Talking about Tuning

I’m scheduled to teach “anything” to my seminar on Wednesday. I though I should tackle tuning. This talk is just written and the sound samples and diagrams have not yet been generated.

what is Just Intonation?

Lou Harrison said that “Just Intonation is the best intonation.” An intonation is a type of tuning. Just tuning is a tuning that uses fractions. In just intonation, pitches are set using whole number ratios. To understand this, let’s look at the harmonic series.

Harmonic Series

The fundamental is the base frequency. (sound sample)
The first overtone is twice the base frequency. It’s relationship to the base frequency is 2/1. In other words, the base frequency * 2 = the first overtone. this makes a perfect (or just) octave. (sound sample)
The second overtone is 3 times the base frequency. It makes an octave plus a fifth. this fifth is perfectly in tune with the base frequency and the first overtone. (sound sample) but is an octave too high to use in a scale between those two notes. We can divide it by 2 to make it an octave lower. this new pitch, the fifth between the base frequency and the octave, is related to the base frequency by a ratio of 3/2. (sound sample) The three comes from it’s place in the harmonic series. The two comes from dividing it down to be in the first octave.

2’s and octaves

All notes in the first octave, will be between the ratios of the base frequency, which is 1/1 and the octave, which is 2/1. If something is too small, it is below 1/1, and if it’s too large it’s over 2/1. We can transpose it to the correct octave by multiplying or dividing by 2.

Therefore 2’s are very important for transposition, but they don’t change a pitch, except by octave. If the base frequency is C and we multiply it by 3, we get a G. If we multiply it by 3/2s, we also get a G, but in the first octave. so 2’s are “for free” you can multiply and divide by them any time you need to change the octave and you will still have the same note as before, just an octave higher or lower.

Inversion

the inversion of a fifth is a fourth. So we can invert the fifth fraction to get a fourth. the inversion of 3/2 is 2/3. But 2/3 is too small. It is not between 1/1 and 2/1. We can multiply it by 2, to get 4/3, a perfect just fourth. (sound sample)

You can do this with any tuning fraction. Invert it to find the inversion, then multiply or divide it by 2 to put it in the correct octave.

Pythagoras

3/2, the perfect, just fifth is a ratio made up of small, whole numbers. Small numbers sound more in tune because they are lower in the harmonic series. 3/2 is the most in-tune sounding note that you can get aside from the perfect octave.

There’s a story that pythagoras was walking by blacksmith shop and heard very harmonius sounds. After experimenting with the smiths, he discovered two excellent intervals, 3/2 and 9/8.

9/8 is a major second and since it still has small numbers, it sounds really good. (sound sample)

From this, he hypothesized that good rations were made up of powers of 3 over powers of 2 and their inversions. You know that the circle of fifths will eventually take you through all 11 notes in an octave. According to pythagoras, you can use this to tune all the notes. First, turn the first two strings as a perfect 3/2 fifth. Then tune from the 3/2 to the next fifth, a 9/8. then tune from the 9/8 to the next fifth, the 27/16. Notice that everyone of these ratios is a power of 3 over a power of 2.

Lattices

You can create a chart of these (pass out handout) called a tuning lattice. a lattice of powers of 3 over powers of 2 is called a Pythagorean tuning lattice. The line on your handouts at the top is a pythagorean tuning lattice. Below that, is chart of them in oder of the notes in the scale. Notice that E, the third is not a small number ratio. It is 81/64. This was considered ok at the time because thirds weren’t considered consonant. Notice also, that the octaves don’t line up. The octave, instead of being 2/1 is 243/128.

this is a sound sample of the tuning lattice going around the circle of fifths. (sound sample). This is a sound sample of it climbing the scale diatonically and then chromatically (sound sample). And this sound sample shows the difference between 243/128 and a 2/1 octave.

N-limit tuning

That last example demonstrates why mixing in other numbers than just three is a good idea. People often use 5’s, 7’s and sometimes higher prime numbers like 11’s. Your tuning system draws it’s name from the largest prime number that you use. A tuning that used 2’s and 3’s is a 3-limit tuning. One that uses 2’3, 3’s and 5’s is a 5-limit tuning.

(if I have time)

5-limit tuning

(Draw on blackboard) this is a tuning lattice of 5’s. This note 5/4 is a just third. The ratio has much smaller numbers than the pythagorean third. This is the pythagorean third (sound sample). This is the 5-limit third (sound sample). there’s almost a quarter-tone difference between them (sound sample).

N-limit lattices

When you are drawing lattices, every new prime number gets a new axis. so a tuning lattice could be thought of as an N-dimensional array, where N is the number of prime numbers. this one, with 5’s and 3’s is a two dimensional array. If we added 7’s, we’d need a new axis and we’d have a three dimensional array.

We can add notes to our lattice that are multiples of 5’s and 3’s. (draw on blackboard) All of these notes are the note right below it multiplied by 3/2s. Remember before, that multiplying fractions was raising them. Like 3/2 * 3/2 makes a note a fifth above. So because all of these notes are multiplies by 3/2s, they are all a fifth higher than the notes below them.

this is useful for two reasons. One is that 6/5 is the smallest numbers ratio we’ve yet seen for d#, the minor third. (sound sample). So adding lines like this helps us find extra ratios. the other thing that it’s good for is transposing. All of these notes are the same as the ones right below it, but raised by a fifth. So you might use this when you modulate to a new key.

Uses of Lattices

composers use lattices like these to figure out what tunings they want to use in their piece and then manipulate them for key changes and transpositions. they then use this to give instructions to instrumentalists or to program them into synthesizers.

One composer that uses tuning lattices to figure out how to tune her instrument is Ellen Fullman. She uses larger prime numbers to tune the many strings of her Long String Instrument. This track is based on a sweep of the harmonics of a C chord. (sound sample)

Denominational

I did a wordlcat search on the mystery play about Joan of Arc. worldcat is a nifty tool that lets you search university libraries all over the world. anyway, I found the text of the original play in old French and new French, a face to face translation from a language that I can’t read to another language that I can’t read. It’s possible that I could make out old French like one could make out middle English. The play was written in 1429 and this was around the time that english and French were seperating into distinct languages. Before that, everyone spoke the Norman language. anyway, the book is in 43 libraries worldwide, but it turns out that one of them is one of the three library consortium that Wesleyan is part of. Trinity College actually has a very large number of books of 15th Century French Catholicism. And modern Catholic stuff too (I was doing a search of the music of Dom Remy, Joan of Arc’s neighborhood cathedral). Their name is “Trinity.” Frankly, I’m becoming suspicious of the secularism of that school.
Speaking of secularism or lack therof, the chapel rennovation here is now done enough to have concerts. the first concert in the Wesleyan chapel (the building with the most seats, in case one wants to give a concert) was a priemere of an organ piece written by Christian Wolff specifically for the new organ. the concert sold out weeks ahead of time and Alvin Lucier ordered his class to go, only to find out that the necessary 50 tickets for them to attend were not available. So he invited Wolff to speak to his class and then walk over to the chapel for a sneak preview.
Half of the music department crammed into the evil basement classroom to hear Wolff speak. He talked about a piano piece that he wrote in the 50’s. for some reason, he prepared the piano in the classroom, maybe just to show how it was done, as he didn’t play anythign once it was prepared. then he talked briefly about the organ piece and we walked over to the chapel.
I had never been in the chapel before, but Alvin had trold me that it was nondemonational and that the organist played student works. Lately, rumors had been flying about the mazing programmable organ. Each stop is separately addressable. (I now know what an organ stop is, but I’ll skip it for now.) We got to the chapel and sat down. It’s so non-demoninational that you can’t even tell which northeastern protestant sect the chapel is dedicated to, but I’ll hazard a guess and say Methodist. the windows are stained glass pictures of Jesus and the apostles (no stars of David, no blessed virgins, no buddhas…). there’s no cross in front put the pews have hymninals (“cof Colleges and schools”) and some prayer books (something that resembles a missal, but is protestant and thus has a different name) with pictures of jesus on the front. Clearly, “non denominational” is a word with different meanings to different people.
the Mills chapel, for example, could be called non-demoninational. The big stone altar is in the shape of a square cross, but it looks very pagan and it’s right in the centre of the round building. There’s a pipe organ. the glass is not stained. AFAIK, there are no pictures of Jesus. the pagan group used to have rituals there when it rained. (cuz who wants to stand around ina field in the rain?). anyway, I’m sensing that this is one of those east coast / west coast things.
So we sat down in the chapel and somebody went looking for the organist. the chapel inside wasn’t finished and there were carpenters with saws, hammers, drills and hardhats busily assembling the alter region. the organist (who is so so so gay) talked about the organ for a few minutes and explained that not all the pipes had arrived. He then began to play Wolff’s piece, sans some of the pipes, while the carpenters continued to work.
the piece had several quiet spots (or maybe just the pipes were missing) that got completely drowned out by the carpentry. It was 20 minutes long and hard to focus on in the din. Also, the audience’s focus was difuse and distracted, further making it harder to concentrate on the peice. not that they were making noise (not that it would have made a difference is they did), but just that the energy wasn’t right. It was a very odd organ concert. We clapped at the end.
It was reminding me of a John Cage story, published in Silence and recited in Indeterminancy. In it, Christian Wolff was playing a piano piece next to an open window. through the window came many loud sounds from passersby and automobiles and boat horns and airplanes that made it hard to hear the music and occasionally drowned it completely out. after he had finished playing, somebody asked him if he could play the peice again, but with the window closed. He replied that he would be happy to do so, but the outside sounds had no interfered with or obstructed the piano piece at all.
so we asked Wolff (the same Wolff as in JC’s story) what he thought of the recital and he said that he thought it was great, didn’t mind the carpentry at all. Maybe he should have a carpentry percussion part to go with it. Some of us (christi) giggled.

Research for this semester – maybe thesis

Celeste Hutchins
1 October 2003
Proseminar
Research Proposal

I would like to research music from around the last section of the 100 years war. I am specifically interested in music somehow related to Joan of Arc. For instance, after she raised the Siege of Orleans, a mystery play was written in her honor, which has been performed nearly every year since. I don’t know anything about the music attached to this play, but would like to find out how much it has changed over time and if early transcriptions are available, so that I can analyze the music. If this music is unavailable, I will instead focus on music she might have heard, including the liturgical music performed at Dom Remy, her local cathedral, or the music played then at Charles’ court.

I am interested in this topic because I would like to write a Mass for Joan of Arc (since she’s a saint) and eventually an opera and I would like to represent her with music that would have been somehow meaningful to her. I don’t know what research has been done on this topic, but I’ll be talking with Professor Alden about it. Also, I will be in France over fall break and hopefully will be able to do some research then.

Moo themes

Ok, so I was looking at modifying the source code of SuperCollider to make network connections . . . what a nightmare! So instead, there will be a perl program, that SC will execute that will handle all of the text stuff and the network stuff and everything that supercollider isn’t good at. so the perl thing will do everything but make sound. (Note: maybe someone ought to add sound capablities to Perl. An OSC lib, maybe?)
Players will have the options of attaching arrays to themselves caled themes. So you would need to create a property on yourself called a theme and then put stuff in it. This theme will look *cough* just like a ring tone on a Nokia phone. Some of you have experience with cell phone ring composers and will be able to hear your theme on a phone before comiting it to yourself. I just wrote a ring tone. It goes 4d1 4a1 4b1 2c2 32d2 32#f2 8a2 8f2 8f2 16d2 16f2. It’s pretty jazzy at the end.

Anatomy of a Nokia Composition

the first number is the note duration. It is the denonominator of a fraction whose numerator is 1. So smaller numbers are faster. In musical terms, a 1 is equivalent to a whole note, 2 a half note, 4 a quarter note, 8 and 8th note, etc.
Then comes an optional # and then the note name. There are no flats. but that’s ok because in equal temperment Bb = A#. but since it’s backwards, in this case it equals #a.
The last number indicates the octave. A low number is a low octave. A d2 is an octave higher than a d1.

Garunteed support changes

You will be able to use any number, not just a power of 2, to denote duration, but it will still be in denominator format. therefore a starting number of 3 will be meaningful (in musical terms, it would be a half note triplet, since our notation isn’t so good at divisions by 3). You will be able to specify b for flat or # for sharp. Equal temperment may not be supported. Because an optional ‘b’ means flat, all the note names must be in upper case. and the final integer specifyin octave will be optional (if you don’t specify, then the program will pick an octave and your theme may get moved around, from high to low). you can pick an arbitrarily large or small octave number. the low threshold of human hearing is around 20 Hz, so octave 0 will be down there someplace. the high threshold of human hearing is around 20000 Hz, so figure out what octave number that is and that’s you upper lmit. Extremely high pitched sounds are likely to be played at a much lower volume.

yeah, yeah, ringtones

I’m not crazy about cell phone, or their ringtones, but I have to admit that this has cross-fertalization possibilities. Some phones allow you to specify different rings for different people in your addressbook. Also, some websites have the ring tones for different songs up on them. My phone plays the Indiana Jones theme song, for example. there are Ives-ian possibilities lurking danegerously around.
telnet happy.xkey.com 3333

Moo song

I’ve decided that my midterm SuperCollider project will be a program that logs into the MOO (telnet xkey.com:3333) and interprets MOO text somehow into sounds. for this, I will need to create a $room, which will be called the Music Studio and a $player for my SuperCollider program to use. It would be better if the MOO and SuperCollider could communicate through some other means, besides a fake player. But I don’t think MOO objects are allowed to make network connections and I can’t think of another way to log in. (If you know otherwise, please let me know.)
anyway, the SuperCollider program will log into the MOO and from the text, create sounds. There’s a few ways to do this. I could scan the text for particular key words, which would cause particular actions. for instance, the word “teleport” usually signifies that a new player has come in, and so when “teleport” comes across the network connection, some particular responce could occur. Or, I could do statistical analysis on the network text, changing the texture of the music based on the frequency and length of the received text. for instance, a lot of text arriving quickly would indicate that there’s a lot of activity on the MOO and perhaps the program could respond by creating a dense texture of sounds. another thing that could happen is that $plays (and other objects) could have an optional “themes” property, which would contain information meaningful to the SuperCollider patch, so it could look at your theme property and, in Wagnerian style, play your theme while you are in the room.
If you have any suggestions or ideas about how you would musically represent MOO communications, you should pass them along. to be fair, the patch output will have t be streamed, so $players can listen. so think also of what musical noises might convey useful information. If you have a desk job and can sit with headphones on, would you like to hear it play something that tells you that somebody just logged on or one of your friends is talking?
One of the local guys is giving me and old PC with linux on it, so I can set up a test streaming server. Alas, it will probably not have a fixed IP address until it passes a certain amount of testing, since xkey must be stable. Or, Mitch could put a stable RedHat PC in the CN kitchen that could get nightly builds put up on it. The PC currently there is having some sort of disk problem.
This project probably wouldn’t work well for Stony Brook, but that’s ok. I have a final project after this one.

Dulcimer Piece

I have no dynamics (loudness) markings nor do I have pedal markings, nor anything else except notes and rests. Long Version, Short Version. I think the long one may be too long, but the piece isn’t over at the short one because the two parts don’t line up yet. So I think I have have to do a retrograde or inversion or a retrograde inversion for the second part. Except that the last notes sound very final, so I may need to open with the retrograde inversion. On Wednesday, I’ll get feedback on this from Alvin Lucier, so, you know, my time here isn’t being wasted by stupid paper and whatnot and even if this piece doesn’t draw on anything I didn’t know before, well, the feedback will.
I will be playing for a half hour at SUNY Stonybrook on Dec 6th. I’ll prolly be doing laptop music (even tho, officially, I hate laptop music) because I’ll prolly be playing a project from my SuperCollider class. Now that I have pressure on my projects, I have absolutely no idea what to do. My ideas flew out of my head or don’t seem robust enough or whatever. Of course, I could (should?) play several songs, or at least multiple movements during the half hour unless I can somehow create something that evolves over time, so it’s changing in an algorythmic way or something. How would you do that? changing notes and rythms here and there, but it would have to be changing in an interesting way, like going from a plainsong kind of thing to Ars Nova to Ars Subiliot. That last word is mispelled. It was an outgrowth of ars Nova that was really popular around the papacy at Avinion, but got dropped when the schism ended, even tho it was more complicated and possibly more interesting. The writer of a book I just read said it was more interesting, but I duno, as I haven’t listened to any yet. the scores and recordings section of the library closes at 5:00 pm on saturdays and today I haven’t left the house today.
Aaron (the housemate) brought a futon back today, so now there is a place for you to sleep when you come visit. Tell me when you’re coming and I promise to buy a second set of sheets! the futon used to belong to Scott Rosenberg who went to Mills as a music student when Christi and I were there. Everyone go hum It’s a Small World to yourselves until you go insane.
If I very clearly picture myself getting a package in the mail containing vegan brownies, maybe one will appear. Christi’s mom was always sending brownies when Christi was a freshwoman at Mills. Brownies . . . Brownies . . . Brownes on a cold fall day . . .
My CD burner just made a Bad Sound….

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>The Office of Public Safety has received reports of thefts of 9 more
>laptop computers from student residences over the past week, bringing the
>total number of reported thefts to more than 60 since the semester
>began. In almost all the cases, unknown person(s) gained access to
>student residences through unlocked doors or windows.

ackackack

Everyone here is much smarter than i am and i have 2791634712364946 things that i’m supossed to do and angela says that this place is “like a spa” compared to her last MFA program (she has a Masters in clarinet performance) and Ron says that if I feel like I’m on the verge of a nervous breakdown by fall break, then I’m on schedule for new MFA students. I have to go look up obscure stuff about the hundred years war and i’m a week behind on my supercollider homework and there’s a big midterm project for supercollider and there’s some big projects in my compositon seminar and i’m suppossed to go to two sessions of a class i “wouldn’t normally take” and they have to be the same section of the same class and then i have to write some short thing about it, which isn’t too hard except that the profs have the right to refuse me admission, which apparently, they like to do and that’s due soon and next week i have to teach the students in the class that i TA how to use digital performer, which i’ve never really used, except i bought a copy three years ago and don’t use it because i HATE it because i can’t get it to do anyhting i want it to do, which is not really the best place to be coming from to teach other people to use it and i already feel like i’m on the verge of a nervous breakdown because this is not “like a spa” to me at all, unless you went to a mean spa that stressed you out instead of putting you in a nice hottub or something.
I just got a fantastic offer over email to play a 30 minute set of improv stuff at SUNY.
deep breaths
I spent many hours on the phone with tech support today. my dsl stuff arrived on monday and today i had some time to install it, but there’s no phone jacks in my room, so i went to the campus puter store to buy an airport base staion, but they didn’t have any and talked me into buying a much cheaper linksys wireless router instead. if i were smart, i would have left as soon as i found out they had no base stations instead of arguing with the manager about whether or not he should sell me the display base station for half price (cheap bastard) and i certainly would not have bought the linksys monstrosity and if i were msart, i would gave given up when it didn’t plug and play instead of debugging it and if i were smart, i would NEVER have called tech support, especially not called them back after they explained that macs were unsupported and i had to work out part of the problem for myself and see, if i had these better time management skills, i would have had time to work a half hour set out. maybe i can write an “Unpredictable Music” proggie in super collider and use it as my midterm project and then play it at SUNY.
So if anyone asks how i’m doing, tell them i’m ahead of schedule, according to my advisor.

Ack

For next week, we have your first personal research roundtable. This
should be a 15-minute presentation and 5-minute question period. The
idea is to air some thoughts about what you might be working on
towards as a final project and get some feedback. E mail me if you
want to talk about topics, though basically I’m open to anything.
We can refine our procedures for next rounds, for example decide if
we want a webpage for conversation, etc.

I’m thinking about saying I want to do something about songs from the 100 years war. But what would I say about it for 15 minutes?