch points than nodes. It is useful for a cellist to know all of the alternative touch points for a particular harmonic, to be able to find an efficient and flexible fingering.
The set of nodes for all harmonics is ‘n1’, where n is the harmonic order (i.e., there is 1 node for the second harmonic, 2 nodes for the third harmonic, etc.). So, the maximum number of touch points for a harmonic is n1. All prime number order harmonics have the maximum, n1 touch points. For example, the second harmonic has one touch point, at ½ of its length and the third harmonic has two, at 1/3 and 2/3 of its length, the fifth has four at 1/5, 2/5, 3/5 and 4/5, etc. The number of touch points of nonprime number harmonics is restricted by another factor: a node is excluded from the set of touch points for a particular harmonic if it coincides with the node of a lower harmonic.
For example, the fourth harmonic is produced when the string is touched at 1/4 of its length or at 3/4 of its length, however not at 2/4 (=1/2) of its length: in this case the second harmonic is produced.
Similarly, for the sixth harmonic: the sixth harmonic has nodes at 1/6, 2/6, 3/6, 4/6 and 5/6 of the string. However, touch points only occur at 1/6 and 5/6 of the length of string since 2/6 (=1/3) and 4/6 (=2/3) are touch points for the third harmonic and 3/6 (=1/2) is a touch point for the second harmonic. In other words, a cellist does not have many options to find alternative fingerings for the sixth harmonic.
Arranging this information in a table for the first sixteen harmonics shows how the number of touch points fluctuates.
Harmonic  Pitch of harmonic above the open string. Interval above the fundamental +/ cents (¢)  Touch points at 1/n of the length of open string  No. of touch points  No. of Nodes  Pitch for A string 
First  Open string  None  0  0  A3 
Second  8ve  1/2  1  1  A4 
Third  8ve +5^{th }+2¢  1/3, 2/3  2  2  E5+2¢ 
Fourth  2 8ves  1/4, 3/4  2  3  A5 
Fifth  2 8ves +maj 3^{rd }14¢  1/5, 2/5, 3/5, 4/5  4  4  C#6 14¢ 
Sixth  2 8ves +5^{th} +2¢  1/6, 5/6  2  5  E6+2¢ 
Seventh  2 8ves +min 7^{th }31¢  1/7, 2/7, 3/7, 4/7, 5/7, 6/7  6  6  G6 31¢ 
Eighth  3 8ves  1/8, 3/8, 5/8, 7/8  4  7  A6 
Ninth  3 8ves +maj 2^{nd }+4¢  1/9, 2/9, 4/9, 5/9, 7/9, 8/9  6  8  B6+4¢ 
Tenth  3 8ves +maj 3^{rd }14¢  1/10, 3/10, 7/10, 9/10  4  9  C#7 14¢ 
Eleventh  3 8ves + 4^{th } +51¢  1/11,2/11,3/11,4/11,5/11, 6/11,7/11,8/11,9/11,10/11  10  10  D7 +51¢ 
Twelfth  3 8ves +5^{th }+2¢  1/12, 5/12, 7/12, 11/12  4  11  E7+2¢ 
Thirteenth  3 8ves + min 6^{th }+41¢  1/13, 2/13, 3/13, 4/13, 5/13, 6/13, 7/13, 8/13, 9/13, 10/13, 11/13, 12/13  12  12  F7 +41¢

Fourteenth  3 8ves + min 7^{th} 31¢  1/14, 3/14, 5/14, 9/14, 11/14, 13/14  6  13  G7 31¢ 
Fifteenth  3 8ves + maj 7^{th} 12¢  1/15, 2/15, 4/15, 7/15, 8/15, 11/15, 13/15, 14/15  8  14  G#7 12¢ 
Sixteenth  4 8ves  1/16, 3/16, 5/16, 7/16, 9/16, 11/16, 13/16, 15/16  8  15  A7 
The pitches shown in the righthand collumn are approximate. Tuning deviations occur because of string stiffness. These deviations are increased for higher harmonics, and on thicker strings.
The pattern of available touch points is always symmetrical about the middle of the string. Describing the above table of possible touch points in pictorial terms demonstrates this.
An algorithm for finding touch points
The easiest way to find the position of touch points for a particular harmonic is by drawing a diagram similar to B2d or by following a simple algorithm:
 List the nodes as fractions: 1/n, 2/n…a/n…(n1)/n
 Remove fractions that can be simplified i.e. 2/(n= 2b), 3/(n=3b)…
The result is the list of touch points in the form a/n. The position of each touch point is the a^{th} node from the nut. For example:
The nodes for the 12^{th} harmonic are:
1/12, 2/12, 3/12, 4/12, 5/12, 6/12, 7/12, 8/12, 9/12, 10/12, 11/12
The following fractions in which can be simplified:
2/12, 3/12, 4/12, 6/12, 8/12, 9/12, 10/12,
Leaving the touch points:
1/12, 5/12, 7/12, 11/12
That is the first, fifth, seventh and eleventh nodes from the nut.
The simplest way to find the position of the harmonic nodes relative to stopped string pitch (see figure B2e) is to start with the node closest to the bridge. This node, always a touch point, is equal to the pitch of the harmonic. By calculating the harmonic series backwards from this touch point, the position of each node can be calculated. The node immediately below this is an octave lower, the subsequent nodal position is an octave and a fifth minus 2 cents higher, the subsequent nodal position is two octaves higher, etc. The deviation in cents of the natural harmonic series must be taken into account. For example:
As above, touch points for a particular harmonic x are written in the form n.x, where x=1 is the touch point closest to the nut and x=n1 is the touch point closest to the bridge. Nodes of a harmonic that are included in the touch points of a lower harmonic are written in the form n.(x).
The pitch of touch point closest to the bridge for the tenth harmonic, 10.9, is C#714¢. The pitch of the nodal point immediately below this, 10.(8), is 10.9 – one 8ve = C#614¢.
Similarly 10.7 = 10.9 – (8ve +5^{th }+2¢) = C#714¢ – (8ve +5^{th }+2¢) = F#516¢
10.(6) = 10.9 – (2 8ves) = C#714¢ – (2 8ves) = C#514¢
10.(5) = C#714¢ – (2 8ves +maj 3^{rd }14¢) = A4
10.(4) = C#714¢ – (2 8ves +5^{th}+2¢) = F#416¢
10.3 = C#714¢ – (2 8ves +min 7^{th }31¢) = D#4+17¢
10.(2) = C#714¢ – (3 8ves) = C#414¢
10.1 = C#714¢ – (3 8ves +maj 2^{nd }+4¢) = B418¢
To double check the result of the above algorithm, the number of touch points for any harmonic can be quickly calculated using the following formula. As explained above, nodes are excluded from the set of touch points if they are already touch points for another harmonic – that is, if the harmonic order has integer factors. All prime number order harmonics, therefore, have the maximum number of touch points, n1 where n is harmonic order. The number of touch points for nonprime harmonics is n1 minus the touch points for each prime factor of n. This can be summarised for primes and nonprimes as:
Π [f^{a}^{ }f^{a1}] a³0
Where f is a prime factor of the harmonic order, n and a is the highest factor of the prime.
For example the sixth harmonic has two prime factors, 2 and 3, its factorisation is 2x3 therefore the number of touch points is:
(2^{1}2^{0}) x (3^{1}3^{0}) = 1x2= 2
The twelfth harmonic has two prime factors, 2 and 3. Its factorisation is 2x2x3
The number of touch points is:
(2^{2}2^{1}) x (3^{1}3^{0}) = 2x2= 4
The seventh harmonic has no factors apart from itself since seven is a prime number. The number of touch points is:
(7^{1}7^{0}) = 71 = 6
Thanks to Prof. Erik Oña for devising this formula.
Touch points for the first 13 harmonics on each string
In the videos below I demonstrate all of the touch points for the first 13 harmonics on each string. The accompanying charts show the number and position of the touch points.
Harmonics ASTRING from Musikforschung Basel on Vimeo.
Harmonics on the Dstring
Harmonics GSTRING from Musikforschung Basel on Vimeo.
Harmonics on the Cstring
Harmonics on the Astring
Harmonics DSTRING from Musikforschung Basel on Vimeo.
Harmonics on the Gstring
Harmonics CSTRING from Musikforschung Basel on Vimeo.