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Andrew Meronek

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« 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
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« 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.
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« 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.
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« 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
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« 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.
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« 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
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« 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.
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« 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.
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« 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!
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« 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. 
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« 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!
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« 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.

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« 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. :/
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« 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.
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« 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.
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« 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
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« 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/
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« 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.
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« 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.
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« 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.

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