Topic: the Sault approach
1 scales
| File | Description | Notes | Period (ยข) | Limit |
|---|---|---|---|---|
| strangeion | 19-limit "dodekaphonic" scale (19 & 17). | 12 | 1200.0 | 19 |
Thread (35 messages)
From: Carl Lumma (2003-12-16)
Subject: the Sault approach
Peter and all,
I'm going to attempt a translation of your 'dodekaphonic
program' into list lingo...
1. We want 5-limit harmony (consonances involve ratios with
numbers up to 5, factors of 2 being ignored).
2. We want a 12-tone scale, for compatibility with Western
music.
3. We want the maximum number of "correlations" (dyads anywhere
in the scale which also occur in a single given mode).
4. No two notes shall be separated by less than a Pythagorean
comma.
All of this seems perfectly reasonable. In general on this
list we do not make assumptions 1 or 2, but we understand
them perfectly well. Assumption 4 is what we might call
"taking the Pythagorean comma as a unison vector" (don't ask).
Assumption 3 is not immediately clear. Around here we have
things like "mean variety" and "Rothenberg efficiency" which
are about making the modes of the scale 'similar' in various
ways, and it seems this would be conducive to correlations...
Anyway, the idea of correlations weaving a path through pitch
space as a piece progresses is a new one on me, and a very
interesting one at that, if what you say is true about too
few (random JI) or too many (equal temperament) being
undesirable.
Ok, now let's find scales that are optimum under the criteria.
1. We restrict ourselves to the 5-limit lattice. You'll
recall that triads = triangles.
2. We restrict ourselves to 12 notes.
3. As I posted earlier, I think scales that are convex and
as symmetrical as possible around 1/1 will have the most
correlations (you seemed to recognize the importance of
symmetry in your program NATURAL).
25/18-25/24-25/16
/ \ / \ / \
/ \ / \ / \
10/9---5/3---5/4---15/8
/ \ / \ / \ / \
/ \ / \ / \ / \
16/9---4/3---1/1---3/2---9/8
\ / \ / \ / \ /
\ / \ / \ / \ /
16/15---8/5---6/5---9/5
\ / \ / \ /
\ / \ / \ /
32/25-48/25-36/25
...This 19-tone structure is simply everything in the 5-limit
that lies within a "taxicab" radius of 2 from the origin. It
is maximally convex and symmetrical. Verify that it contains
198 correlations (each corner pitch has 9, each non-corner
edge pitch has 10, and each inside pitch has 14 -- these are
the number of pitches within a radius of 2 of the given pitch).
4. According to Scala's "discard", the only pairs of pitches
closer than a Pythag. comma are 10/9, 9/8 and 9/5, 16/9. To
keep symmetry we would probably delete pitches with their
inversions. We may favor deleting the 9/8, 16/9 because they
are involved in fewer correlations, or we may delete the 10/9,
9/5 pair so to keep the long chain of fifths. We must also
delete 5 more pitches to get the number down to 12. In the
end, I might suggest...
25/16
/ \
/ \
5/3---5/4---15/8
/ \ / \ / \
/ \ / \ / \
16/9---4/3---1/1---3/2---9/8
\ / \ / \ /
\ / \ / \ /
16/15---8/5---6/5
This scale has 72 correlations. However, it doesn't have a
tritone and isn't what we call "Rothenberg proper". These
weren't part of the initial criteria I listed, but oh well,
let's go ahead with something like...
25/18
\
\
5/3---5/4---15/8
/ \ / \ / \
/ \ / \ / \
16/9---4/3---1/1---3/2---9/8
\ / \ / \ /
\ / \ / \ /
16/15---8/5---6/5
...with 66 correlations and now the scale is "strictly proper".
However, now it looks like maybe we should have deleted the
9/8, 16/9 pair instead of the 10/9, 9/5 pair...
25/18
/ \
/ \
10/9---5/3---5/4---15/8
\ / \ / \ /
\ / \ / \ /
4/3---1/1---3/2
/ \ / \ / \
/ \ / \ / \
16/15---8/5---6/5---9/5
...this too has 66 correlations and is also strictly proper,
but it has 11 (as opposed to 10) triads.
For comparison, your scale is...
5/3---5/4--15/8--45/32
/ \ / \ / \ /
/ \ / \ / \ /
16/9---4/3---1/1---3/2---9/8
\ / \ / \ /
\ / \ / \ /
16/15---8/5---6/5
...in which I count 72 correlations and 11 triads. So to the
best of my reckoning you have found the best scale for your
criteria. Still, I'd be interested to hear something like
"Odeion Natural No. 1-003" repeated with...
!
"Odeion Natural No. 1-003" experimental suggestion.
12
!
16/15
10/9
6/5
5/4
4/3
25/18
3/2
8/5
5/3
9/5
15/8
2/1
!
...as well as Ellis' Duodene...
!
Ellis' Duodene.
12
!
16/15
9/8
6/5
5/4
4/3
45/32
3/2
8/5
5/3
9/5
15/8
2/1
!
...not to mention some scales containing intervals like 7:4.
-Carl
From: [email protected] (2003-12-16) Subject: Re: the Sault approach --- In [email protected], Carl Lumma <ekin@l...> wrote: > Peter and all, > > I'm going to attempt a translation of your 'dodekaphonic > program' into list lingo... > > 1. We want 5-limit harmony (consonances involve ratios with > numbers up to 5, factors of 2 being ignored). > > 2. We want a 12-tone scale, for compatibility with Western > music. > > 3. We want the maximum number of "correlations" (dyads anywhere > in the scale which also occur in a single given mode). > > 4. No two notes shall be separated by less than a Pythagorean > comma. > Waitaminute Carl, first, Peter never said he wanted to have only 5-limit. And his use of the Pyth. comma is not as simple as saying nothing is closer than that. It seems there is a lack of understanding of Peter's passband concept. His concept is to compare a chain of Otonal 2:3's with a chain of Utonal 2:3's (3:4's) and then to define "passbands" such that, as an example, the third diatonic step of the scale could be any note between the Pythagorean diminished 4th (8192/6561 c.384 cents) and the Pythagorean major third (81/64 c.408 cens). While Peter chose his exact scale then for correlative reasons, his passpand concept would theoretically accept any just ratio of any limit that fell within that range. This is why he would not use 7/4, because it does not fit the Pythagorean passband for that scale degree. Now, I am not defending this passband idea as sensible, but please understand what it is. I am very curious if anyone is aware of this sort of idea being suggested by anyone else previously. Does anyone have thoughts on there being a theoretical reason this passband idea would be a sensible limiting factor in scales? Or is it truly arbitrary? Is there any other way to apply this idea? -Aaron
From: [email protected] (2003-12-16) Subject: Re: the Sault approach --- In [email protected], backfromthesilo@y... wrote: > Now, I am not defending this passband idea as sensible, but > please understand what it is. What's not sensible about it? It seems one approach to adaptive tuning would be enforcing passbands; that prevents comma drift and keeps comma jumps inside of specified limits.
From: [email protected] (2003-12-16) Subject: Re: the Sault approach - Carl --- In [email protected], Carl Lumma <ekin@l...> wrote: > Peter and all, > > I'm going to attempt a translation of your 'dodekaphonic > program' into list lingo... > > 1. We want 5-limit harmony (consonances involve ratios with > numbers up to 5, factors of 2 being ignored). > That is not actually a criterion or constraint. It just works out that way. > 2. We want a 12-tone scale, for compatibility with Western > music. > We get the 12 tones from the cycle of 5ths up to the second node of the series of powers of 2 and of 3, where the pythagorean comma occurs. Standard derivation of 12-tone. > 3. We want the maximum number of "correlations" (dyads anywhere > in the scale which also occur in a single given mode). Yes - the mode being I. > > 4. No two notes shall be separated by less than a Pythagorean > comma. No - the pythagorean comma is the width of each passband within which the respective note ratio must fall. Between each passband is a stopband, or forbidden zone. For 12-tone, passband x stopband = 1 semitone (approx) = 1.059. So the width of each stopband is about 1.045 whereas each passband is roughly 1.014 wide. Another way of viewing a passband is as the 'tolerance' (in the engineering sense) allowed each ratio. So you can select any ratio that falls with the compass of the passband, that being a pythagorean comma. Note, however, that the center of each passband does *not* correspond to the value of the respective ET vibration ratio. Although the tonic can never be anything but 1:1 (or 1/1 as seems to be the common usage in this group) and the octave can only be 1:2, the tonic passband stretches from 0.986 to 1.000 and the octave passband stretches from 2.000 to 2.027 so as you can see the centers are a little further apart than the ET vibration ratios, although the centers are equi- proportional. The centers are in fact 2.0273^(1/12) apart (evaluating to 1.0606602) and commence at 0.9932474. Since, in a complete cycle of 12 perfect ascending 5ths, the 'octave' ends up high of the mark so that when transposed down it becomes 2.0273, the passband lower boundaries can be calculated by dividing each upper boundary by a pythagorean comma. The lower boundary values then correspond to the values that result from transposing upwards a cycle of 12 perfect *descending* 5ths. Hope that clarifies the matter a little. The same process applied to the primes 2 and 5 (instead of 2 and 3), taken to the node at 125:128, gives us 'triphony', or the augmented triad. - Peter > > All of this seems perfectly reasonable. In general on this > list we do not make assumptions 1 or 2, but we understand > them perfectly well. Assumption 4 is what we might call > "taking the Pythagorean comma as a unison vector" (don't ask). > > Assumption 3 is not immediately clear. Around here we have > things like "mean variety" and "Rothenberg efficiency" which > are about making the modes of the scale 'similar' in various > ways, and it seems this would be conducive to correlations... > Anyway, the idea of correlations weaving a path through pitch > space as a piece progresses is a new one on me, and a very > interesting one at that, if what you say is true about too > few (random JI) or too many (equal temperament) being > undesirable. > > Ok, now let's find scales that are optimum under the criteria. > > 1. We restrict ourselves to the 5-limit lattice. You'll > recall that triads = triangles. > > 2. We restrict ourselves to 12 notes. > > 3. As I posted earlier, I think scales that are convex and > as symmetrical as possible around 1/1 will have the most > correlations (you seemed to recognize the importance of > symmetry in your program NATURAL). > > 25/18-25/24-25/16 > / \ / \ / \ > / \ / \ / \ > 10/9---5/3---5/4---15/8 > / \ / \ / \ / \ > / \ / \ / \ / \ > 16/9---4/3---1/1---3/2---9/8 > \ / \ / \ / \ / > \ / \ / \ / \ / > 16/15---8/5---6/5---9/5 > \ / \ / \ / > \ / \ / \ / > 32/25-48/25-36/25 > > ...This 19-tone structure is simply everything in the 5-limit > that lies within a "taxicab" radius of 2 from the origin. It > is maximally convex and symmetrical. Verify that it contains > 198 correlations (each corner pitch has 9, each non-corner > edge pitch has 10, and each inside pitch has 14 -- these are > the number of pitches within a radius of 2 of the given pitch). > > 4. According to Scala's "discard", the only pairs of pitches > closer than a Pythag. comma are 10/9, 9/8 and 9/5, 16/9. To > keep symmetry we would probably delete pitches with their > inversions. We may favor deleting the 9/8, 16/9 because they > are involved in fewer correlations, or we may delete the 10/9, > 9/5 pair so to keep the long chain of fifths. We must also > delete 5 more pitches to get the number down to 12. In the > end, I might suggest... > > 25/16 > / \ > / \ > 5/3---5/4---15/8 > / \ / \ / \ > / \ / \ / \ > 16/9---4/3---1/1---3/2---9/8 > \ / \ / \ / > \ / \ / \ / > 16/15---8/5---6/5 > > This scale has 72 correlations. However, it doesn't have a > tritone and isn't what we call "Rothenberg proper". These > weren't part of the initial criteria I listed, but oh well, > let's go ahead with something like... > > 25/18 > \ > \ > 5/3---5/4---15/8 > / \ / \ / \ > / \ / \ / \ > 16/9---4/3---1/1---3/2---9/8 > \ / \ / \ / > \ / \ / \ / > 16/15---8/5---6/5 > > ...with 66 correlations and now the scale is "strictly proper". > However, now it looks like maybe we should have deleted the > 9/8, 16/9 pair instead of the 10/9, 9/5 pair... > > 25/18 > / \ > / \ > 10/9---5/3---5/4---15/8 > \ / \ / \ / > \ / \ / \ / > 4/3---1/1---3/2 > / \ / \ / \ > / \ / \ / \ > 16/15---8/5---6/5---9/5 > > ...this too has 66 correlations and is also strictly proper, > but it has 11 (as opposed to 10) triads. > > For comparison, your scale is... > > 5/3---5/4--15/8--45/32 > / \ / \ / \ / > / \ / \ / \ / > 16/9---4/3---1/1---3/2---9/8 > \ / \ / \ / > \ / \ / \ / > 16/15---8/5---6/5 > > ...in which I count 72 correlations and 11 triads. So to the > best of my reckoning you have found the best scale for your > criteria. Still, I'd be interested to hear something like > "Odeion Natural No. 1-003" repeated with... > > ! > "Odeion Natural No. 1-003" experimental suggestion. > 12 > ! > 16/15 > 10/9 > 6/5 > 5/4 > 4/3 > 25/18 > 3/2 > 8/5 > 5/3 > 9/5 > 15/8 > 2/1 > ! > > ...as well as Ellis' Duodene... > > ! > Ellis' Duodene. > 12 > ! > 16/15 > 9/8 > 6/5 > 5/4 > 4/3 > 45/32 > 3/2 > 8/5 > 5/3 > 9/5 > 15/8 > 2/1 > ! > > ...not to mention some scales containing intervals like 7:4. > > -Carl
From: Carl Lumma (2003-12-16) Subject: Re: [tuning] Re: the Sault approach >Waitaminute Carl, first, Peter never said he wanted to have only >5-limit. He effectively did. >And his use of the Pyth. comma is not as simple as saying nothing >is closer than that. It seems there is a lack of understanding >of Peter's passband concept. His concept is to compare a chain >of Otonal 2:3's with a chain of Utonal 2:3's (3:4's) and then to >define "passbands" such that, as an example, the third diatonic >step of the scale could be any note between the Pythagorean >diminished 4th (8192/6561 c.384 cents) and the Pythagorean major >third (81/64 c.408 cens). So something like... 0-90 cents 90-204 cents 204-294 cents 294-408 cents 408-498 cents 498-588 cents 588-702 cents 702-792 cents 792-906 cents 906-996 cents 996-1110 cents 1110-1200 cents ...through which a mode of... ! "Odeion Natural No. 1-003" experimental suggestion. 12 ! 16/15 10/9 6/5 5/4 4/3 25/18 3/2 8/5 5/3 9/5 15/8 2/1 ! ...fits. The duodene also fits of course, and I neglected to give its correlations last time; 68. So it actually beats the above scale, but not Peter's scale. By the way, I think I'm counting correlations twice compared to Peter, but it won't change anything. >While Peter chose his exact scale then >for correlative reasons, his passpand concept would theoretically >accept any just ratio of any limit that fell within that range. >This is why he would not use 7/4, because it does not fit the >Pythagorean passband for that scale degree. There are an infinite number of > 5-limit ratios that would fit through the passbands as you define them. >Now, I am not defending this passband idea as sensible, but >please understand what it is. I am very curious if anyone is >aware of this sort of idea being suggested by anyone else >previously. Does anyone have thoughts on there being a >theoretical reason this passband idea would be a sensible >limiting factor in scales? Or is it truly arbitrary? Is there >any other way to apply this idea? It's perfectly reasonable if you want to approximate the scale that defined the passbands. -Carl
From: Carl Lumma (2003-12-16) Subject: Re: [tuning] Re: the Sault approach - Carl >> 1. We want 5-limit harmony (consonances involve ratios with >> numbers up to 5, factors of 2 being ignored). > >That is not actually a criterion or constraint. It just works out >that way. A bit later I'll show you a > 5-limit scale that meets the other criteria. >> 2. We want a 12-tone scale, for compatibility with Western >> music. > >We get the 12 tones from the cycle of 5ths up to the second node of >the series of powers of 2 and of 3, where the pythagorean comma >occurs. Standard derivation of 12-tone. Ok, so you'll need a node-finding function which rules out 17, 19, 29, 41, etc. Presumably by penalizing tunings for the number of notes they contain. >> 4. No two notes shall be separated by less than a Pythagorean >> comma. > >No - the pythagorean comma is the width of each passband within >which the respective note ratio must fall. Between each passband is >a stopband, or forbidden zone. For 12-tone, passband x stopband = 1 >semitone (approx) = 1.059. So the width of each stopband is about >1.045 whereas each passband is roughly 1.014 wide. Another way of >viewing a passband is as the 'tolerance' (in the engineering sense) >allowed each ratio. So you can select any ratio that falls with the >compass of the passband, that being a pythagorean comma. Oh, ok, it seems Aaron had this right but I read him wrong... >Note, however, that the center of each passband does *not* correspond >to the value of the respective ET vibration ratio. Although the tonic >can never be anything but 1:1 (or 1/1 as seems to be the common usage >in this group) and the octave can only be 1:2, the tonic passband >stretches from 0.986 to 1.000 and the octave passband stretches from >2.000 to 2.027 so as you can see the centers are a little further >apart than the ET vibration ratios, although the centers are equi- >proportional. The centers are in fact 2.0273^(1/12) apart (evaluating >to 1.0606602) and commence at 0.9932474. Since, in a complete cycle >of 12 perfect ascending 5ths, the 'octave' ends up high of the mark >so that when transposed down it becomes 2.0273, the passband lower >boundaries can be calculated by dividing each upper boundary by a >pythagorean comma. The lower boundary values then correspond to the >values that result from transposing upwards a cycle of 12 perfect >*descending* 5ths. Ya lost me here. Can you give the actual passbands in cents? For a factor F and an equal temperament ET, here is how to find F as a number of steps in ET... round(log2(F) * ET) ...and if you don't have base2 logs handy, you can use whatever log you want... round( (log(F) / log(2)) * ET) ) ...Now for cents, ET = 1200. -Carl
From: [email protected] (2003-12-17) Subject: Re: the Sault approach - Carl --- In [email protected], Carl Lumma <ekin@l...> wrote: > >> 1. We want 5-limit harmony (consonances involve ratios with > >> numbers up to 5, factors of 2 being ignored). > > > >That is not actually a criterion or constraint. It just works out > >that way. > > A bit later I'll show you a > 5-limit scale that meets the other > criteria. > > >> 2. We want a 12-tone scale, for compatibility with Western > >> music. > > > >We get the 12 tones from the cycle of 5ths up to the second node of > >the series of powers of 2 and of 3, where the pythagorean comma > >occurs. Standard derivation of 12-tone. > > Ok, so you'll need a node-finding function which rules out 17, 19, > 29, 41, etc. Presumably by penalizing tunings for the number of > notes they contain. Increasing complexity means increasing inaccessibility, workability, practicality, expense. Same with everything. Apart from that I would suppose that there is an infinity of analogs. However, all must include 2. I have experimented with prime pairs which do exclude 2 and these can give rise to overlapping passbands, leading to the absurdity that scale degree N could be assigned a higher vibration number than scale degree N+1. > > >> 4. No two notes shall be separated by less than a Pythagorean > >> comma. > > > >No - the pythagorean comma is the width of each passband within > >which the respective note ratio must fall. Between each passband is > >a stopband, or forbidden zone. For 12-tone, passband x stopband = 1 > >semitone (approx) = 1.059. So the width of each stopband is about > >1.045 whereas each passband is roughly 1.014 wide. Another way of > >viewing a passband is as the 'tolerance' (in the engineering sense) > >allowed each ratio. So you can select any ratio that falls with the > >compass of the passband, that being a pythagorean comma. > > Oh, ok, it seems Aaron had this right but I read him wrong... > > >Note, however, that the center of each passband does *not* correspond > >to the value of the respective ET vibration ratio. Although the tonic > >can never be anything but 1:1 (or 1/1 as seems to be the common usage > >in this group) and the octave can only be 1:2, the tonic passband > >stretches from 0.986 to 1.000 and the octave passband stretches from > >2.000 to 2.027 so as you can see the centers are a little further > >apart than the ET vibration ratios, although the centers are equi- > >proportional. The centers are in fact 2.0273^(1/12) apart (evaluating > >to 1.0606602) and commence at 0.9932474. Since, in a complete cycle > >of 12 perfect ascending 5ths, the 'octave' ends up high of the mark > >so that when transposed down it becomes 2.0273, the passband lower > >boundaries can be calculated by dividing each upper boundary by a > >pythagorean comma. The lower boundary values then correspond to the > >values that result from transposing upwards a cycle of 12 perfect > >*descending* 5ths. > > Ya lost me here. Can you give the actual passbands in cents? > > For a factor F and an equal temperament ET, here is how to find F > as a number of steps in ET... > > round(log2(F) * ET) > > ...and if you don't have base2 logs handy, you can use whatever > log you want... > > round( (log(F) / log(2)) * ET) ) > > ...Now for cents, ET = 1200. > > -Carl You can easily do that for yourself Carl. My calculator is rather old and simple. The dodekachromatic comb filter is shown at http://www.odeion.org/atlantis/chapter-1.html#table1-4 - Peter
From: Carl Lumma (2003-12-19) Subject: Re: [tuning] Re: the Sault approach - Carl >> Ya lost me here. Can you give the actual passbands in cents? >> >> For a factor F and an equal temperament ET, here is how to find F >> as a number of steps in ET... >> >> round(log2(F) * ET) >> >> ...and if you don't have base2 logs handy, you can use whatever >> log you want... >> >> round( (log(F) / log(2)) * ET) ) >> >> ...Now for cents, ET = 1200. >> >> -Carl > > You can easily do that for yourself Carl. My calculator is rather > old and simple. The dodekachromatic comb filter is shown at > http://www.odeion.org/atlantis/chapter-1.html#table1-4 So the passbands, in cents, are: -24 0 90 114 181 204 294 318 384 408 498 521 588 612 678 702 792 816 882 906 996 1020 1086 1110 1200 1223 >>> 1. We want 5-limit harmony (consonances involve ratios with >>> numbers up to 5, factors of 2 being ignored). >> >>That is not actually a criterion or constraint. It just works out >>that way. > >A bit later I'll show you a > 5-limit scale that meets the other >criteria. Actually, though of course one can be found, as Gene pointed out the traditional Pythagorean scale has more correlations and will naturally fit into the passbands. So 5-limit harmony *is* a criterion. -Carl
From: Carl Lumma (2003-12-19) Subject: Re: [tuning] Re: the Sault approach - Carl I wrote... >So the passbands, in cents, are: > > -24 0 > 90 114 > 181 204 > 294 318 > 384 408 > 498 521 > 588 612 > 678 702 > 792 816 > 882 906 > 996 1020 >1086 1110 >1200 1223 > >>>> 1. We want 5-limit harmony (consonances involve ratios with >>>> numbers up to 5, factors of 2 being ignored). >>> >>>That is not actually a criterion or constraint. It just works out >>>that way. >> >>A bit later I'll show you a > 5-limit scale that meets the other >>criteria. > >Actually, though of course one can be found, Here's an example... ! strangeion.scl ! 19-limit "dodekaphonic" scale (19 & 17). 12 ! 17/16 !.......C# 19/17 !........D 19/16 !.......D# 323/256 !......E 8192/6137 !....F 361/256 !.....F# 6137/4096 !....G 512/323 !.....G# 32/19 !........A 34/19 !.......A# 32/17 !........B 2/1 !..........C ! ! F#--G ! / \ / ! D---D#--E ! / \ / \ / ! B---C---C# ! / \ / \ / ! G#--A---A# ! / ! F ...the above lattice is actually a 19-limit lattice, where / stands for a 19:16 minor third, and --- stands for a 17:16 semitone. The result fits through your passbands, has 72 correlations (the same as your 5-limit scale), and actually is an interesting 'justifying' of 12-tET. The 5ths range from 693 to 710 cents, which is the entire range we usually consider acceptable. -Carl
From: Peter Wakefield Sault (2003-12-20) Subject: Re: the Sault approach - Carl --- In [email protected], Carl Lumma <ekin@l...> wrote: > >> Ya lost me here. Can you give the actual passbands in cents? > >> > >> For a factor F and an equal temperament ET, here is how to find F > >> as a number of steps in ET... > >> > >> round(log2(F) * ET) > >> > >> ...and if you don't have base2 logs handy, you can use whatever > >> log you want... > >> > >> round( (log(F) / log(2)) * ET) ) > >> > >> ...Now for cents, ET = 1200. > >> > >> -Carl > > > > You can easily do that for yourself Carl. My calculator is rather > > old and simple. The dodekachromatic comb filter is shown at > > http://www.odeion.org/atlantis/chapter-1.html#table1-4 > > So the passbands, in cents, are: > > -24 0 > 90 114 > 181 204 > 294 318 > 384 408 > 498 521 > 588 612 > 678 702 > 792 816 > 882 906 > 996 1020 > 1086 1110 > 1200 1223 > Hi Carl That looks about right, at a glance.... > >>> 1. We want 5-limit harmony (consonances involve ratios with > >>> numbers up to 5, factors of 2 being ignored). > >> > >>That is not actually a criterion or constraint. It just works out > >>that way. > > > >A bit later I'll show you a > 5-limit scale that meets the other > >criteria. > > Actually, though of course one can be found, as Gene pointed out > the traditional Pythagorean scale has more correlations and will > naturally fit into the passbands. So 5-limit harmony *is* a > criterion. > Not really;- there is an infinity of infinity-limit ratios that will filter through each passband. It's just turns out that the particular set which maximizes correlations indeed comprises 5-limit ratios. So it is not necessary to specify that as a constraint because you get it anyway. Peter > -Carl
From: Peter Wakefield Sault (2003-12-20) Subject: Re: the Sault approach - Carl --- In [email protected], Carl Lumma <ekin@l...> wrote: > I wrote... > > >So the passbands, in cents, are: > > > > -24 0 > > 90 114 > > 181 204 > > 294 318 > > 384 408 > > 498 521 > > 588 612 > > 678 702 > > 792 816 > > 882 906 > > 996 1020 > >1086 1110 > >1200 1223 > > > >>>> 1. We want 5-limit harmony (consonances involve ratios with > >>>> numbers up to 5, factors of 2 being ignored). > >>> > >>>That is not actually a criterion or constraint. It just works out > >>>that way. > >> > >>A bit later I'll show you a > 5-limit scale that meets the other > >>criteria. > > > >Actually, though of course one can be found, > > Here's an example... > > ! strangeion.scl > ! > 19-limit "dodekaphonic" scale (19 & 17). > 12 > ! > 17/16 !.......C# > 19/17 !........D > 19/16 !.......D# > 323/256 !......E > 8192/6137 !....F > 361/256 !.....F# > 6137/4096 !....G > 512/323 !.....G# > 32/19 !........A > 34/19 !.......A# > 32/17 !........B > 2/1 !..........C > ! > ! F#--G > ! / \ / > ! D---D#--E > ! / \ / \ / > ! B---C---C# > ! / \ / \ / > ! G#--A---A# > ! / > ! F > > ...the above lattice is actually a 19-limit lattice, where / > stands for a 19:16 minor third, and --- stands for a 17:16 > semitone. The result fits through your passbands, has 72 > correlations (the same as your 5-limit scale), and actually > is an interesting 'justifying' of 12-tET. The 5ths range > from 693 to 710 cents, which is the entire range we usually > consider acceptable. > > -Carl Hi Carl Well that looks very interesting. I will have to free up my correlator to allow the entry of user-specified ratios. Those which are built into the program are restricted to 2-digit denominators. There's just one problem with your correlation count. The absolute maximum possible number of correlations is 66. You must be including the definitive set in your total. Since the definitive set is not itself correlative it cannot be included in the count. So the max is 1 + 2 + 3 +... ...+ 10 + 11 = 66 By the way, a version of the correlator which allows complementary intervals to be decoupled, allowing asymmetric inversions, is now available and can be downloaded from http://www.odeion.org/atlantis/natural2.html This reveals that the total number of correlations in the Ellis Scale, after including both aug 4th and dim 5th, is 41. With full symmetry, using 9:16 (the so-called 'grave' 7th) in place of the Ellis minor 7th of 5:9, the total is 44. I additionally need to extend the software to show a count of only correlative 3rds and 5ths, the criterion you were originally using to compare the Ellis scale to the symmetric inversion equivalent. Peter
From: Carl Lumma (2003-12-20) Subject: Re: [tuning] Re: the Sault approach - Carl >> Here's an example... >> >> ! strangeion.scl >> ! >> 19-limit "dodekaphonic" scale (19 & 17). >> 12 >> ! >> 17/16 !.......C# >> 19/17 !........D >> 19/16 !.......D# >> 323/256 !......E >> 8192/6137 !....F >> 361/256 !.....F# >> 6137/4096 !....G >> 512/323 !.....G# >> 32/19 !........A >> 34/19 !.......A# >> 32/17 !........B >> 2/1 !..........C >> ! >> ! F#--G >> ! / \ / >> ! D---D#--E >> ! / \ / \ / >> ! B---C---C# >> ! / \ / \ / >> ! G#--A---A# >> ! / >> ! F >> >> ...the above lattice is actually a 19-limit lattice, where / >> stands for a 19:16 minor third, and --- stands for a 17:16 >> semitone. The result fits through your passbands, has 72 >> correlations (the same as your 5-limit scale), and actually >> is an interesting 'justifying' of 12-tET. The 5ths range >> from 693 to 710 cents, which is the entire range we usually >> consider acceptable. > >Well that looks very interesting. I will have to free up my >correlator to allow the entry of user-specified ratios. Those which >are built into the program are restricted to 2-digit denominators. You could also check the 3-limit Pythagorean scale, and see if I'm right that it does as well as your 5-limit scale. >There's just one problem with your correlation count. The absolute >maximum possible number of correlations is 66. You must be including >the definitive set in your total. As I've said, I think I count each correlation twice. By your way of counting both the above scale and your scale have 37 correlations. >By the way, a version of the correlator which allows complementary >intervals to be decoupled, allowing asymmetric inversions, is now >available and can be downloaded from >http://www.odeion.org/atlantis/natural2.html I thought that's what I have -- the about box says 2.0 -- but I don't see how to decouple inversions. But that was before I understood what you were after. I think we can safely restrict ourselves to inversion-symmetrical scales to satisfy your criteria. >This reveals that the total number of correlations in the Ellis >Scale, after including both aug 4th and dim 5th, is 41. I really don't see how you can include both and call it "dodekaphony". You want a 13-tone scale, then we can talk 13-tone scales. -Carl
From: Peter Wakefield Sault (2003-12-20) Subject: Re: the Sault approach - Carl --- In [email protected], Carl Lumma <ekin@l...> wrote: > >> Here's an example... > >> > >> ! strangeion.scl > >> ! > >> 19-limit "dodekaphonic" scale (19 & 17). > >> 12 > >> ! > >> 17/16 !.......C# > >> 19/17 !........D > >> 19/16 !.......D# > >> 323/256 !......E > >> 8192/6137 !....F > >> 361/256 !.....F# > >> 6137/4096 !....G > >> 512/323 !.....G# > >> 32/19 !........A > >> 34/19 !.......A# > >> 32/17 !........B > >> 2/1 !..........C > >> ! > >> ! F#--G > >> ! / \ / > >> ! D---D#--E > >> ! / \ / \ / > >> ! B---C---C# > >> ! / \ / \ / > >> ! G#--A---A# > >> ! / > >> ! F > >> > >> ...the above lattice is actually a 19-limit lattice, where / > >> stands for a 19:16 minor third, and --- stands for a 17:16 > >> semitone. The result fits through your passbands, has 72 > >> correlations (the same as your 5-limit scale), and actually > >> is an interesting 'justifying' of 12-tET. The 5ths range > >> from 693 to 710 cents, which is the entire range we usually > >> consider acceptable. > > > >Well that looks very interesting. I will have to free up my > >correlator to allow the entry of user-specified ratios. Those which > >are built into the program are restricted to 2-digit denominators. > > You could also check the 3-limit Pythagorean scale, and see if I'm > right that it does as well as your 5-limit scale. > > >There's just one problem with your correlation count. The absolute > >maximum possible number of correlations is 66. You must be including > >the definitive set in your total. > > As I've said, I think I count each correlation twice. By your > way of counting both the above scale and your scale have 37 > correlations. > > >By the way, a version of the correlator which allows complementary > >intervals to be decoupled, allowing asymmetric inversions, is now > >available and can be downloaded from > >http://www.odeion.org/atlantis/natural2.html > > I thought that's what I have -- the about box says 2.0 -- but I > don't see how to decouple inversions. > Hi Carl, I know the webpage still says 'natural2.html' but the latest is actually 2.01 > But that was before I understood what you were after. I think > we can safely restrict ourselves to inversion-symmetrical scales > to satisfy your criteria. > > >This reveals that the total number of correlations in the Ellis > >Scale, after including both aug 4th and dim 5th, is 41. > > I really don't see how you can include both and call it > "dodekaphony". You want a 13-tone scale, then we can talk > 13-tone scales. > > -Carl It's a purely theoretical necessity of 'rational' intervals. The aug 4th and dim 5th both belong to the same pitch class = 6. Everyone since and including The Master of The Moon himself has ignored the necessity for a pair at this position, apparently because of the conceptual difficulty it engenders. In this scheme there is no interval which is not paired with another, its complement. Pitch class 6 is self-complementary. Conceptually there is still a complementary pair in 12ET but they are exactly equal to one another and therefore indistinguishable. Yes it adds up to 13 intervals - but only 12 pitch classes. Peter
From: Carl Lumma (2003-12-20)
Subject: Re: [tuning] Re: the Sault approach - Carl
>> really don't see how you can include both and call it
>> "dodekaphony". You want a 13-tone scale, then we can talk
>> 13-tone scales.
>
>It's a purely theoretical necessity of 'rational' intervals. The aug
>4th and dim 5th both belong to the same pitch class = 6.
Assuming you mean pitch class 6 = passband 6, lots of pairs of
rational intervals fit through the passbands. You mean, both
the interval in question and its inversion fit through the *same*
passband? Good luck justifying that one.
> Everyone
>since and including The Master of The Moon himself has ignored the
>necessity for a pair at this position, apparently because of the
>conceptual difficulty it engenders.
So dodekaphony refers not to the number of pitches, but the number
of passbands ("pitch classes" is taken by another meaning, I'm
afraid). If we're allowed to have multiple pitches in each passband
so long as their inversion appears in the scale, we can do *lots* of
fancy stuff not currently admitted by your paradigm.
-Carl
From: Peter Wakefield Sault (2003-12-21)
Subject: Re: the Sault approach - Carl
--- In [email protected], Carl Lumma <ekin@l...> wrote:
> >> really don't see how you can include both and call it
> >> "dodekaphony". You want a 13-tone scale, then we can talk
> >> 13-tone scales.
> >
> >It's a purely theoretical necessity of 'rational' intervals. The
aug
> >4th and dim 5th both belong to the same pitch class = 6.
>
> Assuming you mean pitch class 6 = passband 6, lots of pairs of
> rational intervals fit through the passbands. You mean, both
> the interval in question and its inversion fit through the *same*
> passband? Good luck justifying that one.
Facts are facts. They need no "justifying". Good luck producing more
spurious arguments.
>
> > Everyone
> >since and including The Master of The Moon himself has ignored the
> >necessity for a pair at this position, apparently because of the
> >conceptual difficulty it engenders.
>
> So dodekaphony refers not to the number of pitches, but the number
> of passbands ("pitch classes" is taken by another meaning, I'm
> afraid). If we're allowed to have multiple pitches in each passband
> so long as their inversion appears in the scale, we can do *lots* of
> fancy stuff not currently admitted by your paradigm.
>
> -Carl
Wrong. There is one passband for each pitch class. Each pitch class
comprises an infinity of pitches whether you like it or not. I take
it that either you do not know what a 'pitch class' is, or that you
are inventing some specious definition of your own (which you appear
unwilling to share) solely for the purpose of trying counter the
facts as I stated them. That is sophistry and not even very good
sophistry.
From: Carl Lumma (2003-12-21)
Subject: Re: [tuning] Re: the Sault approach - Carl
>> Assuming you mean pitch class 6 = passband 6, lots of pairs of
>> rational intervals fit through the passbands. You mean, both
>> the interval in question and its inversion fit through the *same*
>> passband? Good luck justifying that one.
>
>Facts are facts. They need no "justifying". Good luck producing more
>spurious arguments.
Facts are facts, the universe is full of them. Which ones you
apply to music are up to you.
>> > Everyone
>> >since and including The Master of The Moon himself has ignored the
>> >necessity for a pair at this position, apparently because of the
>> >conceptual difficulty it engenders.
>>
>> So dodekaphony refers not to the number of pitches, but the number
>> of passbands ("pitch classes" is taken by another meaning, I'm
>> afraid). If we're allowed to have multiple pitches in each passband
>> so long as their inversion appears in the scale, we can do *lots* of
>> fancy stuff not currently admitted by your paradigm.
>
>Wrong. There is one passband for each pitch class.
That's what I said.
>Each pitch class comprises an infinity of pitches whether you like it
>or not.
That's also what I said.
>I take
>it that either you do not know what a 'pitch class' is, or that you
>are inventing some specious definition of your own (which you appear
>unwilling to share) solely for the purpose of trying counter the
>facts as I stated them. That is sophistry and not even very good
>sophistry.
I'm not trying to counter facts, I'm trying to figure out why
you admit a pair of tritones but only single pitch in all other
pitch classes. Why is that, Peter?
You have not defined pitch class. Rather than worry about whether
I "know what it means", why not just define it?
-Carl
From: Kurt Bigler (2003-12-21)
Subject: Re: [tuning] Re: the Sault approach - Carl
on 12/20/03 9:29 PM, Peter Wakefield Sault <[email protected]> wrote:
> --- In [email protected], Carl Lumma <ekin@l...> wrote:
>>>> really don't see how you can include both and call it
>>>> "dodekaphony". You want a 13-tone scale, then we can talk
>>>> 13-tone scales.
>>>
>>> It's a purely theoretical necessity of 'rational' intervals. The
> aug
>>> 4th and dim 5th both belong to the same pitch class = 6.
>>
>> Assuming you mean pitch class 6 = passband 6, lots of pairs of
>> rational intervals fit through the passbands. You mean, both
>> the interval in question and its inversion fit through the *same*
>> passband? Good luck justifying that one.
>
>
> Facts are facts. They need no "justifying". Good luck producing more
> spurious arguments.
>
>
>>
>>> Everyone
>>> since and including The Master of The Moon himself has ignored the
>>> necessity for a pair at this position, apparently because of the
>>> conceptual difficulty it engenders.
>>
>> So dodekaphony refers not to the number of pitches, but the number
>> of passbands ("pitch classes" is taken by another meaning, I'm
>> afraid). If we're allowed to have multiple pitches in each passband
>> so long as their inversion appears in the scale, we can do *lots* of
>> fancy stuff not currently admitted by your paradigm.
>>
>> -Carl
>
>
> Wrong. There is one passband for each pitch class. Each pitch class
> comprises an infinity of pitches whether you like it or not. I take
> it that either you do not know what a 'pitch class' is, or that you
> are inventing some specious definition of your own (which you appear
> unwilling to share) solely for the purpose of trying counter the
> facts as I stated them. That is sophistry and not even very good
> sophistry.
Peter, isn't this your opinion as to whether it is sophistry? Do you really
believe you understand (beyond any doubt) Carl's intention is this? I
believe Carl is working harder than anyone at helping to interpret your work
on behalf of and for the benfit of the rest of us. I'm glad for the dialog
he is having with you. Please realize Carl is working at something here.
He is not trying to obscure or evade. Like all of us, he might even have
some habits that unknowingly obscure something, but so might you. If
something seems to indicate sophistry to you, can look beyond that
appearance and see Carl's intention? He's trying. Everyone can make
mistakes, including you, right? A mistake can look like something else if
you want to judge it against your personal criteria. And isn't is possible
that you can make a mistake of misinterpretation? I still see not much
flexibility in the forms of dialog that you find admissable. We (people) do
not meet each other specifications. We are alive and have hopes and
limitations, and we need each other's help and forbearance. It may be that
you actually wish to help even when you appear harsh, yet please realize
that may be very hard for us to recognize. As I see it, things you respond
harshly too are often much less harsh in tone than your responses.
Again, this is just "as I see it". That's my own truth in this, not
perfectly stated, but I hope not too far off the mark, and I'd appreciate
your forbearance in reading it, because no doubt I've made some mistakes.
In Carl's response he suggested that you might simply share your definition
of "pitch class". To simply give a definition (and not be offended by the
request), this is a good way to be more direct. Patience is particularly a
virtue in teaching, and like it or not, teaching is one thing you are doing
here, and I think (please correct me if I'm wrong) in a way you *do* like
being a teacher, so why not make the best of it?
-Kurt
From: Peter Wakefield Sault (2003-12-21)
Subject: Re: the Sault approach - Carl
--- In [email protected], Carl Lumma <ekin@l...> wrote:
> >> Assuming you mean pitch class 6 = passband 6, lots of pairs of
> >> rational intervals fit through the passbands. You mean, both
> >> the interval in question and its inversion fit through the *same*
> >> passband? Good luck justifying that one.
> >
> >Facts are facts. They need no "justifying". Good luck producing
more
> >spurious arguments.
>
> Facts are facts, the universe is full of them. Which ones you
> apply to music are up to you.
>
> >> > Everyone
> >> >since and including The Master of The Moon himself has ignored
the
> >> >necessity for a pair at this position, apparently because of
the
> >> >conceptual difficulty it engenders.
> >>
> >> So dodekaphony refers not to the number of pitches, but the
number
> >> of passbands ("pitch classes" is taken by another meaning, I'm
> >> afraid). If we're allowed to have multiple pitches in each
passband
> >> so long as their inversion appears in the scale, we can do
*lots* of
> >> fancy stuff not currently admitted by your paradigm.
> >
> >Wrong. There is one passband for each pitch class.
>
> That's what I said.
>
> >Each pitch class comprises an infinity of pitches whether you like
it
> >or not.
>
> That's also what I said.
>
> >I take
> >it that either you do not know what a 'pitch class' is, or that
you
> >are inventing some specious definition of your own (which you
appear
> >unwilling to share) solely for the purpose of trying counter the
> >facts as I stated them. That is sophistry and not even very good
> >sophistry.
>
> I'm not trying to counter facts, I'm trying to figure out why
> you admit a pair of tritones but only single pitch in all other
> pitch classes. Why is that, Peter?
>
I quote myself from above - "Each pitch class comprises an infinity
of pitches". You appear to have a mental block over plain English, or
a reading difficulty. Get some therapy then maybe afterwards we can
have a meaningful discussion.
> You have not defined pitch class. Rather than worry about whether
> I "know what it means", why not just define it?
>
> -Carl
You appear to have a reading problem. Get some tuition.
From: Peter Wakefield Sault (2003-12-21)
Subject: Re: the Sault approach - Carl
--- In [email protected], Kurt Bigler <kkb@b...> wrote:
> on 12/20/03 9:29 PM, Peter Wakefield Sault <sault@c...> wrote:
>
> > --- In [email protected], Carl Lumma <ekin@l...> wrote:
> >>>> really don't see how you can include both and call it
> >>>> "dodekaphony". You want a 13-tone scale, then we can talk
> >>>> 13-tone scales.
> >>>
> >>> It's a purely theoretical necessity of 'rational' intervals. The
> > aug
> >>> 4th and dim 5th both belong to the same pitch class = 6.
> >>
> >> Assuming you mean pitch class 6 = passband 6, lots of pairs of
> >> rational intervals fit through the passbands. You mean, both
> >> the interval in question and its inversion fit through the *same*
> >> passband? Good luck justifying that one.
> >
> >
> > Facts are facts. They need no "justifying". Good luck producing
more
> > spurious arguments.
> >
> >
> >>
> >>> Everyone
> >>> since and including The Master of The Moon himself has ignored
the
> >>> necessity for a pair at this position, apparently because of the
> >>> conceptual difficulty it engenders.
> >>
> >> So dodekaphony refers not to the number of pitches, but the
number
> >> of passbands ("pitch classes" is taken by another meaning, I'm
> >> afraid). If we're allowed to have multiple pitches in each
passband
> >> so long as their inversion appears in the scale, we can do
*lots* of
> >> fancy stuff not currently admitted by your paradigm.
> >>
> >> -Carl
> >
> >
> > Wrong. There is one passband for each pitch class. Each pitch
class
> > comprises an infinity of pitches whether you like it or not. I
take
> > it that either you do not know what a 'pitch class' is, or that
you
> > are inventing some specious definition of your own (which you
appear
> > unwilling to share) solely for the purpose of trying counter the
> > facts as I stated them. That is sophistry and not even very good
> > sophistry.
>
> Peter, isn't this your opinion as to whether it is sophistry? Do
you really
> believe you understand (beyond any doubt) Carl's intention is
this? I
> believe Carl is working harder than anyone at helping to interpret
your work
> on behalf of and for the benfit of the rest of us. I'm glad for
the dialog
> he is having with you. Please realize Carl is working at something
here.
> He is not trying to obscure or evade. Like all of us, he might
even have
> some habits that unknowingly obscure something, but so might you.
If
> something seems to indicate sophistry to you, can look beyond that
> appearance and see Carl's intention? He's trying. Everyone can
make
> mistakes, including you, right? A mistake can look like something
else if
> you want to judge it against your personal criteria. And isn't is
possible
> that you can make a mistake of misinterpretation? I still see not
much
> flexibility in the forms of dialog that you find admissable. We
(people) do
> not meet each other specifications. We are alive and have hopes and
> limitations, and we need each other's help and forbearance. It may
be that
> you actually wish to help even when you appear harsh, yet please
realize
> that may be very hard for us to recognize. As I see it, things you
respond
> harshly too are often much less harsh in tone than your responses.
>
> Again, this is just "as I see it". That's my own truth in this, not
> perfectly stated, but I hope not too far off the mark, and I'd
appreciate
> your forbearance in reading it, because no doubt I've made some
mistakes.
>
> In Carl's response he suggested that you might simply share your
definition
> of "pitch class". To simply give a definition (and not be offended
by the
> request), this is a good way to be more direct. Patience is
particularly a
> virtue in teaching, and like it or not, teaching is one thing you
are doing
> here, and I think (please correct me if I'm wrong) in a way you
*do* like
> being a teacher, so why not make the best of it?
>
> -Kurt
I'm afraid you're wrong Kurt. I quote from Carl above (scroll up to
verify):-
*** ("pitch classes" is taken by another meaning, I'm afraid). ***
So let's have Carl's exclusive definition first, since he has already
prejudged the matter. Carl implies that he knows what my definition
is because he has already denied it. Do you always have such
difficulty detecting bullshit?
Peter
From: Kurt Bigler (2003-12-21)
Subject: Re: [tuning] Re: the Sault approach - Carl
on 12/20/03 11:00 PM, Peter Wakefield Sault <[email protected]> wrote:
> --- In [email protected], Carl Lumma <ekin@l...> wrote:
>>>> Assuming you mean pitch class 6 = passband 6, lots of pairs of
>>>> rational intervals fit through the passbands. You mean, both
>>>> the interval in question and its inversion fit through the *same*
>>>> passband? Good luck justifying that one.
>>>
>>> Facts are facts. They need no "justifying". Good luck producing
> more
>>> spurious arguments.
>>
>> Facts are facts, the universe is full of them. Which ones you
>> apply to music are up to you.
>>
>>>>> Everyone
>>>>> since and including The Master of The Moon himself has ignored
> the
>>>>> necessity for a pair at this position, apparently because of
> the
>>>>> conceptual difficulty it engenders.
>>>>
>>>> So dodekaphony refers not to the number of pitches, but the
> number
>>>> of passbands ("pitch classes" is taken by another meaning, I'm
>>>> afraid). If we're allowed to have multiple pitches in each
> passband
>>>> so long as their inversion appears in the scale, we can do
> *lots* of
>>>> fancy stuff not currently admitted by your paradigm.
>>>
>>> Wrong. There is one passband for each pitch class.
>>
>> That's what I said.
>>
>>> Each pitch class comprises an infinity of pitches whether you like
> it
>>> or not.
>>
>> That's also what I said.
>>
>>> I take
>>> it that either you do not know what a 'pitch class' is, or that
> you
>>> are inventing some specious definition of your own (which you
> appear
>>> unwilling to share) solely for the purpose of trying counter the
>>> facts as I stated them. That is sophistry and not even very good
>>> sophistry.
>>
>> I'm not trying to counter facts, I'm trying to figure out why
>> you admit a pair of tritones but only single pitch in all other
>> pitch classes. Why is that, Peter?
>>
>
> I quote myself from above - "Each pitch class comprises an infinity
> of pitches".
Is that really the entire definition? Please realize that simple things can
be elusive because of their simplicity, particularly when something more
complex is suspected. When communication fails, I often err on the side of
expecting something more complex. That is possibly a rather bad strategy,
and I realize I have been bitten by it many times. Still I think it may not
be that uncommon of an error for humans to make.
-Kurt
From: Kurt Bigler (2003-12-21)
Subject: Re: [tuning] Re: the Sault approach - Carl
on 12/20/03 11:09 PM, Peter Wakefield Sault <[email protected]> wrote:
> --- In [email protected], Kurt Bigler <kkb@b...> wrote:
>> on 12/20/03 9:29 PM, Peter Wakefield Sault <sault@c...> wrote:
>>
>>> --- In [email protected], Carl Lumma <ekin@l...> wrote:
>>>>>> really don't see how you can include both and call it
>>>>>> "dodekaphony". You want a 13-tone scale, then we can talk
>>>>>> 13-tone scales.
>>>>>
>>>>> It's a purely theoretical necessity of 'rational' intervals. The
>>> aug
>>>>> 4th and dim 5th both belong to the same pitch class = 6.
>>>>
>>>> Assuming you mean pitch class 6 = passband 6, lots of pairs of
>>>> rational intervals fit through the passbands. You mean, both
>>>> the interval in question and its inversion fit through the *same*
>>>> passband? Good luck justifying that one.
>>>
>>>
>>> Facts are facts. They need no "justifying". Good luck producing
> more
>>> spurious arguments.
>>>
>>>
>>>>
>>>>> Everyone
>>>>> since and including The Master of The Moon himself has ignored
> the
>>>>> necessity for a pair at this position, apparently because of the
>>>>> conceptual difficulty it engenders.
>>>>
>>>> So dodekaphony refers not to the number of pitches, but the
> number
>>>> of passbands ("pitch classes" is taken by another meaning, I'm
>>>> afraid). If we're allowed to have multiple pitches in each
> passband
>>>> so long as their inversion appears in the scale, we can do
> *lots* of
>>>> fancy stuff not currently admitted by your paradigm.
>>>>
>>>> -Carl
>>>
>>>
>>> Wrong. There is one passband for each pitch class. Each pitch
> class
>>> comprises an infinity of pitches whether you like it or not. I
> take
>>> it that either you do not know what a 'pitch class' is, or that
> you
>>> are inventing some specious definition of your own (which you
> appear
>>> unwilling to share) solely for the purpose of trying counter the
>>> facts as I stated them. That is sophistry and not even very good
>>> sophistry.
>>
>> Peter, isn't this your opinion as to whether it is sophistry? Do
> you really
>> believe you understand (beyond any doubt) Carl's intention is
> this? I
>> believe Carl is working harder than anyone at helping to interpret
> your work
>> on behalf of and for the benfit of the rest of us. I'm glad for
> the dialog
>> he is having with you. Please realize Carl is working at something
> here.
>> He is not trying to obscure or evade. Like all of us, he might
> even have
>> some habits that unknowingly obscure something, but so might you.
> If
>> something seems to indicate sophistry to you, can look beyond that
>> appearance and see Carl's intention? He's trying. Everyone can
> make
>> mistakes, including you, right? A mistake can look like something
> else if
>> you want to judge it against your personal criteria. And isn't is
> possible
>> that you can make a mistake of misinterpretation? I still see not
> much
>> flexibility in the forms of dialog that you find admissable. We
> (people) do
>> not meet each other specifications. We are alive and have hopes and
>> limitations, and we need each other's help and forbearance. It may
> be that
>> you actually wish to help even when you appear harsh, yet please
> realize
>> that may be very hard for us to recognize. As I see it, things you
> respond
>> harshly too are often much less harsh in tone than your responses.
>>
>> Again, this is just "as I see it". That's my own truth in this, not
>> perfectly stated, but I hope not too far off the mark, and I'd
> appreciate
>> your forbearance in reading it, because no doubt I've made some
> mistakes.
>>
>> In Carl's response he suggested that you might simply share your
> definition
>> of "pitch class". To simply give a definition (and not be offended
> by the
>> request), this is a good way to be more direct. Patience is
> particularly a
>> virtue in teaching, and like it or not, teaching is one thing you
> are doing
>> here, and I think (please correct me if I'm wrong) in a way you
> *do* like
>> being a teacher, so why not make the best of it?
>>
>> -Kurt
>
> I'm afraid you're wrong Kurt. I quote from Carl above (scroll up to
> verify):-
> *** ("pitch classes" is taken by another meaning, I'm afraid). ***
>
> So let's have Carl's exclusive definition first, since he has already
> prejudged the matter. Carl implies that he knows what my definition
> is because he has already denied it. Do you always have such
> difficulty detecting bullshit?
Indeed, I trust his purpose, and trust also that his mistakes can be
arbitrarily complex, just like mine.
-Kurt
>
> Peter
From: Carl Lumma (2003-12-21)
Subject: Re: [tuning] Re: the Sault approach - Carl
>I'm afraid you're wrong Kurt. I quote from Carl above (scroll up to
>verify):-
>*** ("pitch classes" is taken by another meaning, I'm afraid). ***
>
>So let's have Carl's exclusive definition first, since he has already
>prejudged the matter. Carl implies that he knows what my definition
>is because he has already denied it. Do you always have such
>difficulty detecting bullshit?
"Pitch classes" is sometimes used as synonymous with "interval
classes", "generic intervals" and a host of other terms.
I have given the following definitions, which are probably not
optimal, in this forum before...
interval - a distance between two scale members, as the number of
scale members subtended (inclusive of starting and ending positions).
Example: 'thirds of the diatonic scale'
interval class - an ordered list of log-frequency changes produced
by moving a given interval sequentially over all positions in a scale.
Example: 'major thirds and minor thirds in the diatonic scale'
Please understand Peter that I often take a guess at what somebody
is saying as I ask them what they were saying. Because if my guess
is right it saves a round of e-mail, and if it's wrong I've given
the other person an idea of how to explain it to me.
-Carl
From: Dave Keenan (2003-12-21) Subject: Re: the Sault approach - Carl Carl, I must say I admire your patience and perseverence in dealing with Peter's (self-confessed) arrogance, and apparent paranoia. Peter, Relax man. You may have got off to a bad start on this list, but we're not out to get you. It's just that you have been working on your own and so it's gonna take some time to get accurate translations to the common terminology on this list. It would really help if you would not assume the worst about any questions people ask you, and if you could try to be a little less rude and dismissive. This is one the most polite, tolerant and friendly lists I have ever been involved with. -- Dave Keenan
From: Joseph Pehrson (2003-12-21) Subject: Re: the Sault approach - Carl --- In [email protected], "Peter Wakefield Sault" <sault@c...> http://groups.yahoo.com/group/tuning/message/50243 > > I quote myself from above - "Each pitch class comprises an infinity > of pitches". You appear to have a mental block over plain English, or > a reading difficulty. Get some therapy then maybe afterwards we can > have a meaningful discussion. > > > You have not defined pitch class. Rather than worry about whether > > I "know what it means", why not just define it? > > > > -Carl > > You appear to have a reading problem. Get some tuition. ***Yes, this is true. Carl is dumb... [that's just a joke, Carl :) }, but, seriously, pitch class is something very clearly defined in most theory literature. It's any given pitch and its octave equivalents. Do you really consider that an infinite set?? It's finate throughout the range of hearing... we can only hear so many octaves, and not really all that many... Joseph Pehrson
From: Werner Mohrlok (2003-12-21) Subject: AW: [tuning] Re: the Sault approach - Carl > I quote myself from above - "Each pitch class comprises an infinity > of pitches". You appear to have a mental block over plain English, or > a reading difficulty. Get some therapy then maybe afterwards we can > have a meaningful discussion. > > > You have not defined pitch class. Rather than worry about whether > > I "know what it means", why not just define it? > > > > -Carl > > > You appear to have a reading problem. Get some tuition. > ***Yes, this is true. Carl is dumb... [that's just a joke, > Carl :) }, but, seriously, pitch class is something very clearly > defined in most theory literature. It's any given pitch and its > octave equivalents. > Do you really consider that an infinite set?? It's finate throughout > the range of hearing... we can only hear so many octaves, and not > really all that many... > Joseph Pehrson Where is the limit of hearing? I just tested it by occasion, the upper Limt is actually by 11,700 Hz - but I am already 65 years old. Why only thinking at mankind? Think at the whales: These aware our tuning "beats" as tones. Think at the birds, the insects, the angels... Best Werner
From: Joseph Pehrson (2003-12-21)
Subject: Re: the Sault approach - Carl
--- In [email protected], Carl Lumma <ekin@l...> wrote:
http://groups.yahoo.com/group/tuning/message/50256
> >I'm afraid you're wrong Kurt. I quote from Carl above (scroll up
to
> >verify):-
> >*** ("pitch classes" is taken by another meaning, I'm afraid). ***
> >
> >So let's have Carl's exclusive definition first, since he has
already
> >prejudged the matter. Carl implies that he knows what my
definition
> >is because he has already denied it. Do you always have such
> >difficulty detecting bullshit?
>
> "Pitch classes" is sometimes used as synonymous with "interval
> classes", "generic intervals" and a host of other terms.
>
***I don't believe this is correct, Carl, at least for any of the
contemporary music theory courses I've taken.
A pitch class, or P.C. is, say, all the instances of the same pitch
throughout the range (they speak of 12-equal with this, since this is
mostly 12-tone theory... in other words, a C is a C is a C,
throughout the entire range)
An "interval class" or I.C. is, naturally, anything that has the same
interval number. For example, the "perfect" fifth has a number 7
(always starting with 0) and wherever that is found throughout the
range that is the "interval class..."
They are certainly not at all synonymous, at least not in the
contemporary literature, Forte, Perle, etc., etc.
J. Pehrson
From: Joseph Pehrson (2003-12-21) Subject: Re: the Sault approach - Carl --- In [email protected], "Werner Mohrlok" <wmohrlok@h...> wrote: http://groups.yahoo.com/group/tuning/message/50272 > > I quote myself from above - "Each pitch class comprises an infinity > > of pitches". You appear to have a mental block over plain English, > or > > a reading difficulty. Get some therapy then maybe afterwards we can > > have a meaningful discussion. > > > > > You have not defined pitch class. Rather than worry about whether > > > I "know what it means", why not just define it? > > > > > > -Carl > > > > > You appear to have a reading problem. Get some tuition. > > > > ***Yes, this is true. Carl is dumb... [that's just a joke, > > Carl :) }, but, seriously, pitch class is something very clearly > > defined in most theory literature. It's any given pitch and its > > octave equivalents. > > > Do you really consider that an infinite set?? It's finate throughout > > the range of hearing... we can only hear so many octaves, and not > > really all that many... > > > Joseph Pehrson > > Where is the limit of hearing? I just tested > it by occasion, the upper Limt is actually by 11,700 Hz - > but I am already 65 years old. > Why only thinking at mankind? Think at the whales: These aware > our tuning "beats" as tones. Think at the birds, the insects, the > angels... > > Best > > Werner ***Well, that's a good point, Werner, and I certainly am not opposed to a *theoretical* consideration of such theory. But I've never heard "pitch class" and "interval class" being the same thing, at least not in *theory* class... J. Pehrson
From: Gene Ward Smith (2003-12-21) Subject: Re: the Sault approach - Carl --- In [email protected], "Joseph Pehrson" <jpehrson@r...> wrote: > Do you really consider that an infinite set?? It's finate throughout > the range of hearing... we can only hear so many octaves, and not > really all that many... Assuming it is finite sometimes leads to wholly unnesessary problems.
From: Carl Lumma (2003-12-21) Subject: Re: [tuning] Re: the Sault approach - Carl >***Yes, this is true. Carl is dumb... [that's just a joke, >Carl :) }, but, seriously, pitch class is something very clearly >defined in most theory literature. It's any given pitch and its >octave equivalents. That's right, Joseph. It would be incorrect to even use it as synonymous with "interval class", as I was doing. -Carl
From: Carl Lumma (2003-12-21) Subject: Re: [tuning] Re: the Sault approach - Carl >> "Pitch classes" is sometimes used as synonymous with "interval >> classes", "generic intervals" and a host of other terms. > >***I don't believe this is correct, Carl, at least for any of the >contemporary music theory courses I've taken. > >A pitch class, or P.C. is, say, all the instances of the same pitch >throughout the range (they speak of 12-equal with this, since this is >mostly 12-tone theory... in other words, a C is a C is a C, >throughout the entire range) > >An "interval class" or I.C. is, naturally, anything that has the same >interval number. For example, the "perfect" fifth has a number 7 >(always starting with 0) and wherever that is found throughout the >range that is the "interval class..." > >They are certainly not at all synonymous, at least not in the >contemporary literature, Forte, Perle, etc., etc. See my earlier correction. I'm not sure where I got this from... it seems like I've heard it used this way outside of Peter's post, but, you're absolutely right JP, it should be deprecated. -Carl
From: Joseph Pehrson (2003-12-21) Subject: Re: the Sault approach - Carl --- In [email protected], "Gene Ward Smith" <gwsmith@s...> wrote: http://groups.yahoo.com/group/tuning/message/50280 > --- In [email protected], "Joseph Pehrson" <jpehrson@r...> wrote: > > > Do you really consider that an infinite set?? It's finate > throughout > > the range of hearing... we can only hear so many octaves, and not > > really all that many... > > Assuming it is finite sometimes leads to wholly unnesessary problems. ***Well, OK... then we'll just say that *I* can't possibly hear it and even my dog couldn't hear it... (although I don't own a dog... :) J. Pehrson
From: Paul Erlich (2003-12-30) Subject: Re: the Sault approach --- In [email protected], backfromthesilo@y... wrote: > His concept is to > compare a chain of Otonal 2:3's with a chain of Utonal 2:3's > (3:4's) You must be misunderstanding the meaning of Otonal and Utonal, as defined by Partch. Any dyad ratio (such as 2:3 or 3:4) is simultaneously *both* Otonal and Utonal, since it can be found just as easily in an "overtone series" as in an "undertone series". This is Partch's "dual nature of ratios" tenet. Meanwhile, it is most chords of three or more notes that can accurately be described as either "Otonal" or "Utonal", since typically one derivation will be simpler than the other. For example major triads are more easily found in the "overtone series", thus they're "Otonal"; while minor triads are more easily found in the "undertone series", thus their "Utonal". Partch's diamonds all show Otonal harmonies slanting in one direction, and Utonal harmonies slanting in the other direction.
From: Paul Erlich (2003-12-30) Subject: Re: the Sault approach - Carl --- In [email protected], "Werner Mohrlok" <wmohrlok@h...> wrote: > Where is the limit of hearing? I just tested > it by occasion, the upper Limt is actually by 11,700 Hz - > but I am already 65 years old. Unfortunately, as a male over 50, you are actually a member of the group of humans with the greatest loss of higher-frequency hearing. Younger males and most females have an upper limit closer to 20,000 Hz. My own upper limit is somewhere in-between, and I'm sure the loud rehearsals and performances aren't helping.
From: monz (2003-12-31) Subject: Re: the Sault approach - Carl hi paul and Werner, --- In [email protected], "Paul Erlich" <paul@s...> wrote: > --- In [email protected], "Werner Mohrlok" <wmohrlok@h...> wrote: > > > Where is the limit of hearing? I just tested > > it by occasion, the upper Limt is actually by 11,700 Hz - > > but I am already 65 years old. > > Unfortunately, as a male over 50, you are actually a > member of the group of humans with the greatest loss > of higher-frequency hearing. Younger males and most > females have an upper limit closer to 20,000 Hz. > My own upper limit is somewhere in-between, and I'm > sure the loud rehearsals and performances aren't helping. i know what you mean. i tested my frequency response last year, and the highest pitch i can hear is around 13,800 Hz. i was 40 at the time. oddly, the 16-year-old friend on whose computer i did the test, and who has exceptionally good hearing and sight, tested his highest limit with the same numbers in different places, at 18,300 Hz. -monz
From: Peter Wakefield Sault (2003-12-21)
Subject: Re: the Sault approach - Carl
--- In [email protected], Kurt Bigler <kkb@b...> wrote:
> on 12/20/03 11:00 PM, Peter Wakefield Sault <sault@c...> wrote:
>
> > --- In [email protected], Carl Lumma <ekin@l...> wrote:
> >>>> Assuming you mean pitch class 6 = passband 6, lots of pairs of
> >>>> rational intervals fit through the passbands. You mean, both
> >>>> the interval in question and its inversion fit through the
*same*
> >>>> passband? Good luck justifying that one.
> >>>
> >>> Facts are facts. They need no "justifying". Good luck producing
> > more
> >>> spurious arguments.
> >>
> >> Facts are facts, the universe is full of them. Which ones you
> >> apply to music are up to you.
> >>
> >>>>> Everyone
> >>>>> since and including The Master of The Moon himself has ignored
> > the
> >>>>> necessity for a pair at this position, apparently because of
> > the
> >>>>> conceptual difficulty it engenders.
> >>>>
> >>>> So dodekaphony refers not to the number of pitches, but the
> > number
> >>>> of passbands ("pitch classes" is taken by another meaning, I'm
> >>>> afraid). If we're allowed to have multiple pitches in each
> > passband
> >>>> so long as their inversion appears in the scale, we can do
> > *lots* of
> >>>> fancy stuff not currently admitted by your paradigm.
> >>>
> >>> Wrong. There is one passband for each pitch class.
> >>
> >> That's what I said.
> >>
> >>> Each pitch class comprises an infinity of pitches whether you
like
> > it
> >>> or not.
> >>
> >> That's also what I said.
> >>
> >>> I take
> >>> it that either you do not know what a 'pitch class' is, or that
> > you
> >>> are inventing some specious definition of your own (which you
> > appear
> >>> unwilling to share) solely for the purpose of trying counter the
> >>> facts as I stated them. That is sophistry and not even very good
> >>> sophistry.
> >>
> >> I'm not trying to counter facts, I'm trying to figure out why
> >> you admit a pair of tritones but only single pitch in all other
> >> pitch classes. Why is that, Peter?
> >>
> >
> > I quote myself from above - "Each pitch class comprises an
infinity
> > of pitches".
>
> Is that really the entire definition? Please realize that simple
things can
> be elusive because of their simplicity, particularly when something
more
> complex is suspected. When communication fails, I often err on the
side of
> expecting something more complex. That is possibly a rather bad
strategy,
> and I realize I have been bitten by it many times. Still I think
it may not
> be that uncommon of an error for humans to make.
>
> -Kurt
Kurt
I'd be more than happy to answer you since you, unlike Carl, are not
trying to put words into my mouth that I never said. Carl has
obviously missed his vocation and should have become a lawyer.
In any regular division of the octave, there are as many pitch
classes as there are divisions. That should be self-evident, however,
for everyone else's benefit a simple example should suffice. In 17-DO
there are 17 pitch classes. The octave always belongs to the same
pitch class as the tonic and is therefore not counted as a separate
pitch class, so there are not 18 pitch classes in this example but
17. There is however, a theoretically infinite number of ways of
defining specific proportions for each of those pitch classes, each
definition producing a different numerical value. To take 12-DO as an
example, pitch class 7 may be, among an infinity of other values,
designated by either 2^(7/12) or by 3/2. Each produces a separate
*pitch*, but both pitches belong to the same pitch class, which is 7.
Peter