pitch bend/cents translation

Hi friends,
A very simple question. I'm happy to say we've found a
way to work pitch bend rather elegantly into my
53-equal keyboard. I just need to know: if I
understand it correctly, MIDI pitch bend command
numbers are from something like 1 to 4000, and the
default setting is a 12-EQ whole step (200 cents).
Does that mean a pitch bend of, say, 20, equals one
cent of pitch bend? Does pitch bend follow the same
logarithmic curve that the "cents" system does, so
that simple arithmetic is all that's needed to
translate pitch bend numbers to cents? If not, has
anyone figured out the correspondence (I hate
reinventing the wheel).
Thank you in advance!
Chris Mohr, 53-equal Man.

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hi Chris,

> From: "Chris Mohr" <fromtherealmoftheshadow@yahoo.com>
> To: <tuning@yahoogroups.com>
> Sent: Wednesday, January 15, 2003 3:16 PM
> Subject: [tuning] pitch bend/cents translation
>
>
> Hi friends,
> A very simple question. I'm happy to say we've found a
> way to work pitch bend rather elegantly into my
> 53-equal keyboard. I just need to know: if I
> understand it correctly, MIDI pitch bend command
> numbers are from something like 1 to 4000, and the
> default setting is a 12-EQ whole step (200 cents).
> Does that mean a pitch bend of, say, 20, equals one
> cent of pitch bend? Does pitch bend follow the same
> logarithmic curve that the "cents" system does, so
> that simple arithmetic is all that's needed to
> translate pitch bend numbers to cents? If not, has
> anyone figured out the correspondence (I hate
> reinventing the wheel).
> Thank you in advance!
> Chris Mohr, 53-equal Man.

yes, pitch-bend is logarithmic just like cents.

you're thinking of a common pitch-bend specification
such as that used by Cakewalk (a very popular music
sequencer software). i've dubbed one unit of this
pitch-bend measurement a "cawapu":
http://sonic-arts.org/dict/cawapu.htm

there are 4096 of these units per 12edo Semitone.

here's a table of cawapu pitch-bend values for 53edo:

53edo 12edo
degree degree + cawapus

0 0 0
1 0 927
2 0 1855
3 0 2782
4 0 3710

5 1 541
6 1 1468
7 1 2396
8 1 3323

9 2 155
10 2 1082
11 2 2009
12 2 2937
13 2 3864

14 3 696
15 3 1623
16 3 2550
17 3 3478

18 4 309
19 4 1237
20 4 2164
21 4 3091
22 4 4019

23 5 850
24 5 1778
25 5 2705
26 5 3632

27 6 464
28 6 1391
29 6 2318
30 6 3246

31 7 77
32 7 1005
33 7 1932
34 7 2859
35 7 3787

36 8 618
37 8 1546
38 8 2473
39 8 3400

40 9 232
41 9 1159
42 9 2087
43 9 3014
44 9 3941

45 10 773
46 10 1700
47 10 2628
48 10 3555

49 11 386
50 11 1314
51 11 2241
52 11 3169

( 53 12 0 )

in creating an actual MIDI-file tuned in 53edo,
you would:

- replace the integers 0 ... 11 in the "12edo" column
with the MIDI-note letter-names, so that, for example,
if you pick "C" to equal 0, then 1 = C# or Db, 2 = D,
etc.

- then there would be a pitch-bend command at the same
start time using the appropriate number from the
"cawapu" column. (actually, my own preference is to
have the pitch-bend command at the idealized start-time
of the note, then the MIDI note-on command one "tick"
later.)

PS -- you might also want to look at these definitions:

http://sonic-arts.org/dict/midipu.htm
http://sonic-arts.org/dict/cents.htm

and i have some useful stuff on 53edo here:

http://sonic-arts.org/dict/eqtemp.htm#53
http://sonic-arts.org/dict/bingo.htm#53

-monz

monz yer awesome! Thanks!
Chris

--- monz <monz@attglobal.net> wrote:
>
> hi Chris,
>
>
>
> > From: "Chris Mohr"
> <fromtherealmoftheshadow@yahoo.com>
> > To: <tuning@yahoogroups.com>
> > Sent: Wednesday, January 15, 2003 3:16 PM
> > Subject: [tuning] pitch bend/cents translation
> >
> >
> > Hi friends,
> > A very simple question. I'm happy to say we've
> found a
> > way to work pitch bend rather elegantly into my
> > 53-equal keyboard. I just need to know: if I
> > understand it correctly, MIDI pitch bend command
> > numbers are from something like 1 to 4000, and the
> > default setting is a 12-EQ whole step (200 cents).
> > Does that mean a pitch bend of, say, 20, equals
> one
> > cent of pitch bend? Does pitch bend follow the
> same
> > logarithmic curve that the "cents" system does, so
> > that simple arithmetic is all that's needed to
> > translate pitch bend numbers to cents? If not, has
> > anyone figured out the correspondence (I hate
> > reinventing the wheel).
> > Thank you in advance!
> > Chris Mohr, 53-equal Man.
>
>
>
>
> yes, pitch-bend is logarithmic just like cents.
>
>
> you're thinking of a common pitch-bend specification
> such as that used by Cakewalk (a very popular music
> sequencer software). i've dubbed one unit of this
> pitch-bend measurement a "cawapu":
> http://sonic-arts.org/dict/cawapu.htm
>
> there are 4096 of these units per 12edo Semitone.
>
>
>
> here's a table of cawapu pitch-bend values for
> 53edo:
>
>
> 53edo 12edo
> degree degree + cawapus
>
> 0 0 0
> 1 0 927
> 2 0 1855
> 3 0 2782
> 4 0 3710
>
> 5 1 541
> 6 1 1468
> 7 1 2396
> 8 1 3323
>
> 9 2 155
> 10 2 1082
> 11 2 2009
> 12 2 2937
> 13 2 3864
>
> 14 3 696
> 15 3 1623
> 16 3 2550
> 17 3 3478
>
> 18 4 309
> 19 4 1237
> 20 4 2164
> 21 4 3091
> 22 4 4019
>
> 23 5 850
> 24 5 1778
> 25 5 2705
> 26 5 3632
>
> 27 6 464
> 28 6 1391
> 29 6 2318
> 30 6 3246
>
> 31 7 77
> 32 7 1005
> 33 7 1932
> 34 7 2859
> 35 7 3787
>
> 36 8 618
> 37 8 1546
> 38 8 2473
> 39 8 3400
>
> 40 9 232
> 41 9 1159
> 42 9 2087
> 43 9 3014
> 44 9 3941
>
> 45 10 773
> 46 10 1700
> 47 10 2628
> 48 10 3555
>
> 49 11 386
> 50 11 1314
> 51 11 2241
> 52 11 3169
>
> ( 53 12 0 )
>
>
>
> in creating an actual MIDI-file tuned in 53edo,
> you would:
>
> - replace the integers 0 ... 11 in the "12edo"
> column
> with the MIDI-note letter-names, so that, for
> example,
> if you pick "C" to equal 0, then 1 = C# or Db, 2 =
> D,
> etc.
>
> - then there would be a pitch-bend command at the
> same
> start time using the appropriate number from the
> "cawapu" column. (actually, my own preference is
> to
> have the pitch-bend command at the idealized
> start-time
> of the note, then the MIDI note-on command one
> "tick"
> later.)
>
>
>
> PS -- you might also want to look at these
> definitions:
>
> http://sonic-arts.org/dict/midipu.htm
> http://sonic-arts.org/dict/cents.htm
>
>
> and i have some useful stuff on 53edo here:
>
> http://sonic-arts.org/dict/eqtemp.htm#53
> http://sonic-arts.org/dict/bingo.htm#53
>
>
>
>
> -monz
>
>
>
>
>
>
>
>
>
>
>
>
>

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--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:

> and i have some useful stuff on 53edo here:
>
> http://sonic-arts.org/dict/eqtemp.htm#53
> http://sonic-arts.org/dict/bingo.htm#53

i think this stuff could be integrated a little better. i created the
first graph on the eqtemp page above specifically to be compatible
with *hexagonal* bingo cards, as i'll show now.

a while back, i posted such hexagonal bingo cards -- the one for 53-
equal looks like this:

/tuning-math/files/Paul/53p.gif

the key to understanding this lattice is simply to note that the
triangle in the center, 0-17-31, is the "JI" major triad in 53-equal.
in the rest of the lattice, like spatial relationships represent like
intervals. so a perfect fifth is one step east, a major third is one
step northeast, a major sixth is one step northwest, etc.

here you can see that by moving around through the lattice of "JI"
intervals, you can come back to the note you started in ways that are
not possible in true JI (where you'd end up a certain "comma" away
from where you started). this is easily seen by assuming (without
loss of generality) that you start at pitch 0 -- other instances of
pitch 0 are shown in red.

if you connect one 0 with another 0 with a straight line, you are
depicting what is known as a "unison vector"*. for example, if you go
up a major sixth and then five major thirds (in just intonation, the
resulting interval is called a kleisma), you'll end up at the same
note you started. a line connecting your starting note to your ending
note will point basically upwards but lean a little bit to the left.
now if you look at my ET graph again (the first graph on http://sonic-
arts.org/dict/eqtemp.htm), and locate 53 on it, you'll see a bunch of
green lines passing through it. one of them points basically upwards
and leans a little bit to the left. lo and behold, this line is
labeled "kleismic" (if you can read the green print -- monz said he
had trouble reading it on his screen). it also passes through 15, 19,
34, and 72, other ETs where the kleisma is a unison vector. for
example, look at these bingo cards:

/tuning-math/files/Paul/19p.gif

/tuning-math/files/Paul/34p.gif

notice that while these ETs generally have different unison vectors
from one another, they all share a unison vector in common -- the
kleisma.

you may go through this process for the other unison vectors of 53-
equal, if you like. you'll see that each of them that you can find in
the bingo card corresponds to one of the green lines on the big ET
graph, and that graph will show you other ETs which share the same
unison vector

the actual compositional importance of this information is something
i'll be able to more easily explain after future updates to monz's
pages -- though if you're interested, i could try to explain this now.

cheers,
paul

*for the purposes of this post, "unison vector" means "commatic
unison vector".

p.s. monz -- the table of commas and associated linear temperaments
(the one you have directly below my big ET chart on your eqtemp page)
has been updated to include useful information such as the generator
and period of each temperament. see

/tuning/database?
method=reportRows&tbl=10&sortBy=4

or

/tuning/database?
method=reportRows&tbl=10&sortBy=4&sortDir=down&startAt=&prntRpt=1

i tried to sort by denominator but yahoo doesn't appear to recognize
scientific notation -- so if anyone can compute these ratios out to
the last digit, i'd be appreciative . . .