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Resonance: Acoustics of the Narciso Yepes 10-String Guitar

Tuning & Sympathetic String Resonance (Inter-string Transmission of Vibrations)



"The most important reason why I developed the 10-string guitar is because the traditional 6-string guitar has resonance [...] only in four notes of the scale." (Narciso Yepes)
 
"The normal tuning that I use for the resonance, for the overtones: is C, Bb, Ab, Gb." (Narciso Yepes)

 

 

Standard 10-String Guitar Tuning
  Standard Tuning
(The tuning of strings eight to ten could also be written or notated enharmonically as Bb, Ab, Gb.)
 
 
The six-string guitar has sympathetic string resonance from its bass strings for essentially four notes of the treble strings: D, A, E, and B. That is, at the moment of playing any of these four notes on any of the treble strings, at least one open bass string begins to vibrate sympathetically at the same frequency. However, the guitar lacks the same support for the other eight notes ( C, C#, D#, F, F#, G, G# and A#, or their enharmonic equivalents), which do not benefit from the same sustaining and enriching effect of this type of resonance. That is, unlike D, A, E, and B, the other eight notes of the chromatic scale either do not consistently have this sympathetic string resonance for all their treble-range octaves (eg. F# or G), or no such resonance at all (eg. F or Bb). As a consequence, C, C#/Db, D#/Eb, F, F#/Gb, G, G#/Ab, and A#/Bb are naturally more inclined to sound dry and clipped short in relation to the richer and more sustaining notes of D, A, E, and B.

 
In contrast, the 10-string guitar with the particular tuning that Narciso Yepes conceived of in 1963 has sympathetic string resonance in unison with any of the twelve chromatic notes played in any octave of the treble strings. That is, as a fact of acoustics and without exception, only when it is tuned in the standard way as follows:

 
(7) C2 (the string with the lowest pitch)
(8) Bb2 / A#2 (a minor 7th above string seven)
(9) Ab2 / G#2
(10) Gb2 / F#2

(The above scientific notation system is used to indicate the exact pitches of notes on this site. In this notation, Middle C is written as C4; the D above it is D4 and the B below it is B3. The classical guitar is a transposing instrument and in normal musical notation we write the pitches of its strings an octave higher than they actually sound.)

 

 
 
(A video by Rob Beer illustrating the phenomenon of sympathetic resonance, comparing a 6-string guitar with a 10-string guitar in standard tuning.)
 

   
What is Resonance?

Writing in the 2nd century B.C.E., Dong Zhongzu describes the phenomenon of resonance as follows:
"The vital spirits of humankind, tuned to the tones of heaven and earth, express all the tremors of heaven and earth, just as several cithars, all tuned on gong [tonic], all resonate when the note gong sounds. The fact of harmony between heaven and earth and humankind does not come from a physical union, from a direct action; it comes from a tuning on the same note producing vibrations in unison . . . In the universe nothing happens by chance, there is no spontaneity; all is influence and harmony, accord answering accord."

Alternatively, to describe this acoustic phenomenon in a less 'esoteric' manner, resonance is the reason why a musical note can cause a certain object in a room to rattle. Or, to take another example, a truck on a road next to a house might make a certain window in the house vibrate when the truck is moving in a particular gear, producing a particular frequency. However, the external sound and the resonator (the object that responds to that external vibration) need not necessarily be tuned in unison ("on the same note") as long as one of the stronger overtone frequencies of the resonator (its octave or compound fifth) is perfectly in tune with the frequency of the external sound.

 
Scientists working in the field of acoustics define this phenomenon as:
the tendency of a system to vibrate sympathetically at a particular frequency in response to energy induced at that frequency.
Thus, resonance is an acoustic phenomenon wherein a passive system (such as an untouched string) responds to external vibrations to which its tuning (or natural frequency) has a close harmonic relation.


 
How Sympathetic String Resonance Works on the Yepes 10-String Guitar:

 
In the particular case of the 10-string guitar, the 'systems' that resonate are the instrument's bass strings, (4) to (10). These begin to vibrate sympathetically (in other words, without being actively touched) in unison with notes played on the treble strings. But whether a note from a treble string induces a bass string to resonate sympathetically depends on whether the bass string is tuned so that one of its natural frequencies (partials or overtones) corresponds exactly with the frequency of the equal tempered note played on the treble string. To illustrate this, let us consider the case of the six-string guitar.

If we play a note of D on the second string, fret III, (written in scientific pitch notation as D4) and then dampen only the second string's vibration with any finger, the same note of D4 will, in fact, continue to sound audibly. This lingering sound is the sympathetic resonance of D4 on another, untouched string. The energy induced at that frequency has forced the fourth string (D3) to vibrate sympathetically at the frequency of its first overtone (a.k.a. second partial), which is D4. In other words, without being touched, the D-string begins to vibrate not at its fundamental frequency, but at the frequency of the note played on the second string.
 
However, if we were to attempt the same procedure on the C#4 or Eb4 just below and above D4, we would observe that these notes produce no comparable effect. They are naturally inclined to have a secco sound (that is to say, clipped short and dry in timbre) while the D has a more sustained sound that is also richer in overtones.

Now, taking care to mute all but the D-string, if we were to produce in turn each of the chromatic steps of the treble strings, we would observe that the D-string resonates basically with two of the twelve chromatic notes, the D 's and the A 's. (Note that the relationship between these two notes, D to A, is that of the perfect fifth.) If we play any of the notes D4, A4, D5, A5 the fourth string reproduces their frequencies via the phenomenon of resonance. Similarly, the fifth string will resonate in unison with its higher octaves of A and also with its twelfth (or compound fifth), which is E4, and its higher octave. Illustrated more simply, the bass strings of the six-string guitar are shown below, followed by each of the notes that (when plucked on a treble string) will induce that bass string to resonate at the pitch of the plucked note.

 
(4) D3: D4, A4, D5, A5
(5) A2: A3, E4, A4, E5, A5, B5
(6) E2: [E3,] B3, E4, B4, E5, F#5
 
[Sympathetic resonance is present for all treble notes of D, A, E, and B only.]

 
Taking the fifth string as an example, the treble notes (A3, E4, A4, E5, A5, B5) that induce resonance from it correspond to the 1st, 2nd, 3rd, 5th, 7th and 8th overtones above the fundamental note of the fifth string, A2. The reason why no overtones other than these resonate in any significant way is because only the abovementioned overtones correspond sufficiently with the frequencies of the equal tempered (ET) notes played on the fret-board. For example, the 9th overtone of an A2-string (a natural C#5) is, in fact, 14cents flatter than an ET C#5 produced on the fret-board. The only overtones that correspond closely enough to their equal tempered equivalents are the octaves (overtones 1, 3 and 7), the fifths (overtones 2 and 5, which are 2cents sharper than ET), as well as the 8th overtone (ET +4cents). Only these abovementioned overtones of the bass strings resonate in any significant way with fretted notes of the treble strings.

Furthermore, while F#5 (string one, fret fourteen) will make string six resonate slightly at the pitch of F#5, this does not mean that the six-string guitar has resonance for F#. This becomes clear when a lower octave of F# (F#4 or F#3) is played: It yields only an even fainter resonance at the higher pitch of F#5, thus neither a significantly strong resonance nor a unison. In fact, based on all the above, we could say that the six-string guitar has resonance from its bass strings for all the octaves of D, A, E, and B that can be played on the trebles, but not for the other eight notes of the chromatic scale.

It was primarily this imbalance of resonance that, in 1963, prompted Narciso Yepes to add four strings to the classical guitar as tuned resonators. Yepes settled on this particular number of strings and the standard tuning of these strings not on a whim and not based on any historical precedents (i.e. 10-course lutes or 10-stringed harp-guitars), but based on the science of acoustics. In fact, the primary reason for Yepes's innovation is that the standard tuning of the four added strings yields resonance for all octaves of the twelve chromatic notes played on the treble strings.
 
Furthermore, this is achieved by the addition of the minimum number of new strings, and (for the most part) without introducing duplication of resonances over multiple strings. Excessive duplication of resonances was avoided as this would have undermined the sonorous balance that Yepes desired and would, furthermore, have impaired nuanced control of resonances. (If a note causes resonance on one sympathetic string, that individual string can easily be checked, if desired; but if a note causes resonance on multiple strings, checking them becomes either more ponderous or less refined.)

The additional basses thus resonate when their corresponding natural frequencies are played on the trebles, as listed below:

 
(7) C2: [C3,] G3, C4, G4, C5, D5
(8) A#2: A#3, F4, A#4, F5, A#5, C6
(9) G#2: G#3, D#4, G#4, D#5, G#5, A#5
(10) F#2: F#3, C#4, F#4, C#5, F#5, G#5
  • (7) supplies the C and G resonances.
     
     
  • (8) supplies the A# (or Bb) and F resonances.
     
     
  • (9) supplies the G# (or Ab) and D# (or Eb) resonances.
     
     
  • (10) supplies the F# (or Gb) and C# (or Db) resonances.

     
     

Thus, the combination of the traditional six-string guitar plus four more strings tuned in this singular manner gives the guitar a balanced sonority that it otherwise lacks, whether it has six or ten strings. The guitar lacks balance because there is no consistency between the musical 'envelopes' of notes that naturally sustain, and are enriched through resonance, and those other notes that lack the same sympathetic support. This means that, in order to achieve a balanced sound, good guitarists are constantly having to dampen those resonances they have, while trying to compensate for dry ( secco) notes that do not sustain after the left-hand finger is lifted. But on the 10-string guitar, with the particular tuning designed by Yepes, all twelve notes have basically the same resonance. It is thus possible to use or to mute the resonance for contrast, and also to sustain, enrich, check or underplay any note as demanded by the musical context rather than the acoustic whims of the instrument.

Some notes are no longer arbitrarily richer than others. Some notes no longer arbitrarily sustain longer than others when the finger is lifted away from the string or the hand shifts position. These qualities are particularly beneficial in the performance of music with slow tempi, polyphonic music (where lines must be sustained) and 20th century repertoire (where all twelve notes appear frequently). Furthermore, the new possibilities offered by the resonance, analogous to those of the piano's pedals, enable new interpretative nuances in the performance of all traditional seven- or six-string guitar repertoires. It offers expanded possibilities without expanded limitations.

Of course, in order to check individual resonances (if demanded, say, by the harmony of the musical context), it is necessary to know which bass string resonates when a particular treble note is struck. For ease of reference, the following chromatic scale lists the treble-string notes from the lowest G3 to the highest C6 and shows the bass strings that resonate when these treble notes are played:

 
G3 - (7) ... In other words, a played G3 makes string seven vibrate in unison. Etc.
 
G#3 - (9)
A3 - (5)
A#3 - (8)
B3 - (6)
C4 - (7)
C#4 - (10)
D4 - (4)
D#4 - (9)
E4 - (5) and (6)
F4 - (8)
F#4 - (10)
G4 - (7)
G#4 - (9)
A4 - (4) and (5)
A#4 - (8)
B4 - (6)
C5 - (7)
C#5 - (10)
D5 - (4) and (7)
D#5 - (9)
E5 - (5) and (6)
F5 - (8)
F#5 - (6) and (10)
G5 - (7)
G#5 - (9) and (10)
A5 - (4) and (5)
A#5 - (8) and (9)
B5 - (5) and (6)
C6 - [(7) and] (8)
 

 
The sympathetic string resonance can also be heard in this audio clip of the above chromatic scale, containing all notes of the treble strings from G3 to C6. (Note, however, that this is an mp3 file and as such is not truly an accurate representation of the live experience.) This chromatic scale is executed in the manner of the classic example used by Yepes to foreground the sympathetic resonance by playing a note and then immediately stopping the vibration on that same string. The same note is heard continuing to ring softly after being dampened. This sustaining effect is the sympathetic resonance on another string. When G3 is plucked, string (7) or C2 vibrates at the same frequency of G3; when G#3 is plucked, string (9) or G#2 vibrates at the frequency of G#3; etc., as written out in-full above. Attempt the same experiment on a 6-string guitar (or using another tuning of ten or less strings) and some notes will be followed by an abrupt silence (or a muddy, pseudo-resonance of negligible volume and indeterminate pitch) revealing a much greater inconsistency of the musical envelopes (the characteristic way in which the intensity of a note changes through time).

 
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To read about the 10-string guitar's possibilities as a guitar with an augmented tessitura, particularly for the transcription of music originally written for baroque lute or keyboard, click here .
Copyright 2009-2016 by Viktor van Niekerk (All rights reserved in all countries; unauthorized duplication of material on this site is prohibited.) Information about the 10-String Classical Guitar designed by Narciso Yepes