Yahoo Tuning Groups Ultimate Backup tuning-math review requested

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Hey everybody,

I wrote this in a hurry, trying to explain the activity on this list
to a friend is as short a document as possible.

I haven't read *The Forms of Tonality* since early 2001, but I plan
to do that again now. I'm sure it covers much of the same ground.

I wonder what everyone thinks of this? Anything you disagree with?
Errors? Just not worth fixing?

Thanks,

-Carl

________________________________________________________________________

() Assume...

() Music involves repetition. Sometimes, instead of an exact
repetition, a few things are changed while the rest stay the
same.

() A theme played in a different mode keeps generic
intervals (3rds, etc.) the same while pitches change
[only true for Rothenberg-proper scales].

() A theme played in a different key keeps absolute
intervals the same while pitches change [as goes
Rothenberg-efficiency, one relies more on something like
the rules of tonal music to make the key-changes
recognizeable].

() Possible intervals between notes are to be taken from some
fixed set of just or near-just intervals [to exploit the signal-
processing capabilities of the hearing system to deliver
information to the listener].

() So, to build a scale, we take our chosen just intervals and *stack*
them. This generates a lattice. In the 3-limit we get a chain. In the
5-limit we get a planar lattice. 7-limit, we can use the face-centered
cubic lattice. etc.

() Eventually, we will run into pitches that are very close to pitches
we already have. The small intervals between such pairs of pitches
called commas.

() We create a "pun" if we use the same name ("Ab") for both
notes in such a pair.

() We create a "comma pump" by writing a chord progression whose
starting and ending "tonic" involve a "pun". Every time the
chord progression is repeated, our pitch standard moves by the
comma involved. Or...

() We can temper the comma(s) out!

() Doing so collapses the lattice into a finite "block". The
block tiles the lattice. To move between tiles, simply
transpose all the notes in the basic block by some number of
commas.

() It appears that most scales that have been popular around the world
and throughout history correspond fairly well to temperaments where very
"simple" commas have been tempered out.

() "Simple" is measured by the distance on the lattice between
the two pitches that generate the comma. Thus, simple commas
naturally tend to define smaller (fewer pitches) blocks.

() You can think of "simple" as giving more intervals
with fewer tones if the comma is tempered out.

() To further rate temperaments, simple may be balanced against error.
The error is determined by the (log) size of the comma and the number
of notes in the block over which it must be distributed (which simple
already tells us).

() That is, it's easy to find small ratios with arbitrarily
large denominators (1001:1000). We want temperaments based on
the simplest ratios in a given size range.

() As a matter of strange coincidence, the same math is
behind harmonic entropy!

() To see a database of 5-limit temperaments database, go to...

/tuning/database/

...or try the Excel version at...

http://lumma.org/stuff/5-limit_Linear_Temp.xls

...Try sorting by denominator (the Excel version should be by default).
You can see that 81:80 is one of the simplest commas, and is by far the
smallest comma among the few most simple (the list itself is the result
of searching 5-limit ratio space for low size*denominator ratios).

Another comma that looks good is the major diesis (648:625), which leads
to the "diminished" temperament. If we take 8 tones/octave of this
temperament, we get the octatonic scale of Stravinsky and Messiaen.

However, while diminished makes a good showing, you can see that
"porcupine" is better. Note...

() Diminished, but not porcupine, is to be found in 12-et.

() Diminished was not used by composers until after 12-et had
become entrenched.

This suggests that porcupine is a potentially fertile direction for new
music. Indeed, the temperament is named after a rather fetching piece
by composer Herman Miller, the "Mizarian Porcupine Overture"...

http://lumma.org/stuff/Mizarian_Porcupine_Overture.mp3

http://www.io.com/~hmiller/music/temp-porcupine.html

() Reactions...

>>() We can temper the comma(s) out!
>> () Doing so collapses the lattice into a finite "block".
>> The block tiles the lattice.
>
>There you've lost me. I would think that when the lattice collapsed,
>that would be the end of it. What is the nature of the lattice,
>once you've tempered the commas out?

It collapses to a regular tiling. Here...

http://lumma.org/stuff/135-128.gif

...the "unison vectors" (Fokker's term for commas) 81:80 and 32:25
define the tile shown in red, which defines the scale shown in green.

>Is it tempered or untempered (infinite or finite)?

Blocks may be left untempered. In the 5-limit, if you temper one comma
out you get a linear temperament (chain). Temper both commas out and
you get an equal temperament. In the 7-limit, 3 commas are required to
tile the lattice. Temper one out and get a planar temperament, two out
get a linear temperament, three out get an equal temperament.

In the 3-limit, the diatonic scale may be seen as a block defined by the
2187:2048 (apotome), not tempered out. In the 5-limit, it may be
defined by 81:80 and 25:24 with the 81:80 tempered out and the 25:24 not
tempered out.

>If it's tempered, aren't all instances the same? If it's untempered,
>what does it mean to "tile the lattice" with a tempered block?

If all commas are tempered out, you can move from tile to tile without
changing any pitches. For some untempered comma, you must transpose all
the pitches in your tile by it n times to get the pitches for the nth
tile away in that comma's direction (which is why Fokker called it a
vector, though Gene says it's not really a vector).

-Carl

🔗Paul Erlich <perlich@aya.yale.edu>

is the diatonic major second just or near-just?

> () Eventually, we will run into pitches that are very close to
pitches
> we already have. The small intervals between such pairs of pitches
> called commas.
>
> () We create a "pun" if we use the same name ("Ab") for both
> notes in such a pair.

i think "pun" refers to the same pitch being used for different
musical functions.
>
> () We can temper the comma(s) out!
>
> () Doing so collapses the lattice into a finite "block".

or an infinite "strip", "slice", etc., depending on how many
independent commas we temper out.

> The
> block tiles the lattice. To move between tiles, simply
> transpose all the notes in the basic block by some number of
> commas.

if you're tempered the relevant comma out, no transposition is
involved at all.

> () You can think of "simple" as giving more intervals
> with fewer tones if the comma is tempered out.

>
> () As a matter of strange coincidence, the same math
is
> behind harmonic entropy!

behind or in front of?

> However, while diminished makes a good showing, you can see that
> "porcupine" is better. Note...
>
> () Diminished, but not porcupine, is to be found in 12-et.
>
> () Diminished was not used by composers until after 12-et had
> become entrenched.
>
> This suggests that porcupine is a potentially fertile direction for
new
> music. Indeed, the temperament is named after a rather fetching
piece
> by composer Herman Miller, the "Mizarian Porcupine Overture"...
>
> http://lumma.org/stuff/Mizarian_Porcupine_Overture.mp3
>
> http://www.io.com/~hmiller/music/temp-porcupine.html

my piece "glassic" is even more directly based on this temperament,
using its 7-note MOS for long stretches, and was just rebroadcast on
wnyc!

>
> () Reactions...
>
> >>() We can temper the comma(s) out!
> >> () Doing so collapses the lattice into a finite "block".
> >> The block tiles the lattice.
> >
> >There you've lost me. I would think that when the lattice
collapsed,
> >that would be the end of it. What is the nature of the lattice,
> >once you've tempered the commas out?
>
> It collapses to a regular tiling.

or, perhaps better, you could picture it as being wrapped into a
cylinder (in the case of the infinite strip), or a torus (in the case
of the finite block), etc.

🔗Carl Lumma <ekin@lumma.org>

>is the diatonic major second just or near-just?

Not sure where this is pointed.

>> () We create a "pun" if we use the same name ("Ab") for both
>> notes in such a pair.
>
>i think "pun" refers to the same pitch being used for different
>musical functions.

It was my understanding that it was your assumption that two Ab
notes in a score refer to the same pitch, and thus, imply meantone
temperament for common practice music. In which case my phrasing
above is ok. If we don't want to make that assumption I should
change it.

>> () We can temper the comma(s) out!
>>
>> () Doing so collapses the lattice into a finite "block".
>
>or an infinite "strip", "slice", etc., depending on how many
>independent commas we temper out.
>
>> The
>> block tiles the lattice. To move between tiles, simply
>> transpose all the notes in the basic block by some number of
>> commas.
>
>if you're tempered the relevant comma out, no transposition is
>involved at all.

Yes, it seems these do not belong as sub-items to tempering it
out! Obviously, I'm talking about the untempered case. Huge
oversight there.

>> () You can think of "simple" as giving more intervals
>> with fewer tones if the comma is tempered out.
>
>??

Simple commas tend to define small blocks, so if all commas are
tempered out we get all the intervals with fewer notes. Even though
a linear temp. has infinitely many notes, there must be something
similar going on...

>> () As a matter of strange coincidence, the same math
>> is behind harmonic entropy!
>
>behind or in front of?

Anyway, this clearly doesn't belong in the doc. But if you could
write a blurb on this for monz or someone to post, I think it'd
be interesting.

>> http://lumma.org/stuff/Mizarian_Porcupine_Overture.mp3
>>
>> http://www.io.com/~hmiller/music/temp-porcupine.html
>
>my piece "glassic" is even more directly based on this temperament,
>using its 7-note MOS for long stretches, and was just rebroadcast on
>wnyc!

Nice. Do you have a link? If I ever decide to publish this, it
will be as a web page with inline graphics, and I'll ask Herman
for a link to MPO. Or, I'm happy to provide links at lumma.org.

>>I haven't read *The Forms of Tonality* since early 2001, but I plan
>>to do that again now. I'm sure it covers much of the same ground.
>
>not enough of it.

I assume you mean I'm not covering enough of your ground? My goal
is to make a document much shorter than TFOT. Actually, maybe I
haven't even do so. TFOT was pretty short IIRC!

-Carl

🔗monz@attglobal.net

hi Carl and paul,

> From: Carl Lumma [mailto:ekin@lumma.org]
> Sent: Sunday, August 03, 2003 11:24 AM
> To: tuning-math@yahoogroups.com
> Subject: Re: [tuning-math] Re: review requested
>
>
> <snip>
>
> >> () We can temper the comma(s) out!
> >>
> >> () Doing so collapses the lattice into a finite "block".
> >
> >or an infinite "strip", "slice", etc., depending on how many
> >independent commas we temper out.
> >
> >> The
> >> block tiles the lattice. To move between tiles, simply
> >> transpose all the notes in the basic block by some number of
> >> commas.
> >
> >if you're tempered the relevant comma out, no transposition is
> >involved at all.
>
> Yes, it seems these do not belong as sub-items to tempering it
> out! Obviously, I'm talking about the untempered case. Huge
> oversight there.
>
> >> () You can think of "simple" as giving more intervals
> >> with fewer tones if the comma is tempered out.
> >
> >??
>
> Simple commas tend to define small blocks, so if all commas are
> tempered out we get all the intervals with fewer notes. Even though
> a linear temp. has infinitely many notes, there must be something
> similar going on...
>
> <snip>

Carl, it's simply a matter of differing dimensions.

i think i must be misunderstanding this disucssion,
because i'd think that *you* would get this. if i'm
having to explain this to you, then i must be missing
something.

finity does not necessarily imply a dualistic distinction
between finite and infinite. assuming that each prime-factor
is a unique dimension in a multi-dimensional array
characterizing the mathematics of the harmony, as
successive commas are tempered out or ignored, there is
a subsequent successive reduction of the dimensions
by elimination, one dimension at a time for each comma.

... but *you* know that already, no? so then what
what are you referring to by "something similar going on"?

-monz

🔗Carl Lumma <ekin@lumma.org>

Heya monz,

>>>> () You can think of "simple" as giving more
>>>> intervals with fewer tones if the comma is
>>>> tempered out.
>>>
>>>??
>>
>>Simple commas tend to define small blocks, so if all commas are
>>tempered out we get all the intervals with fewer notes. Even
>>though a linear temp. has infinitely many notes, there must be
>>something similar going on...
>
>Carl, it's simply a matter of differing dimensions.
>
>i think i must be misunderstanding this discussion,

I'm not sure which part of the above quote you're referring to.
The last part there is a aggressive abstraction of Paul's
complexity heuristic, which may not be accurate.

>finity does not necessarily imply a dualistic distinction
>between finite and infinite. assuming that each prime-factor
>is a unique dimension in a multi-dimensional array
>characterizing the mathematics of the harmony, as
>successive commas are tempered out or ignored, there is
>a subsequent successive reduction of the dimensions
>by elimination, one dimension at a time for each comma.
>
>... but *you* know that already, no? so then what
>what are you referring to by "something similar going on"?

That part refers to the simple thing. The simpler the comma,
the lower the complexity of the resulting temperament(s).
That comes from Paul's heuristic. Complexity can be defined
as intervals/notes.

-Carl

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >is the diatonic major second just or near-just?
>
> Not sure where this is pointed.

you said something about the lattice being constructed out of the
musical intervals used, or something . . .

> >> () We create a "pun" if we use the same name ("Ab") for both
> >> notes in such a pair.
> >
> >i think "pun" refers to the same pitch being used for different
> >musical functions.
>
> It was my understanding that it was your assumption that two Ab
> notes in a score refer to the same pitch, and thus, imply meantone
> temperament for common practice music.

hmm . . . ok maybe i misread you or something . . . not sure . . .

> In which case my phrasing
> above is ok. If we don't want to make that assumption I should
> change it.

_the forms of tonality_ does not make that assumption. it is intended
to appeal to the just intonation (network) crowd.

>
> >> () You can think of "simple" as giving more intervals
> >> with fewer tones if the comma is tempered out.
> >
> >??
>
> Simple commas tend to define small blocks, so if all commas are
> tempered out we get all the intervals with fewer notes.

but not more intervals.

> Even though
> a linear temp. has infinitely many notes, there must be something
> similar going on...

yes, more per pitch or whatnot.

> >> () As a matter of strange coincidence, the same math
> >> is behind harmonic entropy!
> >
> >behind or in front of?
>
> ?
>
> Anyway, this clearly doesn't belong in the doc. But if you could
> write a blurb on this for monz or someone to post, I think it'd
> be interesting.

i'd love to, but what would i be writing a blurb on?
>
> >> http://lumma.org/stuff/Mizarian_Porcupine_Overture.mp3
> >>
> >> http://www.io.com/~hmiller/music/temp-porcupine.html
> >
> >my piece "glassic" is even more directly based on this temperament,
> >using its 7-note MOS for long stretches, and was just rebroadcast
on
> >wnyc!
>
> Nice. Do you have a link?

it used to be on tuning-punks, chris bailey or someone may have
rescued them . . .

> If I ever decide to publish this, it
> will be as a web page with inline graphics, and I'll ask Herman
> for a link to MPO. Or, I'm happy to provide links at lumma.org.
>
> >>I haven't read *The Forms of Tonality* since early 2001, but I
plan
> >>to do that again now. I'm sure it covers much of the same ground.
> >
> >not enough of it.
>
> I assume you mean I'm not covering enough of your ground?

no, i mean the forms of tonality doesn't cover enough of what you're
covering here. and hoped to make a tuning-math collaborative paper on,
but there were too many disagreements to get started.

> My goal
> is to make a document much shorter than TFOT. Actually, maybe I
> haven't even do so. TFOT was pretty short IIRC!
>
> -Carl

carl, please help us see beyond our differences and produce a document
which will be beautiful and matter to the future of music . . .

🔗Paul Erlich <perlich@aya.yale.edu>

🔗Carl Lumma <ekin@lumma.org>

🔗Paul Erlich <perlich@aya.yale.edu>

🔗Graham Breed <graham@microtonal.co.uk>

Paul Erlich wrote:

> no, i mean the forms of tonality doesn't cover enough of what you're > covering here. and hoped to make a tuning-math collaborative paper on, > but there were too many disagreements to get started.

I got started with this document:

http://x31eq.com/temper/method.html

which covers the parts I'm directly interested in. Now I have more time on my hands, I could write up the method from unison vectors as well. But not this week, because there's good weather forecast.

I don't know about the philosophy behind unison vectors, because that's not really my thing. But I suggest the idea of Constant Structure is important:

http://sonic-arts.org/dict/constant.htm

A full set of unison vectors define a constant structure. If you temper out all commas, all intervals of a particular kind are equally well approximated. If you temper none of them out, some intervals are perfect and some are wolves. With a linear temperament, some are "official" approximations, and others wolves, and so on.

Is the caveat about unison vector sizes necessary? I think a periodicity block will always obey the mapping, but the notes might not be in ascending order!

Graham

review requested

🔗Carl Lumma <ekin@lumma.org>

🔗Paul Erlich <perlich@aya.yale.edu>

🔗Paul Erlich <perlich@aya.yale.edu>

🔗Carl Lumma <ekin@lumma.org>

🔗Carl Lumma <ekin@lumma.org>

🔗monz@attglobal.net

🔗Carl Lumma <ekin@lumma.org>

🔗Paul Erlich <perlich@aya.yale.edu>

🔗Paul Erlich <perlich@aya.yale.edu>

🔗Paul Erlich <perlich@aya.yale.edu>

🔗Carl Lumma <ekin@lumma.org>

🔗Paul Erlich <perlich@aya.yale.edu>

🔗Graham Breed <graham@microtonal.co.uk>

🔗Carl Lumma <ekin@lumma.org>

🔗Gene Ward Smith <gwsmith@svpal.org>