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Eltham Jones, guitar repair and technical services :Bristol : Cardiff : Bridgend : Tel. 07971 240296

The Buzz Feiten Tempered Tuning system is something that has, from it's inception, attracted controversy.

Sadly, most of the controversy has not centred upon the system's technical and logical rationale, about which I have some serious concerns, but about the allegation that Feiten has attempted to patent an idea that is already common currency in the guitar making world. This would seem to me to have more to do with the resentment felt by Feiten's detractors at having been bested by someone who has come from outside the closed world of arcane secrets of luthierie than any constructive critique of the logic and reasoning behind the system itself.

The problem in formulating any such critique however is that before anyone can understand why the Feiten system is such a technical blind alley it's necessary to have an in depth understanding of the physics and psycho-acoustics of the guitar and this is something the average guitarist has no more idea of than the average driver understands the finer details crankshaft stress analysis in internal combustion engines.

So, before beginning to talk about the flaws in the Feiten system it's necessary to understand why it is we perceive the guitar as sounding "out of tune" in the first place.

Dr. Scheitenballs

or how I learned to stop worrying and love equal temperament

Historically, keyboard instruments could be tuned so that natural harmonic intervals were available in at least one key. This is possible because each note can be tuned separately. Even with these systems however, not all intervals worked in all positions. Keys could be "near" or "distant" and the "wolf note" - a particularly dissonant interval named for its howl - was ever present.

The reason this cannot be done on a guitar is because there are six strings and the range of each string overlaps with its neighbour over at least three quarters of it's compass so that the same note can appear in multiple positions. If, for example, you attempt to tune the low E and the A to a perfect fourth, because the strings are divided by the constant ratio scale the A at the fifth fret on the E would sound sharp relative to the open A because the interval defined by the 5th fret is a wider one than the one being employed between the two open strings. You could, of course, move the fret a little way to accommodate the error but it would then be in the wrong place to define the D on the A string and the g on the D string, et cetera...

Furthermore, how would the open strings sound?

If you tuned each course to a perfect harmonic interval you would end up with a very flat top string because each fourth is represented by a ratio of 4/3 and there are four of these in the two octave span of the guitar, plus one major third at a ratio of 5/4 evaluating to an interval between the low and high E of 320/81 and not the mandatory 4/1 that would be a true double octave.

There are guitars that attempt to resolve these discrepancies - the Fretwave and the so-called True Temperament guitars by Anders Thidell - but it's hard to see how they don't cause more problems than they solve. and even harder to see how they overcome the problem of fitting standard tuning into a two octave interval without stretching the fourths and thirds or resorting to a unique tuning or tuning to a chord, requiring a complete reworking of tradtional guitar technique.

Modern music is irrevocably wedded to equal temperament. All keyboards and other fixed pitch instruments are built to use it and there is little place for a modern guitar built to use earlier systems of intonation except in the vanishingly small special interest groups devoted to the performance of early music. In these arenas a modern instrument would have no place since authenticity is the name of the game and a guitar or lute that actually plays true to a key would be anything but authentic...

But there are solutions for the piano; why not for the guitar?

The harmonics exist in a series of whole number multiples of the fundamental in the ratios 1/2/3/4/5/6 and it's the relationship between these harmonics that define our perception of what sounds "right".

Remember that what we think of as a linear "step" in music is actually a ratio of two frequencies; the octave is defined by a doubling of the frequency so the first step ratio in the above series is 1/2. Following on from this is the perfect fifth defined by the ratio 2/3, the perfect fourth is 3/4, and the major and minor thirds by 4/5 and 5/6 respectively.

Interestingly these successive intervals exactly spell out a major chord...

The harmonic series unfortunately gives us no value for the tone or semitone so these need to be derived from the larger intervals of the harmonic series. There is a gap of a tone between the fourth and fifth so dividing one by the other gives us a value for the tone of 8/9. There is also a step of a semitone between the major and minor third so this gives us a value for the semitone of 24/25. Two of these multiplied together then will obviously be 8/9; except it is not, neither is the step between the major third and the perfect fourth the same value as the step between the major and minor third; it's actually 15/16...

We have now encountered the first problem associated with our devotion to the harmonic series; it doesn't add up.

It is, in mathematical terms, irrational.

We perceive it as being "correct" because neighbouring tones whose fundamental frequencies are the same as - or multiples of - a harmonic in another tone will share a range of harmonics and difference tones ( link will open new window) which mesh together to produce few interference effects. However there is no fixed value for any of the smaller intervals nor do the larger harmonic intervals add up to an octave; four minor thirds in sequence evaluate to a ratio of 2.07 while three major thirds evaluate to 1.95. the only intervals which actually combine to create a perfect octave are the fourth and fifth and that's only because a fourth is an inverted fifth.

You may have noticed that we have got this far without even mentioning the guitar; that's because - and take good note of this - it's not a guitar problem! The problem is with music itself...

It is a problem for the guitar because it is a fixed pitch instrument and in common with all other fixed pitch instruments the maker needs to be able to define a series of values for the pitch of each note which, in the harmonic system, don't actually exist.

Why we think some sounds are "in tune" and why this deceives us.

The problem of creating a fixed pitch scale for fretted instruments perplexed musicians for generations. In a sense, there is no actual solution since it is the mathematical equivalent of trying to divide 9 by 4 and have it equal 2; it simply isn't viable. It was Vincenzo Galilei, who first theorised that the problem could be alleviated by making all intervals equal divisions of the octave, eventually codifying the principle as the "rule of 18" in which the string is reduced by progressively by 1/18 of its length to produce a scale of 12 more or less equal semitones. This first "constant ratio" scale was not perfect but for the first time it allowed lutes to play tolerably in tune all over the fingerboard. Previously it had been necessary for lutenists to have an intimate knowledge of the parts of the fingerboard which could and couldn't be used and, in fact, tablature notation had been developed so that information could be conveyed not only about which notes were to be played but where on the fingerboard they had to be played.

In due course, and after much work by the musicologists and mathematicians of the renaissance and the enlightenment eras such as as Andreas Werckmeister, Marin Mersenne and Simon Stevin, the equal temperament system emerged.

Based on a single step ratio for the semitone derived from the 12th root of 2, equal temperament assembles larger intervals as multiples of the semitone. Unlike the harmonic series, which sacrifices the value of individual notes in favour of perfect harmonic consonance, equal temperament sacrifices harmonic perfection to achieve a fixed value for all twelve notes in the scale. It succeeds in this aim because the amounts by which the intervals differ from the harmonic intervals are very small and usually within the threshold of perception of most people.

The solution - equal temperament

The real reason guitars don't sound in tune and why it's not really a problem

In order to properly understand this it's necessary to take a short detour into the physics of vibration.

Ultimately, All sound is vibration of a fluid medium. In our case, as land dwelling mammals, the medium is usually air; although occasionally it can be water and rarely it is a mixture of other gases. There are a number of different other vibrating media that can be employed to generate these airborne vibrations but because this article is aimed at guitarists I'll make reference to the physics of vibrating strings as this also provides the most convenient way to visualise some of the more complex ideas.

The term resonance is an overused term and, like it's stablemate phase, is often used incorrectly and out of context.

In the context of acoustic science resonance refers to a process in which an object is provoked to vibrate continuously by means of a mechanical stimulus to the point where the vibration becomes self-sustaining for a period of time. All vibrating systems have a frequency at which vibrations sustain most easily and this is known as the system's natural frequency. In electronic systems this is also sometimes called the resonant peak.

The building block of resonance is the phenomenon called a standing wave. In an unrestricted medium repeating waves move freely outward from their source but when they encounter a reflective surface and waves are reflected back along their path of origin the waveforms can produce an interference effect where movement of the medium in one direction either cancels or reinforces the movement associated with the opposing wave.

When the point of reflection and the point of origin of the the wave are separated by a distance which is a multiple of the wavelength of the sound then these cancellation and reinforcement points will always occur in the same place and a standing wave results.

A vibrating guitar string is an example of one such standing wave; when it is plucked the tension in the string - which causes the string to occupy the shortest length possible between the two tether points of bridge and nut - forces the string to straighten out, driving a pulse towards each end of the string. When these two pulses are reflected back as a result of the string's momentum and inertia they cause a standing wave to be created.

A standing wave in a guitar string will occur at a specific frequency defined by the tension of the string, it's linear density (gauge) and length. The lowest frequency at which this can develop, the system's "natural" frequency, is called the string's fundamental frequency because a standing wave can develop not at one frequency but many, all of which are exact multiples of the fundamental and these frequencies are called the harmonics.

Illustration of a standing wave

The Feiten System attracted my attention in the late nineties as a direct result of the extraordinary claims made by it's inventor.

The system was unequivocally being touted as a solution to the discrepancy which exists between the natural harmonic intervals and the equal temperament ones. Since any kind of accommodation other than equal temperament is impossible for the guitar it seemed to be offering something that was patently impossible; as impossible for example, as claiming to have discovered a new value for pi, the ratio of a circle's diameter to its circumference.

The problem I have with the Feiten system, after some considerable research into those parts of it that have been released in the public domain, is that it has been based on inadequate and sometimes faulty reasoning.

There are many errors of history and comprehension, some unverifiable assumptions and the entire system appears to have been built on the presumption of these as accepted fact.

Such is the complex of confusion surrounding the system's rationale that it 's actually very difficult to deconstruct and examine its flaws and even harder to explain them to anyone with a lack of in depth knowledge of physics, mathematics and the science of musicology. What follows is a series of quotes drawn from Buzz Feiten's original website (now unfortunately taken down) and from his patent application along with my comments on these which I hope will serve to illustrate my concerns over it.

moving on

The Feiten System

© Eltham Jones, EDGE Guitar Services

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