Paul Erlich on 22

Paul Erlich's newly corrected "Tuning, Tonality, and 22 Tone Equal Temperament" has been posted to my Notes on Microtonality here:
http://math.cudenver.edu/~jstarret/notes.html

Highly recommended.

John Starrett

--- In tuning@y..., "jdstarrett" <jstarret@c...> wrote:
> Paul Erlich's newly corrected "Tuning, Tonality, and 22 Tone Equal
Temperament" has been posted to my Notes on Microtonality here:
> http://math.cudenver.edu/~jstarret/notes.html
>
> Highly recommended.
>
> John Starrett

What changed? In other words, should I re-print the whole
thing, or can I just print certain pages?

Jeff

--- In tuning@y..., "jjensen142000" <jjensen14@h...> wrote:
> --- In tuning@y..., "jdstarrett" <jstarret@c...> wrote:
> > Paul Erlich's newly corrected "Tuning, Tonality, and 22 Tone
Equal
> Temperament" has been posted to my Notes on Microtonality here:
> > http://math.cudenver.edu/~jstarret/notes.html
> >
> > Highly recommended.
> >
> > John Starrett
>
> What changed? In other words, should I re-print the whole
> thing, or can I just print certain pages?
>
> Jeff

the big changes involve an entire page (formerly page 20) which got
deleted, and the appendix of key signatures at the end, in which half
of the key signatures were previously incorrect, and the new
corrected version contains a keyboard diagram along with the key sigs.

lots of small changes too.

for your purposes, jeff, i wouldn't worry about it. it seems we have
some bigger fish to tackle in regard to your webpage:

(1) the idea of periodicity blocks as a natural determinant
of "finity" (i.e., why 7 pitches?);

(2) the historical and theoretical importance of meantone temperament;

(3) the emergence of tonality from modality and the reason two
modes "won out" . . .

cheers,
paul

--- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:
> the big changes involve an entire page (formerly page 20) which got
> deleted,

Hi Paul. You know, I printed the .pdf copy, and I just now
noticed that there are no page numbers...! I wonder if it is just
my system?

>
> for your purposes, jeff, i wouldn't worry about it. it seems we
have
> some bigger fish to tackle in regard to your webpage:
>

I've been working on a couple other things, so I haven't
made it though my stack of reading material yet ( I have been
perusing tuning-math a bit), but here are some fuzzy opinions
that I'm starting to form:

> (1) the idea of periodicity blocks as a natural determinant
> of "finity" (i.e., why 7 pitches?);
>

After reading parts 1 and 2 of the "Gentle Introduction", and
some postings, I'm just not sure that Fokker periodicity blocks
are more than a clever diagram. For example, whether or not
a block is convex would seem, a priori, to have no
relationship to music....

I'm conjecturing that people settled on the 7 pitches (making
the chords C, F, G) because you don't need more to have
"tonality", whatever that is, exactly. If you want more pitches,
you change key. [My working definition of tonality is the
ability of a scale to

(1) define a particular consonant chord (the tonic C!) and
(2) to define other chords like g-b-d-f (or c-f-g, b-f,
b-d-f) that are
(a) close to C and
(b) either intrinsically dissonant, or at least dissonant
against the pitch c that is somehow held in the listener's
mind.
Thus these dissonant chords "resolve" to C]

> (2) the historical and theoretical importance of meantone
temperament;

It still seems to me that one can go from the harmonic purity
of JI to 12tet without needing meantone in the argument; I
guess I need to write up the details and do the calculations
to make sure there are no logical holes.

>
> (3) the emergence of tonality from modality and the reason two
> modes "won out" . . .
>

Yes, this seems to be hitting the nail right on the head...

--Jeff

--- In tuning@y..., "jjensen142000" <jjensen14@h...> wrote:

> After reading parts 1 and 2 of the "Gentle Introduction", and
> some postings, I'm just not sure that Fokker periodicity blocks
> are more than a clever diagram. For example, whether or not
> a block is convex would seem, a priori, to have no
> relationship to music....

If a scale is not convex, there will be notes not in the scale at least as closely related to the notes of the scale as those in it.
Notions of "clique" or "similarity circle" seem to me to relevant musically.

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "jjensen142000" <jjensen14@h...> wrote:
>
> > After reading parts 1 and 2 of the "Gentle Introduction", and
> > some postings, I'm just not sure that Fokker periodicity blocks
> > are more than a clever diagram. For example, whether or not
> > a block is convex would seem, a priori, to have no
> > relationship to music....
>
> If a scale is not convex, there will be notes not in the scale at
least as closely related to the notes of the scale as those in it.
> Notions of "clique" or "similarity circle" seem to me to relevant
musically.

Hi, Gene.

The Periodicity Blocks seems to be a certain mathematical formalism,
and I am wondering what are a few of the major important results
that fall out of this formalism? And to be brutally blunt,
are these results about music, or results about the geometry of
lattices? In other words, do they explain things that occur in
common music, things that are not easily explained other ways?

Also, is the hypothesis "The smaller the whole number ratio, the
greater the consonance" a necessary one for this formalism to
make sense?

thanks,
Jeff

--- In tuning@y..., "jjensen142000" <jjensen14@h...> wrote:
> --- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:
> > the big changes involve an entire page (formerly page 20) which
got
> > deleted,
>
> Hi Paul. You know, I printed the .pdf copy, and I just now
> noticed that there are no page numbers...! I wonder if it is just
> my system?
>
>
> >
> > for your purposes, jeff, i wouldn't worry about it. it seems we
> have
> > some bigger fish to tackle in regard to your webpage:
> >
>
> I've been working on a couple other things, so I haven't
> made it though my stack of reading material yet ( I have been
> perusing tuning-math a bit), but here are some fuzzy opinions
> that I'm starting to form:
>
> > (1) the idea of periodicity blocks as a natural determinant
> > of "finity" (i.e., why 7 pitches?);
> >
>
> After reading parts 1 and 2 of the "Gentle Introduction", and
> some postings, I'm just not sure that Fokker periodicity blocks
> are more than a clever diagram. For example, whether or not
> a block is convex would seem, a priori, to have no
> relationship to music....

though convexity is neither necessary nor sufficient
for 'blockitude', the more convex a scale is, the more consonances
there are between scale tones. if a scale isn't convex, it's very
tempting for musicians to add the tones in the "hole", since they'll
be consonant with so many already existing scale tones. but
blockitude has even deeper desirability, i believe. i can discuss
further if you wish.

> I'm conjecturing that people settled on the 7 pitches (making
> the chords C, F, G) because you don't need more to have
> "tonality", whatever that is, exactly.

well, unfortunately, that's not at all what happened, to the best of
my understanding.

> If you want more pitches,
> you change key. [My working definition of tonality is the
> ability of a scale to
>
> (1) define a particular consonant chord (the tonic C!) and
> (2) to define other chords like g-b-d-f (or c-f-g, b-f,
> b-d-f) that are
> (a) close to C and
> (b) either intrinsically dissonant, or at least dissonant
> against the pitch c that is somehow held in the listener's
> mind.
> Thus these dissonant chords "resolve" to C]

well, this sounds like it's sort of on the right track, but it
doesn't quite narrow the modes down to the ones used in tonality,
does it? nor does it really explain where the diatonic scale comes
from in the first place. in other words, it seems that almost any
scale/mode could be said to satisfy your criteria above -- doesn't it?

> > (2) the historical and theoretical importance of meantone
> temperament;
>
> It still seems to me that one can go from the harmonic purity
> of JI to 12tet without needing meantone in the argument; I
> guess I need to write up the details and do the calculations
> to make sure there are no logical holes.

you can go straight to 12-tET, but meantone supports tonality,
including modulations (except those that "wrap around"), in a much
more acoustically pure way -- it would be a shame to ignore it
entirely. 12-tET introduces much larger errors in order to make G#
the same as Ab, etc. Mozart, and most before him, taught that G# was
different from Ab, Eb was different from D#, etc. -- so going to 12-
tET hardly seems like an "a priori" necessity.

have fun,
paul

--- In tuning@y..., "jjensen142000" <jjensen14@h...> wrote:
> --- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> > --- In tuning@y..., "jjensen142000" <jjensen14@h...> wrote:
> >
> > > After reading parts 1 and 2 of the "Gentle Introduction", and
> > > some postings, I'm just not sure that Fokker periodicity blocks
> > > are more than a clever diagram. For example, whether or not
> > > a block is convex would seem, a priori, to have no
> > > relationship to music....
> >
> > If a scale is not convex, there will be notes not in the scale at
> least as closely related to the notes of the scale as those in it.
> > Notions of "clique" or "similarity circle" seem to me to relevant
> musically.
>
> Hi, Gene.
>
> The Periodicity Blocks seems to be a certain mathematical formalism,
> and I am wondering what are a few of the major important results
> that fall out of this formalism? And to be brutally blunt,
> are these results about music, or results about the geometry of
> lattices? In other words, do they explain things that occur in
> common music, things that are not easily explained other ways?

yes -- it explains a great deal. nearly every culture or theorist who
ever came up with a finite set of tones ended up with a periodicity
block. the mathematical formalism captures the typical desiderata of
scale-builders because of the unison vectors. the unison vectors
determine where you 'cut off' the lattice. if you didn't cut it off
at the width of one unison vector, you'd end up with two tones that
were one unison vector apart, i.e., very close to one another, in
pitch. generally, theorists and musicians avoid such 'twins' in their
scale constructs, but otherwise reach out as far as possible in the
lattice. this means you end up with a periodicity block, not
necessarily a parallelogram as i hope my "excursion" makes clear.

> Also, is the hypothesis "The smaller the whole number ratio, the
> greater the consonance" a necessary one for this formalism to
> make sense?

not really. for gene's convexity argument above, you do need to
assume that the simple-integer ratios that form the "rungs" in the
lattice are consonant or otherwise desirable. i don't think you would
have any objection to this assumption, based on what you've written
so far -- would you?

lata,
paul

--- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:

> though convexity is neither necessary nor sufficient
> for 'blockitude', the more convex a scale is, the more consonances
> there are between scale tones.

That depends on your definition of blockitude, and I thought you had agreed that epimorphic+convex was acceptable to you, in which case convexity would be necessary.

--- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:
> > > for your purposes, jeff, i wouldn't worry about it. it seems we
> > have
> > > some bigger fish to tackle in regard to your webpage:

On the contrary, from the discussion in rec.music.theory, it seems
that "22" talks about a lot of stuff relevant to what I'm thinking
about. I'm going to get it read by this weekend -- also the
remaining periodicity block stuff.

> > I'm conjecturing that people settled on the 7 pitches (making
> > Thus these dissonant chords "resolve" to C]
>
> well, this sounds like it's sort of on the right track, but it
> doesn't quite narrow the modes down to the ones used in tonality,
> does it? nor does it really explain where the diatonic scale comes
> from in the first place. in other words, it seems that almost any
> scale/mode could be said to satisfy your criteria above -- doesn't
it?
>

Yes, that is a good point. My working hypothesis needs more work.

--JEff

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:
>
> > though convexity is neither necessary nor sufficient
> > for 'blockitude', the more convex a scale is, the more
consonances
> > there are between scale tones.
>
> That depends on your definition of blockitude, and I thought you
>had agreed that epimorphic+convex was acceptable to you, in which
>case convexity would be necessary.

well, by 'blockitude' i only meant 'epimorphic', for the purposes of
my discussion with jeff. that the way i thought of it when i wrote
the 'gentle introduction', especially the excursion, so i thought it
best to keep convexity as a separate criterion for the purposes of
said discussion.

--- In tuning@y..., "jjensen142000" <jjensen14@h...> wrote:

/tuning/topicId_36354.html#36427

>
> I'm conjecturing that people settled on the 7 pitches (making
> the chords C, F, G) because you don't need more to have
> "tonality", whatever that is, exactly. If you want more pitches,
> you change key.

****I don't believe the history of this worked out quite this way,
but I will let the "historically inclined" on the list answer this
question...

J. Pehrson