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Thread: A brief history of Equal Temperament and the guitar's place in its development

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    Super Toneologist octavedoctor's Avatar
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    Default A brief history of Equal Temperament and the guitar's place in its development

    Another essay split into three to circumvent the post limit. Hope you find it interesting.
    Part 1
    While much has been written on the subject of equal temperament, most of these academic works have concentrated on the development of tempered tuning systems for the historic range of keyboard instruments. The influence of early fretted instruments has been neglected a little yet the early lutenists were probably the first musicians to encounter the seemingly insurmountable issues of tuning and temperament and were also some of the most important figures in the eventual development of a universal solution.

    The difficulties presented in producing a system of tuning with a range of fixed pitch values had been known since the days of the Babylonian Empire however it was Pythagoras, drawing upon the works of the sophisticated Babylonian mathematicians who first presented us with a mathematical model of the chromatic scale. Pythagoras however was not a musician, but the leader of a monotheistic religious movement whose precepts would later come to influence a number of world religions including Buddhism, Sufi Islam and Christianity. What Pythagoras was investigating was not music but the science of harmonics which he believe held the key to understanding the nature of the universe and ultimately the mind of God. Pythagoras expressed the pitch value of a note in terms of divisions of its length, although it would be another two thousand years before anyone made an accurate connection between tension and pitch.

    Greek music at the time of Pythagoras was centred around the major divisions of the octave, the diapason,the diatessaron and the diapente and was mostly played on wind instruments and multi-strung harp-like instruments such as the lyra and kithara, the latter of which bequeathed us the names for a host of instruments; zither, sitar, cittern, chitarrone, guitarra and ultimately, guitar.

    Fretted instruments were virtually unknown at this time in Europe (although it is believed that they were in use in India and Persia), however by the time of the Crusades traders, pilgrims and crusaders were returning from Asia Minor with examples of an Arabic instrument called the al 'ud. The name means, literally, the wood and soon European makers were copying its design, Al'ud became Lute, from which we get the word luthier which originally meant lute maker but which we now apply to any guitar maker. Its original pronunciation was, incidentally, lutier, with a "t" sound for the "th", and it was as readily applied to lute players as it was to lute makers as the distinction between the two was often blurred.

    The fact that we have a historical word for the maker of lutes (we do not have one for violin makers or harpsichord makers for example) is some indication of the level of respect reserved for this particular artisan.

    A practice that had been in use for some time in Europe involved tying loops of gut string around the neck of a fretted instrument to provide a division point for the string and define the note. Primarily found on folk instruments such as the rebec and the Celtic crwth (spelled crouth and crowd in England and the origin of the name Crowther) these frets initially did little more than divide the string into the major intervals of the fifth and the fourth, or to use the Greek terms, diapente and diatessaron. By about the late mediaeval age however lutenists were quite adventurous and the requirements of full polyphony could only be accommodated by a full chromatic scale. Such a scale existed, courtesy of Pythagoras, but things were not that simple...
    To be continued
    Last edited by octavedoctor; 02-19-2006 at 03:57 PM.
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    Super Toneologist octavedoctor's Avatar
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    Default Re: A brief history of Equal Temperament and the guitar's place in its development

    Part 2
    Anybody who thinks that the idea of fretting a lute was an easy option is not thinking things through from the perspective of the early musician. These days we take the constant ratio of the guitar's progressively contracting fret pattern for granted, but how many of us could generate it from scratch if required to?

    For the musicians of the Renaissance the process of determining the position of each fret was frustrating and complicated. The integer (whole numbers) maths which was all that they knew, in which a fraction could only be expressed as one number divided by another, was cumbersome and didn't lend itself easily to accurate measurement. Added to this was the problem that the step ratios which defined the intervals were known to be irrational; Pythagorean tuning required the use of a small interval called the Pythagorean comma to take up the gap between the two different sized semitones which resulted from attempts to generate a twelve note scale using simple harmonic ratios and it was hard to know what to do with this interval since it could not be incorporated into a fretting schema. The cyclic pattern of alternating fourths and fifths favoured by Pythagoras - up a fifth, a ratio of 3/2, then down a fourth, a ratio of 3/4 - allowed the scale to be completed and the fingerboard filled with a fret for every note, but the fret spacings were not even and while some notes at different points on the fingerboard were consonant, others were very dissonant.

    Nevertheless, there was no real alternative, so lutenists and luthiers struggled on as best they could devising their own notation, tablature, which allowed them to specify not just the note to be played but also the position on the fingerboard where the note was to be found, and part of the lutenists skill was learning where to place the frets in order to give the best consonance for a given piece. This was nothing as simple as tuning to a key, for while a single fret might span all courses and only be out on one note other frets might need to be in half a dozen different positions and this was impossible to achieve.

    Around the early part of the 16th century a Florentine musician and prominent lutenist, Gioseffo Zarlino, began to theorise that it might be possible to create a scale comprised of twelve semitones of equal step ratio. This was cutting edge stuff because even if they had the means to calculate such a ratio, they did not have the mathematical vocabulary to express the result. It must be remembered however that at this time even the intellectual elite were governed more by religion than reason and Zarlino was unable to bring himself to turn away from the apparent perfection of the natural harmonic series, reflecting, as was commonly believed, the perfection of God's design so ultimately he rejected the idea. Perhaps had they known more about the world they might not have taken this view since the geometric progression which ultimately led to our modern equal temperament is actually a far more common phenomenon in nature than the simple harmonic ratios that obsessed early musicians.

    One of Zarlino's students was Vincenzo Galilei (yes, this was the famous Galileo's father). Galilei was less conventional in his outlook and was very taken with the idea of a constant ratio scale and wouldn't give up on the idea, a pursuit which eventually led him into serious dispute with Zarlino when Galilei developed his "Rule of Eighteen". By reducing the string length in steps of 1/18th it was possible to create a twelve note chromatic scale, according to Galiliei. It's tempting to assume that Zarlino's response was as much inspired by jealousy that his former student had been able to solve the problem that had eluded him, but to do so is to deny Zarlino's own considerable contribution to the theory of temperament.
    To be continued
    Last edited by octavedoctor; 02-19-2006 at 04:06 PM.
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    Super Toneologist octavedoctor's Avatar
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    Default Re: A brief history of Equal Temperament and the guitar's place in its development

    Part 3
    Galilei's Rule of Eighteen was, we now know, slightly flawed; had he called it the rule of 16989737/18000000 he would have been almost spot on, however in the context of the time, the 17/18 rule was probably accurate enough. The lutes of the time had gut strings which had uncertain intonation anyway, gut frets which tended to move around a bit under finger pressure and no saddle; the string sounded from the tied knot at the bridge. From a modern perspective it was amazing they got a tune out of the things at all. The small error of just 0. 7% inherent in Galilei's calculations would hardly have been noticeable and would have possibly provided some measure of what we now call intonation compensation as under the rule of eighteen the 12th fret falls slightly on the fingerboard side of the half way mark. In common with most engineers of the renaissance and before, Galilei would have been very dependent on classic Euclidean geometry for a lot of his practical mathematical solutions and the simple geometric technique for calculating and positioning the frets (the reason the equation determining their placement is called a geometric progression) would have lent itself readily to practical adjustment.

    Many sources, including the illustrious Encyclopaedia Britannica, still credit the 18th century organ builder Andreas Werckmeister as the inventor of equal temperament. In fact Werckmeister produced many tempered tunings for organs but none of these were equal temperament as we know it today. Werckmeister described his systems as wohltemperirt, meaning "well tempered", a term which has been corrupted by many sources into the bastardised term "well temperament", a term which didn't actually exist at the time of Werckmeister who referred to his systems by their development numbers of I, II and III. I feel a better name for these systems would perhaps be "refined temperament", representing as they do, successive closer approximations to modern equal temperament.

    The person most commonly credited with deriving the exact value of the step ratio of the semitone is the French mathematician Marin Mersenne. In fact Mersenne's ratio was based on a complicated (and logically obscure) arithmetic manipulation of the difference between the tritave (an interval equal to the span of the fundamental and a frequency three times that of the fundamental, equivalent to an octave + a perf.5th) and the tritone (three intervals of a tone equal to an augmented 4th or diminished 5th and the harmonic mean of the octave). Mersenne's ratio, although more accurate than Galilei's, was still not the 12th root of 2 that we use today.

    In fact it was a Flemish mathematician called Simon Stevin, taking his cue from Galilei's work, who first proposed that the factor determining the semitone should properly be the 12th root of 2, and he was also the first european to calculate it by factorising the exponent (12) into two square roots and a cube root. Around the same time, in China, the prince Chu-Tsai-Yu had independently reached a similar conclusion. What made Stevin's work possible was his use of a decimal notation of his own devising which allowed him to express the results of his calculation in a uniformly recognisable format of a much higher resolution than is possible with fractional notation.

    Sadly, most of Stevin's worksheets are lost to us and his work on temperament was largely ignored by the musical community of the renaissance because, unlike other musicologists of the time he was purely a mathematician, not a musician, and treated the problem as a purely mathematical exercise. Contemporary musicians saw Stevin as having no understanding of their attachment to harmonic intervals, but he remains the progenitor of the modern equal temperament, constant ratio scale which we use today.
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    Stratologist Pierre's Avatar
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    Default Re: A brief history of Equal Temperament and the guitar's place in its development

    Wow. I actually read it all. I didn't get most of it, but it was interesting as hell and I'll be sure to read more about tunings sometimes soon... Thanks Eltham!

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    Super Toneologist octavedoctor's Avatar
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    Default Re: A brief history of Equal Temperament and the guitar's place in its development

    Quote Originally Posted by Pierre
    Wow. I actually read it all. I didn't get most of it, but it was interesting as hell and I'll be sure to read more about tunings sometimes soon... Thanks Eltham!
    Hot Damn that was quick! I've only just finished editing the thing...

    I'm off to bed now, well, once i've finished outlining the presentation I've got to give on Saturday...
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    seafoamer
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    Default Re: A brief history of Equal Temperament and the guitar's place in its development

    part of me wants to read that, but it ain't gonna happen.

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    Super Toneologist octavedoctor's Avatar
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    Default Re: A brief history of Equal Temperament and the guitar's place in its development

    Guess you'll have to wait for the movie then
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    Default Re: A brief history of Equal Temperament and the guitar's place in its development

    I read it and understood it. Outstanding essay OD. Thanks for posting that. (And, I assume you're the author too, correct?)

    Artie
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    Default Re: A brief history of Equal Temperament and the guitar's place in its development

    Very cool. I'm glad we're not doing it the old way anymore.
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    Default Re: A brief history of Equal Temperament and the guitar's place in its development

    Superb- I really appreciate this kind of analysis- How about going on to the underlying beats and expalin why the wolf tone sounds so bad- I never heard a good explanation-

    Also can you point us to other sites with more on tunning- Always have been facinated by piano beat tunning and atlerhantive tunings and would like to understand more
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    Super Toneologist octavedoctor's Avatar
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    Default Re: A brief history of Equal Temperament and the guitar's place in its development

    Quote Originally Posted by ArtieToo
    I read it and understood it. Outstanding essay OD. Thanks for posting that. (And, I assume you're the author too, correct?)

    Artie
    Yes thanks, but i've drawn extensively from academic resources
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    Super Toneologist octavedoctor's Avatar
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    Default Re: A brief history of Equal Temperament and the guitar's place in its development

    Quote Originally Posted by zionstrat
    Superb- I really appreciate this kind of analysis- How about going on to the underlying beats and expalin why the wolf tone sounds so bad- I never heard a good explanation-

    Also can you point us to other sites with more on tunning- Always have been facinated by piano beat tunning and atlerhantive tunings and would like to understand more
    Peter Fraser's www.midicode.com is one of the best i've seen.

    Unfortunately the internet is awash with dodgy information peddling opinion dressed as fact. I found one really interesting (i thought) site on equal temperament which turned out to be a fundamentalist Christian group peddling a right-wing fascist and anti-islamic philosophy

    You'd be better off at your local library, that's where i learned everthing; we didn't have the internet in the olden days

    Check out the other thread here, I'm touching on that now, beat frequencies, difference tones etc. ...

    http://seymourduncan.com/forum/newre...reply&p=792161
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    Default Re: A brief history of Equal Temperament and the guitar's place in its development

    Superb- Will check it out- Thanks!
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    Default Re: A brief history of Equal Temperament and the guitar's place in its development

    Quote Originally Posted by seafoamer
    part of me wants to read that, but it ain't gonna happen.
    You really are missing something Foamy. That's impressive.
    It is always amazing to be told about what lies behind all our so familiar modern technologies.

    Octavedoctor, is it some kind of research you did for uni or for work?

    I am not sure I will look my axes the same way now...

    EDIT: I alsmost forgot: Of course, thanks a lot for this smashing lecture.i quite liked the other thread as well. Will actually pdf them and store this on my hrad drive... No I won't sell it with my name on it !!!
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    seafoamer
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    Default Re: A brief history of Equal Temperament and the guitar's place in its development

    Quote Originally Posted by octavedoctor
    Guess you'll have to wait for the movie then
    I'm not really into movies unless it's a pr0n0

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    seafoamer
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    Default Re: A brief history of Equal Temperament and the guitar's place in its development

    Quote Originally Posted by Jazz Rock
    You really are missing something Foamy.
    I'm sure it's great & all, along with all the classes I took on it when I went to Uni that bored the farts outta me.

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    Super Toneologist octavedoctor's Avatar
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    Default Re: A brief history of Equal Temperament and the guitar's place in its development

    Quote Originally Posted by Jazz Rock
    Octavedoctor, is it some kind of research you did for uni or for work?
    Well...

    My (incomplete) university studies were in psychology, which comes in useful when understand the psychology of perception, which has a strong bearing on how we hear music, but before that I was studying engineering and the rest of it has just been stuff i've accumulated over the years.

    I was interested in ancient instruments from my early teens. I was fascinated by the late David Munrow's writings and broadcast work and I'd always harboured the ambition of being a lute maker. In the end, the closest I came to that was flamenco guitars. As a student guitar maker i was fortunate to have Bernard Richardson as a teacher; Bernard is one of the UK's (and arguably the one of the worlds) principal researchers into the physics of the guitar. Although I have no particular qualifications in this myself, I learnt a lot from Bernards work and his published monographs.

    http://www.astro.cf.ac.uk/cgi-bin/st...hardsonBernard

    Much of my understanding of equal temperament was born out of necessity. As the resident guitar tech at a large distributors I found myself having to explain to players on a regular basis why their guitars didn't sound quite in tune. I found that there was a tremendous lack of knowledge of stuff that I felt should be first principles; the equivalent of driving a car without having the first idea of how it worked. I also found that there were very few sourceworks which successfully explained these principles, so it's become something of a mission for me to propagate understanding.

    I do try, as far as I can, to put this in language that ordinary people without degrees in maths and physics can understand but I don't feel that i succeed in this very often. There is only so far you can distill information before it becomes misleading and you are always treading a fine line between being obscure an being patronising.
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    Morkporkian Vasshu the humanoid typhoon's Avatar
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    Default Re: A brief history of Equal Temperament and the guitar's place in its development

    Very good reading.
    I enjoyed it, thanks for posting it.
    Niels
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    Super Toneologist octavedoctor's Avatar
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    Default Re: A brief history of Equal Temperament and the guitar's place in its development

    Quote Originally Posted by Rid
    Very good reading.
    I enjoyed it, thanks for posting it.
    Niels
    Mange tak, Niels
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