Ah, sorry: The patch is correct, but my explanation is wrong. Here's an update:
In inversion.pd this is realised by walking through the list with list-map. The interval to use next is calculated by taking the difference between the current element and the previous element. This interval is substracted (not added, because we are "retro"-grading) from the previous note, the resulting note is stored for the next step and inserted into the result list. The first element in a list is treated specially as it has no previous element: it's just copied and used as the starting note.
Frank and all,
Attached is an inversion.pd which is a bit simpler -- usually when inverting in pitch (rather than pitch-class) it's easier/simpler/better to invert with respect to an axis of symmetry rather than with respect to the first pitch in the series (of course you can assign the first pitch as the axis).
Meanwhile, if you're doing this with pitch-class instead of pitch (in other words transposing and inverting in a mod 12 universe, which is what you would probably be doing with 12-tone rows before assigning specific registers), instead of inverting with respect to an axis of symmetry, you invert with respect to the sum of the two pitch-classes in the original and inverted row -- sometimes called the "index" of inversion. The group theory is even cleaner if you think of transposition as an addition operator and inversion as a multiplication operator (in this case multiplication by 11, mod 12). If you're doing mod 12 operations, there is one more pitch operator -- multiplication by 5 or 7 -- which maps the chromatic scale to the circle of fifths and vice-versa. Then you can think of retrogression as "order inversion," and rotation, the other standard order-position operator, as "order transposition."
If you think it would be useful I can put together the standard 12-tone operators in mod 12 (or for that matter, an assignable modulus), but of course specific register information disappears. A more interesting but more difficult project would be to write a list abstraction to output the set-class of a given set of pitches, with an assignable modulus.
Matt
A wonderful book. I have emailed the link to many.
Thankyou for providing a wonderful resource
Geoff
Hallo Matt, Matt Barber hat gesagt: // Matt Barber wrote:
Attached is an inversion.pd which is a bit simpler -- usually when inverting in pitch (rather than pitch-class) it's easier/simpler/better to invert with respect to an axis of symmetry rather than with respect to the first pitch in the series (of course you can assign the first pitch as the axis).
Ah, that's much simpler indeed - and thanks a lot for your other explanations. I updated the code on http://footils.org/pkg/list-compose.tgz to include this as inversion-axis.pd and made the old inversion.pd a wrapper around this, which set the first note as the inversion axis.
(I made a little change to save the multiplication by 2 by using [swap] and a loadbang.)
If you think it would be useful I can put together the standard 12-tone operators in mod 12 (or for that matter, an assignable modulus), but of course specific register information disappears. A more interesting but more difficult project would be to write a list abstraction to output the set-class of a given set of pitches, with an assignable modulus.
I think, that would be an interesting project. I took the patches in my example from my port of KHz Essl's RTC lib (in row-modus.pd). Myself I'm not *that* familiar with 12-tone theory ...
Ciao
Frank Barknecht Do You RjDj.me? _ ______footils.org__
On Wed, 18 Mar 2009, Matt Barber wrote:
If you're doing mod 12 operations, there is one more pitch operator -- multiplication by 5 or 7 -- which maps the chromatic scale to the circle of fifths and vice-versa.
The "vice-versa" part is quite cool. Actually, apart from 1,5,7,11, all modulo-multiplications are not undoable, because they forget part of what was the original note, so, the undo would be ambiguous. 5 undoes itself because 5*5=1 in mod 12, 7 undoes itself because 7*7=1 in mod 12, but then 5=-7 and 7=-5 as well, just like 1=-11 and 11=-1.
The undoability depends on whether the greatest common divisor of the modulo and of the multiplicator is 1 or not. If you use the 22 equal temperament, for example, there are 10 invertibles, and with 43 equal temperament, there are 42 of them; the proportion of undoables vs non-undoables varies greatly from modulo to modulo.
I'm not into microtonal stuff, but I studied the modulo theory and I think that people who can care about microtonal music are lucky to have a nice application of that theory in their hands :)
See also http://en.wikipedia.org/wiki/Euler_phi_function
_ _ __ ___ _____ ________ _____________ _____________________ ... | Mathieu Bouchard - tél:+1.514.383.3801, Montréal, Québec
On Thu, Mar 19, 2009 at 12:02 PM, Mathieu Bouchard matju@artengine.ca wrote:
On Wed, 18 Mar 2009, Matt Barber wrote:
If you're doing mod 12 operations, there is one more pitch operator -- multiplication by 5 or 7 -- which maps the chromatic scale to the circle of fifths and vice-versa.
The "vice-versa" part is quite cool. Actually, apart from 1,5,7,11, all modulo-multiplications are not undoable, because they forget part of what was the original note, so, the undo would be ambiguous. 5 undoes itself because 5*5=1 in mod 12, 7 undoes itself because 7*7=1 in mod 12, but then 5=-7 and 7=-5 as well, just like 1=-11 and 11=-1.
The undoability depends on whether the greatest common divisor of the modulo and of the multiplicator is 1 or not. If you use the 22 equal temperament, for example, there are 10 invertibles, and with 43 equal temperament, there are 42 of them; the proportion of undoables vs non-undoables varies greatly from modulo to modulo.
I'm not into microtonal stuff, but I studied the modulo theory and I think that people who can care about microtonal music are lucky to have a nice application of that theory in their hands :)
Right, in mod-12, the other multiplications are not strictly operations (there is no inverse). I used to like to joke with friends that I was "really into the multiplication by 0 mapping."
Recently I've been writing music in various 19-tone equal temperaments, which, since it's prime, has a complete multiplicative group. 19 per octave is nice because you get really "pure" thirds. I've also been experimenting with 19 per perfect 12th (octave and a fifth), the smallest intervals of which work out almost exactly to standard 12-tone half-steps (check the 12th root of 2 and the 19th root of 3).
In addition each modulus has strikingly different voice-leading possibilities.
On Thu, 19 Mar 2009, Matt Barber wrote:
Right, in mod-12, the other multiplications are not strictly operations (there is no inverse).
They are called operations anyway. I don't know your definition of operation.
They're usually called "non-invertible operations", but in a Group (of Group Theory), all elements are invertible.
Group Theory also has an operator (written as a small straight "x" in exponent) that makes a multiplication-wise group from an addition-wise group. For Z/12Z (the mod 12 integers), this gives you a group make of 1,5,7,11, which behaves like (Z/2Z)^2, which is are the 2-D vectors made of Z/2Z (mod 2 integers):
1 -> (0,0) 5 -> (0,1) 7 -> (1,0) 11 -> (1,1)
Recently I've been writing music in various 19-tone equal temperaments, which, since it's prime, has a complete multiplicative group.
yes... and as a bonus, this multiplicative group acts just like Z/18Z !!!
_ _ __ ___ _____ ________ _____________ _____________________ ... | Mathieu Bouchard - tél:+1.514.383.3801, Montréal, Québec
mm. When I was studying music theory we used to reserve "operation" for a function that was 1 to 1 and onto; I think that usage has been pretty standard in music theory since 1987, through the work of David Lewin. Music theorists often screw up standard math terms though, so I never know what to call anything in what company, and always suspect it will be meaningless/wrong... =o)
On Thu, Mar 19, 2009 at 1:09 PM, Mathieu Bouchard matju@artengine.ca wrote:
On Thu, 19 Mar 2009, Matt Barber wrote:
Right, in mod-12, the other multiplications are not strictly operations (there is no inverse).
They are called operations anyway. I don't know your definition of operation.
They're usually called "non-invertible operations", but in a Group (of Group Theory), all elements are invertible.
Group Theory also has an operator (written as a small straight "x" in exponent) that makes a multiplication-wise group from an addition-wise group. For Z/12Z (the mod 12 integers), this gives you a group make of 1,5,7,11, which behaves like (Z/2Z)^2, which is are the 2-D vectors made of Z/2Z (mod 2 integers):
1 -> (0,0) 5 -> (0,1) 7 -> (1,0) 11 -> (1,1)
Recently I've been writing music in various 19-tone equal temperaments, which, since it's prime, has a complete multiplicative group.
yes... and as a bonus, this multiplicative group acts just like Z/18Z !!!
_ _ __ ___ _____ ________ _____________ _____________________ ... | Mathieu Bouchard - tél:+1.514.383.3801, Montréal, Québec
On Thu, 19 Mar 2009, Matt Barber wrote:
mm. When I was studying music theory we used to reserve "operation" for a function that was 1 to 1 and onto; I think that usage has been pretty standard in music theory since 1987, through the work of David Lewin. Music theorists often screw up standard math terms though, so I never know what to call anything in what company, and always suspect it will be meaningless/wrong... =o)
Yeah, it's a tough problem, and actually mathematicians don't say it as "operation" so often, and tends to use "operator" with some special connotations too (but nothing to do with 1-to-1 and onto). It's all a big mess. No way out but to be more verbose; or else, all agree to rely on a specific glossary for all that gets said on pd-list (which is not actually doable...).
_ _ __ ___ _____ ________ _____________ _____________________ ... | Mathieu Bouchard - tél:+1.514.383.3801, Montréal, Québec
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Frank Barknecht -
Geoff -
Mathieu Bouchard -
Matt Barber