Yahoo Tuning Groups Ultimate Backup tuning 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

🔗backfromthesilo@yahoo.com

--- In tuning@yahoogroups.com, 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

🔗gwsmith@svpal.org

🔗sault@cyberware.co.uk

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

🔗Carl Lumma <ekin@lumma.org>

>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

🔗Carl Lumma <ekin@lumma.org>

>> 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

🔗sault@cyberware.co.uk

--- In tuning@yahoogroups.com, 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

🔗Carl Lumma <ekin@lumma.org>

>> 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

🔗Carl Lumma <ekin@lumma.org>

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

🔗Peter Wakefield Sault <sault@cyberware.co.uk>

--- In tuning@yahoogroups.com, 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

🔗Peter Wakefield Sault <sault@cyberware.co.uk>

--- In tuning@yahoogroups.com, 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

🔗Carl Lumma <ekin@lumma.org>

>> 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

🔗Peter Wakefield Sault <sault@cyberware.co.uk>

--- In tuning@yahoogroups.com, 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

🔗Carl Lumma <ekin@lumma.org>

🔗Peter Wakefield Sault <sault@cyberware.co.uk>

--- In tuning@yahoogroups.com, 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.

🔗Carl Lumma <ekin@lumma.org>

>> 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

🔗Kurt Bigler <kkb@breathsense.com>

on 12/20/03 9:29 PM, Peter Wakefield Sault <sault@cyberware.co.uk> wrote:

> --- In tuning@yahoogroups.com, 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

🔗Peter Wakefield Sault <sault@cyberware.co.uk>

--- In tuning@yahoogroups.com, 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.

🔗Peter Wakefield Sault <sault@cyberware.co.uk>

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> on 12/20/03 9:29 PM, Peter Wakefield Sault <sault@c...> wrote:
>
> > --- In tuning@yahoogroups.com, 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

🔗Kurt Bigler <kkb@breathsense.com>

on 12/20/03 11:00 PM, Peter Wakefield Sault <sault@cyberware.co.uk> wrote:

> --- In tuning@yahoogroups.com, 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 Bigler <kkb@breathsense.com>

on 12/20/03 11:09 PM, Peter Wakefield Sault <sault@cyberware.co.uk> wrote:

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>> on 12/20/03 9:29 PM, Peter Wakefield Sault <sault@c...> wrote:
>>
>>> --- In tuning@yahoogroups.com, 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

🔗Carl Lumma <ekin@lumma.org>

>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

🔗Dave Keenan <d.keenan@bigpond.net.au>

🔗Joseph Pehrson <jpehrson@rcn.com>

🔗Werner Mohrlok <wmohrlok@hermode.com>

> 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

🔗Joseph Pehrson <jpehrson@rcn.com>

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

/tuning/topicId_49980.html#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

🔗Joseph Pehrson <jpehrson@rcn.com>

--- In tuning@yahoogroups.com, "Werner Mohrlok" <wmohrlok@h...> wrote:

/tuning/topicId_49980.html#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

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Carl Lumma <ekin@lumma.org>

🔗Joseph Pehrson <jpehrson@rcn.com>

🔗Paul Erlich <paul@stretch-music.com>

🔗monz <monz@attglobal.net>

🔗Peter Wakefield Sault <sault@cyberware.co.uk>

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> on 12/20/03 11:00 PM, Peter Wakefield Sault <sault@c...> wrote:
>
> > --- In tuning@yahoogroups.com, 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