E Arc-En: Fill & Download for Free

Joyce unpacked "L'Arcs en His Cieling Flee Chinx on the Flur" in a letter (26Jul 1927):God's in his heaven All's Right with the WorldThe Rainbow is in the sky (arc-en-ciel) the Chinese (Chinks) live tranquilly on the Chinese meadowplane (China alone almost of the old continent[s] has no record of a Deluge. Flur in this sense is German. It suggests also Flut (flood) and Fluss (river) and could even be used poetically for the expanse of a waterflood. Flee = free)The ceiling of his [E] house is in ruins for you can see the birds flying and the floor is full of cracks which you had better avoidThere is merriment above (larks) why should there not be high jinks below stairs?The electric lamps of the gin palace are lit and the boss Roderick Rex is standing free drinks to all on the 'flure of the house'He is a bit gone in the upper storey, poor jink. Let him lie as he is (Shem, Ham and Japhet)The birds (doves and ravens) (cf the jinnies is a cooin her hair and the jinnies is a ravin her hair) he saved escape from his waterhouse and leave the zooless patriark alone.

BackgroundFor 3 Quarter Tone Piano Pieces instructions for tuning are provided in the score preface and read:The quarter-tone relationship between the two pianos may be accomplished by one of two methods. Ives suggests that one piano be tuned a quarter-tone higher than the other. This procedure is not looked upon with favor by piano technicians. They prefer to take one piano down a quarter-tone. If this latter procedure is adopted, the part written to be played by the quarter-tone higher piano should be played by the piano at normal pitch, while the part written for the normal pitch piano should be taken by the piano tuned down a quarter-tone.In the article Ives' Quarter-Tone Impressions, author Howard Boatwright suggests that the composer's sensibilities regarding temperament were likely influenced early on by his father George Ives' “quarter-tone machine.” This device consisted of twenty-four (or more) violin strings that could be tuned in various ways. The elder Ives used it to “pick out quarter-tone melodies” and “try to get his family to sing them.” Later on in an effort to better train the ear to hear quarter-tone melodies, George Ives developed a way to sustain the strings using weights and violin bows, allowing for quarter-tone harmonies. Nevertheless, it is melodic lines available with this non-traditional temperament that stuck with the young Charles Ives. The composer stated in his writings that after the novelty of the quarter-tone machine wore off it was used “as a means of punishment – though we got to like some of the tunes which kept close to the usual scale and had quarter-tone notes.”What is perhaps even more telling when it comes to what Charles Ives hoped to gain through the use of microtonality is his prediction for the future. In Joseph Yasser's A Theory of Evolving Tonality (1932), the Ives passage reads: “In some century to come, when school children will whistle popular tunes in quarter-tones – when the diatonic scale will be as obsolete as the pentatonic is now – then … boarderland experience may be both easily expressed and readily recognized.” Again, the horizontal intervallic opportunities are given the primary focus. The function, as expressed by Ives in the term “boarderland experience,” may lie in the increased expressivity through greater tonal ambiguity. Microtonality, as seen in 3 Quarter Tones Piano Pieces through the 24 EDO (equal divisions of the octave), is understood to be more nuanced than chromaticism due to the availability of smaller melodic gradations.If one were to continue on with Ives' progression of expressivity via smaller divisions of the octave, pentatonic to diatonic to chromatic to quarter-tone, Georg Friedrich Haas' Limited Approximations would represent two notches further along the continuum; as it utilizes 1/6th and 1/12th tones in addition to quarter-tones. In the score preface Haas outlines how the 72 EDO is to be achieved through the piano temperament, stating (translated into English):1st piano +1/6–Ton (raised a 1/6th tone)2nd piano +1/12–Ton (raised a 1/12th tone)3rd piano Normalstimmung (normal tuning)4th piano -1/12–Ton (lowered a 1/12th tone)5th piano -1/6–Ton (lowered a 1/6th tone)6th piano -1/4–Ton (lowered a quarter-tone)[Image viaDonaueschinger Musiktage entdecken das Streichquartett]The background to Limited Approximations takes a very different focus than Ives had with 3 Quarter Tone Pieces, emphasizing large scale gestures and (as the title implies) the approximation of overtones. Publisher Universal Edition has included statements from the composer regarding the composition on their website. Here Haas remarks that:limited approximations does not tell a story. As with all my compositions, there is also no formal development or traditional formal structure. Contrasting elements alternate with one another – moments of smoothness and friction. “Pseudo-glissandi” in the pianos arrive unexpectedly at overtone chords. Apparently stable constellations of intervals begin to falter as the twelfth-tones merge.The spectral, telescoping chords of the pianos are taken up by the orchestra, over and over again. In my early works I had to limit myself to a few basic tones, out of respect for what was practically realisable: in vain makes do with only the 12 tones of the traditionally tuned scale. Natures mortes uses only six different overtone chords, of which four are based on tones found in the traditionally tuned system. In limited approximations, thanks to the pianos, the whole world of sound is open to me.In his paper “Some Theoretical Attributes of 72-tone Equal Temperament and Their Realization in Georg Friedrich Haas' limited approximations,” Will Mason also details the significance of composer Ivan Wyschnegradsky's arc-en-ciel (1956) as a historical precursor in the six piano 72 EDO realm. Mason contextualizes Limited Approximations as, in part, a homage to Wyschnegradsky and mentions Haas' use of “Wyschnegradsky’s method of modulation in quarter-tone pitch space and his idiosyncratic use of a chord of stacked fourths and tritones.” Returning to the quotations via Universal Edition, Haas also cites the specific influence of Wyschnegradsky on Limited Approximations:At the end, a quotation from Ivan Vishnegradsky’s harmonies (against the relics of an overtone chord in the strings) – not with the clarity of his composition “arc-en-ciel” (I was able to première this work for 6 pianos tuned in twelfth-tone intervals in 1988) but in the clouding over of twelfth-tone chords gliding gently towards the heavens. Even this approximation is only a limited one.ComparisonsWyschnegradsky may then be the link connecting Ives' 3 Quarter Tone Pieces with Haas' Limited Approximations.In the score preface Ives stated: “Of the three short piano pieces that I've composed with the use of quarter-tones, the first and last were originally intended for a quarter-tone piano, two keyboards and for one player.” This proposed setup is clear through the music of each of these movements, however, not realized in typical performances; instead defaulting to the two players, two pianos arrangement (which is reflected in the published score with two grand staves). When this statement from the composer is paired with examination of the notation an argument could be made for Ives' conception of the quarter-tone piano, but settling for the sake of practicality.Excerpt from I. Largo. The phrase, though orchestrated across two pianos, would be playable as a solo if one performer had access to both keyboards.Ives, who was reportedly writing 3 Quarter Tone Pieces from 1923-24, may have been composing with such an instrument in mind. According to Boatwright:[The Czech composer, Alois Haba] wrote for a quarter-tone piano brought out by A. Forster in Prague in 1923; it had two keyboards, one tuned a quarter-tone higher than the other, Ives seems to have been familiar with a similar instrument in New York, as he mentions in his article a quarter-tone piano of this type by Dr. Stoehr, Mehlin & Sons.It is unclear whether Ives had any knowledge of Haba or Wyschnegradsky. The microtonally tuned two manual piano was also used later by Wyschnegradsky, however, his earlier quarter-tone piano works (for example Op.5 Quatre fragments from 1918 and Op.10Variations sur la note from 1918-1920) call for two instruments similar to Ives' pieces for the same reason-- compromise for practicality.Image of Alois Haba with a quarter-tone piano [Page on haba-quartett.de]Though the music clearly necessitates six performers, Haas is also approaching the microtonally tuned piano as one meta-instrument, but takes it further to create one resultant voice. Haas builds a feeling of the single mind generating gestures through the efforts of six individuals. Perhaps a more appropriate imagery for his 72 EDO piano temperament would be a multi-manual piano akin to the Haba and Wyschnegradsky examples, only with six keyboards on the solo instrument rather than two. For as much trouble as Ives had with procuring a quarter-tone piano and the related issues in performance practicalities, Haas' troubles would have been even greater for a hypothetical six manual 1/12th tone piano. Nevertheless, his writing points toward singular goals, often relating to overtone harmonies, achieved through microtonal polyphonic imitation applied to the six pianos.On the aforementioned Universal Edition page Haas describes his approach to the piano writing for the last section of Limited Approximations, stating:[...] A microtonal countermovement is composed into the final third of the piece: from the fifth C‘-G‘ to the neutral second between the E sharp raised by a twelfth-tone and the F lowered by a sixth-tone. Thus 10 different intervals arise, each of which becomes the centre of an overtone chord. [...] Melodies recur again and again, wandering from piano to piano – as tremolo, as individual tones, as overtone chords. [An] interval (for example the fifth C-G) sounds in all octaves – but some of these octaves are expanded by a twelfth of a tone. The ear corrects the chord (or tries to correct it) – looked at horizontally, the intervals oscillate in twelve-tone steps, but stay in the same place after all …The fact that Haas conceived Limited Approximations as a work of three parts may be a coincidence when comparing it to Ives' 3 Quarter Tone Pieces. Despite the composer partitioning of the work into sections, the approach to each is unified in, as Mason expresses, “[its] ability to move between systems within 72tet: traditional 12-tone equal temperament, 24-tone equal temperament (quarter tones), and the pitches of the harmonic series up to the 12th partial are all subsets of 72tet.” The same integration between sections cannot be said about the three Ives pieces. The first two pieces share more in common with the solo piano literature of the 18th and 19th centuries than the 20th and 21st. Ives (again via the score preface) characterizes I. Largo as “[...] primarily diatonic, using quarter-tones as passing tones or suspensions and quarter-tone chords as extensions or variants.” For II. Allegro Ives is even more dismissive, stating “[it] is mostly made of rhythms contrasted or 'split' between the two pianos. From a pure quarter-tone harmonic standpoint it doesn't amount to much.”The parallelisms in the final sections of each work are, however, fortuitous. Like many of Ives' compositions the last movement incorporates a chorale. The composer describes the microtonal functionality here as “[a] cantus firmus, taken by the upper voice in the coda [...] made of a phrase in quarter-tones going to one in half-tones and ending in whole tones, while the harmonic plan remains throughout on a quarter-tone basis.” Though it is not stated if the cantus firmus is original or not, many of his compositions draw upon quotations, in particular hymns (i.e. Three Places in New England). Haas' piano writing at the end of Limited Approximations also functions akin to a chorale, albeit perceptually stretched and intervalically compressed to an extreme. The strongest relationship between these points is the utilization of quotation and the reference of church polyphony. The most disparate aspects lie in the macro treatment of microtonality: spectral techniques without traditional form (Haas) vs. more intuitive quarter-tone treatment of harmonies with traditional form (Ives).

The function [math]y=e^{-x^2}[/math] should be familiar to you. It defines the normal or gaussian distribution function. This is what its curve looks like:-There is no known function which has a derivative equal to [math]e^{-x^2}[/math]. In other words, the function [math]e^{-x^2}[/math] does not have an anti-derivative. How are we then to integrate it?It was Simeon Poisson (I had erroneously attributed this solution to Leonhard Euler, earlier), who showed how its definite integral can still be computed and the area under the curve obtained, using a method that is an all-time classic.Think out of the box - think 3D instead of 2D, and hop from Cartesian to Polar coordinates and back.The Quora question requires us to solve [math]y=\int_{0}^{1}e^{-x^2}dx[/math].Let me first take you on a small detour, to show you how to integrate the curve between the limits [math]x=-\infty\ to\ x= +\infty[/math]. It gives us a very interesting result, well worth the diversion. We will then get back to integrating the function between the limits [math]x=0[/math] [math]to[/math] [math]x=1[/math].Let us do our first jump out of the box. Let us consider not one, but two identical curves [math]z=e^{-x^2}[/math] and [math]z=e^{-y^2}[/math], both standing vertically upon the x-y plane with their peaks pointing in the direction of the z-axis. Let their spreads stretch respectively from [math]x=-\infty\ to\ +\infty[/math] and [math]y=-\infty\ to\ +\infty,[/math] with their orientations perpendicular to one another (left hand sketch below) :-The curves are identical, so the areas under them would be the same. The two areas can be expressed as :- [math]A=\int_{-\infty}^{+\infty}{e^{-x^2}dx}[/math] and [math]A=\int_{-\infty}^{+\infty}{e^{-y^2}dy}[/math]Multiplying the two will give us the volume of a 3-dimensional bell-shaped hump as shown in the right hand sketch above. We get:- [math]A^2=\int_{-\infty}^{+\infty}{e^{-x^2}dx}\times \int_{-\infty}^{+\infty}{e^{-y^2}dy}[/math]This gives a nested double integral like this:- [math]A^2=\int_{-\infty}^{+\infty}\left(\int_{-\infty}^{+\infty}e^{-x^2}e^{-y^2}dx\right)dy[/math]Let us now make our second jump out of the box. Let us move from Cartesian to Polar coordinates.First some simplification[math]e^{-x^2}\times\ e^{-y^2}=e^{-\left(x^2+y^2\right)}\ =\ e^{-r^2}[/math]where r is the polar radius vector of any point (x,y)Let us view that 3-D Bump as a solid of revolution, created by rotating the curve [math]z=e^{-r^2}[/math] stretching from [math]r=0[/math] to [math]\infty[/math], about the z-axis, through an angle equal to [math]2\pi[/math] radians. We will have to change the limits of our double integrals accordingly.We will also have to redefine the differentials [math]dy[/math] & [math]dx[/math] used in that double integral to the polar form using [math]dr[/math] & [math]d\theta[/math]. In the Cartesian World, the volume of a 3D solid is computed by adding the volumes of an infinite number of thin, vertical columns of square cross-section [math]dydx[/math] (somewhat like a 3D map of Manhattan Island). In the polar world, the same solid has to be imagined as being made of an infinite number of very tiny, concentric arc-shaped sections of radial thickness [math]dr[/math] and arc-width [math]rd\theta,[/math] arranged somewhat like at Connaught Place in Delhi.Our new equation will then be:-[math]A^2=\int_{0}^{2\pi}\left(\int_{0}^{\infty}e^{-r^2}dr\right)rd\theta[/math]Now let us make a third jump out of the box. Let us re-appropriate that r from outside the brackets and move it inside.Our first, i.e. inner, integration problem then becomes-[math]\int_{0}^{\infty}{re^{-r^2}dr}[/math]Solving this is a slam dunk.Let [math]r^2=u.[/math] If we differentiate both sides, we get [math]2rdr=du[/math].The first integral then becomes:-[math]\frac{1}{2}\int_{0}^{\infty}{e^{-u}du=-\frac{1}{2}\left[e^{-u}\right]\begin{matrix}\infty\\0\\\end{matrix}}[/math][math]-\frac{1}{2}\left[e^{-\infty}-e^{-0}\right][/math] [math]=[/math] [math]-\frac{1}{2}\left[\frac{1}{e^\infty}-1\right] = -\frac{1}{2}\left[0-1\right][/math][math]=\frac{1}{2}[/math]Let us now perform the second, outer, integration[math]\int_{0}^{2\pi}\frac{1}{2}d\theta=\frac{2\pi}{2}\ =\ \pi[/math]The volume of the bell-shaped solid turns out to be equal to [math]\pi[/math]The area under the curve [math]y=e^{-x^2}[/math] [math]stretching[/math] [math]from[/math] [math]minus[/math] [math]to[/math] [math]plus[/math] [math]infinity,[/math] has to be the square root of the volume, or [math]\sqrt\pi[/math]Wow! What a relationship between the two great mathematical constants [math]e[/math][math][/math] [math]and[/math] [math]\pi[/math] ! I couldn’t resist making this slight diversion en route to our main task.Now let us get down to computing the area under the curve from [math]x=0[/math] [math]to[/math] [math]x=1[/math].Let us resume at the equation [math]A^2=\int_{0}^{2\pi}\left(\int_{0}^{\infty}e^{-r^2}dr\right)rd\theta[/math]Let us switch limits of the first (inner) integral to between [math]r=0[/math] [math]to[/math] [math]1[/math]and also move that [math]r[/math] inside the bracketsWe get the volume of our 3D solid to be [math]A^2=\int_{0}^{2\pi}\left(\int_{0}^{1}re^{-r^2}dr\right)d\theta[/math]let us perform the first integration.[math]\int_{0}^{1}{re^{-r^2}dr}[/math]Let [math]r^2=u.[/math] If we differentiate both sides, we get [math]2rdr=du[/math].The first integral then becomes:-[math]\frac{1}{2}\int_{0}^{1}{e^{-u}du=-\frac{1}{2}\left[e^{-u}\right]\begin{matrix}1\\0\\\end{matrix}}[/math][math]=-\frac{1}{2}\left[\frac{1}{e}-1\right]=\frac{1}{2}\left[1-\frac{1}{e}\right][/math]Now for the second integration[math]\int_{0}^{2\pi}{\frac{1}{2}\left[1-\frac{1}{e}\right]d\theta=\frac{1}{2}\left[1-\frac{1}{e}\right]\left[\theta\right]\begin{matrix}2\pi\\0\\\end{matrix}=}\frac{2\pi}{2}\left[1-\frac{1}{e}\right]=\pi\left[1-\frac{1}{e}\right][/math]This gives us the volume of our 3D bell shaped solid, but truncated at a radius of 1, something like this:-This is a circularly symmetrical solid, so the square root of its volume should give us the area under its vertical cross-sectional curve between the limits [math]x=-1[/math] [math]and[/math] [math]x=+1[/math] , or [math]A=\sqrt{\ \pi\left[1-\frac{1}{e}\right]}[/math]Dividing this by 2 will give us the area under the curve between the limits x=0 and x=+1, that is,[math]\int_{0}^{1}{e^{-x^2}dx}=\frac{1}{2}\sqrt{\ \pi\left[1-\frac{1}{e}\right]}[/math]Which is what we were required to compute.How was that??!!