Tuning, Composition & Structure

Who tunes the instruments guides the future of music
Glenn Gann

To be musical means to have an ear in the musical sense, not in the natural sense. A musical ear must have assimilated the tempered scale. And a singer who produces natural pitches is unmusical, just as someone who acts "natural" on the street may be immoral
Arnold Schoenberg

Introduction

A quick overview of today's musical scene would give as the impression that tuning and equal temperament are synonyms. For most musicians and today's general public the problematics in regards to tuning doesn't exist. They are not present in today's music education or scene. Equal temperament is held dogmatically as if it was a principle of sound itself and not a tuning system that has been recently developed.
The sense that all tones are equal is the novelty and the reason why equal temperament is such a successful tuning system. The chance of modulating, transposing and using the twelve tones in the chromatic scale having register as the only variable, is a possibility that only equal temperament can offer perfectly. This perfection is obtained through equalling the minimum distance (the half tone) in the twelve-tone scale in all its cases. This accomplishes that the scale will have only one kind of half step and this automatically fulfills the same for all the other steps of the scale. All tones have now the same universal quality and are differentiated solely by register. If we think of tones as colours, then equal temperament converted the chromatic scale -by definition, the scale of colours- into the scale of one colour. Or since there is no other colour, it converted the chromatic scale simply into the scale of no colour. As of today's oblivion with the origins of this temperament, we have taken these "colourless" tones as universals, and in doing so we have also admitted universally valid the ways it builds our compositions and perception of musical structure.
Through a short survey of historical tunings and the ideological struggle they entail, the purpose of this paper is to show where has equal temperament taken us in terms of how we build our our understanding of musi. The paper is structured in two main parts. The first part will give us a general view of historical tunings and how they overlap with ideology. This will give us a demonstration of the importance tuning has had in the past and how it has played a fundamental role in the way music has been done in different periods. The second part will focus on the influence of equal temperament and how this influence has given the mistaken impression of a scientific validity that comes prior to artistic decisions.


Short survey of the history of tuning
and the musical ideologies they convey

Tuning theory in the West runs back to Pythagoras and the correlation he established between consonances and his philosophy of the nature of the universe. The link between both is the underlying principle of number. "For Pythagoras, the fundament of the universe (...) was something obvious, yet easy to miss; lacking body, yet clearly defined; beyond sense, but a fulfilment of all man's perceptions. Through it, the hidden structure of the world became transparent. Pythagoras's primordial substance was number." Number symbolized an indissoluble, transcendent and universal unity. Thus human beings should try to become like numbers "as if to become less like (differentiated) human beings and more like (transcendent) numbers."
Pythagoras universe is built in harmony through mathematics, yet as it is a moving universe music consonances acts as the principle for the movement of the universe, that enables the universe as order, that is, as Cosmos. Through experiments with chords of different lengths, Pythagoras is able, from a theoretical standpoint, to determine which sounds are more pleasing to the ear. "As Pythagoras was about to encounter death, he would encourage his disciples the use of the monochord as he would trust only the intellectual and not sensorial appreciation." Through making chords vibrate at the same time, he found out that the chords that had their lengths in simple mathematical proportions sounded good together. Through this discovery, according to Pythagoras the major consonance one could find is that of two chords vibrating in the relation 2:1 (octave). In his system the other consonances are fifths (3:2), fourths (4:3), double-octave (4:1) and octave plus a fifth (3:1). "For the Pythagoreans, though, the importance of these special proportions went well beyond music. They were signs of the natural order, like the laws governing triangles; music's rules were simply the geometry governing things in motion: not only vibrating strings but also celestial bodies and the soul" .
These consonances were the only acceptable ones. "The numbers discovered in the proportions of the consonances belong to the tetractys of the decad (the sum of the first numbers is ten; 1+2+3+4=10) which for Pythagoreans is a sacred concept of their mystical-numeric speculation." The consonant intervals correspond to all the combinations of these four numbers.
This principle brought to life Pythagorean Tuning, which is constructed starting from perfect fifths and it also gave birth to the "music of the spheres". Theory which had a big impact in medieval times, where the word "Cosmos" is replaced by "God" claiming the same consonant intervals now as divine.

Pythagorean tuning gave the framework for a music in which consonant and dissonant intervals were justified rationally according to physical properties of sound, which were thought of as containing the subjacent order of the universe. This tuning was influential in a form of abstract music in which it was appreciated as the harmony of the universe. It was also responsible for the musical practice of the Middle Ages and the ways in which they constructed their music and treated different intervals.
Octaves, fifths and fourths became through cosmic speculation the only consonant intervals considered. Yet, as time went on passing, in folk music and in the work of some English composers, another interval was being held as a consonant interval in its own right: thirds and their inversion sixths. This intervals did not relate to the perfect proportions of Pythagoras, they simply sounded good to people. In this way new tuning systems, which made use of perfect thirds, were necessary. Music ought to be now human and not divine expression.

Pythagoras's universe of simple numbers avoided dealing with a problem that made its foundations collapse: irrational numbers. Instead of symbolizing Cosmos, this numbers symbolized Chaos. They were encounter for the first time inside Pythagoras's most famous discovery: his theorem. Used for calculating the length of any side of a right triangle, it would sometimes bare as a result an infinite number. Far from showing a universe of order and simple mathematical relationships, this revealed an infinite and irrational universe. As an example, this is the case when the length of one of the sides appears as the square root of number two.
This problem arises in his tuning system through the progression of the circle of fifths. As Isacoff puts it, "this progression of fifths move along a great musical circle before eventually reaching a tone with the same name again as the one with which it began. At the end of this experiment, Pythagoras believed he would arrive at the same final do (...) The fact is, octaves and fifths, when created with Pythagoras's pure mathematical ratios, are incommensurate: The further they move away from a common starting point, the more the structures built from these "perfect" intervals diverge."
This was a condition that open the for the use of perfect thirds in tuning. Explorations arose as to make a system that would make compatible perfect thirds and fifths. Not being an easy task, it brought a lot of trouble and it eventually changed the way of understanding tuning and music.
As tuning systems were tried out it became clear that the irregularities between the two different perspectives were not always compatible. For instance, "the interval between fa and la created by means of Pythagorean tuning -that is, through a series of perfect fifths (3:2)- will be slightly wider than the (created through a) "pure" third (5:4) " As a result, a clash between simple proportions, nature and rationality on one side was held against sensuality and taste.
"The mathematics of ideal beauty inherited from Pythagoras was no longer viable. Thirds played in the Pythagorean tuning -that is, the tuning created through a series of perfect fifths- were grating and often unusable (and for that reason long considered dissonances to be avoided). But tuning a keyboard to produce only pure thirds was equally untenable -it created a system at war with itself (sic), unable to render the all-important musical sounds of perfect octaves and fifths. "
At this point in Isacoff's text (as well as in many others) we find him cataloguing certain aspects of a tuning system like "errors" that made it inevitable to continue developong new systems, creating the idea of tuning as a technology that is in progress towards a goal (equal temperament). Tuning is a matter of artistic purpose and it is in itself a part of music creation, whatever that means in a specific time and place. No tuning system is at war with itself; it is the composer's business as a conscious artist to be able to take advantage of it. Is not a pure third tuning system's fault if a composer wrote an interval thinking it was consonant and it did not turn out that way. It is the same logic as blaming a violin for not being able to play a note that a composer writes outside the violin's register. Science and technology are tools used in the process of building tuning systems, nevertheless, they cannot be confused with it. Tuning is inherent to musical language, therefore it is art. And art, as John Blacking writes, does not progress: it changes its emphasis:
"From a distance, the forms, techniques, and building materials of music may seem cumulative, like a technological tradition. But music is not a branch of technology, though it is affected by technological developments. It is more like philosophy, which may also give a superficial impression of being evolutionary. Each apparently new idea in music, like a new idea in philosophy, does not really grow out of previously expressed ideas, though it may well be limited by them. It is a new emphasis which grows out of a composer’s experience of his environment, a realization of certain aspects of the experiences common to all human beings which seem to him to be particularly relevant in the light of contemporary events and personal experiences."
This is a key concept one must keep in mind as we continue our survey of historical tunings.
Going back to Pythagoras, we will remember that the major consonant interval he identifies is the octave. As the XV century was ending, a new approach to tuning was in its dawn. It was develop, among others, by the Spanish theorist Ramos de Parejas. In this system, the first interval to be constructed is the octave, and then the other intervals are filled using pure thirds or fifths. The result of this was the possibility of mixing together intervals that otherwise would be incompatible. Although new to the West, this kind of tuning called Just Intonation was already well known in other musical cultures such as India.

Pareja's tuning system makes it possible to combine different pure thirds and fifths, maintaining a big amount of irregularity between its intervals. "When all the proportions are calculated, it turns out that the distance between do and re, for example, is not the same as that between re and mi. This same inconsistency pops up when measuring the distance between fa and sol against the span between sol and la." As it was made of pure intervals, just intonation presents itself as nature's tuning and at the same time as quite unstable, since "if a experimented singer without any accompaniment tries to sing all the intervals in just intonation, she will face the dilemma of having to sing some intervals out of tune to maintain the same tonal axis, or to sing all the intervals in just intonation and accept tonal instability (...) Just intonation might be natural and desirable, however it is unpractical. There seems to be a discrepancy between art and nature. At the end of XVI century, this opened a bitter debate between Zarlino and V. Galilei."
This debate is typical of renaissance's tuning, and it accounts for an epoch trying to focus in human interests and at the same time trying to understand nature. As to musical composition and comprehension in this period, we will see a flourishing of a repertory, emancipated from liturgy and the contemplation of the divine. Medieval clarity and simple poliphony were replaced with counterpoint and its multiple simultaneous lines that challenged the comprehension of individual ones.
A new approach was developed during the XVI and the XVII century, which expanded in hundreds of versions. As a novelty, tuning was not any longer about keeping intervals pure, modifications to their nature was accepted as a way to make them fit a more homogeneous system. This technique is known as temperament.
Meantone temperament offered a new way of tuning built through the circle of fifths, reducing in a small fraction some fifths in order to make room for thirds. A bigger uniformity between intervals than the one just intonation offered was obtained, irregularities in the system were now found mainly in half steps.
Meantone temperament was a huge success between musicians and it unleashed a multiplicity of different tunings in which every composer would create his own variety and even create temperaments for specific works. Through these centuries, the clash between music as human or divine expression, benefits the first one. Giving way to a baroque music in which the physical principles of tunings were less relevant.
This humanistic music is guided by the expression of affection. Tuning and musical composition in this period will be used accordingly. In this way, musical appreciation will take place not only as an intellectual enterprise, but it will definitively give room to sensuality. As the baroque period starts fading and the classical period starts to unfold, composers start to explore affections in a more psychological way. Whereas in baroque music each piece will depict one affection, in classical music different affections will be confronted in the same piece. This will make the use of modulation a much more necessary tool, therefore for the XVIII century, meantone temperament gives way to well temperament. These are specific meantone temperaments built bearing in mind the possibility of using all the keys.
Early meantone temperaments would result in instruments having to be tuned for every different piece, since each tuning was built around one fundamental tone. This would prevent from many modulations inside one piece if the same quality of intervals would be desirable in different tones.
An interesting case of modulation in an early stage was the one presented by composer Adrian Willaert, whom Isacoff tells, in his 1519 piece Quid non Ebrietas did an intensively modulating music. This piece uses each of the twelve tones available in the scale as the central tone in different moments of the piece. Isacoff sees this piece as destined to failure in any tuning of the period other than equal temperament. Yet, since it is a vocal piece, the singers are not attached to a fix system (as a keyboard is) and they could accommodate their tuning according to the modulations of the piece. Quid non Ebriatas works as an interesting piece of "mobile temperament".
Modulating to distant keys arises as a problem in meantone temperament only in instruments of fix tuning. The difference in the quality of the intervals gave as a result that some intervals would sound dissonant since their proportion was too augmented or diminished from its pure proportion. Composers avoided these intervals and they are known as the wolf intervals. Most commonly, in meantone temperament there would be a specific fifth that would be known as the diminished sixth since its deviated nature. Wolf intervals made it impossible for some tones to function as keys inside the same tuning; this forced to change tuning between pieces with different keys. Surely this was a major practical inconvenience, yet not necessarily an aesthetical one.
At the end of the XVII century, wolf intervals are solved with the appearance of well temperament. The objective of this irregular temperaments that develop during the XVIII and XIX centuries is explained by Goldaráz Gaínza as follows: "The circle of fifths is closed in a irregular manner; all notes are usable but the objective is to make purer consonances in the most common tonalities ("natural") and more deviated in the lesser ones ("transposed"). In this way, the listener receives different impressions from intervals according to their differences. The composer ought to make use of these differences just as Rameau wanted in 1726. Each tonality is different in its quality; changing key does not only bring a change in pitch as in equal temperament but a complete change in sonority destined to exploit the different emotional affections (...)"



Andreas Werckmeister (1645 - 1706) was an organist and music theorist who developed many such tunings. Amongst his most famous ones is Werckmeister III. This temperament was developed working from the Pythagorean comma. "The black notes, the furthest members of the series of fifths and the most troublesome notes of mean tone temperament, are left in Pythagorean just intonation whilst the white notes are tempered."
Among with this tuning, another well-known well-tempered tuning is the one developed by Johan Philipp Kirnberger. In Die Kunst des reinen Satzes in der Musik from 1779 he proposes a temperament which Goldáraz Gaínza comments as "theoretically the most simple one: it divides the comma in two equal parts. He divides the syntonic comma instead of the Pythagorean one and he places it between the fifths G-D-A." For Goldaraz Gaínza this is not a good temperament since "it is clearly a very naive and bad temperament, coming from Ramos or Fogliano. The reduced fifths are diatonic and could as well have been D-A-E. The two cents in the schisma, which is the difference between Pythagorean and syntonic comma should appear in the wolf fifth (F# -Db in this case)"
For the modern mentality, well temperaments have a theoretical difficulty since they do not offer the systematic neutrality equal temperament has. This makes theoretical composition, that is compositions written directly on paper without being tested in an instrument, almost impossible. Irregularities in its intervallic composition would make it necessary for multiple tests of the intervals quality, through theory alone you could get a rough idea of how the piece sounds, but not a definitive idea.
With the exposure of well temperaments, we close the first part of this paper, since only equal temperament is calling for revision.

In this brief survey of the history of tuning we have been able to see how different ways of tuning have play an important role in defining music as a whole, and that, therefore, they frame tuning "inside" music as an art and not as a physical-technological problem that is "before" music as an art.
A baroque doctrine of affections is unattainable without an irregular temperament. Theological justification of consonances in medieval times would not be possible without Pythagorean tuning and its mathematical approach. Renaissance music could not have moved towards an appreciation of human expression if it worked through a tuning that did not use pure thirds.
Understanding tuning in this sense will lead us to show in the second part of this paper, how equal temperament is basically a XX century phenomenon that is intrinsically bound with relevant ways of composing and understanding music in that century.

Equal temperament and its
repercussions in composing and perceiving music

It seems that equal temperament tunings arise for the first time in the West in the XVI century. Frazer sees the beginning of this tuning in the way frets were placed in instruments like the lute. This is possibly true, yet Goldáraz Gaínza explains "fifths (in equal temperament) are very good (...) but thirds are very deviated (...) This last condition as well as the fact that as all semitones are equals, which impoverishes music expressiveness, is what delayed its use at least two centuries from its first appearances in the XVII century." Although it could be attainable from XVI century onwards in some instruments, it was not the tuning the musical practice of those periods would prefer.
In the year 1991 a book written by Owen Jorgensen called "Tuning: Containing The Perfection of Eighteenth-Century Temperament, The Lost Art of Nineteenth-Century Temperament and The Science of Equal Temperament", brought a little into life the forgotten debate about tuning. This was due to his thesis that equal temperament is a particularity of the XX century.
We will follow the debate this thesis opens up through the writings of two authors that participate in the discussion, one defending Jorgensen thesis and the other one claiming its wrong. The first author is Glen Gann and his review for the Village Voice in 1993, and the second author is Isacoff in the epilogue of his already cited book. Both authors coincide that the main point in Jorgensen's book is that equal temperament practically did not exist prior to the XX century; this is argued because it was possible to tune instruments to an equally tempered scale only in instruments that are not tuned by ear like the organ and fret instruments like the lute.
"Equal temperament came into existence only after 1887, when the modern method used by tuners to check the accuracy of their work (by listening to the even progression of beatings throughout the chromatic scale) was first developed. In addition, he claims that tunings with key coloration were preferred in any case by nearly everyone throughout the Baroque, Classical and early Romantic eras." For Glen Gann -following up with Jorgensen's idea- equal temperament for the piano comes into existence beginning from 1917 onwards with the appearance of W.B. White's Modern Piano Tuning.
Isacoff, simplifying, presents three arguments to get rid of these claims, not through a direct negation of them, but lessening their impact. In the first place, he argues that through a revision of German tuning guides from the XVIII and XIX centuries, we see that what Jorgensen calls well tempered tunings in some cases is really a semantic confusion and that they are varieties of equal-tempered tunings. His second argument has to do with ensemble practice where, for instance, a violin would have to adjust itself to the irregular tuning of instruments with a "built-in syntax" like the piano, which would be complex and, in many cases, impossible. As a third argument, Isacoff presents the idea of "psychology and "ideal instrument", following this argument composers like Bach or Chopin composed in a well temperament, but with an equal-temperament in mind.
The first of these arguments is also mentioned in Gann's article, and Jorgensen was well aware of the fact that tuners of the XIX century would refer to their temperaments as "equal" because all keys were usable, although, for him, they were using well temperaments. We are definitively in front of a semantic problem; nevertheless, this confusion is bound to benefit -so to say- well temperament over equal temperament. Well temperament can take equal's place since it can use all the keys, however, equal temperament cannot take well's place in the laters colourful irregularity. To claim for a variety of equal temperaments is a semantic problem for the simple reason that variety means multiplicity and therefore, irregularity. If we are to allow this condition in equal temperament, then we are only giving a different name to well temperament. The basic stance of equal temperament that "in reality, there are not twelve tonalities any longer, just one tonality 11 times transposable" , would be neglected.
The second argument is a practical one. Isacoff refers to the difficulties in ensemble music in well temperament, claiming that instruments without a fix tuning would have to be constantly taking care of maintaining tuning as they would be continually confronting deviated or intolerable intervals. This is indeed true for well temperaments; to keep in tune is a much difficult issue, yet to assume that this makes it undesirable is a different matter. Besides, equal temperament also challenges instruments without a fix tuning. Vibrato in the violin is an effect that makes tolerable equal temperament intervals that otherwise would sound excessively fixed and harsh. Even though fix instruments like the piano are be forced to play in equal temperament intervals, instrumentalist in not fixed instruments make use of small variations in notes allowing a tendency for "sharpness" or "flatness " to add colour and modulation to what otherwise would result in a dull and static tuning. When we get to ensemble practice, all these different variables have to be organized between different musicians, which results in practical problems in ensemble music for equal temperament as well.
The third argument makes a claim for psychology and ideal instrument. The psychological side is advocated through Bach, who is seen by Isacoff as transposing material without any worries as to key colour and doing intense modulations. This case, for Isacoff, means that Bach had accepted psychologically the use of equal temperament, even though if he only had well temperament to work with. It seems to me a fallacious argument to claim what was going on "psychologically" in a composer's mind that died almost three centuries ago. From the fact that Bach's music uses transpositions and modulations, would it be wrong to claim that what he "psychologically" wanted to show was how colours in different keys affects the same material?
Chopin is Isacoff's example to illustrate the case for an "ideal instrument". "Chopin at times intentionally obscured any sense of a key center at all -equal temperament was the frame of reference. (Without a firmly established key center, a piece in well temperament would simply be adrift in a chaos of shifting colour)" Wouldn't it be more chaotic to create tonal confusion when we do not even have colour to give as sense of where we are in the music? What's more, for me, it is not at all clear that Chopin wanted to keep a key center in many of his pieces.
This third argument makes an argument for the ideal and is summarized in Isacoff's claim that equal temperament influences music as a concept even though if in the practical sphere it did not exist. "(...) Equal temperament is unattainable. But to say that it therefore doesn't exist would be like declaring geometry neither real nor powerful, simply because one of its crucial elements, a "point," is by definition dimensionless" This is no doubt a very compelling argument, yet to claim that tuning is like geometry, not in the mystical-physical sense of Pythagoras or Kepler, amounts to an understanding of music as a theoretical activity that is only tangentially concerned with real sound. This is a very powerful stance in the XX century, that was not so in earlier centuries and, as we will show in what follows, is influenced by the importance equal temperament had in that century.
Let us stay with the conclusion that only in the twentieth century, equal temperament influence in music understanding and composition is broad and confirmable. ¿What kind of understanding has this tuning system lead us to have? This even system, in its colourless has freed intervals and harmonies from intrinsic tonal identity. Therefore, equal temperament is in itself an atonal frame for music composition. This has led Glenn Gann to comment, "unconsciously and in the long run, tuning determines what kind of harmonies composers write."
This atonal framework of equal temperament will fit in perfectly with the composition techniques and structural comprehension of music fostered by Schoenberg and spread throughout Europe since the times of Darmstadt School. Based on rational rigour and scientific ways of operation this understanding of music is inherently linked with equal temperament and its "objective" tones that lack any "subjective" colour. This aesthetics understanding of music has influenced not only musical composition, but also interpretation and musical critical thinking.
Theorist Theodor Adorno, who was able to conceptualize this aesthetics, played a major role. Rose Rosengard Subotnik sees Schoenberg-Adorno aesthetics as a way of valorising what she calls "structural listening". Music is seen by them as the mastery of nature by culture, and culture is valorised for its speculative and scientific capacities. "Schoenberg and Adorno define the formal parameter of music as an interconnectedness of structure that is both temporally established, and thus concrete, and also objectively determinable. Consequently, they define structural listening (...) as attentiveness to a concretely unfolding logic that can vouch for the value of the music." This understanding of musical discourse leads Adorno to determine the moral value in music "The more a musical structure approximates the self-contained intelligibility characteristic of logic, the more it can and does free itself from what Adorno sees as the deceptions or falsehoods invariably fostered through social ideology in order to maintain the power of existing institutions." In the same vein, Schoenberg in a spirit of self-discipline finds the most profound musical satisfaction through the "achievement of an intensely expressive structural integrity (which is "independent of style and flourish" and communicable at least to those whose "artistic and ethical culture is on a high level")." Therefore, leaving aside emotions and society, reason alone achieves the biggest musical satisfaction concentrating in musical discourse itself, as if it were a logic that develops independently of all "extra-musical" elements.
Through this formal parameters, music is thought of as acquiring a universal status and the ability to be understand by any rational being. "...Their concept, ultimately demands that musical form, through its uncompromising integrity and renunciation of sensuous distractions, contribute indirectly but concretely, as well as metaphorically, to the betterment of society. In effect, Schoenberg and Adorno offer structural listening as nothing less ambitious than a method for defining and assessing the moral soundness every relationship that bears on music."
Universality, logic and renunciation of sensuality; are key concepts that can also be found in equal temperament. With these concepts , musical composition should be an act of creating an autonomous object, freed from contingency and external conditions. The romantic idea of composition as inspiration is eradicated. Composers become now like engineers that work with music creation theoretically. Directly from their head to notation; analysis is made, formulas and techniques are developed for an structural understanding of music, that legalizes what "functions" inside a work. This structured way of working is directed to creating organic, rational, coherent and self-contained musical objects. Not working according to this parameters leads, this musicians think, to frivolous, amoral or irrelevant music.
Equal temperament helped developing this aesthetics. With this temperament, tuning is seen as a matter of scientific efficiency and not as an artistic resource that gives a special identity to tone relations. With equal temperament tones and intervals become abstract unities independent of particular pitches. Pitches are now accidental; the really important thing is intervallic proportion. Or so to say, the autonomous idea of music, liberated from the empirical and from contingency.
Equal temperament paves the way for composers to relate with their material through abstract and logical parameters alone. This tendency grew from its initial formulations with Schoenberg and Adorno and it spread to cover -generalizing- all of Europe and part of America, and it is to this day the leading ideology inside musical composition schools. Its influence, just like equal temperament did, spread to the understanding not only of twentieth century repertory, but also to music from other periods and the analysis of music in general. (So know we can see from where Isacoff's third argument comes from)
This discourse based in rational pureness and logical autonomy is, nevertheless, rooted in week concepts, hardly defendable through arguments of "pure reason". "The concept of structural listening imagines both composition and listening to be governed by a quasi-Kantian structure of reason that, by virtue of its universal validity, makes possible, at least ideally, the (presumed) ideological neutrality and, hence, something like the epistemological transparency of music. This assumption of a congruence between the underlying principles of composition and those of listening is what lens force to the metaphor of listening to the musical structure "from within"." If the ideal of Structural Listening is valid for music, it should then be for any other human field. It means an affirmation of the human possibility of epistemological transparency, logic universality, and ideological neutrality.
"This ideal of autonomy, I believe, is itself a fiction. In my judgement, Kant's efforts to preserve a universal basis for human communication have proven futile against the power of irreducible differences in perceptions, structuring principles, and values among differently situated individuals and cultures. The ideal amounts to a notion of abstract, ahistorical, meta-physical truth, that is, truth at a level that the relativistic and historicizing culture of the post-Kantian West has long acknowledge as inaccessible."

Far from successfully filling its goal, Structural Listening only accounts for the choices made by one particular musical culture, with a particular vision of things. It lacks universality and neutrality. Structural listening is a metaphor that overlooked its own construction, therefore, its arguments cannot be taken for what they assume. Reason alone is incapable of composing, auditioning, analyzing or interpreting music by itself in an ahistorical and universal way. Reason is intrinsically connected to emotions, personal experience, history, and culture of each human being. They all interact in the way a particular human being makes musical decisions (as well as all decisions).
Equal temperament obtained equality between tones in a search to decolorized the scale and present it in a pure and neutral fashion. Yet, the sounds it offers are not neutral, they amount to an artistic decision of atonality in the scale. Therefore, it presents a subjective sound structure and not a rational and systematic neutrality.
Equal temperament and structural listening are problematic to digest due to their pretensions of universality. Purposes of colour and taste are what guide the election of equal temperament. Not a "being outside" colour and taste in a scientific sphere that comes prior to artistic elections. The same is applied to structural listening.
Both of these musical concepts are useful resources that may be needed in some contexts. Structural listening is an interesting method and the particular colour of equal temperament is clearly necessary for some repertories. This is very different to claim that they are universal.
Scientific justification of artistic decisions and the values of universality and neutral epistemology that entails, undermine musical understanding. Science is a tool music continually uses; to use it should not be confused with having the ability to give universal validation to certain musical resources or ideas. Musical decisions should be always made with a mind/body that does not separate (that does not believe it can separate) its reason from its emotions, experience, culture, and contingency.
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Microtonal tunings that have been used in recent times is not -in principle- different to equal temperament. The only thing that changes is the minimum quantity from half of a tone to quarter, eight or other fraction of a tone. All quantities maintain the same proportion, therefore, they are still equal temperament.
Stuart Isacoff , Temperament, Vintage Books, 2001 pg.27
Idem. pg 31
"Arístides Quintiliano (De Música III, 2) nos dice que Pitágoras, al morir, recomendaba encarecidamente a sus discípulos el uso del monocordio al no fiarse de la apreciación sensorial sino únicamente de la intelectual". J. Javier, Afinación y temperamento en la música occidental, Alianza Editorial, Madrid 1992 pg. 16
Stuart Isacoff , Temperament, Vintage Books, 2001 pg. 38
"para los pitagóricos, los números descubiertos en las razones de las consonancias pertenecen a la tetractys de la década (la suma de los primeros números es diez (1+2+3+4=10), recinto sagrado de la especulación místico-numérica pitagórica." J. Javier, Afinación y temperamento en la música occidental, Alianza Editorial, Madrid 1992 pg. 18
I Stuart Isacoff , Temperament, Vintage Books, 2001pg. 40
Idem pg. 64
Ibid pg. 93
John Blackin, How musical is man?, University of Washington Press, 1973 pg. 72-73
Stuart Isacoff , Temperament, Vintage Books, 2001 pg100
"si un cantante experimentado y cantando sin acompañamiento intenta ejecutar todos los intervalos justos tiene que enfrentarse al dilema de, o bien desafinar determinados intervalos para mantenerse siempre en la misma altura tonal, o si todos los intervalos son justos, aceptar la inestabilidad en la altura tonal (...)
Si la afinación justa es la natural y la deseable, es sin embargo imposible de llevar a la práctica. Parece haber una grave discrepancia entre naturaleza y arte. Ello dio origen a una agria polémica a finales del siglo XVI entre Zarlino y V. Galilei." J. Javier, Afinación y temperamento en la música occidental, Alianza Editorial, Madrid 1992 pg.44-45
"El círculo de quintas queda cerrado de forma irregular; todas las notas son practicables pero el objetivo es producir consonancias más justas en las tonalidades más usuales ("naturales") y más desviadas en las menos ("transpuestas"). De esta forma, el oyente recibe diferentes impresiones de los intervalos de acuerdo a sus diferencias y el compositor saca partido de tales diferencias tal como quería Rameau en 1726. Cada tonalidad tiene una cualidad diferente; el cambio de tonalidad no constituye únicamente un cambio de altura como en el temperamento igual, sino un cambio de sonoridad destinado a explotar los diferentes "afectos" emocionales (...)" J. Javier, Afinación y temperamento en la música occidental, Alianza Editorial, Madrid 1992 pg. 128
Peter A. Frazer, http://www.midicode.com/tunings/temperament.shtml#5.8, 2001
J. Javier, Afinación y temperamento en la música occidental, Alianza Editorial, Madrid 1992 pg. 132
es teóricamente el más sencillo: dividir la comma en dos partes iguales. Sólo que Kirnberger divide la comma sintónica en lugar de la pitagórica y la reparte entre las quintas Sol-Re-La." Sin embargo este temperamento no es de la simpatía de Goldáraz Gaínza para quien "evidentemente se trata de un temperamento muy ingenuo y bastante malo, con antecedentes en Ramos o Fogliano. Las quintas reducidas son diatónicas y podrían haber sido también Re-La-Mi. Los 2 cents del schisma, diferencia entre la comma pitagórica y la sintónica deberán aparecer hacia la quinta del lobo (Fa# - Reb en este caso)."Ibid.
Peter A. Frazer, http://www.midicode.com/tunings/temperament.shtml#5.9, 2001
las quintas (del temperamento igual) son muy buenas (...), pero las terceras mayores están muy desviadas (...). Esta última característica, así como la igualdad de los semitones que empobrecían la expresividad musical, fue lo que hizo que se retrasase su aplicación general al menos dos siglos a partir de sus primeras formulaciones en el siglo XVI." J. Javier, Afinación y temperamento en la música occidental, Alianza Editorial, Madrid 1992 pg. 113
Stuart Isacoff , Temperament, Vintage Books, 2001 pg 241
Dominique Patier en Historia de la Música, Espasa, Madrid 1996 pg.477
Idem, pg 250
For example, see Rose Rosengard Subotnik's How could Chopin's A-Major Prelude Be Deconstructed? compiled in Deconstructive Variations. The debate she opens about what is primary and secondary in the light of the climax of this op.28 prelude should give, at least doubts, about the logical clarity of its tonal unfolding.
Stuart Isacoff , Temperament, Vintage Books, 2001 pg 242
Glenn Gann, E.T. Go Home en Music Downtown University ofCalifornia Press, Los Angeles, 2006. pg.187
Idem pg. 154
IbÍdem
Ídem pg. 155
Ídem pg. 156
Ídem pg. 157
Rose Rosengard Subotnik, Developing Variations, University of Minnesota Press, 1990. pg.266-267