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The Cartesian Product method of multiplying pc sets is simple and straight-forward, but the application of the process as described by Boulez takes on another level of specificity, specifically ordering the sets. This is process that Heinemann describes in his dissertation on pitch multiplication in the music of Boulez. This method involves taking two sets, again X and Y, using set X as the multiplicand and set Y as the multiplier, and determining the product of the two based on an ordered interval series of set X.


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Because this method of pitch class multiplication involves ordering set X, we are able to get different results depending on the order of the multiplicand. For this extension of the technique, concept of initially ordered pc set and initial pitch class r become important. When rotating and changing the value of r, the order of the pitch classes are also reordered. These can now be used as ordered sets for melodic writing, they can be used as unordered sets to determine harmonic structures.

They could be used as unordered sets and be applied free to both melody and harmony to have a consistent pitch structure throughout a section. They could be used as blocked chords and with some working out one could find smooth voice leading to get from one chord to the next. If we were Boulez, we would probably establish some kind of localized ordering scheme and use each of these as unordered sets to be filtered through whatever ordering systems is applied to the overall piece, section of the piece, etc. The example below shows these chords in standard notation to show the sets in a musical context.

Chord spelling used for visual assistance, to keep noteheads from stacking on top of one another; no ordering melodic or vertical ordering is implied by the voicing.

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While the above method uses simple math, it can be a little cumbersome and not always clear to see. Heinemann demonstrates another method for deriving supersets, that might be a preferred method for someone more used to looking at pitch matrices. The example below shows these used in a musical framework to demonstrate how I might go about using the results of the multiplication. There is plenty more to uncover with this technique and Heinemann and Kablyakov do it more justice than I can here.

That said, I strongly encourage you to try this method. The possibilities are endless, and for that reason one could see why Boulez was so drawn to this technique. I understand this is partially because of the wide variety of styles and developments that took place during the 20th century, but I still feel it is problematic that so much is left out. The musical developments of the 20th century, specifically the latter half of the 20th century, shaped our modern approaches and conception of music and art in popular genres and concert music, yet so little time is devoted to the study of that music.

However, there are so many composers that I personally think all undergraduate students should know about, especially if they plan to continue with graduate studies in music. I have narrowed this list down to 35 composers who, if I had my druthers, would required information for all undergraduate history and theory courses. Below is my list of composers as well as a short rationale for each: …but before we get started Honorable Mention These are composers I feel could be on this list, but excluded them because they seem to receive more coverage in undergraduate courses, and are performed more frequently.

This, however, is based solely on my own undergraduate experience and my observation of other undergraduate curriculums I experienced through my graduate and doctoral studies. Luke Passion. Not only are these beatiful works that explore color and timbre as a primary formal framing device, they are all landmark works in sound mass composition. Gyorgi Ligeti - Early pieces utilizing quotation and influence of Bartok and Kodaly, 50s and 60s compositions with micropolyphony, sound-mass composition, textural music based on electronic experimentation; later works showing a multitude of styles.

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Ligeti basically did it all, and did it well. Morton Feldman - major figure in experimental music, developed new systems of graphic notation, influential on music and art of the 60s, 70s and 80s. Feldman is primarily important for his contributions to aleatoric and indeterminate music also developed by his contemporaries Cage, Earle Browne and Christian Wolff.

I think Cage is covered fairly well, and while I would like to see Brown and Wolff covered extensively, I would be fine with undergrads having a more than passing acquaintance with Feldman. He was highly influential on a generation of students including Cindy McTee, David Lang, Laurie Spiegel, Aaron Kernis, Daniel Kellogg and Chris Theofonidis , and he is a great example of a composer who was heavily influenced by electronic music and the potential of new sounds available, more so than many of his post-Modernist contemporaries of the mid-late 20th century.

Davidovsky is one of 3 composers to win a Pulitzer Prize with a piece utilizing electronics as a primary component of the piece Synchronism 6 for piano and tape , the other two being Leon Kirchner and Charles Wuorinen. Gubaidulina has also received numerous awards for her music, and is somehow still not a household name among 20th century composers.

Eastman was an amazing performer singer and pianist and a brilliant composer whose life and work were cut entirely too short.

Oliveros is a highly influential composer and innovator of electronic music who taught at numerous institutions including Mill College, San Diego State University, Oberlin, and the Rensselaer Institute. Terry Riley - I know I just said we all know about Teri Riley, but really most people just know about In C, which is undeniably influential, and is a lot of fun to perform. La Monte Young - Another composer associated with classical minimalism who existed well outside of that aesthetic, Young was very active in experimental music and performance art, specifically his work with Fluxus.

Meredith Monk - Lots of important work in performance art, has received numerous honors and awards for her contributions to contemporary art, music and theater, often works on interdisciplinary multimedia works that stretch genres and has received recognition in the popular music and film world.

His works are not limited to mental gymnastics of serial and post-serial technique, but are full of expressive beauty and nuance. His outdoor pieces are also incredibly creative works that utilize large outdoor spaces and spatialized ensembles to create immersive sonic events.

He was also on faculty at Eastman along with Schwantner and Sam Adler and had his own flock of dedicated and talented pupils. Kaija Saariaho - Arguably one of the most frequently performed composers of the late 20th and early 21st centuries. Saariaho also has many works that are major pieces for instruments and electronics including Pres for cello and electronics, Noanoa for flute and electronics and Nymphea for string quartet and electronics. Part is often considered the originator of this approach to minimalism, and his instrumental and choral works in this style are performed frequently throughout the world.

Gerard Grisey - French composer credited with developing spectral composition along with his colleague Tristan Murail.

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Spectral music was a major development in composition that led to influences in orchestral and chamber music as well as electronic composition with and without instruments and film music. In my personal opinion Grisey should be included with Debussy, Les Six, Messiaen and Boulez in terms of his importance to French music and contemporary composition on the whole.

Andriessen also utilizes odd combinations of instruments instead of more traditional chamber groups or orchestras Joan Tower - A major figure in the development of late 20th century American music. Her music has received numerous awards including a Grawemeyer for Silver Ladders. Other important works include Sequoia, Petroushskates and Fanfare for the Uncommon Woman Frank Zappa - Frank Zappa is, in my opinion, one of the most interesting and talented musical minds of the 20th century.

He never formally studied music or composition and could write chamber and orchestral works on the level of Varese, Stravinsky, Berg, Boulez, you name it. He also wrote interesting genre bending popular music in numerous styles, experimented with electronics having composed numerous fixed media electronic compositions , he worked at IRCAM, was one of the first people to own a personal Synclavier, and arguably influenced music in popular music and contemporary concert music to a greater extent than anyone on this list.

Shulamit Ran teaches at the University of Chicago and has been a leading voice in contemporary music throughout the world and her approach to composition is influenced by her studies with Norman Dello Joio, Ralph Shapey and Elliott Carter. This led to a compositional voice characterized by eclecticism of style, harmonic and rhythmic language. Augusta Reed Thomas - A talented composer who could be described as balancing postmodernism and neo-Romanticism.

Thomas was very talented as a young composer and became a tenured professor at Eastman at the age of 33, but later went on to teach at Northwestern. Thomas has composed for numerous ensembles and genres, has received numerous awards and commissions and is frequently performed throughout the world. It is difficult to deny that she is one of the most celebrated composers of the 21st century.

His music also demonstrates his approach to post-modernism with eclecticism of style, pastiche, quotation and colorful instrumentation.

As we all know, was not an easy year for a multitude of reasons. I always spend the last couple weeks of December reflecting on the year as I imagine a lot of people do , and I thought back to my first KLANG post of posted on this day one year ago. The problem was that it did not stop there.

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Over and over again took from the world one musician after another. Concert music, electronic music, experimental music, popular music, jazz, folk music, nobody seemed off-limits in , and it was definitely a year that took a lot of important musicians and composers who helped shape the musical landscape of While I did not know any of these composers personally on a personal level I had met and spoken with Steven Stucky on multiple occasions, but always under professional circumstances , their music, writings and lectures were a huge influence on me as a young composer searching for my own artistic voice.

Pauline Oliveros and Jean-Claude Risset, who died only a few days apart, were massive influences on me in my early studies of electronic music at Ohio University. Rissets Songes for instruments and electronics is still one of my favorite early works for instruments and electronic sound and Oliveros experiments with electronics are still of particular interest for me. While there was a time in my early studies when I appreciated Glass, Reich and, to the greatest extent, Terry Riley, I was never really taken by minimalism.

Peter Maxwell Davies was one of the most influential composers on my musical thinking as an undergraduate at Ohio University.

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I first found his Eight Songs for a Mad King when I was a sophomore and I listened to the Unicorn recording with Julius Eastman while following along with the score more times than I could possibly even remember. George Martin almost goes without saying. I was never a huge fan of The Beatles I know, blasphemy, right?

It was years later that I found out about the influence of George Martin and the recording and studio techniques he employed that made those records so special. Not to take anything away from The Beatles, but for me it was the studio experimentation that makes some of those songs so special and interesting.

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Additionally, Martin brought the idea of the studio as an instrument an idea that had been in practice in Europe and American universities for years to the world of popular music. But Bassett and Stucky were different. It was also through Bassett that I became familiar with the work of Robert Morris, who would later be very influential on me during my graduate studies.

I went up to the bar to get a refill on my whiskey and I saw Stucky standing, also waiting patiently for a refill. I walked up to him and congratulated him on the performance of his new piece Isabelle Dances and told him it was nice having met him we had already talked times at this point during the festival.

That brings us to Pierre Boulez, definitely the hardest hit for me on this list. I apologize for starting both and with downer posts, but I felt it necessary to do a reflection on the impact of those that were lost in , both for the music world at-large and for me personally. Below is a list of performers and composers in the popular music world I also felt were major losses, at least for me personally.

You will all be missed, and thank you for your contributions to the developments of music in the 20th and 21st centuries. Even a highly chromatic tonal work must, by definition, center around some kind of tonic, and any progression toward or away from that tonic is governed by a series of loose guidelines governed by the hierarchical relationship of scale degrees and chords to the tonic. The same cannot always be said of atonal music, especially atonal music based around pitch-class sets pc sets.

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One method that I believe is lesser-known is the use of sieves, developed by Iannis Xenakis, Greek composer, architect and music-theorist. Xenakis was a pioneer in the use of applying principles of architecture and stochastic processes to his music through mathematically calculated pitch and rhythmic systems. I will discuss what a sieve is, how it is constructed, how it can be mapped to pitch and some of the benefits that can be gained from using this particular method of pitch organization.

In simple terms, a sieve is a means of filtering. Mathematically speaking, it is the filtering of a set of numbers based on specific rules so that only some members of the set remain. Christopher Ariza defines a sieve as a formula consisting of one or more residual classes combined by logic operators. A residual class consists of two integer values, a modulus M and a shift I. The modulus can be any positive integer greater than 0, and the shift can be any integer from 0 to M Another way of notating a sieve is to represent it as as an integer the factor or modulus with a subscript integer to represent the shift.

An example is 3 0 , which represents the series [0, 3, 6, 9, 12…].