Yahoo Tuning Groups Ultimate Backup tuning-math rank complexity explanation updated

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> http://lumma.org/tuning/rank-complexity.txt
>
> -Carl

Of course, you are talking about "external" intervals here, right?
If you take ALL the intervals in a septachord, you get an interval
vector that totals up to 21. (Hexachords, 15, Pentachords, 10) It
would be cool if someone could tie "interval-vectors" (The kind
Jon Wild has compiled- hey that rhymes!) into the main discussion.
I'm still trying to find a good use for them! -And trying to find a
rule for Z-relations...

Paul

🔗Carl Lumma <ekin@lumma.org>

>> http://lumma.org/tuning/rank-complexity.txt
>
>Of course, you are talking about "external" intervals here, right?
>If you take ALL the intervals in a septachord, you get an interval
>vector that totals up to 21. (Hexachords, 15, Pentachords, 10) It
>would be cool if someone could tie "interval-vectors" (The kind
>Jon Wild has compiled- hey that rhymes!) into the main discussion.
>I'm still trying to find a good use for them! -And trying to find
>a rule for Z-relations...

Hi Paul,

I'm afraid I don't know what an "external" interval is. Here's
the interval matrix of the diatonic scale in 12-equal, as given
by Scala...

100.0 : 2 4 5 7 9 11 12
200.0 : 2 3 5 7 9 10 12
400.0 : 1 3 5 7 8 10 12
500.0 : 2 4 6 7 9 11 12
700.0 : 2 4 5 7 9 10 12
900.0 : 2 3 5 7 8 10 12
1100.0: 1 3 5 6 8 10 12

The ruler is...

0...1...2...3...4...5...6...7...8...9...10..11..12

The list of adjacent marks on the ruler:

The rank complexity:

As for the things that Jon Wild compiled, I don't have a clue
what they are... which is akin to saying I don't know of any
use for them.

-Carl

🔗Paul G Hjelmstad <paul.hjelmstad@us.ing.com>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> http://lumma.org/tuning/rank-complexity.txt
> >
> >Of course, you are talking about "external" intervals here, right?
> >If you take ALL the intervals in a septachord, you get an interval
> >vector that totals up to 21. (Hexachords, 15, Pentachords, 10) It
> >would be cool if someone could tie "interval-vectors" (The kind
> >Jon Wild has compiled- hey that rhymes!) into the main discussion.
> >I'm still trying to find a good use for them! -And trying to find
> >a rule for Z-relations...
>
> Hi Paul,
>
> I'm afraid I don't know what an "external" interval is. Here's
> the interval matrix of the diatonic scale in 12-equal, as given
> by Scala...
>
> 100.0 : 2 4 5 7 9 11 12
> 200.0 : 2 3 5 7 9 10 12
> 400.0 : 1 3 5 7 8 10 12
> 500.0 : 2 4 6 7 9 11 12
> 700.0 : 2 4 5 7 9 10 12
> 900.0 : 2 3 5 7 8 10 12
> 1100.0: 1 3 5 6 8 10 12
>
> The ruler is...
>
> 0...1...2...3...4...5...6...7...8...9...10..11..12
>
> The list of adjacent marks on the ruler:
>
> 1
>
> The rank complexity:
>
> 0
>
> As for the things that Jon Wild compiled, I don't have a clue
> what they are... which is akin to saying I don't know of any
> use for them.
>
> -Carl

Interesting. What I meant was really "adjacent, outer" intervals:
This row:

100.0 : 2 4 5 7 9 11 12 Has a vector count of
(2,5,0,0,0,0) "outer" intervals. You can count all the other (inner)
intervals in a set, as well - (outer + inner).So for the diatonic
septachord you get an interval vector of (2,5,4,3,6,1). Jon Wild and
I have been discussing this, and he has compiled interval vector
lists for sets and their subsets all the way up to "C{31,15}-reduced"
(Gene hates that I use the C{m,n} notation, he has proposed a better
way, to name these sets, somewhere in the archives, I'll have to hunt
for it)

- Paul

🔗Carl Lumma <ekin@lumma.org>

>> Here's the interval matrix of the diatonic scale in
>> 12-equal, as given by Scala...
>>
>> 100.0 : 2 4 5 7 9 11 12
>> 200.0 : 2 3 5 7 9 10 12
>> 400.0 : 1 3 5 7 8 10 12
>> 500.0 : 2 4 6 7 9 11 12
>> 700.0 : 2 4 5 7 9 10 12
>> 900.0 : 2 3 5 7 8 10 12
>> 1100.0: 1 3 5 6 8 10 12
//
>Interesting. What I meant was really "adjacent, outer" intervals:
>This row:
>
>100.0 : 2 4 5 7 9 11 12 Has a vector count of
>(2,5,0,0,0,0) "outer" intervals.

I'm lost. There are 2, 5 and 0 of what?

-Carl

🔗Paul G Hjelmstad <paul.hjelmstad@us.ing.com>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> http://lumma.org/tuning/rank-complexity.txt
> >
> >> Hi Paul,
>
> I'm afraid I don't know what an "external" interval is. Here's
> the interval matrix of the diatonic scale in 12-equal, as given
> by Scala...
>
> 100.0 : 2 4 5 7 9 11 12
> 200.0 : 2 3 5 7 9 10 12
> 400.0 : 1 3 5 7 8 10 12
> 500.0 : 2 4 6 7 9 11 12
> 700.0 : 2 4 5 7 9 10 12
> 900.0 : 2 3 5 7 8 10 12
> 1100.0: 1 3 5 6 8 10 12

Q: Shouldn't the first row be 000.0?
>
> The ruler is...
>
> 0...1...2...3...4...5...6...7...8...9...10..11..12
>
> The list of adjacent marks on the ruler:
>
> 1
>
> The rank complexity:
>
> 0
>
> -Carl

Another point,

One can obtain "my" interval vector from "your" interval matrix
by tallying all the intervals from 1 to 6 and ignoring 7 to 12.
You subsequently obtain (2,5,4,3,6,1)

-Paul

🔗Carl Lumma <ekin@lumma.org>

>> >> http://lumma.org/tuning/rank-complexity.txt
>> >
>> >> Hi Paul,
>>
>> I'm afraid I don't know what an "external" interval is. Here's
>> the interval matrix of the diatonic scale in 12-equal, as given
>> by Scala...
>>
>> 100.0 : 2 4 5 7 9 11 12
>> 200.0 : 2 3 5 7 9 10 12
>> 400.0 : 1 3 5 7 8 10 12
>> 500.0 : 2 4 6 7 9 11 12
>> 700.0 : 2 4 5 7 9 10 12
>> 900.0 : 2 3 5 7 8 10 12
>> 1100.0: 1 3 5 6 8 10 12
>
>Q: Shouldn't the first row be 000.0?

It is quite safe to ignore the values before the colon, as they
are merely an artifact of scala's output.

>Another point,
>
>One can obtain "my" interval vector from "your" interval matrix
>by tallying all the intervals from 1 to 6 and ignoring 7 to 12.
>You subsequently obtain (2,5,4,3,6,1)

Sorry, but how does tallying numbers in the above matrix lead to
(2,5,4,3,6,1)?

-Carl

🔗Paul G Hjelmstad <paul.hjelmstad@us.ing.com>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >> http://lumma.org/tuning/rank-complexity.txt
> >> >
> >> >> Hi Paul,
> >>
> >> I'm afraid I don't know what an "external" interval is. Here's
> >> the interval matrix of the diatonic scale in 12-equal, as given
> >> by Scala...
> >>
> >> 100.0 : 2 4 5 7 9 11 12
> >> 200.0 : 2 3 5 7 9 10 12
> >> 400.0 : 1 3 5 7 8 10 12
> >> 500.0 : 2 4 6 7 9 11 12
> >> 700.0 : 2 4 5 7 9 10 12
> >> 900.0 : 2 3 5 7 8 10 12
> >> 1100.0: 1 3 5 6 8 10 12
> >
> >Q: Shouldn't the first row be 000.0?
>
> It is quite safe to ignore the values before the colon, as they
> are merely an artifact of scala's output.
>
> >Another point,
> >
> >One can obtain "my" interval vector from "your" interval matrix
> >by tallying all the intervals from 1 to 6 and ignoring 7 to 12.
> >You subsequently obtain (2,5,4,3,6,1)
>
> Sorry, but how does tallying numbers in the above matrix lead to
> (2,5,4,3,6,1)?
> -Carl

Easy! There are 2 1's, 5 2's 4 3's 3 4's 6 5's and 1 6. (Also works
for 7-11, in reverse from 6:, so the full vector is
(2,5,4,3,6,1,6,3,4,5,2, and 7 if you include '12') (same as '0')
> - Paul

🔗Gene Ward Smith <gwsmith@svpal.org>

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

> 100.0 : 2 4 5 7 9 11 12
> 200.0 : 2 3 5 7 9 10 12
> 400.0 : 1 3 5 7 8 10 12
> 500.0 : 2 4 6 7 9 11 12
> 700.0 : 2 4 5 7 9 10 12
> 900.0 : 2 3 5 7 8 10 12
> 1100.0: 1 3 5 6 8 10 12

Why is the first row 100.0 and not 000.0? I also see I had the wrong
definition of interval matrix; maybe this one would have a more
interesting characteristic polynomial. I'd like something that told me
something about the scale!

🔗Carl Lumma <ekin@lumma.org>

>> >> 100.0 : 2 4 5 7 9 11 12
>> >> 200.0 : 2 3 5 7 9 10 12
>> >> 400.0 : 1 3 5 7 8 10 12
>> >> 500.0 : 2 4 6 7 9 11 12
>> >> 700.0 : 2 4 5 7 9 10 12
>> >> 900.0 : 2 3 5 7 8 10 12
>> >> 1100.0: 1 3 5 6 8 10 12
//
>> >One can obtain "my" interval vector from "your" interval matrix
>> >by tallying all the intervals from 1 to 6 and ignoring 7 to 12.
>> >You subsequently obtain (2,5,4,3,6,1)
>>
>> Sorry, but how does tallying numbers in the above matrix lead to
>> (2,5,4,3,6,1)?
>
>Easy! There are 2 1's, 5 2's 4 3's 3 4's 6 5's and 1 6.

Aha! Are these known as "interval vectors" in the trade? And have
they ever been applied to tunings other than 12-equal?

>Jon Wild and I have been discussing this, and he has compiled
>interval vector lists for sets and their subsets all the way up
>to "C{31,15}-reduced" (Gene hates that I use the C{m,n} notation,
>he has proposed a better way, to name these sets, somewhere in
>the archives, I'll have to hunt for it)

What is this notation? Also, much appreciated if find Gene's
suggestion.

-Carl

🔗Paul G Hjelmstad <paul.hjelmstad@us.ing.com>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >> 100.0 : 2 4 5 7 9 11 12
> >> >> 200.0 : 2 3 5 7 9 10 12
> >> >> 400.0 : 1 3 5 7 8 10 12
> >> >> 500.0 : 2 4 6 7 9 11 12
> >> >> 700.0 : 2 4 5 7 9 10 12
> >> >> 900.0 : 2 3 5 7 8 10 12
> >> >> 1100.0: 1 3 5 6 8 10 12
> //
> >> >One can obtain "my" interval vector from "your" interval matrix
> >> >by tallying all the intervals from 1 to 6 and ignoring 7 to 12.
> >> >You subsequently obtain (2,5,4,3,6,1)
> >>
> >> Sorry, but how does tallying numbers in the above matrix lead to
> >> (2,5,4,3,6,1)?
> >
> >Easy! There are 2 1's, 5 2's 4 3's 3 4's 6 5's and 1 6.
>
> Aha! Are these known as "interval vectors" in the trade? And have
> they ever been applied to tunings other than 12-equal?

Yes. See below. Jon Wild has compiled interval vector tables of ETs
all the way up to 31-et. (I have done some myself, but not that high)
>
> >Jon Wild and I have been discussing this, and he has compiled
> >interval vector lists for sets and their subsets all the way up
> >to "C{31,15}-reduced" (Gene hates that I use the C{m,n} notation,
> >he has proposed a better way, to name these sets, somewhere in
> >the archives, I'll have to hunt for it)
>
> What is this notation? Also, much appreciated if find Gene's
> suggestion.

I will look for it
>
> -Carl

🔗Paul Erlich <perlich@aya.yale.edu>

What kind of artifact are they, if not an error?

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> > >> >> http://lumma.org/tuning/rank-complexity.txt
> > >> >
> > >> >> Hi Paul,
> > >>
> > >> I'm afraid I don't know what an "external" interval is. Here's
> > >> the interval matrix of the diatonic scale in 12-equal, as given
> > >> by Scala...
> > >>
> > >> 100.0 : 2 4 5 7 9 11 12
> > >> 200.0 : 2 3 5 7 9 10 12
> > >> 400.0 : 1 3 5 7 8 10 12
> > >> 500.0 : 2 4 6 7 9 11 12
> > >> 700.0 : 2 4 5 7 9 10 12
> > >> 900.0 : 2 3 5 7 8 10 12
> > >> 1100.0: 1 3 5 6 8 10 12
> > >
> > >Q: Shouldn't the first row be 000.0?
> >
> > It is quite safe to ignore the values before the colon, as they
> > are merely an artifact of scala's output.
> >
> > >Another point,
> > >
> > >One can obtain "my" interval vector from "your" interval matrix
> > >by tallying all the intervals from 1 to 6 and ignoring 7 to 12.
> > >You subsequently obtain (2,5,4,3,6,1)
> >
> > Sorry, but how does tallying numbers in the above matrix lead to
> > (2,5,4,3,6,1)?
> > -Carl
>
> Easy! There are 2 1's, 5 2's 4 3's 3 4's 6 5's and 1 6.

I see 2 6's ;)

>(Also works
> for 7-11, in reverse from 6:, so the full vector is
> (2,5,4,3,6,1,6,3,4,5,2, and 7 if you include '12') (same as '0')
> > - Paul

This time there *really* should be 2 6's.

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> > 100.0 : 2 4 5 7 9 11 12
> > 200.0 : 2 3 5 7 9 10 12
> > 400.0 : 1 3 5 7 8 10 12
> > 500.0 : 2 4 6 7 9 11 12
> > 700.0 : 2 4 5 7 9 10 12
> > 900.0 : 2 3 5 7 8 10 12
> > 1100.0: 1 3 5 6 8 10 12
>
> Why is the first row 100.0 and not 000.0? I also see I had the wrong
> definition of interval matrix; maybe this one would have a more
> interesting characteristic polynomial. I'd like something that told
me
> something about the scale!

The interval matrix often, if not typically, has all unisons/octaves
along the diagonal. This one is merely a reshuffling so that the
diagonal becomes a right vertical.

🔗Paul G Hjelmstad <paul.hjelmstad@us.ing.com>

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul.hjelmstad@u...> wrote:
> > --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> > > >> >> http://lumma.org/tuning/rank-complexity.txt
> > > >> >
> > > >> >> Hi Paul,
> > > >>
> > > >> I'm afraid I don't know what an "external" interval is.
Here's
> > > >> the interval matrix of the diatonic scale in 12-equal, as
given
> > > >> by Scala...
> > > >>
> > > >> 100.0 : 2 4 5 7 9 11 12
> > > >> 200.0 : 2 3 5 7 9 10 12
> > > >> 400.0 : 1 3 5 7 8 10 12
> > > >> 500.0 : 2 4 6 7 9 11 12
> > > >> 700.0 : 2 4 5 7 9 10 12
> > > >> 900.0 : 2 3 5 7 8 10 12
> > > >> 1100.0: 1 3 5 6 8 10 12
> > > >
> > > >Q: Shouldn't the first row be 000.0?
> > >
> > > It is quite safe to ignore the values before the colon, as they
> > > are merely an artifact of scala's output.
> > >
> > > >Another point,
> > > >
> > > >One can obtain "my" interval vector from "your" interval matrix
> > > >by tallying all the intervals from 1 to 6 and ignoring 7 to 12.
> > > >You subsequently obtain (2,5,4,3,6,1)
> > >
> > > Sorry, but how does tallying numbers in the above matrix lead to
> > > (2,5,4,3,6,1)?
> > > -Carl
> >
> > Easy! There are 2 1's, 5 2's 4 3's 3 4's 6 5's and 1 6.
>
> I see 2 6's ;)

Good eye!
>
> >(Also works
> > for 7-11, in reverse from 6:, so the full vector is
> > (2,5,4,3,6,1,6,3,4,5,2, and 7 if you include '12') (same as '0')
> > > - Paul
>
> This time there *really* should be 2 6's.

However, Since this is the same tritone (F-B and B-F) John Rahn
and Allen Forte both divide this by 2. 2/2 =1

🔗Paul G Hjelmstad <paul.hjelmstad@us.ing.com>

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> > <paul.hjelmstad@u...> wrote:
> > > --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...>
wrote:
> > > > >> >> http://lumma.org/tuning/rank-complexity.txt
> > > > >> >
> > > > >> >> Hi Paul,
> > > > >>
> > > > >> I'm afraid I don't know what an "external" interval is.
> Here's
> > > > >> the interval matrix of the diatonic scale in 12-equal, as
> given
> > > > >> by Scala...
> > > > >>
> > > > >> 100.0 : 2 4 5 7 9 11 12
> > > > >> 200.0 : 2 3 5 7 9 10 12
> > > > >> 400.0 : 1 3 5 7 8 10 12
> > > > >> 500.0 : 2 4 6 7 9 11 12
> > > > >> 700.0 : 2 4 5 7 9 10 12
> > > > >> 900.0 : 2 3 5 7 8 10 12
> > > > >> 1100.0: 1 3 5 6 8 10 12
> > > > >
> > > > >Q: Shouldn't the first row be 000.0?
> > > >
> > > > It is quite safe to ignore the values before the colon, as
they
> > > > are merely an artifact of scala's output.
> > > >
> > > > >Another point,
> > > > >
> > > > >One can obtain "my" interval vector from "your" interval
matrix
> > > > >by tallying all the intervals from 1 to 6 and ignoring 7 to
12.
> > > > >You subsequently obtain (2,5,4,3,6,1)
> > > >
> > > > Sorry, but how does tallying numbers in the above matrix lead
to
> > > > (2,5,4,3,6,1)?
> > > > -Carl
> > >
> > > Easy! There are 2 1's, 5 2's 4 3's 3 4's 6 5's and 1 6.
> >
> > I see 2 6's ;)
>
> Good eye!
> >
> > >(Also works
> > > for 7-11, in reverse from 6:, so the full vector is
> > > (2,5,4,3,6,1,6,3,4,5,2, and 7 if you include '12') (same as '0')
> > > > - Paul
> >
> > This time there *really* should be 2 6's.
>
> However, Since this is the same tritone (F-B and B-F) John Rahn
> and Allen Forte both divide this by 2. 2/2 =1

Oops! In this vector there should be 2 6's. Your right, I'm left!

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul.hjelmstad@u...> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> > wrote:
> > > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> > > <paul.hjelmstad@u...> wrote:
> > > > --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...>
> wrote:
> > > > > >> >> http://lumma.org/tuning/rank-complexity.txt
> > > > > >> >
> > > > > >> >> Hi Paul,
> > > > > >>
> > > > > >> I'm afraid I don't know what an "external" interval is.
> > Here's
> > > > > >> the interval matrix of the diatonic scale in 12-equal,
as
> > given
> > > > > >> by Scala...
> > > > > >>
> > > > > >> 100.0 : 2 4 5 7 9 11 12
> > > > > >> 200.0 : 2 3 5 7 9 10 12
> > > > > >> 400.0 : 1 3 5 7 8 10 12
> > > > > >> 500.0 : 2 4 6 7 9 11 12
> > > > > >> 700.0 : 2 4 5 7 9 10 12
> > > > > >> 900.0 : 2 3 5 7 8 10 12
> > > > > >> 1100.0: 1 3 5 6 8 10 12
> > > > > >
> > > > > >Q: Shouldn't the first row be 000.0?
> > > > >
> > > > > It is quite safe to ignore the values before the colon, as
> they
> > > > > are merely an artifact of scala's output.
> > > > >
> > > > > >Another point,
> > > > > >
> > > > > >One can obtain "my" interval vector from "your" interval
> matrix
> > > > > >by tallying all the intervals from 1 to 6 and ignoring 7
to
> 12.
> > > > > >You subsequently obtain (2,5,4,3,6,1)
> > > > >
> > > > > Sorry, but how does tallying numbers in the above matrix
lead
> to
> > > > > (2,5,4,3,6,1)?
> > > > > -Carl
> > > >
> > > > Easy! There are 2 1's, 5 2's 4 3's 3 4's 6 5's and 1 6.
> > >
> > > I see 2 6's ;)
> >
> > Good eye!
> > >
> > > >(Also works
> > > > for 7-11, in reverse from 6:, so the full vector is
> > > > (2,5,4,3,6,1,6,3,4,5,2, and 7 if you include '12') (same
as '0')
> > > > > - Paul
> > >
> > > This time there *really* should be 2 6's.
> >
> > However, Since this is the same tritone (F-B and B-F) John Rahn
> > and Allen Forte both divide this by 2. 2/2 =1
>
> Oops! In this vector there should be 2 6's. Your right, I'm left!

Looks like we're fully agreed, then.

🔗Paul Erlich <perlich@aya.yale.edu>

I didn't know they ever used the "full vector", but if they did, I
would disagree. Only in the tally of interval *classes*, which go
from 1 to 6, should you divide the tritone count by 2. That's why I
wrote a smiley-face in the first case above -- I was winking to you
because in that case, we both know the tritone count should be
divided by 2.

🔗Carl Lumma <ekin@lumma.org>

>> >> >> http://lumma.org/tuning/rank-complexity.txt
>> >> >
>> >> >> Hi Paul,
>> >>
>> >> I'm afraid I don't know what an "external" interval is. Here's
>> >> the interval matrix of the diatonic scale in 12-equal, as given
>> >> by Scala...
>> >>
>> >> 100.0 : 2 4 5 7 9 11 12
>> >> 200.0 : 2 3 5 7 9 10 12
>> >> 400.0 : 1 3 5 7 8 10 12
>> >> 500.0 : 2 4 6 7 9 11 12
>> >> 700.0 : 2 4 5 7 9 10 12
>> >> 900.0 : 2 3 5 7 8 10 12
>> >> 1100.0: 1 3 5 6 8 10 12
>> >
>> >Q: Shouldn't the first row be 000.0?
>>
>> It is quite safe to ignore the values before the colon, as they
>> are merely an artifact of scala's output.
>
>What kind of artifact are they, if not an error?

They indicate the interval between 1/1 and the degree of the
original 'native' mode, on which the mode shown on the particular
line is based.

In this case they are artifactal only because they are given in
different units than is the interval matrix itself.

This happened because I wanted to give the interval matrix in
'steps of 12-tET' units. Unfortunately (and one of my biggest
desired features) Scala does not offer 'degrees of n-ET' units.
So what you see above is actually the RANK ORDER MATRIX of the
diatonic scale in 12-tET. Because the scale covers every degree
of that tuning the two are the same.

-Carl

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >> >> http://lumma.org/tuning/rank-complexity.txt
> >> >> >
> >> >> >> Hi Paul,
> >> >>
> >> >> I'm afraid I don't know what an "external" interval is.
Here's
> >> >> the interval matrix of the diatonic scale in 12-equal, as
given
> >> >> by Scala...
> >> >>
> >> >> 100.0 : 2 4 5 7 9 11 12
> >> >> 200.0 : 2 3 5 7 9 10 12
> >> >> 400.0 : 1 3 5 7 8 10 12
> >> >> 500.0 : 2 4 6 7 9 11 12
> >> >> 700.0 : 2 4 5 7 9 10 12
> >> >> 900.0 : 2 3 5 7 8 10 12
> >> >> 1100.0: 1 3 5 6 8 10 12
> >> >
> >> >Q: Shouldn't the first row be 000.0?
> >>
> >> It is quite safe to ignore the values before the colon, as they
> >> are merely an artifact of scala's output.
> >
> >What kind of artifact are they, if not an error?
>
> They indicate the interval between 1/1 and the degree of the
> original 'native' mode, on which the mode shown on the particular
> line is based.

In that case, shouldn't the first entry be 000.0?

> In this case they are artifactal only because they are given in
> different units than is the interval matrix itself.

OK, but more serious would seem to be the actual error.

🔗Carl Lumma <ekin@lumma.org>

>> > 100.0 : 2 4 5 7 9 11 12
>> > 200.0 : 2 3 5 7 9 10 12
>> > 400.0 : 1 3 5 7 8 10 12
>> > 500.0 : 2 4 6 7 9 11 12
>> > 700.0 : 2 4 5 7 9 10 12
>> > 900.0 : 2 3 5 7 8 10 12
>> > 1100.0: 1 3 5 6 8 10 12

Gene wrote...

>> Why is the first row 100.0 and not 000.0?

Hopefully you've seen the bit about ignoring these numbers
by now.

>> I also see I had the wrong definition of interval matrix;

That's pretty incredible considering that you've got the
Rothenberg papers on the matter.

Paul wrote...

>The interval matrix often, if not typically, has all unisons/octaves
>along the diagonal. This one is merely a reshuffling so that the
>diagonal becomes a right vertical.

No, the interval matrix is always written as above (the values
after the colons, at least). You're thinking maybe of the
tonality diamond.

-Carl

🔗Paul Erlich <perlich@aya.yale.edu>

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

> >The interval matrix often, if not typically, has all
unisons/octaves
> >along the diagonal. This one is merely a reshuffling so that the
> >diagonal becomes a right vertical.
>
> No, the interval matrix is always written as above (the values
> after the colons, at least).

Where do you get *always*??

BTW, In matrix algebra, which is where you find characteristic
polynomials and the like, the *diagonal*, not the *right vertical*,
represents the relations between like elements.

🔗Carl Lumma <ekin@lumma.org>

>> >> >> 100.0 : 2 4 5 7 9 11 12
>> >> >> 200.0 : 2 3 5 7 9 10 12
>> >> >> 400.0 : 1 3 5 7 8 10 12
>> >> >> 500.0 : 2 4 6 7 9 11 12
>> >> >> 700.0 : 2 4 5 7 9 10 12
>> >> >> 900.0 : 2 3 5 7 8 10 12
>> >> >> 1100.0: 1 3 5 6 8 10 12
>> >> >
>> >> >Q: Shouldn't the first row be 000.0?
>> >>
>> >> It is quite safe to ignore the values before the colon, as they
>> >> are merely an artifact of scala's output.
>> >
>> >What kind of artifact are they, if not an error?
>>
>> They indicate the interval between 1/1 and the degree of the
>> original 'native' mode, on which the mode shown on the particular
>> line is based.
>
>In that case, shouldn't the first entry be 000.0?

It should be "1/1". Looks like this was a typo on my part.

>> In this case they are artifactal only because they are given in
>> different units than is the interval matrix itself.
>
>OK, but more serious would seem to be the actual error.

Maybe I clipped this bit when I copied and pasted, and then
filled in by hand (incorrectly) in subconscious mode.

-Carl

🔗Carl Lumma <ekin@lumma.org>

>> >The interval matrix often, if not typically, has all
>> >unisons/octaves along the diagonal. This one is merely
>> >a reshuffling so that the diagonal becomes a right vertical.
>>
>> No, the interval matrix is always written as above (the values
>> after the colons, at least).
>
>Where do you get *always*??

I have never seen it printed this way. Where do you get
"often, if not typically"?

-Carl

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >The interval matrix often, if not typically, has all
> >> >unisons/octaves along the diagonal. This one is merely
> >> >a reshuffling so that the diagonal becomes a right vertical.
> >>
> >> No, the interval matrix is always written as above (the values
> >> after the colons, at least).
> >
> >Where do you get *always*??
>
> I have never seen it printed this way. Where do you get
> "often, if not typically"?
>
> -Carl

Here's just one example:

http://tonalsoft.com/enc/intervalmatrix.htm

🔗Carl Lumma <ekin@lumma.org>

>> >> >The interval matrix often, if not typically, has all
>> >> >unisons/octaves along the diagonal. This one is merely
>> >> >a reshuffling so that the diagonal becomes a right vertical.
>> >>
>> >> No, the interval matrix is always written as above (the values
>> >> after the colons, at least).
>> >
>> >Where do you get *always*??
>>
>> I have never seen it printed this way. Where do you get
>> "often, if not typically"?
>>
>> -Carl
>
>Here's just one example:
>
>http://tonalsoft.com/enc/intervalmatrix.htm

Oh, maybe that is how R. printed them. I'll look when I get home.

-Carl

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> The interval matrix often, if not typically, has all
unisons/octaves
> along the diagonal. This one is merely a reshuffling so that the
> diagonal becomes a right vertical.

That's what I used. Manuel's way is more useful for seeing how the
harmony works out in the scale, however.

🔗Gene Ward Smith <gwsmith@svpal.org>

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

> >> I also see I had the wrong definition of interval matrix;
>
> That's pretty incredible considering that you've got the
> Rothenberg papers on the matter.

Paul now tells us I had a correct definition. Did Rothenberg invent
the idea of an interval matrix?

🔗Carl Lumma <ekin@lumma.org>

>> >> I also see I had the wrong definition of interval matrix;
>>
>> That's pretty incredible considering that you've got the
>> Rothenberg papers on the matter.
>
>Paul now tells us I had a correct definition.

If the difference between Paul's way and Manuel's way affects
the definition, somebody's smoking crack.

Wait, let me guess, it affects the def. if you intend to
interpret them as some sort of algebraic thing. Well
Rothenberg never did this to my knowledge (unless you see
otherwise in his papers) but I'm all ears as to why the
heck anyone would do such a thing.

>Did Rothenberg invent the idea of an interval matrix?

As I am using the term in this thread, yes. If by "invented"
you mean "defined".

-Carl

🔗Manuel Op de Coul <manuel.op.de.coul@eon-benelux.com>

Carl wrote on 29-1:
>This happened because I wanted to give the interval matrix in
>'steps of 12-tET' units. Unfortunately (and one of my biggest
>desired features) Scala does not offer 'degrees of n-ET' units.

Fortunately it does, use
set attribute et_step <steps/oct>

To see the intervals in terms of these units, do
show/attribute intervals

Manuel

🔗Carl Lumma <ekin@lumma.org>

Welcome back, Manuel.

>Carl wrote on 29-1:
>>This happened because I wanted to give the interval matrix in
>>'steps of 12-tET' units. Unfortunately (and one of my biggest
>>desired features) Scala does not offer 'degrees of n-ET' units.
>
>Fortunately it does, use
>set attribute et_step <steps/oct>
>
>To see the intervals in terms of these units, do
>show/attribute intervals

These don't look like units to me, but some secondary abstraction.
Can I author a scl file using them? Does Scala display all its
output in them? No, it still displays cents, with these in a
separate column.

-Carl

🔗Manuel Op de Coul <manuel.op.de.coul@eon-benelux.com>

Carl wrote:
>These don't look like units to me, but some secondary abstraction.

Ah, if you mean you expected non-integer numbers, that's also
possible. Then the command is:
set attribute <steps/oct>
For example: set attribute 12.0

>Does Scala display all its output in them?

No, that's not possible.
Note that you can use the input command to enter a scale in
any logarithmic unit if you convert it afterwards with
mult/abs 2/1, assuming 2/1 is the period you want.

>No, it still displays cents, with these in a
>separate column.

It gave me an idea for an enhancement though, showing the
attributes for the intervals of the interval matrix. I'll put that
in the next version under the command
show/attribute/line intervals.

Manuel