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What this statement suggests is that Buzz has never really understood how to tune a guitar properly. In common with many guitarists who don’t understand the nature of harmony, where it comes from and how we perceive it, Buzz had been attempting to tune his guitar so that all individual chords sound correct only to discover that this distorts the intervals comprising other chord shapes, a simple error but one that is very common in guitarists with no formal musical education but who nevertheless have a sensitive ear for harmony. Such mistakes stem from a lack of awareness that the only way of achieving consonance on the guitar is to tune it so that each individual note at all points on the fingerboard are the same or octave multiples of each other.

A method of doing this, published by the Guild of American Luthiers can be found here: Guild of American Luthiers tuning protocol

You play one chord shape, like A, and it plays in tune, but when you play another chord shape, like D, it's not quite in tune; close interval chords often sound wildly out-of-tune; and no matter how carefully you tune your guitar, some chords still sound only slightly better than fingernails on a chalkboard”

“For centuries guitars have been designed and built using unchanging formulas for nut placement and intonation, and for centuries intonation problems have been accepted as an inevitable deficiency of the instrument. But incorrect nut placement and improper intonation methods directly influence intonation."

Here Buzz has misidentified the problems he has experienced in tuning the guitar as a design or technical deficiency in the instrument, rather than the true cause; the inability of the natural harmonic series (which are the only intervals our brains will allow us to recognise as “correct’) to resolve to the octave.

"In 'even tempered' tuning, notes are tuned relative to the root of a scale. So an instrument set up for, say, the key of C might sound acceptable in G, but downright awful in E or B. To play in those keys, you'd have to retune from scratch."

Although this statement is largely correct He’s clearly very confused about what it relates to. He seems to think that Just Intonation, where tuning in one key produces dissonances in others, is equal temperament!

Or "even temperament", whatever that may be...

"First, some history. Fretted instruments have traditionally been tuned to "equal temperament," a system based on a formula going all the way back to Pythagoras. Problem is, some keys are created more "equal" than others. ”

Here, Buzz’s grasp of history seems as shaky as his understanding of music.

He attributes the development of Equal Temperament to Pythagoras, but there is nothing “equal” about the Pythagorean scale; Pythagoras devised the “Pythagorean comma” - a tiny interval which could be added to other intervals in the scale - to resolve the discrepancies in his scale which had several unequal intervals and a “greater” and “lesser” semitone.

The various pre-equal temperament systems had no analog in the world of fretted instruments which is why the lute was at the forefront of developing systems of tuning which sought to close the gap between the aesthetics of music and the practicalities of musical instrument construction.

Fretted instruments have used a system of “Constant Ratio” proportional fretting since the 1600s but this was not Equal Temperament as we know it today and was not universally adopted until the later Baroque era and the development of fixed fretting techniques. Many early instruments used frets formed of loops of gut tied around the fingerboard and pushed into place to tune by ear and many lutenists and viol players developed their own favoured rules for fret positioning, many of which were nowhere near Galilei’s constant ratio “rule of 18”

The Orwellian reference to some keys being created more “equal” than others is meaningless.

"The historical method of intonating each string to itself results in "equal temperament" - an intonation formula abandoned by piano tuners in the 1600's."

This statement is simply not true, and highlight's Buzz’s apparent lack of understanding and knowledge of equal temperament. Not only was Equal Temperament never abandoned by piano tuners, it wasn’t actually fully adopted until Broadwood began using it it on all their pianos in the late 1800s. In fact, the piano as an instrument didn’t even exist in 1600, so there were no piano tuners! Keyboards did exist and at the time were tuned to a variety of temperaments including Mean Tone and Well Tempered, both of which were practical steps on the road to equal temperament, the evolution of which was dependent on the ability to calculate complex roots. It was the work of Simon Stevin in the late 16th century that allowed this to become possible.

Today all keyboard instruments except those specifically intended for the performance of early music are tuned to equal temperament by default.

"Piano tuners started recognizing this problem 400 years ago. Unlike the piano, the guitar traces its roots to the less formal demands of folk and popular music. It didn't emerge as an important instrument until the middle of the last century. Yet with starring role comes heavier scrutiny: Guitars might be better built than ever, but we expect more of them too. For one thing, we expect them to play in tune. A trait shared by good players is ability to adjust tuning on the fly or mask dissonance with techniques like vibrato or bending. But even good players still face tuning woes, especially when playing chords alongside a piano"

Buzz has a thing about pianos, which he appears to regard as the apotheosis of perfection. The problem with this is that the piano is an equal temperament instrument so on the one hand he is rejecting equal temperament while at the same time he is holding up the piano, which depends on the system he rejects, as a suitable reference. In fact the piano is a very unreliable reference as it is tuned subjectively and then continues to drift from the optimum tuning until the next time it is tuned so a piano is really only a suitable reference immediately after it has been expertly tuned and at no other time. What Buzz fails to understand is that the reason for guitarists' "tuning woes" stems not from their use of equal temperament, but their failure to understand and observe it when tuning their guitars.

And while the guitar did not emerge until the late nineteenth century in the form we know it now its predecessors, the lute, viol and the vihuela were all important fretted instruments which were at the forefront of the development of tempered tunings. Both lute and viol were part of the armoury of the court musician and there was nothing folky or informal about the music played on them.

And there were no piano tuners 400 years ago , because there were no pianos.

"Buzz Feiten was experiencing operational difficulties of this nature when he decided he was mad as hell and wasn't gonna take it anymore. "The guitar was set up perfectly, I was doing everything right, but still, the first three frets were terribly sharp," ....Curious, he started checking his other instruments on an accurate tuner - he found them consistently sharp near the headstock, and flat above the 12th fret. "That meant that the nut was in the wrong place, and has been for over 400 years," he says."

This is one of the more peculiar claims Buzz makes without offering any verifiable evidence in support. The only thing that could cause this is if the nut is too high. I work on around six hundred guitars a year and I’ve never heard one in which the first three frets sound sharp if it had been properly built and set up. In fact, the reverse is true. The fixed value for compensation at the 12th fret tends to result in the first fret receiving more compensation than it needs, resulting in a slight flattening of the note. The same thing causes the flattening above the 12th fret (see my explanation here) The only explanation for this is either that Buzz's guitar wasn't properly set up or that he was mistaking the sharpness of the A-flat in the E major chord - which is normal in a guitar properly tuned to equal temperament - for an error in the placement of the first fret. Compensating nut systems such as Earvana (based on the same principles employed by Stephen Delft are designed to overcome this specific problem but cannot have any effect on the sound of chords in any position other than the first position, so they are really only suitable for strummers who never play more than three open chords. To achieve the same effect on barred chords with a capo would require dividing the fret up into six sections and moving them around each time you changed chords...

There are guitars that attempt to do this but because their makers also do not seem to understand the principles of equal temperament it seems to me they are up the same technological blind alley since such guitars could be designed only to play in one key.

A more sensible approach to guitar intonation, based on sound thinking without overtones of commercial exploitation can be found here:

Improvement of Intonation and Playability of Guitar Fingerboards

"If you had a great-sounding barred A, your barred D sounded awful. There was no tuning around the problem."

“Rather than tune each string to its accurate pitch, he borrowed a trick from piano tuners and tuned subjectively, getting the instrument to sound pleasant.”

"I started doing what I called ‘intonation modeling,’ modeling intonation schemes, and I documented what I was doing. I discovered that there is a window that exists in terms of intervals before they start to sound out of tune."

Again, this points to Buzz chasing the natural harmonic scale on an instrument that can’t resolve it.

Now, this sounds to me as though Buzz is actually starting to tune his guitar to equal temperament almost by trial and error, by reference to the piano.

This “window” Feiten refers to is called the “difference limen”. It is this which equal temperament exploits by tweaking every interval slightly. The largest discrepancy in equal temperament is between the harmonic scale’s minor third and equal temperament’s minor third. The difference in pitch is just under 1% (approximately 15.5 cents). The difference between the fourths is just over a tenth of that. It is beginning to sound as though Feiten is re-inventing Equal Temperament from the perspective of someone who doesn’t realise that it’s already been done!

"A perfect fourth or fifth tends to be beatless, so you have a little area in which to play around. The same with octaves. I’ve narrowed the fourths and fifths slightly in order to improve the thirds and sixths dramatically.”

What does he mean by this? A perfect fourth or fifth is beatless? Fourths and fifths have a beat rate that is in the audible range; in the case of the fifth it’s one-third of the frequency of the higher note and half the frequency of the lower note; the fourth has a beat rate a quarter of the higher note and a third the frequency of the lower note. In both cases they manifest as difference tones (link will open in new window) and it’s the difference tone’s modulation of the summed output of the two waveforms that give the interval its character. Only true unisons and octaves have a zero beat rate.

The claim to improve the thirds by narrowing the fourths and fifths is particularly bizarre because what he is suggesting is not only retrograde, but actually impossible.

The guitar is tuned in four steps of a fourth and one of a third but if we follow the whole number ratios which our ears tell us is “correct” we arrive at a top E which is noticeably flat relative to the low E. This is because the interval ratios of 4:3 for the fourths and 5:4 for the third, when multiplied together in five steps are too “short” to resolve into the two octave interval of 4:1 (the interval created,in fact, has a ratio of 320:81).

The guitarist often attempts to resolve the problem by retuning the top E only to find that the interval between the B and E has opened up and is now dissonant. Attempting to close the gap opens the major third between the B and G strings (this is why people often refer to difficulties in tuning these two strings). Tuning the guitar with natural harmonic fourths between the E,A,D and G, and the B and top E strings results in a stretched third between the G and B, which always sounds out of tune. Equal temperament stretches the fourths by a tiny amount reducing the stretching of the major third needed to close the gap and keep the integrity of the octave intervals. It’s fairly obvious from this that shortening the fourths would only result in the third being stretched more, especially since Feiten also offsets the the bass strings slightly, opening out the octave!

"He (Feiten) was not the first musician to be unhappy with prevailing intonation theory. In 1939 musical acoustician L.S. Lloyd wrote in Intervals, Scales, and Temperaments (St. Martin’s Press), “What [musicians] particularly dislike about equal temperament . . . is the false intonation of the thirds and sixths.” Unfortunately luthiers didn’t listen, maybe because tackling the series of harmonic relationships on a guitar, which are vastly more complex than on the one-string/one-note piano, seemed too daunting."

The first part of this statement is absolutely correct, and impossible to resolve, any more than you can make 2+2=5. The natural harmonic major third is represented by the ratio 4:5 and the harmonic minor third is the ratio 5:6. However neither of these intervals will resolve as multiples within an octave, so it’s impossible to create a scale of fixed pitch which incorporates correct thirds relative to all other notes in the scale. The equal temperament intervals are slightly expanded and compressed respectively. Musicians sensitive to dissonance have always expressed dissatisfaction with equal temperament while nevertheless recognising that it is the only practical solution to the the problems of creating a fixed pitch scale in a hostile harmonic environment.

What Buzz has failed to identify correctly is that the problem is one of music, harmony and perception rather than of guitar construction. All equal temperament instruments are affected by the discrepancy between the natural harmonic intervals which we strive to hear and the equal temperament intervals which sound slightly less than perfect to us. Equal temperament succeeds if its rules are respected, as pianists and other keyboard players do, and the instrument is tuned correctly. The problem for the guitarist is that they are saddled with the problem of tuning their instruments themselves and as most guitarists get by with a minimal musical education they lack the knowledge and understanding of what is happening when they follow the advice of their ears and everything then sounds wrong...

"The concept of tempered tuning (that is, tuning the note spacings so that the pitch relationships in different keys remain constant) isn't all that new. In fact, it's been a common practice applied to tuning instruments like the piano and harpsichord since the 1600s.

In relation to a piano, this means it's actually tuned a little bit sharp from middle C up, and tuned flat from middle C down in order to make the sounds more pleasing.

There are at least 12 or more ways to temper a piano, which vary according to the taste of the player, and different names for each method."

Here again, Feiten’s confusion is evident. He’s confusing Equal Temperament (the blue highlighted statement, which is also incorrect as it happens) with octave stretching (highlighted in red) and the earlier systems of tuning such as mean-tone and Just Intonation, which required retuning of keyboard instruments to suit different keys (highlighted in green). Equal temperament is the default standard tuning for all modern fixed pitch instruments including the guitar and piano, octave stretching is a practice applied only to pianos to counter inharmonicity (an explanation of this will follow) and Mean-tone and Just Intonation have been obsolete for centuries, being resurrected occasionally for authentic performances of music from the pre-classical eras.

One of the more interesting claims made by Feiten is to have followed the practice of “octave-stretching” used by piano tuners. He seems to have naively assumed that this practice is a departure from equal temperament but, yet again, he hasn’t done his homework on the subject.

Inharmonicity is a phenomenon which affects instruments whose primary source for the generation of sound is a vibrating string or bar. Find out about inharmonicity here

Piano strings operate at extreme tension and are of a very heavy gauge. The higher strings are behaving more like bars than strings, and pulling up the true pitch of the higher notes tricks the brain into assessing it as being close to the equal temperament value.

Far from being a departure from equal temperament, this form of tempering tuning is a way of compensating for the instrument's physical limitations so that the illusion of equal temperament pitch is created. The mistake Feiten is making in regard to this is in assuming that the same process needs to be applied to the guitar. In the guitar, inharmonicity is found on the higher frets and predominantly the lower strings, the G and low E being most affected, but a properly undertaken intonation procedure, done by ear, rather than with a tuner, incorporates compensation for any effects of inharmonicity by default. Errors only occur when too much reliance is placed on digital measurement.

"My partner, Greg Back, was taking a course at California State University at Northridge called 'The Physics Of Music.' Included in the course was a description of the model Pythagoras used to figure out the rule of 18, the formula that determines fret placement. That model implies that a string is under constant tension. However, a string is not under constant tension on a guitar neck, and he neglected to take in account that it's easier to press down the string in the middle than right next to the nut. That increase in tension directly translates to pitch, so notes will always be sharp at the first two frets because the model he used did not include that fact."

There are two errors here. He repeats his assertion that Pythagoras devised the rule of 18 and implies that he was working on the physics of fretted instruments when, in fact, fretted instruments didn’t exist. Pythagoras’s monochord was a device for investigating the harmonic relationships in a vibrating string and he used it to develop the Pythagorean scale which formed the foundation for subsequent development of the idea of harmony but which was quite different from the equal temperament scale.

The second error (highlighted in red) is a piece of complete nonsense displaying a massive ignorance of basic physics.

He’s correctly observed that the strings feel tighter closer to the nut and quite wrongly inferred from this that the tension is greater closer to the nut!

Tension is a constant load; it can’t be greater closer to the nut. The strings feel stiffer close to the nut because a displacement here subtends a steeper angle than at the middle of the string so more of the vector component of the string’s tension is directed against the guitarist’s finger. This means that the resistance to displacement can be greater here although the actual stretching effect may be less.

The increase in pitch is actually linked not to these subjective observations but only to the increase in strain applied to the string.

Although a given displacement close to the nut does result in a more rapid increase in strain than the same displacement towards the middle of the string this is actually an irrelevance because in a properly set up guitar the string presents a shallow angle to the fingerboard plane; as a result of this it always experiences more displacement when fretted at the octave fret than at the first fret where clearance is very small.

When a string is fretted it's forced into a longer path between nut and saddle, stretching it, and the percentage by which it's stretched is an indicator of strain. On a properly adjusted guitar the string clearance is always greater at the high frets than the low ones the so the string is always stretched more the higher up the fingerboard you go.

An average guitar has a string clearance at the first fret about five times less than at the twelfth and a brief calculation (courtesy of Pythagoras) gives us an estimate of the strain around a quarter to a fifth of that at the twelfth; the traditional method of intonating a guitar means that any displacement of the saddle to correct the intonation at the twelfth has about half the effect at the first fret, so the first fret is actually being over-compensated.

The spreadsheet shown at left uses data from two guitars that I recently set up to arrive at this conclusion.

Ockham’s Razor urges us that when multiple solutions to a problem present themselves one should select the simplest one and it seems to me that when Buzz discovered that his first fret was sounding sharp the simplest answer was that his guitar hadn’t been set up and tuned properly!

Click image to see spreadsheet

© Eltham Jones, EDGE Guitar Services

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