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LargeHadron
May 19, 2009

They say, "you mean it's just sounds?" thinking that for something to just be a sound is to be useless, whereas I love sounds just as they are, and I have no need for them to be anything more than what they are.
What is spectralism?
Spectral music, or spectralism, is an approach to music composition that began in Paris in the 1970s, and it is most closely associated with the composers Gerard Grisey and Tristan Murail. Most sources will tell you that spectral music is that which uses timbre as the primary source of musical material. However, I believe that the terminology, as well as the actual habits of composers who are considered to be spectralists, allow for a much broader definition. More on this later.

I understand what you mean by timbre, but what *is* timbre?
I’m sure the vast majority of you already understand this concept, but I’ll explain anyways in case the details are unclear. Timbre is what makes a violin sound different from a trumpet (well, that and ADSR envelope characteristics…but let’s ignore that part of it, you can read about it on your own if that’s unfamiliar to you). The actual acoustic phenomenon of timbre is explained by overtones (also called partials and, sometimes, harmonics), which are higher pitches that sound simultaneously with the note that you perceive (i.e. the fundamental pitch). So, when a string on a guitar is plucked, you are actually hearing several pitches at the same time. If you want to convince yourself of this, take out a guitar and play an open string. Then, play the natural harmonic at the twelfth fret. Do it a few times to really get the sound in your ear. Now play the open string again. You should be able to clearly hear the previous sound mixed in with the lower tone. This will work for all of the harmonic positions of the open string.

The timbre of a guitar string is an example of a harmonic spectrum. A harmonic spectrum is one in which the frequencies of the overtones are whole-number ratios to the fundamental. So if the fundamental frequency is a variable f, the overtones in a harmonic spectrum would be 2f, 3f, 4f, 5f, and so forth. Calling a string a harmonic spectrum is a little bit disingenuous, because the fact that a string has mass means that the physics don’t work out perfectly and the harmonics are skewed slightly. This can be demonstrated with a comparison of the low (thick and heavy) strings of a piano and the high (thin and light) strings. The higher strings have overtones which are much closer to the ideal harmonic spectrum than the lower strings.

Sometimes, an instrument’s timbre is similar to a harmonic spectrum, but it is missing some overtones. This is called a defective harmonic spectrum. The clarinet is a perfect example of a defective harmonic spectrum. The way a clarinet is built results in a timbre that is defined by only odd-numbered harmonics. So if a clarinet is playing a pitch with a frequency of f, the overtones that sound above f are 3f, 5f, 7f, and so on.

The other category of timbre is called an inharmonic spectrum which basically just means anything that doesn’t fit into the two categories above. We are just as familiar with these sounds as we are with harmonic and defective spectra. A bell’s timbre is entirely inharmonic. The overtones that sound above a bell’s fundamental pitch do not correspond to frequencies of the harmonic series. This is partly why it is sometimes difficult to tell what pitch a large bell is ringing at, and why a melody played by a group of large bells can sound like a cacophony of noise if your ear is focused on the overtones rather than the fundamentals.

So timbre is just the same thing as harmony?
Sort of, but not really. Timbre-harmony is more of a continuum in which the distinction between the two is often unclear. In fact, that ambiguity itself can be an interesting idea to explore in a piece of music (and it has been). A simple way to look at it is that a harmony is experienced as a timbre if it can be heard as a single sound rather than as a collection of discrete pitches. The relative amplitudes (volumes) of the overtones can have a large effect on whether the sound is perceived as a timbre or a harmony. A loud fundamental tone with quiet overtones will blend into a single sound much more gracefully than if all of the tones were the same volume. But I already said that a trained ear can distinguish individual frequencies within a timbre, so that distinction isn’t perfect. If anyone has experience in the nitty-gritty of acoustics and can shed some light on this, I would love to hear about it.

Let’s get back to the music. What do spectral composers actually *do*?
I mentioned that spectral music is often characterized by its use of timbre as the primary source of musical material. All composers use timbre, as they are using sound in order to make music, but much spectral music (especially early spectral music) focuses on the development of timbre in lieu of familiar things like melodies, chord progressions, and rhythms. What it actually means to develop timbre varies from piece to piece, but the common approach is to create a timbre and progressively modify it so that gradually changes over time. I think it’s a good time to present some actual spectral music. Here is Partiels, by Gerard Grisey: https://www.youtube.com/watch?v=jqzukP_BtW8. This piece serves as the textbook example of spectral music. What Grisey is doing here is reproducing the timbre (as well as the attack characteristics) of a low trombone note, and then very gradually transforming it. Each time the sound comes back, it is slightly distorted by octave-displaced harmonic partials, inharmonic partials, or noise elements like string overpressure. If you have a music textbook that mentions spectralism, it will probably talk about this piece. Partiels is the third movement in a monumental six-movement work called Les espaces acoustiques. It is by no means the first spectral piece, nor does it represent the wide variety of techniques spectral composers of the time were using. Even other movements of the same work do very similar things as Partiels. My own theory as to why it became the quintessential spectral piece is that its title (“partials”, as in “overtones”) makes it easy to remember. It is a very good piece though, check it out.

What does it mean to create or reproduce a timbre?
Remember that a timbre is a collection of overtones that sound simultaneously with a fundamental. Reproducing a timbre using an ensemble of instruments is as simple as figuring out through analysis (an FFT analysis, if you are curious) what are the frequencies of the overtones in a sound and how loud they are, and then assigning those frequencies and volumes to different instruments (this is a technique called additive synthesis). When the instruments play the frequencies together, a unified timbre is heard. Now, a keen reader will recognize two very serious problems with this approach. The first, and most obvious one, is that it’s pointless to reproduce an existing timbre using a combination of instruments. Why bother going through the trouble to recreate a trombone’s timbre, in the above example by Grisey, when a trombone can do it all by itself? The answer to this is that spectral composers aren’t interested only in presenting a timbre as it is, but in using or manipulating it in some way. It doesn’t do much to just have someone blow into a trombone because that timbre is read-only – there is no way to systematically modify it to create a coherent thread of timbre-based music. Only by mimicking the trombone sound in an ensemble can Grisey then gradually change each component of the timbre to develop new timbres and move the music forward.

The second, more subtle problem with using an ensemble to reproduce a timbre is that it is impossible to recreate an instrument’s timbre using other instruments. Remember that each instrument in the ensemble has its own characteristic set of overtones that sound simultaneously with the fundamental. So, when you assign an instrument to an overtone frequency you’re not just getting a single pitch; you are also getting all of the overtones inherent to the instrument. What this means is that, again in the Grisey example, the trombone sound created by the ensemble isn’t actually a trombone sound. It’s a trombone sound plus a bunch of extra overtones that don’t necessarily agree with it. I’ve read various explanations of how early spectral composers felt about this. One is that spectral composers were simply “naïve”. I don’t buy this, as they all very clearly understood acoustics and would have realized this limitation immediately. The better explanation is that they didn’t care. The goal wasn’t to actually reproduce sounds, but rather just to use them as a starting point for a composition. How perfectly the synthesized sounds matched their sources wasn’t as important as the music created by using them.

You mentioned that spectralism isn’t just about timbre though. What else do spectral composers do?
Early spectralism was born of the study of acoustics and sound analysis, and timbral synthesis is only a part of the story. Spectral composers might also use the study of sound envelopes in order to design unified sound gestures that are built of component parts in an ensemble (imagine a group of instruments playing in such a way that they sound like one giant meta-instrument). Early spectralism often simply tried to imitate electronic processes like feedback loops, FM synthesis, and ring modulation using acoustic instruments. In more recent years, composer Tristan Murail has moved onto the study of highly complex sounds that can’t be reduced into simple terms like timbre and ADSR envelope characteristics. I’ll link to an example of this later. In general, the spectralist approach just means that the composer puts more emphasis on creating interesting sounds (informed by the study of acoustics) than on creating abstract relationships between higher-order musical parameters.

This is just strange. People don’t listen to music this way and it doesn’t make any sense.
While I do agree that the spectral approach is at odds with the way we are conditioned to experience most classical music (e.g. following melodies, rhythms, the development of motives, the tension created by movement to distant key areas, etc.), I believe that there is a little spectral listener inside most of us and not everybody realizes it. The best way to support this point is to turn to vernacular music. Let’s say you love Tom Waits and you hate Nickelback (which I’m sure is true for many of you out there). Would you rather hear Nickelback cover a Tom Waits song, or Tom Waits cover a Nickelback song? If you answer this like I’m anticipating, you will have chosen the latter. Why is that? A huge part of what makes Tom Waits interesting is the nuance in his vocal delivery, the gritty timbre of his voice, and the way he orchestrates his music. In contrast, many of us probably think of Chad Kroeger’s vocal delivery as cliché, his vocal timbre as bland or even obnoxious, and the orchestration of the songs to be unremarkable. So, even if Nickelback is covering our favorite song “Downtown Train,” staying true to the lyrics, melodies, and chords, it’ll probably sound like garbage. This is because we are unknowingly engaging in spectral listening, focusing on the subtleties of the actual sounds that are being created. Of course, this is being done in addition to attention to the relationships of musical material. I can’t deny the wonderful feeling of hearing the tune of a favorite chorus or the power of a chromatic harmony at just the right moment. The example was meant to show that we put a great deal of value on timbre and nuance.

If the Tom Waits/Nickelback example doesn’t work for you, try this next one. Compare Bob Dylan to your uncle Bob, a trained singer who can hit every note perfectly on pitch, in perfect rhythm, and with perfectly clear enunciation. I imagine most of you would still prefer to hear Bob Dylan sing “Like a Rolling Stone” instead of your uncle Bob. For those of us who agree, a favorite singer’s characteristic style is more important than a flawless reproduction of musical material.
Now I’ve gotten pretty far off point. What does this have to do with spectralism? Spectralism is based on the study and manipulation of the basic characteristics of sound, those which have for so long been overshadowed by greater attention to higher-order relationships like melody, chord progressions, and rhythm. The kind of listening required to comprehend a piece of spectral music comes quite naturally to us, because it is not very far off from what we are already used to doing subconsciously.

Ok, so what is some good spectral music to listen to?
I already mentioned Gerard Grisey’s Partiels: https://www.youtube.com/watch?v=jqzukP_BtW8. It might interest you to check out the entirety of Les espaces acoustiques, but as a fair warning it is very long.

Tristan Murail
Les Partage des Eaux: https://www.youtube.com/watch?v=p1jrxDTi2Fw
This one is an example of what I mentioned earlier, Murail deriving material from the analysis of highly complex sounds. Here, the sounds are water breaking on the shoreline.

Legendes Urbaines: https://www.youtube.com/watch?v=-cV0DhcZ-ws

Le Lac: https://www.youtube.com/watch?v=56sN6zYUcvk

Georg Friedrich Haas
My understand of Haas’s music is that he is less into analysis and more into inventing neat sounds and deriving harmony from the just-intoned intervals of the harmonic series. Very cool and prolific composer. He currently teaches at Columbia University.

limited approximations: https://www.youtube.com/watch?v=BoqvGLdjUhE
This piece is ridiculous. It’s scored for six pianos tuned to quarter, sixth, and twelfth tones, and orchestra. If you’re not going to listen to the whole thing, at least listen to the arpeggiated chords at the end.

Jonathan Harvey
Advaya: https://www.youtube.com/watch?v=9oUSgXGbuHU
Spectral music isn’t just for large ensembles. Many pieces are for small ensembles or even solo instruments. Composers will often incorporate electronics to create more interest or to help fill out the sound of a small ensemble.

Horatiu Radulescu
Radulescu is similar to Haas in that he is more about inventing engaging sound worlds and using just intonation. He often eschews the common spectralist aesthetic of gradual development in favor of static music that doesn’t change much over time.

Clepsydra: https://www.youtube.com/watch?v=icJNmcSwHk0
Radulescu was also batshit insane and somehow manage to convince the performers of the above piece to lay 16 grand pianos on their sides with the lids removed so they could scratch, strike, and bow the strings in various ways. He considered a piano prepared in this way to be a new instrument which he called a “sound icon”.

Why am I talking so much about this?
If you haven’t guessed it yet, I’m a composer myself and I am very inspired by the richness of sound associated with the spectral approach. I simply wish to share my passion with you, in the hopes that you will also be able to see the beauty in a body of music that is for the most part hidden away from most music enthusiasts. What kind of artist would I be if I didn’t self-promote at least a little bit? Here are some pieces I’ve composed using a spectralist approach:

Inspiration/Expiration for orchestra: https://www.youtube.com/watch?v=ffCBCrUGnIQ
The only piece for full orchestra that I’ve done in this way. Much of the much is derived from timbral synthesis, and some of it is based on the electronic process of time dilation/contraction. This can be most clearly heard in the last five minutes, during which time the musical material stays pretty much the same, but it is slowing down and dropping in pitch proportionally to give the impression of a tape being slowed down.

Artellia for two string quartets: https://www.youtube.com/watch?v=Hld01wd-7sg
Not much to say here, more of the Haas approach where I imagined a sound world and derived harmony from the harmonic series.

The Bells of Cologne Cathedral for two pianos: https://www.youtube.com/watch?v=dCU8RBU5geE
For this piece I analyzed recordings of eleven bells at the Cologne Cathedral in Germany and assigned overtones to the closest possible equal-tempered piano notes. The result is meant to exploit the ambiguity between timbre and harmony that I mentioned much earlier. It should sound like bells and like chords on the piano at the same time. I created various processes to move from one bell timbre to the next. Sometimes the movement is very gradual, with new partials being introduced slowly while old partials drop out. Sometimes two timbres are combined before the first is removed. Other times, the change from one timbre to the next is abrupt and unprepared.

Boronea for a bunch of low clarinets: https://www.youtube.com/watch?v=0cy__4kC-P8
An imagined meta-instrument inspired by the muddiness of the bass/contrabass clarinet’s low register, alternating with sections that explore and expand upon the throaty sound of the clarinet’s higher register.

Cyraenes for chamber ensemble and electronics: https://www.youtube.com/watch?v=l67u9wX8dUg
This one is admittedly the least friendly of the bunch. It’s very long and has a certain nagging quality that might be annoying to some people. The very general idea of the piece is a gradual evolution of timbre from a clarinet spectrum, to a harmonic spectrum, to an inharmonic spectrum, and finally to a period of spectral saturation. I think the local sound events drive the music more than the underlying process (this is, for the most part, true for most spectral music).

Also, my soundcloud (https://soundcloud.com/paul-fake) and my personal website (http://www.paulfake.com/music/) which has links to scores if you are interested.

LargeHadron fucked around with this message at 17:41 on May 22, 2015

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LargeHadron
May 19, 2009

They say, "you mean it's just sounds?" thinking that for something to just be a sound is to be useless, whereas I love sounds just as they are, and I have no need for them to be anything more than what they are.
I'm going to use this space later to go into more detail about the history of spectralism: why it came about and what composers/pieces led up to it. Will also include some good reading resources.

its curtains for Kevin
Nov 14, 2011

Fruit is proof that the gods exist and love us.

Just kidding!

Life is meaningless
I almost but didn't quite pass out listening to limited approximations and what i experienced was pretty surreal watching my half asleep brain try to absorb all the sound. Its definitely something I got more out of when halfway unconscious than when i was fully awake and paying attention to it. Its like a intense theatrical soundtrack that never stops. Quite interesting as a concept, although I doubt it will find its way onto my itunes any time soon.

snorch
Jul 27, 2009
So what makes this a movement worthy of an "-ism" label as opposed to just a sonic tool available as part of a greater palette? Is it just that it's still in its experimental phase or what? To me the examples are sort of reminiscent of John Chowning's early FM synthesis pieces, which are mostly just nerding out about this ONE WEIRD TECHNIQUE, or "exploring the medium" if you'd prefer to put it that way.

snorch fucked around with this message at 17:09 on May 12, 2015

LargeHadron
May 19, 2009

They say, "you mean it's just sounds?" thinking that for something to just be a sound is to be useless, whereas I love sounds just as they are, and I have no need for them to be anything more than what they are.

snorch posted:

So what makes this a movement worthy of an "-ism" label as opposed to just a sonic tool available as part of a greater palette? Is it just that it's still in its experimental phase or what? To me the examples are sort of reminiscent of John Chowning's early FM synthesis pieces, which are mostly just nerding out about this ONE WEIRD TECHNIQUE, or "exploring the medium" if you'd prefer to put it that way.

I prefer to say "spectral approach" because it is, like you said, a sort of compositional tool. I think it's been given an -ism because music that uses this approach tends to have certain hallmark features: glacial harmonic motion, aperiodicity, lack of melody/motivic development, microtonality, attention to subtle detail in instrumental technique. Or maybe there is an important distinction to be made between music that uses a spectral approach (imagine intricate sound design that provides the sonic material for a piece composed in a more traditional way) and music in which the sonic material is the primary focus. Then you could apply the term "spectralism" to the latter, music that is about the sounds rather than structures created using the sounds. Even this distinction gets hairy when you scrutinize it.

abske_fides
Apr 20, 2010
Thanks for the lovely thread! I've been delving deeper into spectralism after I first encounted music from the IRCAM composers, and especially when I had the chance to work with Kaija Saariaho for two concerts.

I'll have to check your compositions a bit later on LargeHadron. Looking forward to it! I myself recently wrote a piece inspired by some of the spectral concepts and using live processing of a grand piano. I'll see if I can post it a bit later, applied to a composition competition so I might not be allowed to post it before I get a result from them.

LargeHadron
May 19, 2009

They say, "you mean it's just sounds?" thinking that for something to just be a sound is to be useless, whereas I love sounds just as they are, and I have no need for them to be anything more than what they are.

abske_fides posted:

Thanks for the lovely thread! I've been delving deeper into spectralism after I first encounted music from the IRCAM composers, and especially when I had the chance to work with Kaija Saariaho for two concerts.

I'll have to check your compositions a bit later on LargeHadron. Looking forward to it! I myself recently wrote a piece inspired by some of the spectral concepts and using live processing of a grand piano. I'll see if I can post it a bit later, applied to a composition competition so I might not be allowed to post it before I get a result from them.

Wow, I'm jealous that you got to work with Saariaho. What is your background in composition? I am definitely curious to hear your piece, and good luck on the competition!

abske_fides
Apr 20, 2010

LargeHadron posted:

Wow, I'm jealous that you got to work with Saariaho. What is your background in composition? I am definitely curious to hear your piece, and good luck on the competition!

First off, you have some really lovely compositions. Inspiration/Expiration had some very interesting use of orchestration, and the processing in Cyraenes was very interesting.

My background in composition actually comes from working with music technology. During my bachelor's I just got more and more interested in the avant-garde of jazz and classical, and started studying it a lot more seriously. Then I had the chance to take an intesive composition year at the conservatory. Currently working on my master's in music technology although my thesis really is a mix of music technology and composition. Hopefully I'll be able to get a few extra classes.

I got the chance to work with Saariaho under a festival here. They needed someone that was familiar with contemporary classical and live processing so I was called up. Getting to meet her, and talk with her completely changed my view on quite a bit of music. I had the chance to work on Lichtbogen and NoaNoa. It was quite interesting to hack into her Max patches haha

Agrinja
Nov 30, 2013

Praise the Sun!

Total Clam
Just posting in to say that this was one of the best reads and listens I've had in a while. I'm a bit of a newbie at all this, but it was a very clear and interesting presentation of good material, and I hope you do more of these. Limited approximations was a stunning listen, and of your compositions, I particularly enjoyed Boronea.

LargeHadron
May 19, 2009

They say, "you mean it's just sounds?" thinking that for something to just be a sound is to be useless, whereas I love sounds just as they are, and I have no need for them to be anything more than what they are.

Agrinja posted:

Just posting in to say that this was one of the best reads and listens I've had in a while. I'm a bit of a newbie at all this, but it was a very clear and interesting presentation of good material, and I hope you do more of these. Limited approximations was a stunning listen, and of your compositions, I particularly enjoyed Boronea.

Thank you :). I'm happy for all of the love of limited approximations; I am head over heels for that one. I still intend to fill in the second post with some more historical info but I just haven't gotten around to it yet. Soon.

Sizone
Sep 13, 2007

by LadyAmbien
This thread is wonderful. It's like orchestral industrial music.

Currently reading Tuning, Timbre, Spectrum, Scale by Sethares. The basic premise of which is that different timbres are germane to different tunings.

LargeHadron
May 19, 2009

They say, "you mean it's just sounds?" thinking that for something to just be a sound is to be useless, whereas I love sounds just as they are, and I have no need for them to be anything more than what they are.

Sizone posted:

This thread is wonderful. It's like orchestral industrial music.

Currently reading Tuning, Timbre, Spectrum, Scale by Sethares. The basic premise of which is that different timbres are germane to different tunings.

Haha totally. I love it when the music sounds mechanical and oppressive. That reminds me of Haas's Violin Concerto: https://www.youtube.com/watch?v=vqneV9s245s. The pounding chords starting at 11:30 really test my ears' endurance.

I'm curious, does the Sethares book mention the Bohlen-Pierce scale? It's a scale that suits well to the clarinet (no coincidence that a specialized Bohlen-Pierce clarinet exists) because it spans a perfect 12th interval rather than an octave. Normally we divide the octave into 12 equal parts and then repeat - the BP scale divides the perfect 12th* into 13 equal parts, so there are no octaves at all in the scale, just like there are no octaves in the clarinet timbre (and, the first partial of the clarinet timbre is a perfect 12th).

*Called the tritave because the perfect 12th is the third harmonic.

Sizone
Sep 13, 2007

by LadyAmbien
It most certainly does. Appropos, that was the section I was reading before I posted. The stretchy triative tuning works because half closed pipes only produce odd harmonics and their consonance/dissonance patterns match better.

Really wish he would set up a legit digital publishing of that book. It can be stunningly neat if you can decipher the sort of odd writing style and organization of it.

Sizone fucked around with this message at 05:08 on May 29, 2015

o.m. 94
Nov 23, 2009

Clepysdra is def. one of my favorite spectral peices. If you want to know more about Radulescu's compositional approach, see if you can get your hands on his written peice "Sound Plasma". Roger Heaton offers a window into his approach in this summary article:

http://researchspace.bathspa.ac.uk/3377/1/Heaton%20full%20text%20-%20Sound%20Plasma.pdf

LargeHadron
May 19, 2009

They say, "you mean it's just sounds?" thinking that for something to just be a sound is to be useless, whereas I love sounds just as they are, and I have no need for them to be anything more than what they are.

o.m. 94 posted:

Clepysdra is def. one of my favorite spectral peices. If you want to know more about Radulescu's compositional approach, see if you can get your hands on his written peice "Sound Plasma". Roger Heaton offers a window into his approach in this summary article:

http://researchspace.bathspa.ac.uk/3377/1/Heaton%20full%20text%20-%20Sound%20Plasma.pdf

Awesome, thanks! I'll add that to the OP/2 when I finally get around to writing it up. Which will be soon, I promise.

coolskull
Nov 11, 2007

Fantastic stuff. This was something I'd heard of but never really dug into. Thanks for the information!

e: Are there any composers/works focused primarily on electronic sources?

coolskull fucked around with this message at 23:02 on May 31, 2015

LargeHadron
May 19, 2009

They say, "you mean it's just sounds?" thinking that for something to just be a sound is to be useless, whereas I love sounds just as they are, and I have no need for them to be anything more than what they are.

LOVE LOVE SKELETON posted:

Fantastic stuff. This was something I'd heard of but never really dug into. Thanks for the information!

e: Are there any composers/works focused primarily on electronic sources?

I'd have to put more thought into it, but one composer that comes to mind is Phill Niblock: https://www.youtube.com/watch?v=UUxoHVHX0II. He's like the Scelsi of electronic music.

abske_fides
Apr 20, 2010

Sizone posted:

This thread is wonderful. It's like orchestral industrial music.

Currently reading Tuning, Timbre, Spectrum, Scale by Sethares. The basic premise of which is that different timbres are germane to different tunings.

I'm about to dive into this book from a copy I found at a library but it doesn't include the CD. How essential is it?

Sizone
Sep 13, 2007

by LadyAmbien

abske_fides posted:

I'm about to dive into this book from a copy I found at a library but it doesn't include the CD. How essential is it?

I don't have the cd either so I can't really say. Obviously it's pretty essential if you want to hear what the examples sound like. The book is...structurally it's weird. Read up the part on dissonance curves, then read the appendix on dissonance curves then continue. All the matlab stuff is on Sethares' website, I need to double check, I bet a lot of the sound examples are up there too.

abske_fides
Apr 20, 2010
If it's mostly matlab stuff that's on the CD for sound examples then it's pretty easy to just compile the code here either way. A few years ago I read Musimathics Vol. 1 but I was pretty disappointed by how the book stays at a very fundamental level and never really goes above basic stuff. I'll probably check out Vol. 2 this summer, hopefully it has a bit more useful/advanced information.

If you guys haven't read it, Saariaho's Timbre and harmony: interpolations of timbral structures is a pretty interesting read. The English translation is awkward in a few places but it's still readable.

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coolskull
Nov 11, 2007

I read all of musimathics 1 but couldn't get far into 2, so you're probably in luck.

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