Welcome, Guest. Please login or register.

Login with username, password and session length

 
Advanced search

1076412 Posts in 71252 Topics- by 18949 Members - Latest Member: sfdubbler
Jump to:  
Pages: 1 2 3 4 5 [All]   Go Down
Print
Author Topic: Just Intonation Composition  (Read 55383 times)
0 Members and 1 Guest are viewing this topic.
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« on: Dec 31, 2007, 08:01AM »

I'm currently doing some research and experimentation with justly tuned composition. Is there anyone else on here who has experience with this stuff?

If you don't know who Harry Partch is, you probably don't know what I'm talking about.
« Last Edit: Apr 10, 2008, 12:58PM by Andrew Meronek » Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
Dave Tatro
The Python's Python
*
Offline Offline

Location: St. Pete, Florida
Joined: May 10, 2006
Posts: 4935

View Profile
« Reply #1 on: Dec 31, 2007, 09:45AM »

I have experience with it only to the degree that most proficient players of non-keyboard instruments and vocalists use it naturally. So any composition written for them would almost by default be written for just temperament.

If you are referring to writing compositions specifically for fixed-intonation instruments like piano but having them tuned in just temperament, then no, I have never done that.

I suppose that one could write for winds/strings and rather than trusting the players' ears to guide them into acoustic perfection, could use some kind of symbols in the music to suggest pitch alterations that would bring the pitch more into line with just temperament for whatever chord/scale was happening at the time. Seems to me that if your players were good enough to pull that off, they would probably be playing in tune already.....
Logged

"He also inevitably discovered the similarities between glass doors and forcefields."- marchingknight
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #2 on: Dec 31, 2007, 10:58AM »

The trick is that just temperament allows for a lot more pitches per octave than 12, and there are a couple other issues I've come upon as well.

For example, the ratios 6:5, 7:6, 11:9, and 19:16 all refer to minor thirds of varying types. Traditional notation doesn't really help distinguish these.

And modulating can get thorny, too. There are at least three basic routes to take there: modulate to the new pitch without adusting any of the ratios, like old practice before equal temperament hit it big (the main reason why each key is thought to have it's own character - in an unequal temperament, they literally do), play chords justly but modulate via equal temperament as most instrumentalists do nowadays, or modulate via going to the new pitch, keeping the justly tempered tone and adjust all the ratios to that tone, as a capella choirs tend to do. Each will have it's own unique sound.

 . . . so how do you notate all of this?
Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
SilverSonic

*
Offline Offline

Location:
Joined: Sep 29, 2006
Posts: 405

View Profile
« Reply #3 on: Dec 31, 2007, 11:00AM »


If you don't know who Harry Partch is, you probably don't know what I'm talking about.

There might have been a few around before Harry hit the scene...
Logged
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #4 on: Dec 31, 2007, 11:02AM »

There might have been a few around before Harry hit the scene...

LOL

Touche!
Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
Dave Tatro
The Python's Python
*
Offline Offline

Location: St. Pete, Florida
Joined: May 10, 2006
Posts: 4935

View Profile
« Reply #5 on: Dec 31, 2007, 01:13PM »

The trick is that just temperament allows for a lot more pitches per octave than 12, and there are a couple other issues I've come upon as well.......

 . . . so how do you notate all of this?

Okay, I see what you are getting at now. You want all the possible permutations of just temperament available on call. The only way I see this happening accurately in a performance situation would be either with tunable instruments (piano, harp, etc.) or electronics. The notation would be to specify the exact tunings in Hz in the score. Then in performance, players would have to switch instruments at the appropriate times. Certain instruments might be able to be re-tuned on the fly.

That's the real world. In theory, you absolutely could come up with a reasonably concise and specific notation system for performance by "ear-tuned" instruments or voice. I'm not sure what that would look like but I'm sure it could be done. But I'm not sure it would ever be able to be performed accurately unless it was very simple stuff musically.
Logged

"He also inevitably discovered the similarities between glass doors and forcefields."- marchingknight
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #6 on: Jan 01, 2008, 03:08AM »

 . . . so performances by the Kronos Quartet and the Kepler Quartet don't count?

 :-P


I do know that string quartets are more than capable of handling some just temperament systems having, say, 16 to 20 pitches per octave. As is a certain brass instrument . . .

Oh, and notating the tuning isn't quite as simle an issue. There are two approaches that I've found: pitches based on fequencies, similar to our five-line staff system, or pitches based on their relationship to other pitches.

What little stuff I've been able to find still uses the five line staff, but, depending on the composer, combined the staff with using both ideas: adding more symbols besides sharp signs, flat signs, double sharps, etc. before notes, or literally writing fractions above notes. The former tells you where in your instrument to play a note; the latter tells you what to listen for. The former can get really messy if there are a whole lot of different pitches to deal with; the latter is a lot cleaner with numerous pitches but is completely foreign to most performers and is thus very hard to read.

I've been thinking of a compromise, writing cent adjustments over each note, which makes the fractions much easier to understand in terms of where to play a note on your instrument . . . but you no longer know what to listen for.

Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
Dave Tatro
The Python's Python
*
Offline Offline

Location: St. Pete, Florida
Joined: May 10, 2006
Posts: 4935

View Profile
« Reply #7 on: Jan 01, 2008, 10:57AM »

. . . so performances by the Kronos Quartet and the Kepler Quartet don't count?

 :-P

Touche.

Okay, apart from very elite small chamber groups that play together constantly, good luck! ;-)

Quote
I do know that string quartets are more than capable of handling some just temperament systems having, say, 16 to 20 pitches per octave. As is a certain brass instrument . . .

Which may answer my next question, which is what your intended instrumentation was going to be.

Quote
I've been thinking of a compromise, writing cent adjustments over each note, which makes the fractions much easier to understand in terms of where to play a note on your instrument . . . but you no longer know what to listen for.

Maybe a system of arrows up or down next to the notes, different lengths of arrows indicating how sharp/flat to play the given note. Makes the piece writable in standard notation, which makes it easier to get readings. Then, with practice, the performers can refine the intonations. Might work. Don't know
Logged

"He also inevitably discovered the similarities between glass doors and forcefields."- marchingknight
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #8 on: Jan 01, 2008, 08:33PM »

Touche.

Okay, apart from very elite small chamber groups that play together constantly, good luck! ;-)

Which may answer my next question, which is what your intended instrumentation was going to be.

Maybe a system of arrows up or down next to the notes, different lengths of arrows indicating how sharp/flat to play the given note. Makes the piece writable in standard notation, which makes it easier to get readings. Then, with practice, the performers can refine the intonations. Might work. Don't know

Yeah; I've seen the arrow idea, too. Positively, it's a very straight-foreward, simple tool; negatively, it still doesn't tell the performer what to listen for. And I think it may also be a tool that some composers have tried with some success.

I've even considered using something besides a five line staff; however I know that whatever solution I get, I have to be able to write the music using five line staff notation if I ever want it to be performed, even if it won't be the most ideal way of writing it.
Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
rlb
Demiurge

*
*
Offline Offline

Location: Inferno, level 7
Joined: Apr 15, 2000
Posts: 7983
"Hails of derisive laughter, Bruce!"


View Profile WWW
« Reply #9 on: Jan 01, 2008, 09:01PM »

Pass me down my diamond marimba...
Logged

Dr. Richard L. Byrd, Forum Director
Every man, wherever he goes, is encompassed by a cloud of comforting convictions, which move with him like flies on a summer day.
   --Bertrand Russell, 1950
Dave Tatro
The Python's Python
*
Offline Offline

Location: St. Pete, Florida
Joined: May 10, 2006
Posts: 4935

View Profile
« Reply #10 on: Jan 02, 2008, 11:42AM »

Pass me down my diamond marimba...

I will, as soon as I dig out my trusty 16 position slide chart! :D
Logged

"He also inevitably discovered the similarities between glass doors and forcefields."- marchingknight
Dave Tatro
The Python's Python
*
Offline Offline

Location: St. Pete, Florida
Joined: May 10, 2006
Posts: 4935

View Profile
« Reply #11 on: Jan 02, 2008, 11:48AM »

Yeah; I've seen the arrow idea, too. Positively, it's a very straight-foreward, simple tool; negatively, it still doesn't tell the performer what to listen for. And I think it may also be a tool that some composers have tried with some success.

I've even considered using something besides a five line staff; however I know that whatever solution I get, I have to be able to write the music using five line staff notation if I ever want it to be performed, even if it won't be the most ideal way of writing it.


You might have to create two versions, a "readable" one and a truly specific one. I feel your pain, man! 

I'm a little confused about the phrase "negatively, it still doesn't tell the performer what to listen for." What DO you want the performer to listen for? (Sorry if this is a stupid question, but I don't want to erroneously assume the obvious when discussing something nontraditional. ;-))
Logged

"He also inevitably discovered the similarities between glass doors and forcefields."- marchingknight
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #12 on: Jan 02, 2008, 08:18PM »

You might have to create two versions, a "readable" one and a truly specific one. I feel your pain, man! 

I'm a little confused about the phrase "negatively, it still doesn't tell the performer what to listen for." What DO you want the performer to listen for? (Sorry if this is a stupid question, but I don't want to erroneously assume the obvious when discussing something nontraditional. ;-))

Five-line staff notation for a single instrument (i.e. you don't see all the notes everyone is playing on your part) cannot tell you what part of the chord you're playing, even with up/down arrows or cent adjustments. I just think it would be neat to be able to tell the performer something like: "your D here is the 5/4 ratio major third." That way, the performer knows what kind of harmonic color to listen for when playing the note. I think that'll ultimately be a more effective way of thinking about controlling tuning and color in ensemble playing than just showing cent adjustments or arrows.

Does that make sense?
Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #13 on: Jan 03, 2008, 09:07AM »

Pass me down my diamond marimba...

It just so happens that one of the recordings I'm waiting for has as part of the instrumentation a diamond marimba . . .

 Eeek!
Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
Dave Tatro
The Python's Python
*
Offline Offline

Location: St. Pete, Florida
Joined: May 10, 2006
Posts: 4935

View Profile
« Reply #14 on: Jan 03, 2008, 02:59PM »

Five-line staff notation for a single instrument (i.e. you don't see all the notes everyone is playing on your part) cannot tell you what part of the chord you're playing, even with up/down arrows or cent adjustments. I just think it would be neat to be able to tell the performer something like: "your D here is the 5/4 ratio major third." That way, the performer knows what kind of harmonic color to listen for when playing the note. I think that'll ultimately be a more effective way of thinking about controlling tuning and color in ensemble playing than just showing cent adjustments or arrows.

Does that make sense?

Okay, so it was the obvious after all! Embarrassed!

So, how to impart this much info note by note... hmmm... Well, I'm not sure that you should have to with players of the caliber that you would be aiming to write for. When I play music, anything from solo to orchestral to jazz to big band, I am constantly aware of what chord tone I am playing if the music is based on relatively traditional tertiary harmony. So for myself, a small arrow or a 5/4 next to the note would suffice. You don't have to tell me I am playing a major third because I already know that. Maybe I am wrong here, but I have to imagine that most good chamber musicians would share that ability.

Sometimes less is more when it comes to notation. Today, composers tend to micro-manage players a little to much IMHO. In the past, much more was left to the players, who were trusted to understand the stylistic conventions of the music. Of course, you want to write stuff that is NOT conventional, at least in a modern way. Perhaps instead of specific markings on every pitch you might be able to achieve the same effect more simply with some good performance instructions and then very minimal markings on the music itself. I am all for simple, direct, efficient communication from composer to performer!

Don't forget, just because you figure out a way to mark every pitch as you want it down to the cent does not mean anybody will be able to play it to that level of precision....


Logged

"He also inevitably discovered the similarities between glass doors and forcefields."- marchingknight
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #15 on: Jan 03, 2008, 09:47PM »

. . . if the music is based on relatively traditional tertiary harmony.

Good key point.

Even in traditional tertiary harmony, there are some grey areas.

For example, there are two major thirds, one 14 cents flat and one 35 cents sharp. Arrows might help here.

And there are more than 3 minor thirds, but the most commonly used ones are 6:5 (16 cents sharp), 7:6 (33 cents flat), and 19:16 (2 cents flat). In this case, a downward arrow simply won't do because there are two downward options. And each tuning creates a unique color, so this can be pretty important.
Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
timothy42b
*
Offline Offline

Location: Colonial Heights, Virginia, US
Joined: Dec 7, 2000
Posts: 12039

View Profile
« Reply #16 on: Jan 03, 2008, 11:32PM »



I do know that string quartets are more than capable of handling some just temperament systems having, say, 16 to 20 pitches per octave. As is a certain brass instrument . . .


I do not think this is the case.  String quartets, the good ones anyway, will adjust some chords to have simple ratios.  You can call those chords just tuning, most people probably would. 

But temperament really means adjusting the scale to remove the comma, and I know of no string players who can play a tempered scale.

And part of the reason this even works for stringed instruments is that they are driven rather than struck or plucked, affecting the way the harmonics line up. 

A lot of digital pianos have a button for changing temperaments.  Have you tried playing some familiar pieces in various ways?  If you don't have one, sneak into a music store and use the display models. 
Logged

Tim Richardson
fluor

*
Offline Offline

Location: norway
Joined: Oct 3, 2005
Posts: 1021

View Profile
« Reply #17 on: Jan 23, 2008, 12:53PM »

since this is a trombone forum, i know that roy nagorcka has written some JI pieces for trombone. when he does he divides each half note into ten equal parts. he writes in regular notes, but with an arrow and a number over each note... the arrow for which way to adjust, and the number for how many cents that way.

ie 3V means 30 cents down.

i think that some of his pieces for trombone are recorded.

clarinet is another instrument which is easy to play micro tones on.
Logged
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #18 on: Jan 23, 2008, 07:14PM »

Hmmm . . .

I looked up Roy Nagorcka on Google, Amazon, and Wikipedia, and came up with nothing. Is the spelling correct?

FYI, I am using a solution similar to the one you describe, in order to create notation that is easier for performers to interpret. Except I just give cent adjustments, like +18 or -31. My thought is that a performer, once playing through the piece, will start to recognize that +18 is the adjustment for only a couple possible chords and will have a very specific sound, even compared to +16, which would normally be indistinguishable, but which would be used in other chords thus giving the performer an expectation of a different sound.

Oh, and if that is a picture of you, hooray for trombone chicks! We need more of 'em! That, or I've been in the Army way too long.
Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
BFW
Pun Gent

*
*
Offline Offline

Location: Alabamor
Joined: Aug 24, 2002
Posts: 21975
"Paronomasiacs Homonymous"


View Profile
« Reply #19 on: Jan 23, 2008, 08:38PM »

Ron Nagorcka, perhaps?

http://www.ronnagorcka.id.au/
Logged

Brian

Our supreme responsibility is the moral obligation to be intelligent. -- Oliver L. Reiser
fluor

*
Offline Offline

Location: norway
Joined: Oct 3, 2005
Posts: 1021

View Profile
« Reply #20 on: Jan 24, 2008, 02:26AM »

ooops. ron was the right name yeah :)
i've been to a composition class with him once. his a cool guy

and heh no it's not me. it's a norwegian singer called siri gjære :p
Logged
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #21 on: Jan 24, 2008, 03:46AM »

siri gjære

Wow; I don't think I could pronounce that if I tried.
Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #22 on: Jan 24, 2008, 03:49AM »

Ron Nagorcka, perhaps?

http://www.ronnagorcka.id.au/

Hmmm . . .

At first glance, he seems to take to some of John Cage's ideas. Or perhaps native Japanese music. Reflecting nature and stuff.
Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #23 on: Jan 24, 2008, 04:01AM »

By the way, here are some useful websites I've found concerning just intonation:

http://www.kylegann.com/microtonality.html

The crash course on Mr. Gann's page is excellent.  Wikipedia has some good stuff, too:

http://en.wikipedia.org/wiki/Microtonal_music
http://en.wikipedia.org/wiki/Just_intonation
http://en.wikipedia.org/wiki/Tonality_diamond
http://en.wikipedia.org/wiki/Harry_Partch%27s_43-tone_scale

The Microtonal Music article has a huge list of composers at the end, and for many tells what general category of theory they used, whether it was just intonation or some form of equal temperament or something else.
Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
David Schwartz

*
Offline Offline

Location: Belmont, MA
Joined: Aug 13, 2001
Posts: 1181

View Profile WWW
« Reply #24 on: Jan 24, 2008, 04:19AM »

. . . so how do you notate all of this?

I'm late joining this conversation, but the same question came up when I was putting together my little just intonation scale and arpeggio practice book, Breakfast.  There I show adjustments in just one place, in a major scale and in a minor scale on page 13, expressed as cent adjustments, like -14, relative to the equal tempered scale. Elsewhere the performer trusts his ear and the accompaniment.

Fine barbershop quartets - for example the Buffalo Bills in the television production of The Music Man a few years ago - certainly don't need notational guidance to sing in just intonation. And surely that is how they sing. So, as Dave T. said up top, trust the performers.

On the computer your music notation program may have a pitch bend function which lets you adjust each note. That function may not print the adjustments on paper, but you can see them on the screen. The program takes care of the documentation and notation for you.

For a thorough discussion of the topic I highly recommend W.A. Mathieu's Harmonic Experience.  At the link, select table of contents to see the broad scope of this 576 page book.  In an appendix, Glossary of Singable Tones in Just Intonation, Mathieu identifies sixty intervals within the octave by ratios to the tonic!

Good luck, Andrew.

David

Logged

David A. Schwartz, Belmont, Massachusetts
Bordogni and Breakfast Website
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #25 on: Jan 24, 2008, 07:43AM »

For a thorough discussion of the topic I highly recommend W.A. Mathieu's Harmonic Experience.  At the link, select table of contents to see the broad scope of this 576 page book.  In an appendix, Glossary of Singable Tones in Just Intonation, Mathieu identifies sixty intervals within the octave by ratios to the tonic!

Good luck, Andrew.

David

Well, well. That book looks like exactly what I'm looking for.

Thanks! Good!
Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #26 on: Feb 01, 2008, 10:46AM »

Here's a slight update:

I received some just-tuning composed music, and some of it is rather awful, and some of it is absolutely awesome. I'll cover the awesome:

The Bells of New Albion by Terry Riley. Riley is known as one of the important minimalists, but he really doesn't deserve the title. This piece is a 2 hour long solo piano work, but it involves a whole lot of improvisation as well as the tune structures which do involve repetitive elements (but not strict like in early Steve Reich or Phillip Glass works) - I really like it a lot. You don't need to listen to the entire two hours at once; he divides the piece into many discernable chunks, and you can listen to each chunk (5-20 minutes) at your leasure. That just-tuned piano is really something else!

And the other masterpiece I found is a set of string quartets written by Ben Johnston and performed by the Kepler Quartet. Man oh man, is this a great album, and man oh man will this album challenge your ears if you're not ready for it! Unlike a piano, in which no matter how creative you get can only really use twelve different pitches, stringed instruments are a lot more flexible in their tuning, and Johnston really exploits that. Not only is the music harmonically and rhythmically challenging, he writes great melodies and gets a huge palette of tone color out of the strings, and the Kepler Quartet really put forth a passionate performance of this stuff; I'd say it's probably the best recorded string quartet performance I've ever heard.
Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
timothy42b
*
Offline Offline

Location: Colonial Heights, Virginia, US
Joined: Dec 7, 2000
Posts: 12039

View Profile
« Reply #27 on: Feb 08, 2008, 06:44AM »

How do you suppose he notated that string quartet music? 
Logged

Tim Richardson
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #28 on: Feb 08, 2008, 09:38AM »

I did get an explanation from Kyle Gann of the notation used in the links below:

Quote
FAC, CEG, and GBD are purely tuned 4:5:6 triads

# = 25/24
b = 24/25
+ = 81/80, adjusting for the syntonic comma
- = 80/81
7 = 35/36
upside-down 7 = 36/35
^ (arrow pointing up) = 33/32
v (arrow pointing down) = 32/33
13 = 65/64
upside-down 13 = 64/65

It's not really possible to do justice to this in typing. The arrows are really arrows, and the sevens are sometimes combined with sharps and flats as a little diagonal line hanging from the top or rising from the bottom. I've been looking around the web for a sample of one of Ben's scores, and all I've found is this one, which doesn't use many of the accidentals:

http://www.smith-publications.com/catalog/samples.pl?id=29

Actually, here's another, which has a little more going on:

http://www.smith-publications.com/catalog/samples.pl?id=11

What people object to are the use of pluses and minuses, which sometimes seem counterintuitive; for instance, C-G, E-B, and F-C are perfect fifths, but D-A is not - the A needs a + or the D a -. There's a competing notation called HEWM using the same symbols but starting with the Pythagorean scale instead of pure triads, which I don't like. And in any case, while the notation makes the relationships perfectly precise, it still doesn't provide much easily digested information for the performer, who has to pretty much memorize where the pitches are anyway.
Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #29 on: Feb 27, 2008, 12:19AM »

Just a useless random addition to this whole idea:

I was thinking about how 3/2 is the perfect fifth ratio, which puts a pitch exactly between the octaves, but our ears tell us that the equal tempered tritone is actually right in between, based on how many steps there are to get to the tritone.

So I was thinking this, then I ended up doing some math.

For starters, the equal tempered tritone is simple: starting pitch * square root of 2.

And I was thinking that it's funny how the various just tuned tritone intervals seem to evenly spread around this equal tempered pitch. For example, 7/5 is 17 cents flat, and 10/7 is 17 cents sharp. And then I thought some more, and in fact there are a whole bunch of just tuned pitches that do the same thing, and cents away from the tritone also correspond to cents away from the octave root. For example, that 7/5 tritone, being 17 cents flatter than the equal tempered tritone, is also 583 cents away from the lower octave root, and the 10/7 tritone is 583 cents away from the upper octave. Major thirds are the same way: a major third is 14 cents flatter than the equal tempered third and 386 cents above the lower octave and the minor sixth is 14 cents sharp, 386 cents below the upper octave.

So, I was thinking this, and I was thinking that it might be cool if I could find out why this pattern was happening. I did a bit of basic math remembering (and a little research to fill in the gaps), and I came up with this:

log2(x/y)=1-log2(2y/x)

This is log base 2, and x and y are the numerator and denominator of the pitch ratio in question. It's actually pretty simple; simply multiply both sides by 1200 and each side represents the number of cents toward the octave. And seeing how this example requires the log base to be 2, it becomes easier to see why the pitches also "revolve" around the square root of 2. Oh, and 2y/x is how you get the inverse interval: a major third, being 5/4 has as it's inverse a minor sixth: 4*2/5 or 8/5. Same with the above-mentioned tritone: 7/5 and 5*2/7=10/7

Another way of thinking about this is that our ears hear intervals via multiplication of frequencies, not by addition of frequencies. Multiply the square root of 2 (the E.Q. tritone) by itself, and you get 2, which DOES give you double the starting pitch. Multiple 3/2 (the perfect fifth) by itself and you get 9/4, which does NOT double your starting pitch. Duh.

Hooray for basic algebra!!! Idea! Idea! Idea! Good! Clever Clever Amazed Eeek!
Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
BFW
Pun Gent

*
*
Offline Offline

Location: Alabamor
Joined: Aug 24, 2002
Posts: 21975
"Paronomasiacs Homonymous"


View Profile
« Reply #30 on: Feb 27, 2008, 07:56AM »

Another way of thinking about this is that our ears hear intervals via multiplication of frequencies, not by addition of frequencies.

Doesn't this result just pop right out of the fact that intervals are ratios?

I think what you've done, interestingly enough, is start from the calculations for equal temperament and derive the octave, rather than the other way around.  You've also run into some of the wonderful properties of exponents and logarithms.  It's great stuff.

This kind of exercise is why I'd like to teach math someday.  I think these results are pure beauty, and I want to help other people see it.
Logged

Brian

Our supreme responsibility is the moral obligation to be intelligent. -- Oliver L. Reiser
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #31 on: Feb 27, 2008, 09:14PM »

Doesn't this result just pop right out of the fact that intervals are ratios?

Yeah; I suppose that is a pretty good clue.


Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
dwdraw
*
Offline Offline

Location:
Joined: Jun 8, 2005
Posts: 95

View Profile WWW
« Reply #32 on: Mar 02, 2008, 11:20AM »

A piece of software you can use to help experimentation (create a MIDI with performance or notation software) is Scala:
http://www.xs4all.nl/~huygensf/scala/

(I mentioned it before, but I'll explain how I used it to experiment with Just Intonation.)

The feature I liked about it was that it Can retune existing MIDI files. You can convert a standard MIDI file to be in any tuning via pitch bend commands or a MIDI Tuning Standard tuning specification.

This feature makes it a useful tools for sort of quickly previewing simple music using Just Intonation. Basically, you have to export each tonal center as a different MIDI, create 12+ scale files (one for each key), apply appropriate scale files to the appropriate sections, and then find a way to playback the sections consecutively. For complex music, or note perfect music, you would still have to find a way to manually adjust for the different types of intervals (e.g. grave minor seventh versus minor seventh), but at least a bulk of the work could become automated when you're producing simple previews.

- - -

If you're looking for beatless music, Just Intonation won't go that far in all situations. (For example, the ensemble trick is to play a just dominant chord, but to weaken the volume of the dominant seventh to mask the beats produced by the seventh in a just setting.)

To make reading easier, most composers make use of the five-line notation system and create a way to notate deviations to that system. Of course, in the realm of easy-to-play, one could use regular notation with a prepared piano or one could write music playable by computer using any exotic notation that can eventually be read by the computer.

- - -

The math used to calculate this stuff is pretty simple. A spreadsheet I lost did this, but it calculated for any given intonation standard (e.g. 440) and base frequency a list of the cents from root for ET, cents from root for Just (by converting the frequencies to the relative [per]cent notation), and found the exact adjustments through subtraction. This is still all abstraction and is a few steps removed from being able to calculate how to adjust any set of frequencies to be as beatless as possible (or the math--probably simple math considering sound revolves around multiples--needed to describe beatless combinations of frequencies).
Logged
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #33 on: Mar 03, 2008, 12:44AM »

If you're looking for beatless music, Just Intonation won't go that far in all situations.

Actually, no. I'm looking for more direct control over consonance vs. dissonance.

That program does look interesting. I'll have to check it out the next time I have a chance.
Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #34 on: Mar 30, 2008, 07:53AM »

Just a slight update on my literature search.

As some of you may know, a lot of folk music (not all) is tuned to just intonation. Perhaps the most commonly heard of those is bagpipe music.

My roommate is a big fan of early European music, and he showed me a really interesting CD that uses just intonation:

"Myths from Medieval Iceland," by a group called "Edda Sequentia." Sung in Icelandic, with some traditional instruments; the most identifiable to me being a kind of viol. Some of the voice leadings are very unique; it caught my ear. I thought some of the voice crossing resulting in major second intervals was particularly striking, given that those seconds were still tuned to the original just tuned scale, meaning that each second interval was not necessarily the same.
Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
timothy42b
*
Offline Offline

Location: Colonial Heights, Virginia, US
Joined: Dec 7, 2000
Posts: 12039

View Profile
« Reply #35 on: Mar 31, 2008, 11:58PM »

As some of you may know, a lot of folk music (not all) is tuned to just intonation. Perhaps the most commonly heard of those is bagpipe music.


Just idly wondering about something.

Inharmonicity.

For example, the piano is not actually tuned to equal temperament.  Because of inharmonicity (the stiffness of steel strings causes the overtones to not quite line up) the piano sounds bad if the fundamental of each pitch is tuned to ET, because then the overtones clash.  So the piano is stretched, moving the fundamentals away from ET, so that the overtones are close enough that the piano sounds like it is ET even though it is not quite. 

The same thing must apply to just intervals (not sure just temperament itself is possible.  But intervals are.) 

But not all instruments have this problem.  An organ wouldn't, for example.  The pipes are driven by constant input, therefore the overtones line up.  Bowed strings would be the same.  Plucked strings would be different. 

So, we have folk music with bagpipes or hurdy gurdy, no inharmonicity.  But with guitar or hammered dulcimer, large inharmonicity.  Accordions with steel reeds, even more. 

I'm not asserting any point here, just bringing up something I started wondering about. 
Logged

Tim Richardson
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #36 on: Apr 01, 2008, 12:51AM »

 Don't know

I'm not quite sure you're correct about piano tuning not quite being equal tempered. Obviously, a piano tuned by ear alone will not be. But symthesized pianos definitely are. And pianos tuned with the aid of a tuner should be pretty darn close. And I'm not sure I agree mathematically the reason for adjusting away from equal temperament, either. By definition, nothing besides an octave is in tune in equal temperament, so you're not necessarily making things more "out of tune." Overtones of non-just tuned strings should clash regardless of whether it's equal tempered or not.

Are you talking about a 19th century tuning which was really a refinement of the well temperament that Bach's famous piece was written for? In that temperament, notes aren't quite evenly spaced; in the link below, look for Thomas Young's system:

http://www.kylegann.com/histune.html
Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
timothy42b
*
Offline Offline

Location: Colonial Heights, Virginia, US
Joined: Dec 7, 2000
Posts: 12039

View Profile
« Reply #37 on: Apr 01, 2008, 03:40AM »

Don't know

Obviously, a piano tuned by ear alone will not be.

Pianos tuned by ear are tuned to have a specific number of beats sounding for various intervals.

But you listen for these beats in the upper partials, not in the fundamentals.

Pianos tuned electronically (ETDs, Electronic Tuning Devices) actually mimic the ear.  They have software that detects different partials, measures the inharmonicity, and calculates the stretch - which is how far off you have to tune the fundamental.  That is, how far off from ET.  You can set this software to detect whatever partial you want.  TuneLab has a free download if you want to play with it. 
Logged

Tim Richardson
timothy42b
*
Offline Offline

Location: Colonial Heights, Virginia, US
Joined: Dec 7, 2000
Posts: 12039

View Profile
« Reply #38 on: Apr 03, 2008, 11:37PM »

Just to add something I learned yesterday:

Even octaves are not tuned ET on the piano.  Yes, that was a surprise to me. 

Octaves are tuned to a compromise between 4:2 and 6:3.  That terminology was new to me.  An octave has a nominal frequency ration of 2:1, obviously, for the fundamental.  But the tuner doesn't listen to the fundamental.  He can make the 4th partial of the lower octave correspond to the 2cnd partial of the upper octave, OR he can make the 6th partial of the lower octave correspond to the 3rd partial of the upper octave, but not both.  But the articles I read say that the best results come from tuning the octave somewhere between those choices. 
Logged

Tim Richardson
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #39 on: Apr 04, 2008, 02:23AM »

Hmmmm . . .

Something tells me that I won't actually understand what's going on until I start doing wave analyses with the sound.

But, upon thinking about what you've told me so far, I am reminded that inharmonicity may make the octave partials not exactly correspond to the whole number ratios. Wierd, given that octaves are pretty much THE simplest interval besides a unison.

Do you know if this is a specific cent or microcent difference between the pitches to tune a proper octave on the piano due to this effect?

This reminds me that on the trombone, the partials aren't exact integer multiples of each other either, primarily because the trombone is not a perfectly cylindrical instrument.
Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
timothy42b
*
Offline Offline

Location: Colonial Heights, Virginia, US
Joined: Dec 7, 2000
Posts: 12039

View Profile
« Reply #40 on: Apr 04, 2008, 04:46AM »

Do you know if this is a specific cent or microcent difference between the pitches to tune a proper octave on the piano due to this effect?

This reminds me that on the trombone, the partials aren't exact integer multiples of each other either, primarily because the trombone is not a perfectly cylindrical instrument.

Much of this is new to me, and I'm going to find a tuning wrench and play with the old pianos at church.  I only have a digital at home.  The church has a nice grand which I won't dare touch, but two beater uprights that haven't been tuned in decades, there's no harm I can do.  So I don't claim I really know what i'm talking about, but you've intrigued me, and some of this is fascinating.  to us geeks anyway. 

So here's what I think.  I can defend it logically but don't claim 100% certainty here. 

The fundamental for an octave on the piano should be 2:1, whether you're ET or most other temperaments, but it won't sound right because the overtones will beat against each other.  So you compromise by making the octave a tiny bit wider than 2:1, that's called stretch.  There isn't a specific amount that fits all cases, because every piano is slightly different in how much inharmonicity exists.  I thought the amount should be relatively stable for any given piano, but a technician told me this morning that humidity affects it.  ETD software calculates inharmonicity when you do the tuning and gives you recommended amounts to stretch.  Technicians use that recommendation or not, as their ears tell them.  They apparently store tuning sets for reference. 

It may not be obvious, but this problem exists whether you do ET or any of the historic temperaments.

Now, the trombone.  Be careful with terminology.  You are correct that the partials are not integer multiples, and you are correct that the reason is the construction of the trombone with cylinders, cones, curves, constrictions, and other c words.  However.  When playing, the overtones are forced mathematically to be pure integer ratios, unlike the piano.  That is why I often say partials are NOT overtones.    If you lip slur up a partial series, you do not get integral multiples.  But if you play the fundamental and only listen up the partial series, you will. 
Logged

Tim Richardson
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #41 on: Apr 04, 2008, 07:35AM »

Much of this is new to me, and I'm going to find a tuning wrench and play with the old pianos at church.  I only have a digital at home.  The church has a nice grand which I won't dare touch, but two beater uprights that haven't been tuned in decades, there's no harm I can do.  So I don't claim I really know what i'm talking about, but you've intrigued me, and some of this is fascinating.  to us geeks anyway. 

So here's what I think.  I can defend it logically but don't claim 100% certainty here. 

The fundamental for an octave on the piano should be 2:1, whether you're ET or most other temperaments, but it won't sound right because the overtones will beat against each other.  So you compromise by making the octave a tiny bit wider than 2:1, that's called stretch.  There isn't a specific amount that fits all cases, because every piano is slightly different in how much inharmonicity exists.  I thought the amount should be relatively stable for any given piano, but a technician told me this morning that humidity affects it.  ETD software calculates inharmonicity when you do the tuning and gives you recommended amounts to stretch.  Technicians use that recommendation or not, as their ears tell them.  They apparently store tuning sets for reference. 

It may not be obvious, but this problem exists whether you do ET or any of the historic temperaments.

Now, the trombone.  Be careful with terminology.  You are correct that the partials are not integer multiples, and you are correct that the reason is the construction of the trombone with cylinders, cones, curves, constrictions, and other c words.  However.  When playing, the overtones are forced mathematically to be pure integer ratios, unlike the piano.  That is why I often say partials are NOT overtones.    If you lip slur up a partial series, you do not get integral multiples.  But if you play the fundamental and only listen up the partial series, you will. 

Yeah; I realized that about terminology.

And, yes, this problem would obviously exist whatever temperament one uses. It makes me wonder how composers who used really wild just intonation based systems tuned their pianos: Terry Riley on "The Bells of New Albion" and especially LaMonte Young on his masterpiece, "The Well Tuned Piano."
« Last Edit: Apr 04, 2008, 12:27PM by Andrew Meronek » Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
svenlarsson

*
Offline Offline

Location: Enskede, Sweden.
Joined: Sep 15, 2001
Posts: 4473

View Profile WWW
« Reply #42 on: Apr 04, 2008, 12:11PM »

This reminds me that on the trombone, the partials aren't exact integer multiples of each other either, primarily because the trombone is not a perfectly cylindrical instrument.
I confess, I am not following this thread, just came by to see what’s up.
No horns are perfect cylindrical or conical.
The saxes are very conical, but necessarily there are some compromises.
The most cylindrical horns are flutes, clarinets and didgeridoo’s.
The flutes are open in both ends when played, there horns are not.
Clarinets and didgeridoos series of partials looks like this:  8:va basso  locco      b #
But the overtones from any of the partials look like this,    you can transpose the series to fit any of the partials.

Logged

Kanstul 1662. Bach 45B. Kanstul 1555. Besson Euphonium. Kanstul 66-S Tuba. Sackbuts in F/E/Eb Bb/A
And several horns I should sell.
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #43 on: Apr 04, 2008, 09:05PM »

Hey there, Sven! Hi

On a slightly related but different note, I just finished my first little experimentation: a quartet fo trombone quartet entitled "Toccata in Bb" using this just-intonation based system. You can see the music here:

http://www.sibeliusmusic.com/cgi-bin/show_score.pl?scoreid=122086

and although I have that site set to force people to pay for printouts, I'm perfectly okay with sending out some parts so I can get some feedback from some of you here on the forum. I'm particularly focused on making sure that it is clear what I want without having to explain much in the music. That, and whether it sucks or not. :D
Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #44 on: Apr 13, 2008, 01:21PM »

I just uploaded an edited version with MIDI pitch bends, so anyone who wants to can hear the intended tuning. It'll bend your ears!

FYI, I ended up using a grand total of 26 pitches per octave, although a couple of them appear pretty rarely. The most complex one is the -47 cent B-natural at the climax, which is technically a 33/32 above the pitch center Bb.
Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
JacobGarchik

*
Offline Offline

Location: Brooklyn, NY
Joined: Aug 30, 2005
Posts: 355
"NYC Freelancer and omnivore"


View Profile WWW
« Reply #45 on: Apr 13, 2008, 02:13PM »

Hello Conversation,
I've been using and learning about microtones for a few years, but am by no means an expert. However I have had success using the so-called "Sims-Maneri" system of Equal-temperment and notation, which uses 72 notes to the octave. It was created by Ezra Sims and popularized through Joe Maneri's microtonal class at the New England Conservatory.
Although it's an equal-tempered system, it can be learned and notated quite easily, hundreds of musicians already know it, and the smallest interval (16 2/3 cents or a twelfth-tone) is small enough that the system can be used as an easy approximation of just-tempered systems. Most performers use the Sims-Maneri system in conjunction with cent markings anyway, so if you wanted +15 instead of 16 2/3 you could just use the symbol for 16 2/3 with an asterisk or something. There are fonts for Sibelius and Finale available.
Students of the system have become so familiar with it that they can actually sight-read using it.
You can order the workbook which explains the whole system, along with exercises, here:
http://bostonmicrotonalsociety.org/Pages/Workbook.html

Here in a nutshell is the notation:

 
Logged

Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #46 on: Apr 13, 2008, 10:28PM »

Hello Conversation,
I've been using and learning about microtones for a few years, but am by no means an expert. However I have had success using the so-called "Sims-Maneri" system of Equal-temperment and notation, which uses 72 notes to the octave. It was created by Ezra Sims and popularized through Joe Maneri's microtonal class at the New England Conservatory.
Although it's an equal-tempered system, it can be learned and notated quite easily, hundreds of musicians already know it, and the smallest interval (16 2/3 cents or a twelfth-tone) is small enough that the system can be used as an easy approximation of just-tempered systems. Most performers use the Sims-Maneri system in conjunction with cent markings anyway, so if you wanted +15 instead of 16 2/3 you could just use the symbol for 16 2/3 with an asterisk or something. There are fonts for Sibelius and Finale available.

Those equal temperament systems based on multiples of 12 do tend to be fairly easy to convert to a variant of standard notation. Cool stuff.

FYI, I also found on Wikipedia this excellent little teaser on a turkish classical notation system based on dividing the whole tone into 9 commas, giving 54 pitches per octave:

http://en.wikipedia.org/wiki/Makam

Given how many different notation systems are out there to approximate various just-tuned scales, I still think it's better to just go with a just-tuned scale with just-tuned notation rather than closer and more complex approximations.
Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #47 on: Apr 16, 2008, 05:37AM »

A piece of software you can use to help experimentation (create a MIDI with performance or notation software) is Scala:
http://www.xs4all.nl/~huygensf/scala/

(I mentioned it before, but I'll explain how I used it to experiment with Just Intonation.)

The feature I liked about it was that it Can retune existing MIDI files. You can convert a standard MIDI file to be in any tuning via pitch bend commands or a MIDI Tuning Standard tuning specification.

This feature makes it a useful tools for sort of quickly previewing simple music using Just Intonation. Basically, you have to export each tonal center as a different MIDI, create 12+ scale files (one for each key), apply appropriate scale files to the appropriate sections, and then find a way to playback the sections consecutively. For complex music, or note perfect music, you would still have to find a way to manually adjust for the different types of intervals (e.g. grave minor seventh versus minor seventh), but at least a bulk of the work could become automated when you're producing simple previews.


Along these lines, as I said above, I added MIDI pitch bends to my Sibelius file. The best solution I've found so far is to create the most common 12 note scale used in the piece with Scala, re-tune the MIDI file, then go into Sibelius and manually change all the pitches that didn't get correct right. Sibelius apparently uses a standardized MIDI script language; I'm not sure HOW standardized it is.

Does the phrase "~B0,64" mean anything to anyone? This should bend the pitch to the 12-note equal tempered frequency. Different numbers bend it differently, although it's a bit convoluted to figure out how to convert between this and a microcent adjustment.
Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #48 on: Apr 23, 2008, 11:43AM »

Here are some free downloads of just tuned music that I've found on the web. Some of it is good, some of it is not.

http://www.ubu.com/sound/tellus_14.html I like the "Opening Kyrie" and "Tocatta for Violoncello." A lot of it is not that great - an acquired taste, I guess.

http://www.avantgardeproject.org/AGP9/index.htm Some Ben Johnston music. A very large download, and the site features a lot of other avant garde recordings as well.

http://www.youtube.com/results?search_query=michael+harrison+revelation&search_type= Excerpts from a very recent piece for just tuned piano by Michael Harrison. Interesting that I found it on YouTube.

Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
David Schwartz

*
Offline Offline

Location: Belmont, MA
Joined: Aug 13, 2001
Posts: 1181

View Profile WWW
« Reply #49 on: Apr 23, 2008, 08:51PM »

A couple of years before he died (2004) Fred Nachbaur posted some utilities and some performances of music in both just intonation and tempered.  To listen to his fascinating samples, scroll down to "Demonstrations" and the buttons for Original and Tempered at this link.
Logged

David A. Schwartz, Belmont, Massachusetts
Bordogni and Breakfast Website
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #50 on: Apr 27, 2008, 12:13AM »

Wow, great website, David!

Yesterday, I was pondering some other odd ways of constructing scales, and I decided to try to build a scale that defines an octave as being 3:1 instead of 2:1, that is, I triple the starting pitch. Thus, if I start this scale on a piano's C and ascend, the next octave of the scale starts on a piano's G, a twelveth above. I got the idea from this: http://en.wikipedia.org/wiki/Bohlen-Pierce_scale

Although, unlike the example given in the website, I deliberately made a scale that maps onto a standard piano keyboard, and I used a lot of 5-limit intervals. Therefore, it's still a very consonant, pretty scale with lots of thirds - but chord inversions get pretty interesting, and the way I ended up structuring it makes for creating a lot of "jazz" chords that all fall within one octave. Cool stuff.
Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
timothy42b
*
Offline Offline

Location: Colonial Heights, Virginia, US
Joined: Dec 7, 2000
Posts: 12039

View Profile
« Reply #51 on: Apr 27, 2008, 04:54AM »

A little serendipity this weekend:

I was reading Benade, trying to find the explanation for strange results in one of my kids physics lab writeups:  a cylindrical tube resonating far below predicted frequency based on length.

I came across an interesting comment he offhandedly threw away:  talking about why violin strings don't produce beats when wind instruments do (has to do with the uneven stick and slide bowing mechanism), he mentioned that because of these factors string partials end up 20 cents wide, causing string quartets and wind quartets to have totally different intonation characteristics.  Winds can produce both purer consonances and purer dissonances.  Strings cannot but can vary color more.

If so, there would surely be some implications for composing just interval music for different groups. 

Oh, the physics lab:  Benade didn't cover it, my analysis suggests the kids shortened the tube enough to get a Helmholtz resonator effect. 
Logged

Tim Richardson
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #52 on: Apr 27, 2008, 06:59AM »

Benade who?
Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
timothy42b
*
Offline Offline

Location: Colonial Heights, Virginia, US
Joined: Dec 7, 2000
Posts: 12039

View Profile
« Reply #53 on: Apr 27, 2008, 10:34PM »

While there are many texts on musical acoustics, THE classic work is Fundamentals of Musical Acoustics by Benade.  The paperback edition is about $12 on Amazon, every musician should have a copy of this work. 

He did his research before computers and some of the modern measurement technology, stuff like RTA equipment.  He deleted the math in order to make the text more accessible to lay readers.  So there are some limitations, and other authors to read.  But this is still the one to start with. 
Logged

Tim Richardson
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #54 on: Apr 28, 2008, 12:48AM »

Ah, cool. I'll check him out. Good!
Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #55 on: May 16, 2008, 10:33AM »


http://www.youtube.com/results?search_query=michael+harrison+revelation&search_type= Excerpts from a very recent piece for just tuned piano by Michael Harrison. Interesting that I found it on YouTube.


Well, I just got the recording, and this is a real gem. He tuned his piano to have both consonant and dissonant intervals, and his dissonant intervals, what he refers to as "commas," create specific interference patterns, i.e., beats and he uses these patterns as rhythmic devices. This makes for a really interesting, unique sound. I highly recommend checking this out! Good!
Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
savio

*
Offline Offline

Location: Norway
Joined: Aug 10, 2006
Posts: 5056

View Profile WWW
« Reply #56 on: May 17, 2008, 11:03AM »

I know someone has maked an organ to play the way we do. Make the major 3rd lower and minor 3rd higher.
Logged

Bass Trombone - Conn, Holton
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #57 on: Jun 04, 2008, 07:46AM »

One more literature update:

I managed to my hands on a relatively rare CD: Ben Johnston Microtonal Piano. I'm listening to it right now, and boy is it something! Amazed

The two microtonal pieces on it are a suite and a sonata; the suite is tonal, although complex, and is tuned entirely in the 5th octave harmonic series pitches, and the sonata is completely atonal, with each key on the piano tuned almost arbitrarily accoring to relationships to other notes, but ends up having very few consonant octaves. Both approaches result in a really, really unusual, colorful sound - although I think I like the overtone-based Suite better, at least upon my first listenings.
Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
William Lang
*
Offline Offline

Location: New York City
Joined: Jul 31, 2006
Posts: 147

View Profile
« Reply #58 on: Jun 22, 2008, 11:20AM »

Hello,

I just got done playing a Maneri piece a few months ago, and I have to say that the Sims-Maneri system works very very well for microtonal writing. Lots of musicians know it, and there's even the Boston Microtonal Society, which plays mostly 72 note ET music, but uses other systems as well depending on the composer.

I also took a class from Michael Harrison this year where he explained how he tuned the piano for Revelations. Michael tended to do a lot of tunings by ear, and he was heavily influenced by Indian Raga music and the alternate tuning systems found there.

Hope this helps, and if you ever have a solo trombone work, please let me know!

William Lang
Logged
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #59 on: Oct 23, 2008, 06:34AM »

Upon doing some fresh googling of just intonation, I came across this excellent site:

http://tonalsoft.com/enc/encyclopedia.aspx

And includes:

A Microtonal Analysis of Robert Johnston's "Drunken Hearted Man"
HEWM Notation

and of course lots of other really useful articles. Good!
Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #60 on: Oct 30, 2008, 12:24AM »

It's amazing what one can find with a little googling:

http://www.chrysalis-foundation.org/musical_mathematics.htm
Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #61 on: Nov 09, 2008, 10:35PM »

Redgarding the HEWM notation discussed on the Tonalsoft website I posted on the previous page: I think that I just figured out for myself a really slick way to write key signatures in just intonation.

It turns out to be stupidly simple. I workedout the math, and it turns out that I can use exactly the same notation of sharps/flats as standard practice. However, I add other symbols for the other kinds of commas.

Essentially, I start with a Pythagorean tuned C major scale. Now, given this scale, there are no sharps or flats, so none of those go in the key signature for this key. But in just intonation, the Zarlino (or Ptolemic) major scale is based on the more pure sounding 5-limit intervals, being (a review from previous pages):

1:1
9:8 major second
5:4 major third
4:3 perfect fourth
3:2 perfect fifth
5:3 major sixth
15:8 major seventh
2:1 octave

Only theee notes of that scale are not in Pythagorean tuning: the third, the sixth, and the seventh. But each of them happens to be exactly 80:81 (a syntonic comma) away from it (for example, 5:4 is exactly 80:81 below 81:64, the Pythagorean major third), so the key signature for C major is: zero sharps or flats, but add the syntonic comma symbol for E, A, and B and you have your perfefctly tuned major scale. Other keys will move around the notes to be adjusted by the syntonic comma, but that number of commas never exceeds 4. Additionally, further accidentals added within the melody have to be considered additive; that is, they add their tuning to the existing key signature instead of replacing whatever is in the key signature as happens with modern 12-note equal tempered notation. And the natural sign returns the pitfch value of a note to the key signature, thus E natural in C major means the E is the 5:4 E, not the 81:64 E.

Anyway, this seems like a really slick system, and given how similar it is to standard notation, I bet it'll be fairly easy for performers to adjust to. I do find it funny that the issue of key signatures isn't discussed anywhere I've found in the 'net so far. Don't know

I can post the details here if anyone is interested.
Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #62 on: Nov 12, 2008, 03:09AM »

A small update: it turns out that the commas listed in the key signature are simpler if I use the Ptolemic scale as the base instead of the Pythagorean, even for the more distant key signatures like B major.

And accidentals add to the key signature indications instead of trumping them. This means that in the key of Bb, for example, the Eb with an additional accidental means the note is changed based on Eb, not E natural. This makes it possible to use the same modifying accidentals in every key, which should make a lot more sense to the ear than otherwise. And thus, the natural sign simply returns the note to the key signature indication, NOT to C major.

This is a slight departure from standard notation, but I think it's necessary to clean up the barrage of symbols that would otherwise need to be written on the page. Sight readability is the goal, and the fewer symbols to process, the better.
Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #63 on: Jan 01, 2009, 08:37AM »

Just a random thought for those who might be interested:

Many of the more complex jazz chords have multiple interpretations in JI. One chord is very problematic in JI: the major 6/9 chord. There are several possible spellings, each with it's problems:

1
5/4
3/2
5/3 (major sixth)
9/8

(creates a wolf interval between 9/8 and 5/3)

or

1
5/4
3/2
5/3
10/9

(creates a wold interval between 10/9 and 3/2)

or

1
5/4
3/2
27/16 (pythagorean major sixth)
9/8

(creates a wolf interval between 5/4 and 27/16)

or

1
81/64 (pythagorean major third)
3/2
27/16
9/8

(eliminates the wolf intervals but the third is quite unpleasant)

Listening to big band music from the 1930s, I'm pretty sure that for the most part the musicians tended to play the last one, with the completely pythagorean tuning. However, using one of the tunings with a wolf interval creates some very colorful high overtones in addition to the "wolf howling" which is somewhat tempered by the combination of the 5 pitches, although definitely still audible. I suspect that the second one listed above is also a fairly popular "adjustment" in performance practice.

Thoughts? Don't know
Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #64 on: Apr 13, 2009, 08:37AM »

Just in case anyone might have the answer, what symbol does Johnston use for the 17th partial comma? This is 51/50, the interval needed to turn 25/24, or a sharp, into 17/16, the 17th overtone. Used primarily in the dominant 7, flat 9 chord.

I might just have to make one up, but I'd perfer not to if there's already an established practice.
Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #65 on: Aug 18, 2009, 05:11AM »

I found that someone posted the first 10 minutes of LaMonte Young's masterpiece, "The Well-Tuned Piano" on youtube. Enjoy:

http://www.youtube.com/watch?v=a7tmxHhcH0w
Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #66 on: Mar 02, 2010, 11:54PM »

Bach Prelude in C Major set to just intonation with a new kind of keyboard: the Tonal Plexus

And to answer this one:

Just in case anyone might have the answer, what symbol does Johnston use for the 17th partial comma? This is 51/50, the interval needed to turn 25/24, or a sharp, into 17/16, the 17th overtone. Used primarily in the dominant 7, flat 9 chord.

I might just have to make one up, but I'd perfer not to if there's already an established practice.

Johnston, according to an essay of his I found, simply put "17" or the same upside-down in front of the note.  I think I'll come up with a symbol that looks like "17" but is one symbol, not two.
Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
mbarbier
*
*
Offline Offline

Location: Los Angeles, Ca
Joined: Jul 25, 2010
Posts: 172

View Profile WWW
« Reply #67 on: Jul 29, 2010, 05:55PM »

I'd suggest actually checking out www.plainsound.org It's a self-publishing collective of composers working in Just Intonation (composition and instrumental research). Especially the work from Marc Sabat, Wolfgang von Schweinitz, and Douglas Wadle. Outside of having excellent research on the subject, theoretical and performance-wise, they have a really excellent notation system.
       It takes a bit of learning but I've really found it to be the clearest. It's the "Helmholtz-Ellis JI Pitch Notation System." Yea...the names of some of the stuff is kinda funny, but the system is really great. Rather than being based off of cent deviations the accidentals are in reference to the harmonic series. Therefore when you read the notation (once you get used to it) you know where the note needs to be, but also, and more importantly, it's harmonic function. Which helps solve the difficulty of things like Partch scores where everything is notated by ratio or Ben Johnston's music where everything is based off of a C-Major Scales, creating un-notated comma issues.
      A part of the Helmholtz-Ellis system that is really useful as a trombonist is the fact that the notation literally gives you an overtone number so it has a very clear application for use of alternate positions.

There are also several excellent books on acoustics by Arthur Benade which go into the tuning issues that arise on trombone because of the 'impurity' of instruments being neither conical or cylindrical, but a combination of both. His 'Fundamentals of Musical Acoustics' goes into it, and acoustics in general, very thoroughly. He also has a book, "Horns, Strings and Harmony," which is much shorter (maybe 180 pages instead of 500) and covers the issue quite well. It's really interesting stuff.
Logged
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #68 on: Jul 29, 2010, 11:42PM »

I've checked out the Helmholtz-Ellis system.  From what I can tell, it is very similar to the Johnston system that I like, except it starts from the base Pythagorean scale, instead of the Ptolemic scale.  The common (relatively) reason why people like it that they say that using the Pythagorean system gets rid of an imerfect fifth - but it actually doesn't.  It just moves it.  It also requires more syntonic comma accidentals to indicate in-tune thirds than the Johnston system, which makes it a bit more cluttered-looking. 

It does make the G-major-related key signatures (plus or minus a couple flats) look more similar on the page, which may or may not be a good thing, depending on how one looks at things.  I think that ultimately, if one applies the symtonic commas to the key signature in addition to the normal sharps and flats, the Johnston system (based on the 5-limit Ptolemic scale) is actually a bit simpler to notate in all 12 keys.

Good food for thought, though. Good!

And a good website. Good!
Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
timothy42b
*
Offline Offline

Location: Colonial Heights, Virginia, US
Joined: Dec 7, 2000
Posts: 12039

View Profile
« Reply #69 on: Jul 30, 2010, 09:04AM »

On a piano forum I visit, we had a thread on tuning temperament choices.

Someone posted several pieces tuned two ways so you could do A-B comparisons between ET and an alternate.

If anyone is interested, I could post a link.

I took a quick listen and was embarassed to admit I couldn't tell the difference.  I meant to go back with a better sound card and headphones but never got around to it. 
Logged

Tim Richardson
mbarbier
*
*
Offline Offline

Location: Los Angeles, Ca
Joined: Jul 25, 2010
Posts: 172

View Profile WWW
« Reply #70 on: Jul 30, 2010, 09:12AM »

Yea that's the slight problem with both. The Helmholtz one looks a bit complex till you get used to it, but the Johnston one actually really gets messed up the more you do it. I'd been playing his music for a while but I spent a good bit of time working with him last year and he pointed out alot of comma issues that aren't apparent when you get away from C- Major. He's actually a big fan of the Helmholtz notation, he's just used to his.
 
   It also does get rid of the fifth issue. A# is not Bflat in the tuning, F# isn't Gflat and so on.They worked really carefully to make sure and remove the comma issues from Ben's notation. It's definitely written with modulation in mind so the syntonic commas must be notated for clarity's sake because with most of their music you'll have alot of almost the same note in a very short period.
So, at least to me, if you're gonna be in C Major (or utonal harmonies closely related to C) the Johnston system is great. When you get away from it and start using more pitch models based on James Tenney's theory of harmonic space (the article is on Plainsound) the Johnston gets un-usable quickly and the Helmholtz makes alot of sense. It  kinda becomes a debate of where you're going with it. If you're not going to modulate the Johnston remains simple, but when you start moving through harmonic space (as with most of Ben's music) it gets super complex. Just my thought from playing and composing with it.

It's nice to find such a subject on a trombone board!
Logged
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #71 on: Jul 30, 2010, 09:17AM »

If anyone is interested, I could post a link.

Post it, please! :)

In general, those differences aren't very audible in the styles of classical piano where there are a lot of runs/ornaments and not a lot of sustained notes.  I've seen some stuff like that on Youtube.  There are also some temperaments which are in reality very similar and virtually indistinguishable, especially for stuff like Bach.  For example, modern equal temperament and it's predecessor, Wernsteiner (sp?) which Bach ended up using are very close to each other, within a few cents.  But the Renaissance meantone temperament is a bit farther away and it's easier to hear the differences, especially as you hear different keys sounding different in that temperament.  It's also easier to hear in organ music than in keyboard music.

Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #72 on: Jul 30, 2010, 10:09AM »

Yea that's the slight problem with both. The Helmholtz one looks a bit complex till you get used to it, but the Johnston one actually really gets messed up the more you do it. I'd been playing his music for a while but I spent a good bit of time working with him last year and he pointed out a lot of comma issues that aren't apparent when you get away from C- Major. He's actually a big fan of the Helmholtz notation, he's just used to his.
 
   It also does get rid of the fifth issue. A# is not Bflat in the tuning, F# isn't Gflat and so on.They worked really carefully to make sure and remove the comma issues from Ben's notation. It's definitely written with modulation in mind so the syntonic commas must be notated for clarity's sake because with most of their music you'll have alot of almost the same note in a very short period.
So, at least to me, if you're gonna be in C Major (or utonal harmonies closely related to C) the Johnston system is great. When you get away from it and start using more pitch models based on James Tenney's theory of harmonic space (the article is on Plainsound) the Johnston gets un-usable quickly and the Helmholtz makes alot of sense. It  kinda becomes a debate of where you're going with it. If you're not going to modulate the Johnston remains simple, but when you start moving through harmonic space (as with most of Ben's music) it gets super complex. Just my thought from playing and composing with it.

It's nice to find such a subject on a trombone board!

Well, I am a bit professionally jealous that you're able to study with Mr. Johnston.  I love some of his music, and think a lot of his decision to write for traditional instruments instead of going the electronic route.  Both have their points, but with complex microtonal music, the gravity seems to be more electronic than otherwise, except for a few composers like him, Terry Riley, Michael Harrison, etc.  I am in the Army, and have not been stationed in areas where I could study with a microtonal composer along these lines on a regular basis, at least, I don't know of any.  I have had to work out a lot of the stuff myself (some of which I detailed in this thread, as a way to get random (and sometimes very useful) feedback.  I got my information about HEWM notation here:

http://tonalsoft.com/enc/h/hewm.aspx

and worked out the Johnston system based on some of Johnston's writings (and what score snippets I could find) and an early helpful email from Kyle Gann.

I did discover that, while the Johnston system has the imperfect fifth at the ii chord, that if I notated the syntonic comma in the key signature, the syntonic commas cancelled +s and -s in a similar way that key signatures cancel #s and bs (Cb major vs. B major, for example, goes from 7 flats to a slightly simpler 5 sharps), and the pattern of adding more and more syntonic commas as I moved away from C major can be largely cancelled out.  Because this over-accidentalizing problem had this solution, and because I overwhelmingly want to write music based more on the 5-limit major scale than the Pythagorean major scale, I decided to focus more on Johnston's system.

Since you're more "in the loop" than I am, how standardized is HEWM notation becoming?  Seeing as how I obviously have made one false assumption about it (the imperfect fifth just being moved), I may not understand the system as well as I hoped, and it may be worth it to study it more. :/
Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
mbarbier
*
*
Offline Offline

Location: Los Angeles, Ca
Joined: Jul 25, 2010
Posts: 172

View Profile WWW
« Reply #73 on: Jul 30, 2010, 06:15PM »

I don't study with Ben; he doesn't teach anymore. I just got to work with him last year for a concert and a new edition of one of his pieces I made. We mostly just talked about issues of notation. Ironically though he's probably closer to an Army base than a center for microtonal experimentation- he lives with relatives in rural Wisconsin now.. He's also a super amiable guy. It might take him a few days to answer but he's always very happy to get e-mails.

Some of his music is really beautiful, but I really have found his writings and process to be of real value. I don't have any interest in doing it myself, but his application of serial techniques to JI is really enlightening. It really makes flexible notation on central issue, but is also just a pretty cool idea. I will admit I like his music after that break alot more, however. I definitely agree about his use of traditional instruments. I'd really suggest checking out Tenney. They were students together and both kind of went in different directions, but both working with JI mostly for traditional instruments in very interesting ways.


For what you're doing it raises a question to me as to how complex of 5-limit you want to use. Whether its simple,  clear harmonies are getting into more extended use of things like Major or minor dieses and also what you prefer to read. I find the commas to be quite nice because you only have to read it if it's a third or you choose to use a comma shift, therefore its clear if you want a 5:4 or a Pythagorean third and so on. But then again you might be keeping it fairly simple so those kind of issues might not be making their way into the music. Which definitely isn't bad, there's alot more super complex ji music than simple, clearly pure stuff! 
I've also found the lack of a key signature to be a fairly helpful. I've found people to get freaked out by trying to remember that there's a comma in the key. Not that it's that hard, performers just seem to get freaked when you say JI. But then again it's all about if you're staying in one key, modulating or not using a specific key at all, but moving through Harmonic Space.

The Helmholtz notation definitely has a solid foothold in certain circles in LA and Berlin. There are alot of composers (in relation to the number of microtonal composers out here) outside of Plainsound that are using it and I can think of maybe 15-20 players in LA (with a similar number in Berlin) who know the system well. It helps that Plainsound has a decent presence, along with JI in general, at CalArts and is used as the main notation system in the tuning theory class. It's a bit overwhelming of a system cause is gets complex but it really makes alot of sense after a while, but it definitely took me a while to really get it.

So from taking the class and working with alot of the Plainsound people I have a really obvious bias towards the system so I hope I don't come across as too overbearing.
Logged
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #74 on: Aug 01, 2010, 10:42AM »

Cool.  Some of our preference for our favorite systems may have to do with our compositional goals.

I enjoy writing tonal music, and I like to write both heavily modulating music and very monotonally centered music.  And I like using a range of harmonic color, from open fifth power chords and major/minor triads through 17th (and higher) harmonic color chords, although melodically I tend to try to keep it digestible.  Once I figured out what the 7 primary 5-limit (Ptolemic) major scales would be in JN, I found that modulation is no problem.

Otherwise, I don't see how one system better or worse than the other.  When getting to more than 5-limit harmony (and not using a tempered approximation) in any notation system, more accidentals become necessary.

And I'll very rarely use a Pythagorean major third in harmony.  The only cases that I've worked out that I'd accept is the ever-unstable major 6/9 chord, or if I'm writing something medieval stylistically, a.k.a. Perotin.
Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #75 on: Nov 09, 2010, 11:12AM »

I posted this link in the "other performers" section, and I just realized that it may fit here even better:

LaMonte Young - Just Charles & The Romantic Chord

Similar in many ways to LaMonte Young's Well Tuned Piano, it is very long and is performed in just intonation.  From what I understand, the piece is a solo piece, but the cellist plays cello with pre-recorded (by himself) cello drones controlled by foot pedals.  The cellist, Charles Curtis, has a phenomenal ear! Amazed Pant
Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #76 on: Mar 04, 2011, 01:28PM »

In case anyone is interested, this slipped by my notice until quite recently:

http://newdissonance.com/2011/01/20/newcd/

The Kepler String Quartet just got released their second CD of Ben Johnston string quartets, this time Nos. 1, 5, and 10. Johnston is something of a giant in just intonation music, and his style is quite classical, incorporating serialism, romanticism, and some other traditional styles throughout the course of his career. There's a promotional video here that includes a short interview with Ben and some musical samples:

http://newdissonance.com/2011/02/25/promo-video-for-new-kepler-cd/
Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #77 on: Mar 16, 2011, 09:02PM »

Okay, sometimes I'm a little slow, but concerning the above post, I just found the publisher's web page for this recording.

http://www.newworldrecords.org/album.cgi?rm=view&album_id=87338

I include it here because it contains some music samples for anyone who might be interested.
Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #78 on: Jan 01, 2013, 03:15PM »

Attached to this post should be a calculator I made in Open Office to help myself with some painful calculations. The idea is to easily convert a just-intonation ratio into a MIDI pitch-bending command of the format ~Bx,y which is what Sibelius uses.

Specifically, let's say I want to tune more exactly a perfect fifth, which is normally about 2 cents sharp. In just intonation, this translates to a ratio of 3/2. So, go to the spreadsheet and look for the row that begins with the numerator column as 3 and the denominator column as 2. This one I already have in the list. To use it in Sibelius (at least Sibelius 6 which I use) enter as text attached to the note you want to bend: ~B80,64

Note that this will pitch-bend every note on the same stave until the next pitch bend, so it will work best on single-line staves like trombone staves.  So, notes of a chord to be tuned will have to be one-note-per-stave. It also assumes that the MIDI pitch-bend range is set to a semitone, which I think is standard.

Note that the third column, 12ET Half-Steps, is not a formula and is, like the first two columns, must be manually inputted. This is to indicate from which relative pitch to calculate the pitch bend. Most of the time, you will want to just enter the normal equal-tempered approximation of half-steps for a frequency ratio. For example, 3/2 is well-approximated as 7 half-steps away from the root pitch. But, sometimes a pitch is sufficiently in-between the piano keys that it can be reasonably bent to either from above or below. For example, 11/8 is 51.318 cents above 5 half-steps and 48.682 cents below 6 half-steps. This makes it easy to calculate the pitch bend from either direction.

I figure that anyone who might like to experiment a bit will find this useful, as not a whole lot of information is out there dealing with how to do this.
Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #79 on: Mar 05, 2013, 02:47PM »

An updated version of the file is attached to this post.

I look at my post above and realize that it doesn't really explain clearly how to use the file.

So . . .

JI Numerator and JI Denominator represent the upper and lower parts of a fraction which describes a relative pitch.

12ET Half Steps tells the algorithm how many half-steps to subtract from the pitch bend. So, a 3:2 is bent by 1.955 cents instead of 701.955 cents by subtracting 7 half-steps.

~B(*x*,y) and ~B(x,*y*) represent the scripting that you would add to Sibelius to affect the pitch bend. The number under ~B(*x*,y) is the *x*, and the number under ~B(x,*y*) is the *y*. Thus, if under ~B(*x*,y) you see 80 and under ~B(x,*y*) you see 64, the script you want is ~B(80,64). You then enter the text attached to the note you want bent. Sibelius will then bend every subsequent pitch on the same stave by the same amount until you tell it to change the pitch bend again. And, capitalization counts.

The rest of the spreadsheet really is just parts of the calculation functions.

To test and see if this spreadsheet works with your program, test a major third. On one stave, input a held C and on another stave, input a held E a major third above where the C is on the other stave. Above the E, enter in the text
Quote
~B(79,59)
Play it back. If it sounds skunky, something is wrong, and you may need to tweak the function in the 'Cents to Pitch Bend Units' column. Ask on this thread if you need to and can't figure it out yourself. If it sounds clear, vibrant, and resonant, it's working as intended.

Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #80 on: Mar 07, 2013, 11:33AM »

Silly me - having trouble keeping up with the times . . .

I found another website that looks cool for people who might be interested in not only just intonation, but microtonality in general:

http://xenharmonic.wikispaces.com/

Various topics covered include just intonation, historical temperaments, ethnic music tuning systems, notations, mathematics - lots of good stuff.
Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #81 on: Mar 19, 2013, 02:58PM »

For anyone who is interested, I just finished up a little editing project based on a tune I wrote a couple of years ago:

http://andrewmeronek.com/2013/03/19/sagittal-notation-score-and-computer-playback/

Included is a score and a .mp3 recording realized by Sibelius string quartet playback. The goal was to use this piece for myself to see how versatile Sagittal notation truly is for notating just intonation, and I dig it. Give it a listen and see if the arrows make sense with how you hear the pitch moving.  :)
Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #82 on: Jun 01, 2013, 09:19AM »

Another blog update, pertinent to this thread:

http://andrewmeronek.com/2013/06/01/unintentional-pitch-drifting/

I hope that it is informational in particular to those of us who play in trombone quartets or brass quintets; smaller groups make these kinds of issues more apparent. But the concepts apply to pretty much everything.
Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #83 on: Aug 06, 2013, 11:14PM »

I just finished a first draft of a Sagittal chord chart:

http://andrewmeronek.com/music-tools/sagittal-chord-chart/

This one is in Pure Sagittal; I'll add a Mixed Sagittal in a bit, with the normal sharps and flats instead of the double and triple arrow accidentals. Even without knowing exactly what all the symbols mean, it's very clear to see how often a purely-tuned chord sequence can result in a lot of small pitch movement. If the accidental changes, the pitch changes. Kind of brings the question to mind: "how much music do I listen to (and even how much music do I write) which is really in tune?"
Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
Strussman

*
Offline Offline

Location: Hattiesburg Ms
Joined: Oct 1, 2013
Posts: 31

View Profile
« Reply #84 on: May 07, 2014, 11:02AM »

I think there is a good place for this in tbone pedagogy.  Have you tried two sizes and/or widths and/or boldness of the arrows to indicate degree and direction and place the base of the arrow  :clever:on the line or space of the intended note?
Logged
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #85 on: May 12, 2014, 03:40PM »

I think there is a good place for this in tbone pedagogy.  Have you tried two sizes and/or widths and/or boldness of the arrows to indicate degree and direction and place the base of the arrow  Clever on the line or space of the intended note?

There's all of that already in Sagittal notation. It's pretty slick, and IMHO pretty useful. Next time I really want to nail a trombone soli in an orchestra performance, I have a mind to write in at least the syntonic commas with the help of a bit of score study.
Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #86 on: Nov 24, 2014, 02:35PM »

It's been a little while, but I finally got another blog post on this subject up. I hope some of you find it interesting. This one is a bit more technical on the music theory level than my others thus far; be fairly warned.

http://andrewmeronek.com/music-tools/musings-on-otonality-and-utonality/
Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
Strussman

*
Offline Offline

Location: Hattiesburg Ms
Joined: Oct 1, 2013
Posts: 31

View Profile
« Reply #87 on: Dec 23, 2015, 07:09AM »

I wonder if you could use the old shape note methods of triangles, squares, etc. to prevent from having to read a note and an arrow.  Maybe add a new shape like a teardrop that points in the direction of the proper intonation.
Thanks,
Bill
Logged
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #88 on: Dec 23, 2015, 10:34AM »

I wonder if you could use the old shape note methods of triangles, squares, etc. to prevent from having to read a note and an arrow.  Maybe add a new shape like a teardrop that points in the direction of the proper intonation.
Thanks,
Bill

It's been done. The problem is one of legibility, that unless the noteheads are really big sometimes it's hard to distinguish between a circle and a square notehead. A teardrop would be worse in that respect. Why not just use a  notehead as a triangle pointed up as a sharpened note and pointed down as a flatted note? Because it's easier to read sharps and flats when the sign is a different symbol in a close but different spot than the notehead itself which already has to indicate staff position and duration.
Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
Andrew Meronek

*
Offline Offline

Location: Livonia, MI
Joined: Sep 30, 2001
Posts: 6840
"Justly Intoned"


View Profile
« Reply #89 on: Sep 09, 2016, 03:44AM »

I've still been picking up little tidbits here and there . . . this time:

https://xenharmonic.wikispaces.com/harmonic+entropy

Harmonic entropy is pretty interesting stuff. Compare the graph examples given to what you hear when you do a glissando against a drone, paying attention to where the intervals becomes more and less dissonant. Pretty similar. I believe it's a good practical connection between how harmony works psychoacoustically and with standard musical practices. There be good reasons why we choose temperaments that get pretty close to just intervals even when a temperament isn't "perfect".
Logged

"All musicians are subconsciously mathematicians."

- Thelonious Monk
Pages: 1 2 3 4 5 [All]   Go Up
Print
Jump to: