deriving classical major/minor scales

Hello.

I hope this is not too far off-topic. I recently was
was wondering about why the classical diatonic major
and minor scales have the notes that they do, what
principles define these scales, and so forth.

I did some reading and I invented some answers to some
music theory questions that I did not find ready answers to,
the result was the following brief online HTML paper and
some java applets (e.g. Sethares' dissonance curve, ...):

http://home.austin.rr.com/jmjensen/musicTheory.html

Perhaps some people here have thought through these issues
before, and more deeply than I have? I suspect so.

I would be interested in reading any comments that you might have.

regards,
Jeff

In-Reply-To: <a7ucn2+1t7t@eGroups.com>
jjensen142000 wrote:

> I hope this is not too far off-topic. I recently was
> was wondering about why the classical diatonic major
> and minor scales have the notes that they do, what
> principles define these scales, and so forth.
>
> I did some reading and I invented some answers to some
> music theory questions that I did not find ready answers to,
> the result was the following brief online HTML paper and
> some java applets (e.g. Sethares' dissonance curve, ...):
>
> http://home.austin.rr.com/jmjensen/musicTheory.html

The dissonance curve applet is good! I notice you seem to be using
straight amplitudes for the weighting. The thing Sethares calls
"amplitude" is more like SPL, and so should be measured in decibels. That
means you'll be underestimating the higher partials.

> Perhaps some people here have thought through these issues
> before, and more deeply than I have? I suspect so.

Yes, and one thing we all find is that the explanation of a major scale in
terms of 4:5:6 triads is historically incorrect. The major scale
originated from the spiral of fifths before triadic harmony was
recognized.

It isn't specifically "European" either. It appears in Mesopotamia, from
Sumerian to Arabic times, and in India.

> I would be interested in reading any comments that you might have.

Where you say of tuning minor keys "we do not have an adequate explanation
for this without appealing to equal temperament" that isn't right. You
only have to appeal to meantone temperament.

You can link to <http://www.mmk.ei.tum.de/persons/ter/top/basse.html> for
Terhardt on the fundamental bass.

For dynamic retuning, see <http://www.adaptune.com/>.

The plural of "maximum" is "maxima". There is no word "maximas".

Graham

Graham,
Thanks for your response!

--- In tuning@y..., graham@m... wrote:
> The dissonance curve applet is good! I notice you seem to be using
> straight amplitudes for the weighting. The thing Sethares calls
> "amplitude" is more like SPL, and so should be measured in
decibels. That means you'll be underestimating the higher partials.

What is SPL? Is that related to the methodology of Kameoka and
Kuriyagawa in 1969?

> > Perhaps some people here have thought through these issues
> > before, and more deeply than I have? I suspect so.
>
> Yes, and one thing we all find is that the explanation of a major
scale in
> terms of 4:5:6 triads is historically incorrect. The major scale
> originated from the spiral of fifths before triadic harmony was
> recognized.

I think I'm going to stay restricted to a "logical" discussion,
rather than go into what happened historically. Maybe I should
add a remark though, of what you just said.

> It isn't specifically "European" either. It appears in
Mesopotamia, from
> Sumerian to Arabic times, and in India.

I think those music systems are rather different though (?)
Probably Helmholtz discusses this, but since I'm not a specialist
in this field, I haven't had the gumption to plough through it.

> Where you say of tuning minor keys "we do not have an adequate
explanation
> for this without appealing to equal temperament" that isn't right.
You
> only have to appeal to meantone temperament.

Are you talking about where I'm trying to explain what is
a relative minor?

> You can link to
<http://www.mmk.ei.tum.de/persons/ter/top/basse.html> for
> Terhardt on the fundamental bass.
>
> For dynamic retuning, see <http://www.adaptune.com/>.
>
> The plural of "maximum" is "maxima". There is no word "maximas".

Thanks!

Jeff

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

> I would be interested in reading any comments that you might have.

I think you need to read up on meantone intonation. Your derivation of the major and minor scales is well-known, but ahistorical. You leave the impression that 12-et was invented in an attempt to approximate JI diatonic, which is not the case and which leaves open the question, why 12? A discussion of meantone would be more to the point.

--- In tuning@y..., "jjensen142000" <jjensen14@h...> wrote:
>
> I did some reading and I invented some answers to some
> music theory questions that I did not find ready answers to,
> the result was the following brief online HTML paper and
> some java applets (e.g. Sethares' dissonance curve, ...):
>
> http://home.austin.rr.com/jmjensen/musicTheory.html
>
> Perhaps some people here have thought through these issues
> before, and more deeply than I have? I suspect so.
>
> I would be interested in reading any comments that you might have.
>

***Very cool Applets! I suspect you may wish to do a little
more "historical" study of some of these questions, though, since
there *are* answers!

That's the nice thing about *history.* There actually *is* an
answer, if only we can find it.

Now predicting the *future??* That's a bit trickier... :)

J. Pehrson

Hello Jeff,

>I hope this is not too far off-topic.

I'd say it's perfectly on-topic.

>Perhaps some people here have thought through these issues
>before, and more deeply than I have? I suspect so.

Well, others have thought them through. One of my favorite
treatments is by David Rothenberg, though his treatment is
incomplete (only addresses melodic aspects, for starters) and
his papers may be hard to find. Citations are in the tuning
bibliography...

http://www.xs4all.nl/~huygensf/doc/bib.html

I found most of these in the New York public library. I've
summarized them here and on the tuning-math list a number of
times -- searching the archives for "Rothenberg" might be of
some use.

Also I like Paul Erlich's treatment, both in his "Tuning,
Tonality, and 22-tone Temperament" paper...

http://www-math.cudenver.edu/~jstarret/Erlich.html

...and in his more recent paper "The Forms of Tonality",
which is available by mail (I think).

My own shopping list is here...

http://lumma.org/gd.txt

>I would be interested in reading any comments that you might have.
/.../
>http://home.austin.rr.com/jmjensen/musicTheory.html

Well, I'm one of the people Graham mentions who agrees that the
harmonic derrivation of the scale is historically backward.

Given the constraints of a melodic-harmonic music, as you take
for granted in your paper, however, a number of people have been
able to derrive the diatonic scale and/or meantone temperament
by minimizing pairwise (dyadic) dissonance (Erlich, Sethares), by
maximizing connectivity by dyads (Gene Smith), over 7-tone scales.
There's clearly something to this, but derrivation does equal a
complete understanding where I'm from, nor does it necessarily
mean you can repeat your procedure for (say) 8-tone scales and get
something 'just as good' as the diatonic scale.

Graham wrote...
>Where you say of tuning minor keys "we do not have an adequate
>explanation for this without appealing to equal temperament" that
>isn't right. You only have to appeal to meantone temperament.

You have to appeal to the same things for the major scale as you
do any minor scale which is a mode of it, I'd reckon.

>>It isn't specifically "European" either. It appears in
>>Mesopotamia, from Sumerian to Arabic times, and in India.
>
>I think those music systems are rather different though (?)

What's a music system? The use of the scale differs in Western
culture from anywhere else, but the scale does appear world-wide,
and its origins are lost in antiquity.

-Carl

jjensen142000 wrote:

> What is SPL? Is that related to the methodology of Kameoka and
> Kuriyagawa in 1969?

Yes, whatever K&K used ;)

We haven't really nailed down what the inputs to Sethares' algorithm
should be, but they look more like decibels than amplitude.

> > Where you say of tuning minor keys "we do not have an adequate
> explanation
> > for this without appealing to equal temperament" that isn't right.
> You
> > only have to appeal to meantone temperament.
>
> Are you talking about where I'm trying to explain what is
> a relative minor?

I think so. There's another bit where you get the correct logic for equal
temperament: that it gives infinite modulation without any keys coming out
"bad". But meantone can handle any case where different JI pitches are
implied for notes that are spelled the same.

Graham

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

> > Yes, and one thing we all find is that the explanation of a major
> scale in
> > terms of 4:5:6 triads is historically incorrect. The major scale
> > originated from the spiral of fifths before triadic harmony was
> > recognized.
>
> I think I'm going to stay restricted to a "logical" discussion,
> rather than go into what happened historically. Maybe I should
> add a remark though, of what you just said.

well, the advent of the major and minor modes was a historical
phenomenon (around 1670), and before that the other modes were used
perfectly 'logically', and just as frequently, if not more so. it is
my strong belief that the *tritone* is the reason the major and minor
modes 'won out' -- the tritone is disjoint from the tonic triad
*only* in the major and minor modes. the style that crystallized by
1670 is known commonly as 'tonality' or 'western common-practice',
and the resolution tritone played an all-important role in making
this new kind of 'logic' really work.

i don't think you can understand music without understanding its
evolution. stick around and you may learn a lot about historical
scales, tuning systems, and musical style -- particularly if margo
chooses to join in our discussion (hint hint).

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "jjensen142000" <jjensen14@h...> wrote:
>
> > I would be interested in reading any comments that you might have.
>
> I think you need to read up on meantone intonation. Your derivation
>of the major and minor scales is well-known, but ahistorical. You
>leave the impression that 12-et was invented in an attempt to
>approximate JI diatonic, which is not the case and which leaves open
>the question, why 12? A discussion of meantone would be more to the
>point.

i agree completely. meantone was considered the 'correct intonation'
in most of europe from about 1480 until the late 18th century -- in
england and spain, through the mid 19th century -- so may be
considered 'europe's most successful tuning system' (to quote kyle
gann).

--- In tuning@y..., Carl Lumma <carl@l...> wrote:

> Given the constraints of a melodic-harmonic music, as you take
> for granted in your paper, however, a number of people have been
> able to derrive the diatonic scale and/or meantone temperament
> by minimizing pairwise (dyadic) dissonance (Erlich, Sethares),

note that i pretty much got a meantone-tempered diatonic scale with
this procedure.

> by
> maximizing connectivity by dyads (Gene Smith), over 7-tone scales.
> There's clearly something to this, but derrivation does equal a
> complete understanding where I'm from, nor does it necessarily
> mean you can repeat your procedure for (say) 8-tone scales and get
> something 'just as good' as the diatonic scale.

and don't forget the periodicity block derivation of the diatonic
scale:

http://www.ixpres.com/interval/td/erlich/intropblock1.htm

http://www.ixpres.com/interval/td/erlich/intropblock2.htm

--- In tuning@y..., graham@m... wrote:
> In-Reply-To: <a7ucn2+1t7t@e...>
> jjensen142000 wrote:
>
> The plural of "maximum" is "maxima". There is no word "maximas".

Bob Wendell:
And while we're taking care of that little problem of plurals, let's
add some other common ones to the list:

Singular: Plural:

medium media ( NEVER can we say "The media is". The
medium of radio "is", but if we include other media, such as TV and
newsprint, then the media "are"! I'm amazed at how many national news
anchors and journalists make this semi-literate boo-boo, as if they
had no idea that communication media constitute the plural of a
single medium of communication! They don't even seem to notice what
the word means or that it is related to the word "medium"!

criterion criteria
phenomenon phenomena
crisis crises (rhymes with "cry seize")
analysis analyses
process processes (pronounced to rhyme with "process says"
as a normal English plural, which it *is*, and NOT to rhyme with the
Greek-derived "crises" or "analyses", as so many semi-literate pseudo-
academics pronounce it today)

I will now stand down from my soapbox and continue to wonder how
incoherent our language will become, as if it weren't already
inconsistent enough...

--- In tuning@y..., "robert_wendell" <rwendell@c...> wrote:
> --- In tuning@y..., graham@m... wrote:
> > In-Reply-To: <a7ucn2+1t7t@e...>
> > jjensen142000 wrote:
> >
> > The plural of "maximum" is "maxima". There is no word "maximas".
>
> Bob Wendell:
> And while we're taking care of that little problem of plurals,
let's
> add some other common ones to the list:

oddly, these don't bother me as much as the frequent misspellings of
proper names around here:

it's vicentino, *not* vincentino

it's lamothe, *not* lamonthe

really, though, this is a topic for metatuning (where you can also
learn that hitler is gaining quite a bit of popularity lately).

I wrote...

> My own shopping list is here...
>
> http://lumma.org/gd.txt

I've just updated this; you may have to hit refresh
in your browser.

-Carl

--- In tuning@y..., "jjensen142000" <jjensen14@h...> wrote:
[message 35912]

> > [Graham Breed]
> > It isn't specifically "European" either. It appears in
> > Mesopotamia, from Sumerian to Arabic times, and in India.
>
> I think those music systems are rather different though (?)
> Probably Helmholtz discusses this, but since I'm not a specialist
> in this field, I haven't had the gumption to plough through it.

my webpage "Speculations on Sumerian Tuning" at least covers
my ideas on Mesopotamian music c. 3000 - 1000 BC.

http://www.ixpres.com/interval/monzo/sumerian/sumeriantuning.htm

-monz
(whose own computer is still offline, but will be back on April 1)

Gene, thanks for your response.

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "jjensen142000" <jjensen14@h...> wrote:
>
> > I would be interested in reading any comments that you might have.
>
> I think you need to read up on meantone intonation. Your derivation
of the major and minor scales is well-known, but ahistorical. You
leave the impression that 12-et was invented in an attempt to
approximate JI diatonic, which is not the case and which leaves open
the question, why 12? A discussion of meantone would be more to the
point.

I am going to read up on meantone.

I don't think my derivation is unique, since it is quite simple,
and there are far too many people writing about this subject over
the last 300+ years to have missed it. But do you know of a
non-obscure book that discusses it *explicitly*? I don't.

I think it should not be necessary to follow an historical path
(however interesting!) to derive basic music theory. I probably
should give an argument as to why 12 notes, but that would seem
to follow from a reasonably elementary argument given on
Dave Rusin's math.niu.edu web page.

--Jeff

Hi Carl,
thanks for your reply! Looks like a lot to read.

--- In tuning@y..., Carl Lumma <carl@l...> wrote:

> Well, I'm one of the people Graham mentions who agrees that the
> harmonic derrivation of the scale is historically backward.
>
> Given the constraints of a melodic-harmonic music, as you take
> for granted in your paper, however, a number of people have been
> able to derrive the diatonic scale and/or meantone temperament
> by minimizing pairwise (dyadic) dissonance (Erlich, Sethares), by
> maximizing connectivity by dyads (Gene Smith), over 7-tone scales.
> There's clearly something to this, but derrivation does equal a
> complete understanding where I'm from, nor does it necessarily
> mean you can repeat your procedure for (say) 8-tone scales and get
> something 'just as good' as the diatonic scale.

I printed your "Diatonicity in a nutshell" and Paul Erlich's
"22" paper, which I am guessing he discusses the minimization
of pairwise dissonance. Where does Sethares discuss it?

Also, what/where is the Gene Smith paper you are referring to?

Could you tell me what references are very easy to read?

thanks,
Jeff

>I think it should not be necessary to follow an historical path
>(however interesting!) to derive basic music theory.

I agree. But see my earlier message.

>I probably should give an argument as to why 12 notes, but that
>would seem to follow from a reasonably elementary argument given
>on Dave Rusin's math.niu.edu web page.

I can't get DNS for that right now, but I have been to Dave's
site. It's quite large, and some of it looked pretty advanced.
Does Dave's argument explain why 12 and *not* 19 or 31?

-Carl

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

/tuning/topicId_35909.html#35936

>
> I think it should not be necessary to follow an historical path
> (however interesting!) to derive basic music theory. I probably
> should give an argument as to why 12 notes, but that would seem
> to follow from a reasonably elementary argument given on
> Dave Rusin's math.niu.edu web page.
>
> --Jeff

***I'm not sure I'm getting this, Jeff, but I've been enjoying your
pages so far.

The title of your post refers to the derivation of the "classical"
major and minor scales, not *your* diatonic major and minor scales.

If you want to consider the "classical" diatonic collection, how can
you possibly ignore the history and developments that caused it to
come about?

In fact, on your Website you state:

"Hence this discussion, whose goal is to derive the standard Western
musical scales, and to make the derivation as simple and natural as
possible."

I see the words "the standard Western musical scales, and to make the
derivation..."

How are you going to do that without referencing history?? Don't get
it.

I admit you can make your *own* major and minor scales out of just
intervals any way you wish, and create any systems you want...

But I don't think it's possible to tie them in to the "standard
Western musical scales" or the "classical major/minor scales" without
taking history into account??

??

Joe Pehrson

--- In tuning@y..., Carl Lumma <carl@l...> wrote:

> I can't get DNS for that right now, but I have been to Dave's
> site. It's quite large, and some of it looked pretty advanced.
> Does Dave's argument explain why 12 and *not* 19 or 31?

dave never gets past the 3-limit! thus 19 and 31 don't show up as
viable candidates for the 'next step' after 12, while 41 does.

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:

> I see the words "the standard Western musical scales, and to make
the
> derivation..."
>
> How are you going to do that without referencing history?? Don't
get
> it.

well, as an example, in the second part of my 'gentle introduction to
periodicity blocks'

http://www.ixpres.com/interval/td/erlich/intropblock2.htm

i derive the diatonic scale *ex nihilo*, as it were -- history plays
no role.

Paul,

thanks for your reply. See, I did post to this group...

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> > [Jeff:]
> > I think I'm going to stay restricted to a "logical" discussion,
> > rather than go into what happened historically. Maybe I should
> > add a remark though, of what you just said.
>
> well, the advent of the major and minor modes was a historical
> phenomenon (around 1670), and before that the other modes were used
> perfectly 'logically', and just as frequently, if not more so.

I meant "logic" in the sense of a deductive argument

Harmonics ==> JI,
JI + (uncontrollable urge to change keys) ==> 12tet

not whether or not it makes sense in the context of history
to use the Church modes.

it is
> my strong belief that the *tritone* is the reason the major and
minor
> modes 'won out' -- the tritone is disjoint from the tonic triad
> *only* in the major and minor modes. the style that crystallized by
> 1670 is known commonly as 'tonality' or 'western common-practice',
> and the resolution tritone played an all-important role in making
> this new kind of 'logic' really work.
>

Is this belief summarized somewhere in an easily readable way?

--Jeff

> i don't think you can understand music without understanding its
> evolution. stick around and you may learn a lot about historical
> scales, tuning systems, and musical style -- particularly if margo
> chooses to join in our discussion (hint hint).

--- In tuning@y..., Carl Lumma <carl@l...> wrote:
>> >on Dave Rusin's math.niu.edu web page.
>
> I can't get DNS for that right now, but I have been to Dave's
> site. It's quite large, and some of it looked pretty advanced.
> Does Dave's argument explain why 12 and *not* 19 or 31?
>
> -Carl

server math.niu.edu has ip addr 131.156.3.4

I think you go to ~rusin and look for "math and music";
it also comes up under a yahoo search for that phrase.

I've got to go now; I'll read any replies tomorrow or sat.

Jeff

--- In tuning@y..., "jjensen142000" <jjensen14@h...> wrote:
> Paul,
>
> thanks for your reply. See, I did post to this group...
>
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
> > > [Jeff:]
> > > I think I'm going to stay restricted to a "logical" discussion,
> > > rather than go into what happened historically. Maybe I should
> > > add a remark though, of what you just said.
> >
> > well, the advent of the major and minor modes was a historical
> > phenomenon (around 1670), and before that the other modes were
used
> > perfectly 'logically', and just as frequently, if not more so.
>
> I meant "logic" in the sense of a deductive argument
>
> Harmonics ==> JI,
> JI + (uncontrollable urge to change keys) ==> 12tet

right, but i thought you were talking about explaining why the major
and minor modes were preferred . . . ?

> it is
> > my strong belief that the *tritone* is the reason the major and
> minor
> > modes 'won out' -- the tritone is disjoint from the tonic triad
> > *only* in the major and minor modes. the style that crystallized
by
> > 1670 is known commonly as 'tonality' or 'western common-
practice',
> > and the resolution tritone played an all-important role in making
> > this new kind of 'logic' really work.
> >
>
> Is this belief summarized somewhere in an easily readable way?

well, you can see if it makes sense in my paper, but as far as the
musical-historical examples involved, a bit of that has been
discussed here, especially by margo -- stay tuned and keep asking
questions.

On 3/28/02 8:07 PM, "tuning@yahoogroups.com" <tuning@yahoogroups.com> wrote:

> Message: 2
> Date: Thu, 28 Mar 2002 21:13:11 -0000
> From: "jpehrson2" <jpehrson@rcn.com>
> Subject: Re: deriving classical major/minor scales
>
> --- In tuning@y..., "jjensen142000" <jjensen14@h...> wrote:
>>
>> I did some reading and I invented some answers to some
>> music theory questions that I did not find ready answers to,
>> the result was the following brief online HTML paper and
>> some java applets (e.g. Sethares' dissonance curve, ...):
>>
>> http://home.austin.rr.com/jmjensen/musicTheory.html
>>
>> Perhaps some people here have thought through these issues
>> before, and more deeply than I have? I suspect so.
>>
>> I would be interested in reading any comments that you might have.
>>
>
> ***Very cool Applets! I suspect you may wish to do a little
> more "historical" study of some of these questions, though, since
> there *are* answers!
>
> That's the nice thing about *history.* There actually *is* an
> answer, if only we can find it.
>
> Now predicting the *future??* That's a bit trickier... :)
>
> J. Pehrson

Joseph, you certainly bring a refreshing flair to the tuning list. I enjoy
your sense of humor, not to mention your musical insights.

In my opinion, one should beware of swallowing "history" uncritically. I
think we should temper everything we read by comparing it to our own
experience. There certainly were some cool dudes back then; but there is no
guarantee they totally understood what they were describing; at least, no
more than we do at the moment. ;-) Remember that one of the "answers" from
the past is that the world is flat. Hmmmmm.

Jerry

On 3/28/02 8:07 PM, "tuning@yahoogroups.com" <tuning@yahoogroups.com> wrote:

> What's a music system? The use of the scale differs in Western
> culture from anywhere else, but the scale does appear world-wide,
> and its origins are lost in antiquity.
>
> -Carl

One of the important things we have in common with the ancients is the
physical nature of musical sound. I think we can find much about "origins"
by observing our own perceptions and insights. It may not "prove" anything,
but it might provide insights about how music "works."

Jerry

Hi Joe, thanks for your reply.

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> The title of your post refers to the derivation of the "classical"
> major and minor scales, not *your* diatonic major and minor scales.
>
> If you want to consider the "classical" diatonic collection, how can
> you possibly ignore the history and developments that caused it to
> come about?

Because these scales exists today, right now. We can explore
certain properties that they have, like consonant chords,
leading tones, and so on. Even if I invented a scale, you could
still play music with it and study its properties without knowing
anything about me.

Now, I fully agree that studying the history of music adds a
great deal of insight! I have done a little reading along
those lines. But I want to keep this particular online paper
focused on the non-historical aspects.

Maybe you are interpreting the word "classical" differently
than I meant it. Maybe I should have said "the diatonic
major and minor scales that are most commonly used today in
America" but that is a mouthful...

--Jeff

Hi Jeff,

>I printed your "Diatonicity in a nutshell" and Paul Erlich's
>"22" paper, which I am guessing he discusses the minimization
>of pairwise dissonance.

That never made it in a paper -- it was done here on this list,
two years ago maybe. The paper covers the discovery of the
remarkable decatonic system.

>Where does Sethares discuss it?

In his book. It may also be reprinted on his web page.

>Also, what/where is the Gene Smith paper you are referring to?

Gene is on this list, and on tuning-math (for a complete list
of the lists here, try /tuning2/),
where he cooked up a large number of scales based on
connectivity only a few months ago.

> Could you tell me what references are very easy to read?

My nutshell list will only be clear after you've got some
of the arcane terminology we use around here under your
belt. There's no real FAQ, unfortunately, but Joe Monzo has
put together a dictionary at:

http://www.ixpres.com/interval/dict/

And Graham has a fantastic page at:

http://x31eq.com/

You've already been to John Starrett's site:

http://www-math.cudenver.edu/~jstarret/microtone.html

Rothenberg's papers are some of the hardest to read I've ever
seen, unfortunately. On the other hand, I find both of Paul's
papers easy to read, and Sethares' book.

Partch's book _Genesis of a Music_ is the one must, must, must
read I can think of no matter what you're in to, and it's a
great deal of fun to read.

>Harmonics ==> JI,
>JI + (uncontrollable urge to change keys) ==> 12tet

Sure, though to be more complete:

JI + modal modulation ==> meantone (12, 19, 31...)
meantone + convenience ==> 12-tone subset
12-tone subset + key modulation ==> 12-equal

Another (and maybe more historically-accurate) version:

Tetrachordality + Pythagorean comma ==> 12-tone Pythagorean
12-tone Pythag. + 5-limit harmony => 12-tone meantone
12-tone meantone + augmented and diminished "puns" => 12-equal

-Carl

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

> I don't think my derivation is unique, since it is quite simple,
> and there are far too many people writing about this subject over
> the last 300+ years to have missed it. But do you know of a
> non-obscure book that discusses it *explicitly*? I don't.

If I recall correctly both Redfield and Schoenberg discuss it.

> I think it should not be necessary to follow an historical path
> (however interesting!) to derive basic music theory.

It isn't, but the alternative of a mathematical path does not spare you all difficulties, it just broadens the scope of the question.

I probably
> should give an argument as to why 12 notes, but that would seem
> to follow from a reasonably elementary argument given on
> Dave Rusin's math.niu.edu web page.

I looked at it, and it has a bunch of stuff, including my notion of relating it to the Riemann Zeta function. I don't know what elementary argument you mean in the midst of all that, since it is clear enough from this page that 12-et is one choice among many. If you mean his discussion of how 7/12 is a convergent for log base 2 of 3/2, then I'm afraid that doesn't really take us far enough, since we are discussing "5-limit" harmonies, involving thirds as well as fifths.

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

> Harmonics ==> JI,

This should be Harmonics ==> things are not too awfully out of tune.

> JI + (uncontrollable urge to change keys) ==> 12tet

This should be "meantone + (uncontrollable urge to change keys) ==>
12, 19, 31, 50, 55-et among other possibilities".

--- In tuning@y..., "clumma" <carl@l...> wrote:

> Gene is on this list, and on tuning-math (for a complete list
> of the lists here, try /tuning2/),
> where he cooked up a large number of scales based on
> connectivity only a few months ago.

Gene now thinks he should have used the characteristic polynomial to count triads, not just dyads, and may get back to this.

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

/tuning/topicId_35909.html#35941

> --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
>
> > I see the words "the standard Western musical scales, and to make
> the
> > derivation..."
> >
> > How are you going to do that without referencing history?? Don't
> get
> > it.
>
> well, as an example, in the second part of my 'gentle introduction
to
> periodicity blocks'
>
> http://www.ixpres.com/interval/td/erlich/intropblock2.htm
>
> i derive the diatonic scale *ex nihilo*, as it were -- history
plays no role.

***Hi Paul!

But isn't that a bit like "reverse engineering" where you know the
result you want to get to and just work backwards?

Would the diatonic "12" have any significance to anybody if it
weren't for the historical practice??

jp

--- In tuning@y..., Gerald Eskelin <stg3music@e...> wrote:

/tuning/topicId_35909.html#35948

>>
> Joseph, you certainly bring a refreshing flair to the tuning list.
I enjoy your sense of humor, not to mention your musical insights.
>
> In my opinion, one should beware of swallowing "history"
uncritically. I

****Hi Jerry.

Well, *personally* I believe this list can easily use
my "personality...", but I'm sure it also annoys some people. :)

In all seriousness, though, I find the idea of the past-present-
future timeline a bit mystifying. If it's all somewhat the same
thing, certainly there is more *certainty* on the earlier end!

However, if it's a timeline, like a number line, it means that the
present is infinitely small, no??

What exactly *is* the present, anyway? It must be some incredibly
small number.

Definitely "microtonal..." :)

jp

Hello, there, everyone, and please let me begin by thanking Paul for
inviting me to join in this discussion. While my contribution might or
might not fit the expected agenda as implied by the dialogue I've seen
so far, I hope that at least it may lend another perspective both on
history and current practices.

Part of the discussion, as Paul and Joe have indicated, is an attempt
to understand history and to "get things in the right order," so to
speak. Here I hope I can help, as invited, but should caution that as
has been well said, viewing history is a matter of one's frame of
reference.

From my own perspective, "classical" often means the music of Gothic
Europe in the 13th and 14th centuries based on Pythagorean tuning, and
an outstanding size for a relatively small tuning circle or
quasi-circle in the modern fashion is 17. Thus one of my favorite
questions is "Why 17?" -- and with the benefit of some enlightening
discussions with George Secor, whose superb 17-tone well-temperament
is a topic on which we look forward to publishing forthcoming
articles, here are some brief answers:

(1) Melodically, there's an ideal contrast of sizes
between whole-tones and diatonic semitones;

(2) Harmonically, there's an excellent contrast
between perfect concords (fifths and fourths)
and imperfect concords (thirds and sixths, and
also major seconds and minor sevenths), promoting
ideally efficient and compelling cadential action.

(3) Intervals such as thirds and sixths have three
basic types -- major, minor, and middle or
neutral -- for lots of satisfying variety with
relatively few notes, especially in an unequal
circle or "orbit," either a just intonation
scheme or a well-temperament.

Note that this explanation, as applied either to a 17-note
well-temperament or equal temperament, or to a more eccentric "17-note
orbit" where each whole-tone is divided into three diatonic semitones
but some fifths might be up to a 64:63 comma wider than 3:2, assumes
certain musical and stylistic patterns.

The evident musical and stylistic assumptions of a Western European
musician such as Corelli or Werckmeister around 1680 are quite
different, and logically result in a different approach to tuning
systems.

Having alerted readers to some of my biases, as well as emphasized
that musical and tuning solutions are as much fashion statements as
deductions of logic, I'll try to cover a few points about historical
and modern scales and tunings.

First, for the questions "Why 7?" and "Why 12?" I would say that one
short answer has been provided by Ervin Wilson: these sizes are among
those defining a "Moment of Symmetry" (MOS) where a tuning system has
reached a point of poise with two sizes of adjacent steps.

As has been pointed out, diatonic modes or octave species with seven
notes have played a role in various traditions, with the relationship
between two similar or identical tetrachords an important factor.
The ancient Greek and medieval Near Eastern traditions, as well as the
modes of Byzantine and Gregorian chant, illustrate this point.

As the variety of interval sizes and ratios for diatonic scales in
these traditions might also illustrate, the "7-note" concept can have
appeal in a variety of tuning systems.

However, as I'll be repeatedly emphasizing, "Why 7?" doesn't
necessarily imply "Why _only_ 7?" Even in Gregorian chant, a kind of
ideal type for a music based on diatonic modality, pieces may often
involve _8_ notes per octave, with B/Bb alternating as forms of
a fluid degree, with both forms to be considered part of the regular
medieval gamut (_musica recta_).

In the Western European compositional tradition, the development from
7 or 8 to 12 and beyond took place largely in the setting of
Pythagorean tuning, which I'd call the most successful just intonation
system in the history of this tradition.

By around 1200, composers of polyphonic music such as Perotin were
using an expanded gamut of 11 notes per octave, not only the _musica
recta_ gamut of Bb-B, but the additional accidentals Eb, F#, and C#.
These additional notes came to be termed _musica ficta_, which one
might translate as "invented" music or notes.

From one perspective, we might regard the 13th century as the choice
era of "modal" polyphony, since many sophisticated compositions for
three or four voices draw their sonorities from the notes of a single
octave species or "mode" of 7 or 8 notes, with the B/Bb alternation
often colorfully in evidence.

However, this shouldn't be taken to imply that composers felt
themselves strictly bound to a given 7-note or 8-note set. Around
1300, in fact, Johannes de Grocheio emphasizes the vertical structure
of polyphony and tells us that it is based indeed on octave species,
but not on "modes" in the proper definition of formulas indicating the
beginning, middle, and end of a melody, as in chant.

Now we come to the 14th century, which marks the expansion of
Pythagorean tuning in this tradition to and beyond 12 notes.

In the decades after 1300, one vital factor mandating the regular and
routine use of accidentals is the ideal of "closest approach," with an
unstable interval resolving by stepwise contrary motion to the
"nearest" stable concord, one voice moving by a whole-tone and the
other by a compact diatonic semitone. Thus a major third "strives" to
expand to a fifth, and a major sixth to an octave, while a minor third
seeks contraction to a unison.

As early 14th-century theorists themselves emphasize, obtaining these
closest approach progressions on various degrees requires lots of
accidentalism, as in the following cadence (with C4 as middle C):

C#4 D4
G#3 A3
E3 D3

Here we have the lower pair of voices expanding from major third to
fifth, and the outer pair from major sixth to octave -- requiring the
use of C# and G#.

The historical tuning system for this music, Pythagorean intonation,
very nicely serves this kind of style: the pure fifths and fourths,
active thirds and sixths, and rather compact diatonic semitones all
make "closest approach" effective on the fine level of structure, as
it were. A modern system like a 17-note circle or quasi-circle further
accentuates the latter two qualities, albeit with compromises
regarding the purity of at least some fifths and fourths, and could be
considered as a bit of tasty icing on the intonational cake.

By sometime around the early to middle 14th century, keyboards were
standardizing on a 12-note layout. A tuning set such as Eb-G# nicely
accommodates most of the vocal repertory of the era, and the earliest
known keyboard music from the Robertsbridge Codex (with proposed dates
ranging from around 1325 to 1365) beautifully fits this complete
12-note range.

An attraction of 12 is that it is an MOS, with visual symmetry on a
keyboard: each of the five whole-tone steps in a diatonic scale is
divided into two semitones: the diatonic or minor semitone at 256:243
(~90.22 cents) also present at E-F and B-C, and the chromatic or major
semitone at 2187:2048 (~113.69 cents). Thus there are two adjacent
step sizes: the diatonic and chromatic semitones.

However, by the late 14th century, some composers were not only using
the full resources of 12 but going beyond it: Solage, in his famous
_Fumeux fume_, uses a 15-note set of Gb-G#.

Two Italian theorists of the early 15th century, Prosdocimus of
Beldemandis and Ugolino of Orvieto, propose a 17-note Pythagorean
tuning of Gb-A# in order to accommodate closest approach cadences
with regular Pythagorean thirds and sixths on a maximum number of
degrees, with the latter advocating a 17-note organ permitting a
discerning player to achieve a perfect intonation of these cadences.

Like 7 and 12, 17 is an MOS in Pythagorean tuning, with two sizes of
adjacent steps shown in the division of whole-tones (e.g. C-Db-C#-D),
the diatonic semitone (C-Db, C#-D) and the Pythagorean comma at a
ratio of 531441:524288 (~23.46 cents), for example Db-C#.

An important point about the 14th-century practice I'm discussing -- a
"classic" point of departure for the 21st-century approach in which
I'm involved -- is that beautiful vertical progressions are not
confined to a given "mode" of 7 or 8 notes:

D4 C#4 D4
A3 G#3 A3
F3 E3 D3

This progression has a clear and compelling focus of D, but the
sonorities are not all drawn from any 7-note or 8-note diatonic
set. The fluidity of degrees, and flexibility of inflections, is
an important feature both of this historical music and of 21st-century
offshoots in tuning systems such as a 17-note circle or quasi-circle.

Of course, we might well view such a passage as expressing a kind of
"mode of D," with inflections to achieve the harmonic goal of closest
approach, and I have no problem in saying that I am playing at once in
"D Dorian" (very freely defined) and in "a focality of D intensive"
(meaning that a final cadence like the above involves ascending
semitones and descending whole-tones).

An important point is that either the accidentalism of the 14th
century or the chromaticism and enharmonicism of the late 16th century
is much less restrictive than a classical 18th-century key system. All
three approaches, and their 21st-century variations, reflect certain
artistic choices.

Let's now consider the order of events for some developments raised in
earlier dialogue.

During the early and middle 15th century, there is a trend toward the
use of thirds and sixths at or near ratios of 5 -- as reflected and
promoted first by the modified Pythagoraen tunings of around 1400-1450
with written sharps realized as Pythagorean flats (specifically the
12-note keyboard tuning of Gb-B), and then, by around 1450-1480, in
the growing adoption of meantone temperament.

By around 1500, composers such as Josquin and Isaac are starting
frequently to conclude pieces with sonorities including thirds, and by
the 1520's, a theorist such as Pietro Aaron is noting a preference
specifically for the major third at certain points, using accidentals
to obtain this interval above the lowest part in closing sonorities.

Vicentino (1555) and Zarlino (1558) both advocate the use of "rich" or
"perfect" harmony including the third plus the fifth above the lowest
voice -- Zarlino's _harmonia perfetta_ which he considers basic to a
well-organized composition for three or more voices. In his writings
of 1610 and 1612, Johannes Lippius extols this sonority as the _trias
harmonica_, or "harmonic triad."

Here it should be noted that both 7-note and 12-note sets, and indeed
the 17-note set of Prosdocimus and Ugolino, were recognized in a
Pythagorean setting; the concept of saturated 5-limit concord arises,
not surprisingly, in an intonational environment of meantone. Zarlino,
a leading theorist of meantone temperaments for keyboard, suggests
some kind of adaptive 5-limit just intonation is being practiced by
singers or other performers in flexible-pitch ensembles when he
observes that they seem to seek pure concords while avoiding the
problems with "least intervals" such as commas which arise on
keyboards in the syntonic diatonic.

In a meantone system, MOS sizes are 12, 19, or 31 notes, and in fact
the equal or near-equal 31-note keyboards instruments of Vicentino
(1555), and of Stella and Colonna documented in the latter's treatise
(1618), as well as the keyboard of Costeley based on 19-note equal
temperament (1570), illustrate the latter sizes for a circulating
system.

However, while Colonna includes in his treatise an "example of
circulation" cadencing on all 31 steps of his instrument by
circumnavigating the circle of fifths, circulation is not a theme of
most Renaissance and Manneristic music. Thus 19-note keyboards tuned
in 1/4-comma meantone or the like (where 31 notes would be required
for a circulating system) were popular in Naples around 1600.

Additionally, various keyboards had a usual 12-note design enhanced by
"split keys" for certain additional meantone accidentals: the
accidental keys might be divided so that front and back portions
produce G#/Ab and Eb/D#.

As already mentioned, a vital feature of music around 1600, whether
routine or highly "experimental," is the flexible use of accidentals,
with the fluidity of steps such as B/Bb or C/C# routine.

Thus I might explain the late 17th-century practice of major/minor
tonality, codified in the early 18th century by theorists such as
Rameau, as reflecting a certain choice of materials -- in some ways a
considerably more restricted choice than the chromaticism of the
Manneristic era around 1540-1640, let alone the fifthtone
enharmonicism of a Vicentino or Colonna.

As Paul and Jerry have discussed, a central feature of major/minor
tonality is the role of the tritone or diminished fifth as an interval
resolving by stepwise contrary motion to a third or sixth. This kind
of resolution, routine in 16th-century practice and theory, becomes
the defining element of key.

An important point about 14th-century accidentalism, as well as
16th-century chromaticism or 18th-century tonality, is that these
practices are defined as much by characteristic vertical progressions
as by a choice of "scales."

What happens intonationally around 1680, as others have noted, is that
the desire for free transpositions _with a limited number of notes_
leads to the popularity of 12-note well-temperaments such as
Werckmeister's.

Obviously a 31-note meantone system such as that of Vicentino or
Colonna in the Manneristic era, or of Huygens in this same late
17th-century era, would permit free transpositions also -- with such
added possibilities as closely approximated ratios of 7, and various
neutral types of intervals.

One lesson of world musics is that there are a wide range of natural
and beautiful solutions, with Western Europe in the 14th or 18th
century merely as one example.

Trying to articulate some of the "explanations" for intonational
choices may help us better to understand not only the roads taken, but
the roads not taken and open to fruitful exploration.

Of course, an acquaintance with the natural and beautiful scalar and
intonational systems of a range of world musics, for example the many
neutral intervals of medieval or modern Near Eastern music and the
slendro and pelog scales of Javanese or Balinese gamelan, brings home
the point that there are many artistically compelling choices.

Most appreciatively,

Margo Schulter
mschulter@value.net

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

/tuning/topicId_35909.html#35909

>>
> I would be interested in reading any comments that you might have.
>
> regards,
> Jeff

Hello Jeff!

Finally I had a chance to study through your page, and it was really
interesting for me to see it from the perspective you were driving
at. Although it might sometimes seem a little like "reinventing the
wheel" it was still refreshing, and the applets were truly terrific,
as I mentioned before.

I was a little troubled by some of the *implied* conclusions of this
method, such as what I perceived to be an "implied" superiority of
harmony over melody, such as in this statement:

"Definition: A scale or key is the set of individual notes taken from
a related family of chords. Thus a melody can wander around in this
set of tones and pleasantly harmonize with the chords, or another
melodic line. Of course, the scale we just defined also allows for
the dissonant tension and resolution of a cadence. Remark: This is
what we propose as the basic definition of a musical scale."

Historically it was, I believe, the other way around, with chords
being derived from *melodies.* So the *melody* was really the "a
priori" and the harmonies gradually evolved from that.

At least, that's *my* understanding, and I believe it is significant
to understanding a scale. After all, many cultures use scales
without any harmonies at all!

It seemed to me that your puzzlement on why certain notes like C# and
F# were more "dissonant" to C than notes like F and G was a little on
the "baroque" side, and I don't mean the *period!*

I believe a simple reference to the circle of fifths and the
increasingly complicated ratios of the "black notes" would
easily "solve" this riddle without a necessary reference to sensory
consonance curves although, admittedly, this was interesting [and the
applets, again, the best part! :) ]

Finally, since you cited the Backus book, you might want to look on
page 126 where he presents a just intonation chromatic scale in a
somewhat similar way to the one *you* presented, maybe with a couple
of different conventions.

His "set up" uses a just major third between D-F#, and just minor
triads on C and G and just major triads on E and A. He shows the
varying syntonic comma differences from Pythagorean as plusses and
minuses as such:

C0, C#-2, D0, Eb+1, E-1, F0, F#-1, G0, [G#-2,Ab+1], A-1, Bb+1, B-1, C0

Of course, one of the big "bugbears" is the two choices between G#
and Ab but, as you mention yourself, even the fifth D0:A-1 is "out of
tune" in this setup.

By the way, these notations correspond pretty much to the kind of
notation used by American Just Intonation composer Ben Johnston.

He uses plusses and minuses for commas in this way.

HOWEVER, these are used in a *more consistent* way by Joe Monzo, Dan
Wolf in a notation called HEWM, based upon a *Pythagorean* derivation.

What that means is that a simple just intonation triad based on C
would be written C0:E-1:G0, showing the comma right "off the bat" in
reference to the scale degrees shown on our traditional staff, which
is descended from Pythagorean tuning.

The *minor* in this case would be the "reverse" as far as the commas
go, and would be more consistent in HEWM as well, being shown as:

E-1:G0:B-1

The fifths in the HEWM system, would always have the same sign, which
is important for consistency's sake.

More information on this interesting system partially developed by
Joe Monzo can be found here:

http://www.ixpres.com/interval/dict/hewm.htm

Well... Joe *tried* to make it understandable. It's just *very,
very* deep...:)

Oh, and while I'm on the topic of Joe's very substantial
contributions, you might want to listen to a page where he tunes up
various just intonation chords in a simple I-IV-V7-I progression.

This is one of my *own* very favorite pages of all time.

I think theory teachers should be *required* to listen to this page
before they enter a classroom! (and repeatedly!)

http://www.ixpres.com/interval/td/monzo/i-iv-v7-i/I-IV-V-I.htm

Thanks so much again, Jeff, for your interesting contribution and
point of view!

Joseph Pehrson

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

/tuning/topicId_35909.html#35952

> Hi Joe, thanks for your reply.
>
>
> --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> > The title of your post refers to the derivation of
the "classical"
> > major and minor scales, not *your* diatonic major and minor
scales.
> >
> > If you want to consider the "classical" diatonic collection, how
can
> > you possibly ignore the history and developments that caused it
to
> > come about?
>
> Because these scales exists today, right now. We can explore
> certain properties that they have, like consonant chords,
> leading tones, and so on. Even if I invented a scale, you could
> still play music with it and study its properties without knowing
> anything about me.
>
> Now, I fully agree that studying the history of music adds a
> great deal of insight! I have done a little reading along
> those lines. But I want to keep this particular online paper
> focused on the non-historical aspects.
>
> Maybe you are interpreting the word "classical" differently
> than I meant it. Maybe I should have said "the diatonic
> major and minor scales that are most commonly used today in
> America" but that is a mouthful...
>
> --Jeff

***Hi Jeff!

Actually, the lack of historical perspective didn't bother me so much
after I bothered to actually *read* the article carefully!

jp

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
>
> /tuning/topicId_35909.html#35941
>
>
> > --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> >
> > > I see the words "the standard Western musical scales, and to
make
> > the
> > > derivation..."
> > >
> > > How are you going to do that without referencing history??
Don't
> > get
> > > it.
> >
> > well, as an example, in the second part of my 'gentle
introduction
> to
> > periodicity blocks'
> >
> > http://www.ixpres.com/interval/td/erlich/intropblock2.htm
> >
> > i derive the diatonic scale *ex nihilo*, as it were -- history
> plays no role.
>
>
> ***Hi Paul!
>
> But isn't that a bit like "reverse engineering" where you know the
> result you want to get to and just work backwards?

i would say that something like this could be true of balzano's
theory and explanation of the diatonic scale. but the principles
behind periodicity blocks, i feel, are very powerful, and of rather
universal validity. note that other periodicity blocks are possible,
besides 7- and 12-tone ones (remember all the 'zoos' i posted)?

> Would the diatonic "12"

what's that?

> have any significance to anybody if it
> weren't for the historical practice??

the point is to try to explain why certain alternatives were heavily
favored, even in disparate cultures, among the vast array of possible
scales one could imagine.

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:

/tuning/topicId_35909.html#35967

> --- In tuning@y..., Gerald Eskelin <stg3music@e...> wrote:
>
> /tuning/topicId_35909.html#35948
>
> >>
> > Joseph, you certainly bring a refreshing flair to the tuning
list.
> I enjoy your sense of humor, not to mention your musical insights.
> >
> > In my opinion, one should beware of swallowing "history"
> uncritically. I
>
>
> ****Hi Jerry.
>
> Well, *personally* I believe this list can easily use
> my "personality...", but I'm sure it also annoys some people. :)
>
> In all seriousness, though, I find the idea of the past-present-
> future timeline a bit mystifying. If it's all somewhat the same
> thing, certainly there is more *certainty* on the earlier end!
>
> However, if it's a timeline, like a number line, it means that the
> present is infinitely small, no??
>
> What exactly *is* the present, anyway? It must be some incredibly
> small number.
>
> Definitely "microtonal..." :)
>
> jp

***This is actually verging on a "metatuning" subject, but, perhaps
the "present" could be defined as the instant we perceive something...

??

jp

--- In tuning@y..., "M. Schulter" <MSCHULTER@V...> wrote:

/tuning/topicId_35909.html#35971

> Hello, there, everyone, and please let me begin by thanking Paul for
> inviting me to join in this discussion.

***Many thanks to Margo Schulter for her contribution to the
question "why 12??"

I promptly saved that one...

>>
> Note that this explanation, as applied either to a 17-note
> well-temperament or equal temperament, or to a more eccentric "17-
note orbit" where each whole-tone is divided into three diatonic
semitones but some fifths might be up to a 64:63 comma wider than
3:2, assumes certain musical and stylistic patterns.
>

*** I really like the idea of calling a non-equal system within the
octave an "orbit."

Calling Monz.

Maybe something for the dictionary??

Joe P.

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

/tuning/topicId_35909.html#35978

> >
> > ***Hi Paul!
> >
> > But isn't that a bit like "reverse engineering" where you know
the
> > result you want to get to and just work backwards?
>
> i would say that something like this could be true of balzano's
> theory and explanation of the diatonic scale. but the principles
> behind periodicity blocks, i feel, are very powerful, and of rather
> universal validity. note that other periodicity blocks are
possible, besides 7- and 12-tone ones (remember all the 'zoos' i
posted)?

***Hi Paul.

I read this over again, and I admit, there is nothing "a priori"
about it! The method you describe really *does* generate those 12
just pitches.

>
> > Would the diatonic "12"
>
> what's that?
>

***Sorry, I just meant the 12 just pitches within the parallelogram.

> > have any significance to anybody if it
> > weren't for the historical practice??
>
> the point is to try to explain why certain alternatives were
heavily favored, even in disparate cultures, among the vast array of
possible scales one could imagine.

***Yes, the method works, and your presentation seems much more
sophisticated than just "stacking up" just triads...

jp

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> --- In tuning@y..., "M. Schulter" <MSCHULTER@V...> wrote:
>
> /tuning/topicId_35909.html#35971
>
>
> > Hello, there, everyone, and please let me begin by thanking Paul
for
> > inviting me to join in this discussion.
>
> ***Many thanks to Margo Schulter for her contribution to the
> question "why 12??"
>
> I promptly saved that one...
>
> >>
> > Note that this explanation, as applied either to a 17-note
> > well-temperament or equal temperament, or to a more eccentric "17-
> note orbit" where each whole-tone is divided into three diatonic
> semitones but some fifths might be up to a 64:63 comma wider than
> 3:2, assumes certain musical and stylistic patterns.
> >
>
> *** I really like the idea of calling a non-equal system within the
> octave an "orbit."

i have a feeling margo meant something a little more specific than
any non-equal system within the octave . . .

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

> i would say that something like this could be true of balzano's
> theory and explanation of the diatonic scale. but the principles
> behind periodicity blocks, i feel, are very powerful, and of rather
> universal validity.

The periodicity block business, in suitable generality, ends up saying that scales tend to be convex, which seems reasonable.

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:

> ***This is actually verging on a "metatuning" subject, but, perhaps
> the "present" could be defined as the instant we perceive something...

I've percieved things for years, so this won't work. I think this is philosophy, and belongs in metatuning.

On 3/29/02 7:27 PM, "tuning@yahoogroups.com" <tuning@yahoogroups.com> wrote:

> Message: 5
> Date: Fri, 29 Mar 2002 17:44:40 -0000
> From: "jpehrson2" <jpehrson@rcn.com>
> Subject: Re: deriving classical major/minor scales
>
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
>
> /tuning/topicId_35909.html#35941
>
>
>> --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
>>
>>> I see the words "the standard Western musical scales, and to make the
>>> derivation..."
>>>
>>> How are you going to do that without referencing history?? Don't get
>>> it.
>>
>> well, as an example, in the second part of my 'gentle introduction to
>> periodicity blocks'
>>
>> http://www.ixpres.com/interval/td/erlich/intropblock2.htm
>>
>> i derive the diatonic scale *ex nihilo*, as it were -- history plays no role.
>
>
> ***Hi Paul!
>
> But isn't that a bit like "reverse engineering" where you know the
> result you want to get to and just work backwards?
>
> Would the diatonic "12" have any significance to anybody if it
> weren't for the historical practice??
>
> jp

(Pardon the intrusion, Joe.)

Keep in mind that historical practice is based on the same raw acoustical
input we "hear" today. Consider pentatonic tradition (12 tones minus the
tritone), modal tradition (12 tones including the tritone with warning about
its problems) and major/minor tradition (finally celebrating the tritone).
Suppose we had no historical record available to us. Would we "discover" the
same pentatonic, modal and functional characteristics of *natural* sound?
It's not out of the realm of possibility, I would think. Of course,
historical practice helps; but that may not totally account for the reason
we hear what we hear today.

Jerry

On 3/29/02 7:27 PM, "tuning@yahoogroups.com" <tuning@yahoogroups.com> wrote:

> Message: 6
> Date: Fri, 29 Mar 2002 18:01:04 -0000
> From: "jpehrson2" <jpehrson@rcn.com>
> Subject: Re: deriving classical major/minor scales
>
> --- In tuning@y..., Gerald Eskelin <stg3music@e...> wrote:
>
> /tuning/topicId_35909.html#35948
>
>> Joseph, you certainly bring a refreshing flair to the tuning list. I enjoy
>>your sense of humor, not to mention your musical insights.
>>
>> In my opinion, one should beware of swallowing "history" uncritically. I
>
>
> ****Hi Jerry.
>
> Well, *personally* I believe this list can easily use
> my "personality...", but I'm sure it also annoys some people. :)
>
> In all seriousness, though, I find the idea of the past-present-
> future timeline a bit mystifying. If it's all somewhat the same
> thing, certainly there is more *certainty* on the earlier end!

Some people (including me) view *certainty* as dangerous. (Look at the
middle east.) We now can see that the "certainty" of earlier writers have
often proven inadequate in terms of understanding our world.
>
> However, if it's a timeline, like a number line, it means that the
> present is infinitely small, no??

Huh? Turn your timeline around, man. The numbers get *bigger* the farther
you go. I think I know more than my dad did (in some areas, at least) and my
kids will know more than I do (in a lot more areas than I even know are
areas to know something about). What they do with it is another question, of
course.
>
> What exactly *is* the present, anyway? It must be some incredibly
> small number.

Joe, I think we're going the opposite direction on the escalators of life.
:-)
>
> Definitely "microtonal..." :)

Oh, my god! We *are* in trouble.
>
> jp
>
gre

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
>
> > i would say that something like this could be true of balzano's
> > theory and explanation of the diatonic scale. but the principles
> > behind periodicity blocks, i feel, are very powerful, and of
rather
> > universal validity.
>
> The periodicity block business, in suitable generality, ends up
>saying that scales tend to be convex, which seems reasonable.

is that what it ends up saying? lots of periodicity blocks are not
convex, and lots of convex scales are not periodicity blocks . . .

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

> is that what it ends up saying? lots of periodicity blocks are not
> convex, and lots of convex scales are not periodicity blocks . . .

Maybe you should migrate over to tuning math and give some examples; this sounds as if we are not on the same page here.

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
>
> > is that what it ends up saying? lots of periodicity blocks are
not
> > convex, and lots of convex scales are not periodicity blocks . . .
>
> Maybe you should migrate over to tuning math and give some
>examples; this sounds as if we are not on the same page here.

this is not terribly advanced mathematics. for example: draw a circle
randomly in the just lattice. all the notes inside that circle form a
convex scale. but do they form a periodicity block (with reasonably
small unison vectors)? probably not.

the melodic minor scale is an example of a non-convex periodicity
block (at least some ji renditions of it).

--- In tuning@y..., "clumma" <carl@l...> wrote:
> >I printed your "Diatonicity in a nutshell"

You know, I think it would be great if "diatonicity in a nutshell"
had a one or two sentence definition of phrases like
"Rothenberg efficient". ( I will check out Joe Monzo's dictionary).

>
> >Where does Sethares discuss it?
>
> In his book. It may also be reprinted on his web page.

I can't quite find where...

> And Graham has a fantastic page at:
>
> http://x31eq.com/
>
I am planning to read the article about meantone ASAP.

--Jeff

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "jjensen142000" <jjensen14@h...> wrote:
>
> > Harmonics ==> JI,
>
> This should be Harmonics ==> things are not too awfully out of tune.
>

I don't quite understand that. Chords ( 3 tone chords anyway)
constructed from harmonics
should be as "in tune" as you can get. Are you talking about
inharmonicity due to real strings with stiffness?

> > JI + (uncontrollable urge to change keys) ==> 12tet
>
> This should be "meantone + (uncontrollable urge to change keys) ==>
> 12, 19, 31, 50, 55-et among other possibilities".

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:

> ***Yes, the method works, and your presentation seems much more
> sophisticated than just "stacking up" just triads...

thanks joseph. i hope jeff (jjensen142000) will see this and 'bone
up' on periodicity blocks. i think fokker had a great idea with them
(though he didn't mention that the diatonic scale is a periodicity
block, as far as i know), and i thank paul hahn for telling me about
them (i don't recall seeing them mentioned in rasch's fokker book)
and joe monzo for inspiring me to look into them more deeply. it
seems, though, that gene may have recognized their significance quite
a long time ago, in the guise of the epimorphic property . . .

Hi Margo.

Thank you very much for the detailed exposition.
I had almost no idea of what happened between Pythagoras
and J.S. Bach!

regards,
Jeff

Hi Joe,

>Although it might sometimes seem a little like "reinventing the
> wheel"

I was perhaps trying to invent it for myself for the first time!

> I was a little troubled by some of the *implied* conclusions of
this
> method, such as what I perceived to be an "implied" superiority of
> harmony over melody, such as in this statement:
>
> "Definition: A scale or key is the set of individual notes taken
from
> a related family of chords. Thus a melody can wander around in this
> set of tones and pleasantly harmonize with the chords, or another
> melodic line. Of course, the scale we just defined also allows for
> the dissonant tension and resolution of a cadence. Remark: This is
> what we propose as the basic definition of a musical scale."

No, I'm saying they work together.

Rhetorical question: What actually *is* the definition
of a scale? Most generally, I guess an arbitrary collection
of tones? I'm not sure I'd find that useful. There probably isn't
an agreed upon definition...

> Historically it was, I believe, the other way around, with chords
> being derived from *melodies.* So the *melody* was really the "a
> priori" and the harmonies gradually evolved from that.
>
I agree with that.

> It seemed to me that your puzzlement on why certain notes like C#
and
> F# were more "dissonant" to C than notes like F and G was a little
on
> the "baroque" side, and I don't mean the *period!*
>
> I believe a simple reference to the circle of fifths and the
> increasingly complicated ratios of the "black notes" would
> easily "solve" this riddle without a necessary reference to sensory
> consonance curves although, admittedly, this was interesting [and
the
> applets, again, the best part! :) ]

We may have to agree to disagree on this... I'd argue that
the ratios are considered consonant only because of the
sensory phenomena. Otherwise, how does the human ear know
about fractions, etc...?

>
> Finally, since you cited the Backus book, you might want to look on
> page 126 where he presents a just intonation chromatic scale in a
> somewhat similar way to the one *you* presented, maybe with a
couple
> of different conventions.
>

Yes, I see that. I think different conventions are going to cause
me worry...

--JEff

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
>
> > ***Yes, the method works, and your presentation seems much more
> > sophisticated than just "stacking up" just triads...
>
> thanks joseph. i hope jeff (jjensen142000) will see this and 'bone
> up' on periodicity blocks. i think fokker had a great idea with
them
> (though he didn't mention that the diatonic scale is a periodicity
> block, as far as i know)

Hi Paul

Yes, I'm planning to read the "Gentle Introduction" article,
but it is going to take some time to get to all these
things...

Jeff

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

> this is not terribly advanced mathematics. for example: draw a circle
> randomly in the just lattice. all the notes inside that circle form a
> convex scale. but do they form a periodicity block (with reasonably
> small unison vectors)? probably not.

Sorry; what I meant was that epimorphic+convex <==> block so far as I can see, because from convexity you can construct an appropriate metric.

> the melodic minor scale is an example of a non-convex periodicity
> block (at least some ji renditions of it).

Eh? What's non-convex about it--where is the element of the convex hull which is not already in the scale?

--- In tuning@y..., "jjensen142000" <jjensen14@h...> wrote:
> --- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> > --- In tuning@y..., "jjensen142000" <jjensen14@h...> wrote:
> >
> > > Harmonics ==> JI,
> >
> > This should be Harmonics ==> things are not too awfully out of tune.

> I don't quite understand that. Chords ( 3 tone chords anyway)
> constructed from harmonics
> should be as "in tune" as you can get.

That certainly does not describe 12-et, so you are never going to be able to derive 12-et from this starting point. You either get rational intonation or at least something closer to it than 12-et, which you may or may not want to call JI.

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

it
> seems, though, that gene may have recognized their significance quite
> a long time ago, in the guise of the epimorphic property . . .

I had noticed paralleopided fundamental regions; I didn't attach too much significance to them, and missed the fun games you were playing by using the paralleopided fundamental regions (Fokker blocks) to construct scales. Your hypothesis was way of out of left field for me, partly because I looked at it the other way around, starting from mappings. I still think convex fundamental regions are a less specialized, and hence better, approach; and am hoping we can agree on a definition of a block which involves a convex fundamental region.
It would subsume "Paul" and "Fokker" blocks.

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

> Rhetorical question: What actually *is* the definition
> of a scale? Most generally, I guess an arbitrary collection
> of tones? I'm not sure I'd find that useful. There probably isn't
> an agreed upon definition...

I proposed a definition which everyone hated:

Scale

A discrete set of real numbers, containing 0 and regarded as representing intervals logarithmically (e.g., in terms of cents), and such that the distance between sucessive elements of the scale is bounded both below and above by positive real numbers. The least upper bound of the intervals between successive elements of the scale is the maximum scale step, and the greatest lower bound is the minimum scale step. The element of the scale obtained by counting up n scale steps is the nth degree, by counting down is the -nth degree; 0 is the 0th degree.

--- In tuning@y..., Gerald Eskelin <stg3music@e...> wrote:

/tuning/topicId_35909.html#35995

> On 3/29/02 7:27 PM, "tuning@y..." <tuning@y...> wrote:
>
> > Message: 6
> > Date: Fri, 29 Mar 2002 18:01:04 -0000
> > From: "jpehrson2" <jpehrson@r...>
> > Subject: Re: deriving classical major/minor scales
> >
> > --- In tuning@y..., Gerald Eskelin <stg3music@e...> wrote:
> >
> > /tuning/topicId_35909.html#35948
> >
> >> Joseph, you certainly bring a refreshing flair to the tuning
list. I enjoy
> >>your sense of humor, not to mention your musical insights.
> >>
> >> In my opinion, one should beware of swallowing "history"
uncritically. I
> >
> >
> > ****Hi Jerry.
> >
> > Well, *personally* I believe this list can easily use
> > my "personality...", but I'm sure it also annoys some people. :)
> >
> > In all seriousness, though, I find the idea of the past-present-
> > future timeline a bit mystifying. If it's all somewhat the same
> > thing, certainly there is more *certainty* on the earlier end!
>
> Some people (including me) view *certainty* as dangerous. (Look at
the
> middle east.) We now can see that the "certainty" of earlier
writers have
> often proven inadequate in terms of understanding our world.
> >
> > However, if it's a timeline, like a number line, it means that the
> > present is infinitely small, no??
>
> Huh? Turn your timeline around, man. The numbers get *bigger* the
farther
> you go. I think I know more than my dad did (in some areas, at
least) and my
> kids will know more than I do (in a lot more areas than I even know
are
> areas to know something about). What they do with it is another
question, of
> course.
> >
> > What exactly *is* the present, anyway? It must be some incredibly
> > small number.
>
> Joe, I think we're going the opposite direction on the escalators
of life.
> :-)
> >
> > Definitely "microtonal..." :)
>
> Oh, my god! We *are* in trouble.
> >
> > jp
> >
> gre

***I'm going to have to continue these discussion, as Gene suggests,
on *Metatuning.* I wish there was more of a *scientific* basis and
study of such things. Probably people have discussed them, and I
just don't know the references!

jp

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

/tuning/topicId_35909.html#36006

> Hi Joe,
>
> >Although it might sometimes seem a little like "reinventing the
> > wheel"
>
> I was perhaps trying to invent it for myself for the first time!
>

****Quite frankly, Jeff, I see nothing wrong with that. In fact, it
might lead to differing insights from the expected, so it may even be
best in some cases. However, it does expose one to possible
criticism.

The more "accepted" way of proceeding is to study up on a topic first
and *then* make commentary. :) However, like I said, your approach
can yield results that the other way maybe never will, in some cases.

> > I was a little troubled by some of the *implied* conclusions of
> this method, such as what I perceived to be an "implied"
superiority of harmony over melody, such as in this statement:
> >
> > "Definition: A scale or key is the set of individual notes taken
> from a related family of chords. Thus a melody can wander around in
this set of tones and pleasantly harmonize with the chords, or
another melodic line. Of course, the scale we just defined also
allows for the dissonant tension and resolution of a cadence. Remark:
This is what we propose as the basic definition of a musical scale."
>
> No, I'm saying they work together.
>

***Right. But the *derivation* you present suggests a "harmony
first" attitude, at least it did to me...

> Rhetorical question: What actually *is* the definition
> of a scale? Most generally, I guess an arbitrary collection
> of tones? I'm not sure I'd find that useful. There probably isn't
> an agreed upon definition...
>

***I'm sure Somebody has come up with one, and my guess is that it
would differ substantially from *your* derivation, dunno...

Calling Somebody?

>
>
> > Historically it was, I believe, the other way around, with chords
> > being derived from *melodies.* So the *melody* was really the "a
> > priori" and the harmonies gradually evolved from that.
> >
> I agree with that.
>

***That's not implied at all from your discussion, Jeff, which is
decidedly "harmonics first" and "harmony first." I'm not going to
lose any sleep over this, though... :)

>
>
> > It seemed to me that your puzzlement on why certain notes like C#
> and F# were more "dissonant" to C than notes like F and G was a
little n
> > the "baroque" side, and I don't mean the *period!*
> >
> > I believe a simple reference to the circle of fifths and the
> > increasingly complicated ratios of the "black notes" would
> > easily "solve" this riddle without a necessary reference to
sensory consonance curves although, admittedly, this was interesting
[and the
> > applets, again, the best part! :) ]
>
> We may have to agree to disagree on this... I'd argue that
> the ratios are considered consonant only because of the
> sensory phenomena. Otherwise, how does the human ear know
> about fractions, etc...?
>

***Well, maybe you're right on this point. Am I allowed to change my
mind? I think I'll do that... :)

>
> >
> > Finally, since you cited the Backus book, you might want to look
on page 126 where he presents a just intonation chromatic scale in a
> > somewhat similar way to the one *you* presented, maybe with a
> couple of different conventions.
> >
>
> Yes, I see that. I think different conventions are going to cause
> me worry...
>

****I believe that's a big problem with Just Intonation on the
overall, if I understand it correctly.

However, I would prefer not to offend my many friends who are JI
Fanatics... :)

Joe

>You know, I think it would be great if "diatonicity in a nutshell"
>had a one or two sentence definition of phrases like
>"Rothenberg efficient". ( I will check out Joe Monzo's dictionary).

I'd like to write a facing document that would explain each of
the metrics and why I think they do what they do. I don't feel
like doing that at the momement, but I'll put it on the list.

>>>Where does Sethares discuss it?
>>
>>In his book. It may also be reprinted on his web page.
>
>I can't quite find where...

Can't find where in the book, or on the site? I don't have my
copy of the book handy, I'm afraid.

-Carl

>though, that gene may have recognized their significance quite
>a long time ago, in the guise of the epimorphic property . . .

Does epimorphic = block?

-Carl

>I proposed a definition which everyone hated:
>
>Scale
>
>A discrete set of real numbers, containing 0 and regarded as representing
>intervals logarithmically (e.g., in terms of cents), and such that the
>distance between sucessive elements of the scale is bounded both below and
>above by positive real numbers. The least upper bound of the intervals
>between successive elements of the scale is the maximum scale step, and
>the greatest lower bound is the minimum scale step. The element of the
>scale obtained by counting up n scale steps is the nth degree, by counting
>down is the -nth degree; 0 is the 0th degree.

I don't hate this. I think it's good. But unless you go on to make
use of this kind of detail... also, many people write scales as
frequency, not log-frequency, so you're proposing they change what
they're doing or stop calling them scales... howabout:

Scale

Any ordered set of pitches, expressed in units of frequency or log-
frequency, that are intended to describe the output of a musical
instrument.

-Carl

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:

> Sorry; what I meant was that epimorphic+convex <==> block

aha, that's very different!

> > the melodic minor scale is an example of a non-convex periodicity
> > block (at least some ji renditions of it).
>
> Eh? What's non-convex about it--where is the element of the convex
>hull which is not already in the scale?

in F melodic minor in the following ji rendition . . .

.........E
......../.\
......./...\
F-----C-----G-----D
.\.../.......\.../
..\./.........\./
...Ab..........Bb

. . . Eb is in the convex hull but not already in the scale.

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:

> and am hoping we can agree on a definition of a block which
>involves a convex fundamental region.

i don't think i objected to your last attempt in this regard on
tuning-math.

--- In tuning@y..., Carl Lumma <carl@l...> wrote:
> >though, that gene may have recognized their significance quite
> >a long time ago, in the guise of the epimorphic property . . .
>
> Does epimorphic = block?
>
> -Carl

well, an epimorphic scale is a periodicity block in the most general
sense, where you're allowed to transpose all the notes by arbitrary
combinations of unison vectors, even to the point where the whole
thing becomes disconnected in the lattice. gene would like to make
the definition of 'block' more restrictive, by requiring a 'block' to
be convex as well as epimorphic . . . i don't think that's
unreasonable.

Carl Lumma wrote:

> >>>Where does Sethares discuss it?
> >>
> >>In his book. It may also be reprinted on his web page.
> >
> >I can't quite find where...
>
> Can't find where in the book, or on the site? I don't have my
> copy of the book handy, I'm afraid.

Minimising pairwise dissonance? Chapters 5 to 7 seem to cover it. See
also 3.6 for meantones, and 9.2 (p.200) "Reconstruction of Historical
Tunings".

Graham

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

> in F melodic minor in the following ji rendition . . .

Sorry, I was thinking of harmonic minor. The scale below isn't convex, of course. It isn't a block either, is it?

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
>
> > in F melodic minor in the following ji rendition . . .
>
> Sorry, I was thinking of harmonic minor. The scale below isn't
convex, of course. It isn't a block either, is it?

well, it is a periodicity block (as i define it), as it tiles the
lattice with translations by 25:24 and 81:80.

-paul

>>Does epimorphic = block?
>
>well, an epimorphic scale is a periodicity block in the most general
>sense, where you're allowed to transpose all the notes by arbitrary
>combinations of unison vectors, even to the point where the whole
>thing becomes disconnected in the lattice. gene would like to make
>the definition of 'block' more restrictive, by requiring a 'block' to
>be convex as well as epimorphic . . . i don't think that's
>unreasonable.

Thanks, I understand now. And I agree that adding convexivity sounds
reasonable. You can still transpose by uvs, but you can view it as
borrowing from neighboring copies of the convex version.

-Carl

>The periodicity block business, in suitable generality, ends
>up saying that scales tend to be convex, which seems reasonable.

Gene,

Will this suitable generality be covered in your paper? Or
would you care to explain it on these lists?

-Carl

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning@y..., Carl Lumma <carl@l...> wrote:
>
> > Given the constraints of a melodic-harmonic music, as you take
> > for granted in your paper, however, a number of people have been
> > able to derrive the diatonic scale and/or meantone temperament
> > by minimizing pairwise (dyadic) dissonance (Erlich, Sethares),
>
> note that i pretty much got a meantone-tempered diatonic scale with
> this procedure.

How exactly do I find this?

More generally, I'm not having much luck
searching the group archives either...the titles of postings are
not very descriptive, and no ordering by relevance...

I can't find the derivation in Sethare's book either by thumbing
through the pages or by the index. I assert: It's not there!

>
> > by
> > maximizing connectivity by dyads (Gene Smith), over 7-tone scales.
> > There's clearly something to this, but derrivation does equal a
> > complete understanding where I'm from, nor does it necessarily
> > mean you can repeat your procedure for (say) 8-tone scales and get
> > something 'just as good' as the diatonic scale.
>
> and don't forget the periodicity block derivation of the diatonic
> scale:
>
> http://www.ixpres.com/interval/td/erlich/intropblock1.htm
>
> http://www.ixpres.com/interval/td/erlich/intropblock2.htm

I have printed these, and I am going to read them this week.

Jeff

--- In tuning@y..., "jjensen142000" <jjensen14@h...> wrote:
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
> > --- In tuning@y..., Carl Lumma <carl@l...> wrote:
> >
> > > Given the constraints of a melodic-harmonic music, as you
take
> > > for granted in your paper, however, a number of people
have been
> > > able to derrive the diatonic scale and/or meantone
temperament
> > > by minimizing pairwise (dyadic) dissonance (Erlich,
Sethares),
> >
> > note that i pretty much got a meantone-tempered diatonic
scale with
> > this procedure.
>
> How exactly do I find this?

hi jeff! paul here. i am away from my office right now, but when i
get back (probably tomorrow), i will be more than happy to
reproduce this from scratch, in case you still can't find this in the
archives from a couple of years ago. basically, i have a dyadic
dissonance model (called harmonic entropy -- there's a group
about it called harmonic_entropy@yahoogroups.com). then i can
create a 21-dimensional function [(7*6)/2] representing the total
dyadic dissonance of a 7-tone scale. i can also find local minima
of this function using matlab. what i did was to start the 'ball' on
the 7-equal 'hill', and let it 'roll down' -- what i end up with is the
approximately meantone-tempered diatonic scale. i may have
tried other starting points too -- i don't remember -- but i'm willing
to bet that the approximately meantone-tempered diatonic scale
is the global minimum of total dyadic dissonance (if you don't
allow degenerate solutions such as 6-tone scales).

peace,
paul

>>>Given the constraints of a melodic-harmonic music, as you take
>>>for granted in your paper, however, a number of people have been
>>>able to derrive the diatonic scale and/or meantone temperament
>>>by minimizing pairwise (dyadic) dissonance (Erlich, Sethares),
>>
>>note that i pretty much got a meantone-tempered diatonic scale with
>>this procedure.
>
>How exactly do I find this?

There are threads that start on message # 11779, and 12026.
The only way I was able to find this was by remembering that
Paul had said "eat my shoe", search my mailbox in my mail
client (Eudora) for that phrase, and then go to that date in
the archives and retrieve the message numbers.

The details of Paul's procedure were never completely clear
to me in this thread.

>More generally, I'm not having much luck searching the group
>archives either...the titles of postings are not very
>descriptive, and no ordering by relevance...

Since when are subject titles very descriptive in internet
discussion? Otherwise, you're preaching to the choir around
here on the deficiencies of yahoo's search.

>I can't find the derivation in Sethare's book either by
>thumbing through the pages or by the index. I assert: It's
>not there!

Well, like I said, I don't have a copy handy. But he
definitely derrives some gamelan scales in there, and I
thought sure the diatonic, too.

-Carl

--- In tuning@y..., Carl Lumma <carl@l...> wrote:
> >>>Given the constraints of a melodic-harmonic music, as you
take
> >>>for granted in your paper, however, a number of people
have been
> >>>able to derrive the diatonic scale and/or meantone
temperament
> >>>by minimizing pairwise (dyadic) dissonance (Erlich,
Sethares),
> >>
> >>note that i pretty much got a meantone-tempered diatonic
scale with
> >>this procedure.
> >
> >How exactly do I find this?
>
> There are threads that start on message # 11779, and 12026.
> The only way I was able to find this was by remembering that
> Paul had said "eat my shoe", search my mailbox in my mail
> client (Eudora) for that phrase, and then go to that date in
> the archives and retrieve the message numbers.

great memory, carl!

> The details of Paul's procedure were never completely clear
> to me in this thread.

hopefully my explanation today, that i posted for jeff, will help.
please ask if there are any outstanding questions.

--- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:
> hi jeff! paul here. i am away from my office right now, but when i
> get back (probably tomorrow), i will be more than happy to
> reproduce this from scratch, in case you still can't find this in
the
> archives from a couple of years ago.

Thanks, Paul. Carl gave me a couple of explicit message numbers,
so I will read though those threads over the next few days.
I also printed out the dictionary article on harmonic entropy.

Don't trouble yourself you reproduce those calculations
(at least not yet! :-)

Jeff

--- In tuning@y..., Carl Lumma <carl@l...> wrote:

> There are threads that start on message # 11779, and 12026.
> The only way I was able to find this was by remembering that
> Paul had said "eat my shoe", search my mailbox in my mail
> client (Eudora) for that phrase, and then go to that date in
> the archives and retrieve the message numbers.
>
THANKS, Carl! That is a tremendous help! I was getting frustrated
enough to eat my own shoe, or at least kick the computer with it :-)

> >I can't find the derivation in Sethare's book either by
> >thumbing through the pages or by the index. I assert: It's
> >not there!
>
> Well, like I said, I don't have a copy handy. But he
> definitely derrives some gamelan scales in there, and I
> thought sure the diatonic, too.

When I was in school, I had an advisor who would claim
"I proved this; I proved that -- check your notes!" and he
hadn't (verified by asking everyone else). So I guess its
a bit of a sore subject with me :-)

Jeff

Hello,

I am no professional musician nor scholar, but I find this discussion
exciting. As an outsider to the discussion, I have the impression that in
the Jeff Jensen's paper there is a strong teleological bias that remind me
the (false) view of "evolution ladder" in evolution theory. Many authors
such as Stephen J. Gould (in books like "The Panda Thumb") point out the
fact that such a bias impede a clear view of biological evolution mechanism.
History of music may also suffer too simple causal view such as :

> Harmonics ==> JI,
> JI + (uncontrollable urge to change keys) ==> 12tet

We should take care to not oversimplify causal chains, otherwise, we may as
well conclude that cooling of the climate killed the dinosaurs as well as JI
(see below ;-) ).

It would be like stating (has it has been) that a big brain is clear
tendency of evolution, or that dinosaur disappeared because they were not
fit etc. etc. The "punctuated equilibrium" theory states that brutal changes
in environment may kill extremely well adapted life forms (like dinosaurs)
and give an unexpected chance to up-to-then marginal organism for reason
that are more linked to mere luck that to any simple causal relationship
(warm blood mammal where "better" fitted for climate change).

Punctual events (at the geological scale) can change course of evolution in
unpredictable ways. That should be a lesson for history of music and history
in general. In this sense, the Pythagorean scale is a very "successful"
system, but that does not necessarily imply that it is "better" than any
other scale. The fact that Pythagorean scale leads, by the circle of fifth,
to "out-of-tune" thirds, had a clear and decisive influence on development
of medieval polyphony. Nevertheless, the Pythagorean scale is an
intellectual construction based on the Pythagorean faith stating that the
real world is just an imperfect shadow of a "perfect" mathematical universe.
Other clever intellectual constructs, based on other philosophical view
could have lead to a completely different musical system as Margo Schulter
conclude in her wonderful posting:

> One lesson of world musics is that there are a wide range of natural
> and beautiful solutions, with Western Europe in the 14th or 18th
> century merely as one example.
>
> Trying to articulate some of the "explanations" for intonational
> choices may help us better to understand not only the roads taken, but
> the roads not taken and open to fruitful exploration.
>
> Of course, an acquaintance with the natural and beautiful scalar and
> intonational systems of a range of world musics, for example the many
> neutral intervals of medieval or modern Near Eastern music and the
> slendro and pelog scales of Javanese or Balinese gamelan, brings home
> the point that there are many artistically compelling choices.

There is more than simple causal relationship to explain history. There is a
sort of "co-evolution" (to borrow a term of evolution theory) between
musical theory, musical practice, musical taste, instrument technique and
instrument making.

For instance, if the keyboard instruments have not been the preeminent
instruments for many centuries in Europe, tuning and temperament would have
been a lesser issue. E contrario, a cappella Italian madrigal of around 1600
was a laboratory for quite audacious modulation; those "experimentations"
disappeared as a cappella genres went out of fashion (by the way, try to
play Gesualdo "morro lasso" on a 12ET piano, it give an insight why it is a
cappella).

Other example: throughout 19th century, instrument makers had to cope with
demand for power and roundness (e.g.: look at the history of piano making).
Matter of factly, concert halls became bigger and bigger, but it is
difficult to state if it was because instruments were more powerful or the
other way around. Singing technique also evolved to keep in pace with those
big powerful orchestras: eventually singers learned to sing slightly
out-of-tune to not be totally masked by the orchestra. New singing technique
came with a large vibrato that blur the pitch and make the "out-of-tunness"
much less obvious. But is there is a simple causal relationship that goes
from climate change to 12ET tuning ? :-) :

Climate cooling in Europe => starvation in France => French Revolution =>
democratization of music => larger concert hall => larger orchestras => new
singing technique => need to sing below the pitch in a controlled manner =>
large vibrato to blur the pitch => lack of taste for just intonation => 12ET
tuning !!!

That is certainly not the way it goes. Co-evolution seems here to be a more
productive concept (e.g. large vibrato is a side effect of the new singing
technique, not a goal). Everything change little by little but many effects
(of more or less musical nature) can change course of things.

Other example: There were many attempt throughout the last centuries up to
now to design keyboards with more than 12 keys by octave. In parallel,
fingering technique co-evolved with standard 7 whites / 5 black keyboard up
to a point that by the time of Bach (or little afterward) the technical
standard for keyboard players became so high that it become less and less
likely that the same virtuosity could be achieved with alternate keyboard
design (vita brevis, ars longa). In this case, the taste for virtuosity
provided an "environment" where the 7W/5B keyboard is so well "fitted" that
there is no "niche" for other life-forms.... Further there are probably some
economical reasons for standardized keyboard design.

Other example: Throughout 19th century, piano makers "optimized" the design
in a way that mostly kill the harmonics greater than 7 or so (by choosing
cleverly the place where the hammer hit the string). This had three major
effect:
- The sound is subjectively "rounder".
- The energy of "killed" high harmonics is not lost but transferred to lower
harmonics to which the human ear is much more sensitive. Thus the kinetic
energy of the hammer is efficiently transformed in a subjectively "louder"
sound.
- As a side effect, in chords, the relative energy of the beating harmonics
is lesser than for, let say, harpsichord.

Roundness is matter of taste. Search of power is a general trend in
instrument making trough 19th an 20th century. The side effect on chords
makes ET much more tolerable on piano than on harpsichord (at least to my
opinion). If that is true, the tone of the new dominant keyboard may had an
influence (but not as a single cause) in emergence of 12ET.

My viewpoint is from a quite naive dilettante, especially when compared to
the deep, comprehensive postings of Margo Schulter. Nevertheless, Margo's
posting shows that we should not give up understanding history of tuning
even though there is no simple causal relationship.

yours truly

Fran�ois Laferri�re

Hi Francois. Thanks for your comments

--- In tuning@y..., LAFERRIERE François <francois.laferriere@c...>
wrote:
, I have the impression that in
> the Jeff Jensen's paper there is a strong teleological bias that
remind me
> the (false) view of "evolution ladder" in evolution theory

I hope that you looked at the actual paper :-)
http://home.austin.rr.com/jmjensen/musicTheory.html

I guess I am attempting to determine what fundamental properties
these standard scales have (and figure out some other things too).

The existence of harmonics and reproducible laboratory experiments
on human hearing are independent of arbitrary choices made by
various musicians at various times in history. Indeed, history
is people stumbling around trying things, making mistakes, and
gradually holding onto ideas that seem useful. (Of course,
sometimes good ideas are discarded and bad ones kept, at least for
some period of time).

I like music history and I think it is important. However,
engineers don't need to study James Watt and the steam engine and
Henry Ford and the model T to understand how my car works or
to design a more fuel efficient model. (and engineering is an Art
as well as a science)

Anyway, I'm trying to see whether the hugely successful diatonic
scales are based on fundamental principles i.e. these scales
enable musicians to do things (like make dissonant chords resolve
into consonant ones) that many people want music to do, once
they have experienced it.

Note also that I didn't conclude these scales are inevitable for
"optimal" tonal harmony (maybe they are!) but I don't know how
to answer that question, or if the question is even well defined.

I think it is fun to think about such questions, though, and
see what conclusions fall out...

JEff

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

> Anyway, I'm trying to see whether the hugely successful diatonic
> scales are based on fundamental principles i.e. these scales
> enable musicians to do things (like make dissonant chords resolve
> into consonant ones) that many people want music to do, once
> they have experienced it.
>
> Note also that I didn't conclude these scales are inevitable for
> "optimal" tonal harmony (maybe they are!) but I don't know how
> to answer that question, or if the question is even well defined.
>
> I think it is fun to think about such questions, though, and
> see what conclusions fall out...
>
> JEff

i agree with you completely, and have many similar interests, as you
probably know by now from my 22-tone paper, my periodicity block
article, and the optimization calculations carl found. perhaps even
more interesting to you might be the work going on on the tuning-math
list, currently at

tuning-math@yahoogroups.com

i'm hoping to publish some friendly expositions of some of the work
going on there at some point in the future -- for now, the discovery
continues!

of course, others may come back with the viewpoint that, despite all
appearances, there are no fundamental principles at work, just pure
artistic intuition -- and still others (believe it or not, the recent
academic tack is this) claim that there are abstract mathematical
principles governing scale and chord structure, but these principles
do not arise whatsoever from the harmonic series or acoustics. it
sounds like you and i both belong to (or at least flirt with) a third
school of thought, that suggests that 'nature' has at least something
to do with how certain styles of music end up developing . . .

>hopefully my explanation today, that i posted for jeff, will
>help. please ask if there are any outstanding questions.

Did this show up? Message number?

-Carl

--- In tuning@y..., "clumma" <carl@l...> wrote:
> >hopefully my explanation today, that i posted for jeff, will
> >help. please ask if there are any outstanding questions.
>
> Did this show up? Message number?
>
> -Carl

search for "21-dimensional".

btw, as long as the columbia server has searchable archives, i support
the move to it.

>>>hopefully my explanation today, that i posted for jeff, will
>>>help. please ask if there are any outstanding questions.
>>
>>Did this show up? Message number?
>
>search for "21-dimensional".

I find the above message, and message # 36086, which is just a
pointer to the harmonic entropy list.

Anyway, my outstanding issues were:

() You found that seeding the process with ets didn't always
find the global minimum, and wanted to try seeding random
scales, but I'm not sure I saw the results of that effort.

() I remember 5- and 7-tone results, but not 6, 8, 9, and 10.

() I was never clear on what the "observations" column meant.

-Carl

--- In tuning@y..., Carl Lumma <carl@l...> wrote:

> Anyway, my outstanding issues were:
>
> () You found that seeding the process with ets didn't always
> find the global minimum, and wanted to try seeding random
> scales, but I'm not sure I saw the results of that effort.

any results with an 'observations' column was the result of such an
effort.

> () I remember 5- and 7-tone results, but not 6, 8, 9, and 10.

i most certainly posted those results as well. 8 gave a meantone
diatonic scale, sometimes with a G#, and sometimes with an Ab.

> () I was never clear on what the "observations" column meant.

the number of times that that particular scale was the local minimum
converged upon, out of hundreds or thousands of random seeding
operations.

>> () I was never clear on what the "observations" column meant.
>
>the number of times that that particular scale was the local minimum
>converged upon, out of hundreds or thousands of random seeding
>operations.

Aha!

>> () You found that seeding the process with ets didn't always
>> find the global minimum, and wanted to try seeding random
>> scales, but I'm not sure I saw the results of that effort.
>
>any results with an 'observations' column was the result of such an
>effort.

Aha!

>> () I remember 5- and 7-tone results, but not 6, 8, 9, and 10.
>
>i most certainly posted those results as well. 8 gave a meantone
>diatonic scale, sometimes with a G#, and sometimes with an Ab.

I can't find them in the archives. When you get your computer
back on-line, can you repost these results?

-Carl

Hello, there, Francois Laferriere, and thank you for your comments on
evolutionary concepts in music history and the dangers of a "ladder of
evolution" kind of metaphor.

This kind of dialogue is a special pleasure, and for both of us, it
would seem, the era of Gesualdo around 1600 is an especially
fascinating one in considering the different directions that musical
"co-evolution," as you nicely term it (an eloquent metaphor!), might
have taken then, and can still take now.

For me, the 14th century is another such era; thank you for remarks
which both enrich and stimulate this kind of discourse both on the
specifics of intonation and the general use of evolutionary concepts
in music history.

Please let me comment on a few points which may further illustrate
these themes, while providing a link to your complete article:

/tuning/message/

> We should take care to not oversimplify causal chains, otherwise, we
> may as well conclude that cooling of the climate killed the
> dinosaurs as well as JI (see below ;-) ).

This is wise advice, and from a "historical/evolutionary" point of
view, one might also ask, "What kind of JI is assumed?"

A common assumption is that "JI" in the setting of this kind of
historical discussion means mainly ratios of 3 and 5, since this was
indeed the kind of scheme which, as it happened, followed the
Pythagorean form of just tuning in Western Europe. We find it
used in the monochord of Ramos (1482), who recommends its simpler
ratios for students who find the Pythagorean proportions difficult to
follow, and by the great Italian philosopher Ficino shortly
thereafter, who praises the "sweetness" of the third in the proportion
sesquiquarta (5:4).

However, from practice as well as theory, I might ask, "Why not JI
based, for example, on ratios of 3 and 7, with imperfect concords such
as 7:6, 7:4, 9:7, and 12:7?"

Whether or not the system of vocal intonation for cadences described
and advocated by Marchettus of Padua in his _Lucidarium_ (1318) was
anything like this -- unfortunately there are not any tape recordings
or computer audio files for your adept analysis -- his ideal of
"closest approach" certainly could result in some performers
approximating these ratios of 7.

Here I can at least speak for myself. In 1998, when John Chalmers
remarked to me that a Pythagorean tuning sufficiently extended will
emulate ratios of 7, I associated these ratios with the cadences of
Marchettus, which seem to call for narrower semitones, and wider major
thirds and sixths, than the usual Pythagorean intervals. This meeting
of early 14th-century theory and a late 20th-century observation by a
leading xenharmonic scholar together led me to practical 24-note
tuning systems based on a Pythagorean chain.

Here, too, there has been a bit of "evolutionary branching." I started
with a regular 24-note Pythagorean tuning on two keyboards a
Pythagorean comma apart (531441:524288, ~23.46 cents), where the
near-7 steps and intervals differ from their pure 7-based ratios by
about 3.80 cents.

Later the idea came to me, with some inspiration from participants
here such as Graham Breed, of slightly modifying the distance between
the keyboards to a 64:63 (~27.26 cents), thus obtaining pure ratios of
3 and 7.

Then I came to experiment with other distances, including a system
with the two keyboards a pure 7:6 (~266.87 cents) apart, one leading
to a kind of 21st-century variation on early 15th-century
fauxbourdon, with chains of parallel 12:14:18:21 or 14:18:21:24
sonorities eventually resolving cadentially to a pure 3:2 fifth or
complete 2:3:4.

There are many open questions: for example, to what degree might my
choice of electronic timbres (or, to a considerable degree, the choice
of the synthesizer designer to include certain preset voices)
facilitate the use of certain interval sizes or ratios that might be
less palatable with the prevailing timbres in Europe around 1400, for
example?

> Punctual events (at the geological scale) can change course of
> evolution in unpredictable ways. That should be a lesson for history
> of music and history in general.

Of course, with musical evolution, we are dealing with a form of what
might be termed "artificial selection," or more specifically, artistic
selection, also known as fashion.

Certainly fashion has its vagaries, a theme of much wisdom and humor.

At the same time, to use your geological and biological metaphor, I
would say that there can indeed be "punctuated evolution" in music and
intonation also. During the last couple of years, I seem to be living
in a kind of Ediacarian/Cambrian "explosion" of tuning systems,
intervals, and progressions.

One might possibly say that I happen to be exploring a largely open
"niche": tunings systems and musics where major thirds of around
408-440 cents, and minor thirds of around 260-294 cents, are the usual
and highly esteemed sizes, with progressions often following Gothic
patterns of the kind described by Marchettus and others.

Such systems could have evolved in a 15th-16th century European milieu
with different musical tastes, or in an early 20th-century xenharmonic
setting with more of a "neo-medieval" inclination. Why this should
actually happen around the end of the 20th century, I'm not sure.

However, whatever the reasons, the kind of "unpredictable" changes you
mention are also something I have experienced.

In June of 2000, for example, I got the idea of a tuning in which the
whole-tone and diatonic semitone would have a ratio of sizes equal to
Euler's e, about 2.71828, leading to a fifth of around 704.61 cents.
This was just a pleasant mathematical emblem, and also a kind of
intuition that this might be an interesting region of the spectrum.

By accident, I discovered that this tuning had augmented seconds and
diminished fourths not too far from the ratios of 17:14 (~336.13
cents) and 21:17 (~365.83 cents), and that I loved the effect of these
thirds in cadential progressions. These intervals then became part of
my intonational practice and theory, and a highly valued attraction of
a new or unfamiliar tuning system.

Such fortuitous developments can occur in various musical
"bioenvironments." For example, Mark Lindley has suggested that the
French _temperament ordinaire_ of the later 17th and 18th centuries
may have grown out of a misunderstanding as to the technique for
tuning the flats in a regular meantone scheme.

Lindley observes in tuning a chain of meantone fifths such as
C-G-D-A-E, the upper note must be lowered to make the fifth slightly
narrower than pure. In adding the usual flats of an Eb-G# tuning,
however, one must similarly temper the fifths in the narrow direction
by _raising_ the _lower_ note of the fifth (Bb-F, Eb-Bb). However,
some tuners may have mistaken the instructions to call for a downward
adjustment, making the fifths Eb-Bb-F _wider_ than pure, and
mitigating the usual meantone Wolf G#-Eb, or even making it a
comfortable fifth.

However it may have originated, this system led to the artful
exploitation of a composer such as Couperin, analyzed by Lindley --
for example, the special treatment of some remote major thirds rather
larger than Pythagorean. In a neo-Gothic tuning of the early 21st
century, thirds of this size might define the routine and ideal norm,
but in a late 17th-century setting, they are "strange" and thus a
notable special effect contrasting with the norm of meantone-like
major thirds closer to 5:4.

> There is more than simple causal relationship to explain
> history. There is a sort of "co-evolution" (to borrow a term of
> evolution theory) between musical theory, musical practice, musical
> taste, instrument technique and instrument making.

As noted above, I find this a beautiful metaphor, and the question of
Gesualdo and keyboard instruments that you raise is something I would
like to comment on a bit in order further to support your main
thesis. In my view, the somewhat different conclusions I tend to draw
about the possible role of vocal and instrumental performers may serve
to support your basic concept of "co-evolution" yet more strongly.

First, however, on your vital elements of "musical practice" and
"musical taste," I might share an amusing story which happened on this
list in September of 2000.

At that time, the young mathematician and theorist Keenan Pepper
described a regular tuning in which the whole-tone and chromatic
semitone have a ratio of the Golden Section, or Phi, about 1.618034.
This produces a fifth of around 704.10 cents.

Not so surprisingly, I found this tuning quite delightful, and indeed
a most excellent optimization for four of my favorite ratios (14:11,
13:11, 17:14, 21:17). Delighted both by the ingenious mathematical
concept, and by the result in musical practice, I applauded "Keenan
Pepper's wonderful tuning" in many posts exploring its felicities.

Then, however, Keenan Pepper himself responded by saying that for his
own musical purposes, he didn't like it: he wanted ratios such 5:4 and
7:4, rather than the more complex ones in a 12-note tuning that moved
me to such enthusiasm.

> For instance, if the keyboard instruments have not been the
> preeminent instruments for many centuries in Europe, tuning and
> temperament would have been a lesser issue. E contrario, a cappella
> Italian madrigal of around 1600 was a laboratory for quite audacious
> modulation; those "experimentations" disappeared as a cappella
> genres went out of fashion (by the way, try to play Gesualdo "morro
> lasso" on a 12ET piano, it give an insight why it is a cappella).

Please let me begin by warmly agreeing that Gesualdo's vocal music
might be ideally performed by an a cappella ensemble, possibly using
some kind of adaptive just intonation. However, to observe that
performances on instruments were reported by writers of the time such
as Doni, and that these pieces might be quite pleasantly performed on
keyboards of the period with 19 or 31 notes per octave in extended
meantone (likely at or near 1/4-comma with pure major thirds), is only
to strengthen your point that musical styles, tuning systems, and
instruments indeed "co-evolve."

For example, Doni tells us that in dramatic performances, music might
be played to fit the mood, noting that for a melacholy setting, one
plays a madrigal of the Prince [i.e. Gesualdo, Prince of Venosa] on
the viols.

Viols, although they are fretted instruments, permit flexible
intonation, rather like voices. Mark Lindley, for example, that the
trick of producing some just sonorities is something mastered by every
"first rate" viol consort that he has heard, and that it is both
possible and often desirable to match a keyboard temperament such as
meantone or a _temperament ordinaire_, as French players in the later
17th century may have done.

While some kind of adaptive tuning for singers or viols, possibly
mixing characteristics of meantone and just intonation, is one
possibility, the extended meantone keyboards especially popular in the
region of Naples around 1600 would also nicely fit Gesualdo's music.

Indeed, although you are quite right to emphasize that music of this
era is much less "keyboard-centered" than that of the 19th century,
for example, Gesualdo himself and others in his general milieu such as
Trabaci were composing keyboard music for instruments such as the
_cembalo chromatico_ or "chromatic harpsichord" with 19 notes per
octave (generally Gb-B#) in 1/4-comma meantone or the like.

As it happens, Scipione Stella, who built keyboard instruments in
31-note meantone, entered Gesualdo's service in 1594. His instruments
seem to have influenced Fabio Colonna, who in 1618 described his own
harpsichord with a kind of what might now be called a generalized
keyboard, and included in his treatise a musical piece entitled
"Example of Circulation" in which he cadences on all 31 steps of the
circle of fifths, returning to the opening sonority.

This makes your main point about co-evolution all the more dramatic: a
12-tET piano can accurately represent the intonational nuances neither
of Gesualdo's madrigals nor of harpsichord music in extended meantone,
where, in addition to pure or near-pure 5-limit thirds, the unequal
semitones and distinctions between B# and C (the latter a diesis or
fifthtone higher, about 128:125 or 41.06 cents) are important
elements.

However, as you also may be suggesting here, the limitation of
"standard" keyboards to only 12 notes per octave could play a very
significant role in shaping 17th-century developments. Around
1550-1620, keyboards in extended meantone were a kind of special
delicacy for what might now be described as more "experimental" music.

When, for whatever reasons, more remote transpositions became a
routine imperative in the major/minor key system established in
practice by around 1670-1700, there were two obvious options: either
keyboards with a large number of notes per octave would become a new
standard (e.g. the 31-note cycle of Huygens), or 12-note tunings would
be adjusted to achieve a largely or completely circulating system, as
with Werckmeister's well-temperaments.

As it happens, the taste of the times -- or possibly the economics and
habits of keyboard playing which you describe in the next quoted
passage of your article -- largely sided with Werckmeister rather than
Huygens. This not only set the stage for the music of a genius such as
Bach, but promoted a "closed" 12-note cycle of a kind excluding many
of the creative possibilities of extended meantone as earlier explored
by people such as Vicentino and Gesualdo.

There are other scenarios one could postulate also in order to
appreciate the evolutionary possibilities. Suppose, for example, that
a 19-note equal temperament had become the norm in the early or middle
17th century, as proposed in 1570 by Costeley -- more notes than 12,
but considerably fewer than the 31 of Vicentino or Stella and Colonna
required to make a 1/4-comma temperament a circulating system.

While such a 19-note system (either 19-tET or the almost identical
1/3-comma meantone with pure 6:5 minor thirds) would not have the
diesis or fifthtone nuances of 31, it might provide a rather nice
representation of some of Gesualdo's madrigals, for example --
although 1/4-comma in 19 or more notes seems to me the best choice,
and what was evidently favored on the Neapolitan chromatic
harpsichords with their open 19-note tuning (Gb-B#).

One could even playfully argue that Zarlino (1571) finds 1/3-comma
somewhat "languid," maybe nicely fitting the melancholy quality of
Gesualdo's madrigals noted by Doni, one of this composer's great
admirers.

> Other example: There were many attempt throughout the last centuries
> up to now to design keyboards with more than 12 keys by octave. In
> parallel, fingering technique co-evolved with standard 7 whites / 5
> black keyboard up to a point that by the time of Bach (or little
> afterward) the technical standard for keyboard players became so
> high that it become less and less likely that the same virtuosity
> could be achieved with alternate keyboard design (vita brevis, ars
> longa). In this case, the taste for virtuosity provided an
> "environment" where the 7W/5B keyboard is so well "fitted" that
> there is no "niche" for other life-forms.... Further there are
> probably some economical reasons for standardized keyboard design.

Here, indeed, we have what has sometimes called an "adaptive sink" --
a kind of stasis, in which things seem in such equilibrium that there
is little opportunity for dramatic change -- at least until the next
evolutionary "punctuation."

Writing in 1594, Bottrigari tells us that lots of keyboard player's
were intimidated by Vicentino's _archicembalo_ with its large number
of notes per octave, although we know that keyboards with some split
accidental keys were more common, and 19-note instruments in 1/4-comma
meantone or the like rather the fashion around Naples.

There are also modern compromises like using two standard 12-note MIDI
keyboards to map tunings of between 13 and 24 notes. Here it's
interesting that the schemes I tend to use a lot generally have at
least one of the keyboards in a conventional 12-note arrangement of
some kind (e.g. Pythagorean, meantone, or a regular tuning with wide
fifths), or at least something quite close to it.

People with generalized keyboards are less likely to be biased in this
way.

> Roundness is matter of taste. Search of power is a general trend in
> instrument making trough 19th an 20th century. The side effect on
> chords makes ET much more tolerable on piano than on harpsichord (at
> least to my opinion). If that is true, the tone of the new dominant
> keyboard may had an influence (but not as a single cause) in
> emergence of 12ET.

Here I might just add a couple of related examples. In 1581, Vincenzo
Galilei indeed found 12-tET much more pleasant on lute (where it was a
standard tuning) than on harpsichord, and he discussed some
differences between the material and plucking action on the strings of
the two instruments that might account for this difference.

Also, it has been suggested that the standard of 12-tET in the late
19th and 20th centuries may have accounted for the limited popularity
of the harmonium with its very prominent partials -- the same
instrument proving a delight, for this very reason, to advocates and
designers of just intonation keyboards.

Again, thank you for this most gracious and rewarding dialogue.

Most appreciatively,

Margo Schulter
mschulter@value.net

--- In tuning@y..., "M. Schulter" <MSCHULTER@V...> wrote:

> First, for the questions "Why 7?" and "Why 12?" I would say that one
> short answer has been provided by Ervin Wilson: these sizes are
among
> those defining a "Moment of Symmetry" (MOS) where a tuning system
has
> reached a point of poise with two sizes of adjacent steps.

I'm sorry to say I don't understand this part. What special
properties
do 2 different step sizes imply?

> An attraction of 12 is that it is an MOS, with visual symmetry on a
> keyboard: each of the five whole-tone steps in a diatonic scale is
> divided into two semitones: the diatonic or minor semitone at
256:243
> (~90.22 cents) also present at E-F and B-C, and the chromatic or
major
> semitone at 2187:2048 (~113.69 cents). Thus there are two adjacent
> step sizes: the diatonic and chromatic semitones.
>

> Thus I might explain the late 17th-century practice of major/minor
> tonality, codified in the early 18th century by theorists such as
> Rameau, as reflecting a certain choice of materials -- in some ways
a
> considerably more restricted choice than the chromaticism of the
> Manneristic era around 1540-1640, let alone the fifthtone
> enharmonicism of a Vicentino or Colonna.
>
> As Paul and Jerry have discussed, a central feature of major/minor
> tonality is the role of the tritone or diminished fifth as an
interval
> resolving by stepwise contrary motion to a third or sixth. This kind
> of resolution, routine in 16th-century practice and theory, becomes
> the defining element of key.
>

OK, now the mysterious "tritone", which is curiously absent from
the music books I have (or is given one or two sentences). I did
read the full article in Joe's dictionary, but I still don't
see the significance of the tritone...

Very basic question: in C major, the tritone is the interval
C-Gb and/or C-F# (assuming a tuning which distinguishes Gb/F#)?

Now, I really want to understand the defining properties of
major/minor tonality and what turns an arbitrary bunch of pitches
into a "Key". I suspect it is the ability to form a cadence
like G^7 ---> C major, but then where is the tritone?

A related question is what properties of the Ionian mode and
Aeolian mode made them special? What can they do that they
others can't? Is the Ionian mode the same thing as the modern
standard major scale?

Thanks!
Jeff

--- In tuning@y..., "jjensen142000" <jjensen14@h...> wrote:
> --- In tuning@y..., "M. Schulter" <MSCHULTER@V...> wrote:
>
> > First, for the questions "Why 7?" and "Why 12?" I would say that
one
> > short answer has been provided by Ervin Wilson: these sizes are
> among
> > those defining a "Moment of Symmetry" (MOS) where a tuning system
> has
> > reached a point of poise with two sizes of adjacent steps.
>
> I'm sorry to say I don't understand this part. What special
> properties
> do 2 different step sizes imply?

there is a huge academic literature on this very topic -- clough
might be a good name to begin with, or clampitt . . . i myself long
had little appreciation for the musical attractiveness of such
structures . . . until i realized that they (or at least the many
specific ones that theorists of 'our school' ["start with nature"]
have proposed for the last millenium) could all be understood as:

*****fokker periodicity blocks with all but one of the unison vectors
tempered out*****

when i discovered this, i sacrificed a bull in its honor (well, not
really, but i call it simply 'the hypothesis' because i feel it's so
unique).

don't worry if you don't understand this -- this is pretty advanced
stuff, in that i couldn't possibly have grasped it without playing
with lots of tuning systems for years. if you're interested, though,
i'd love to help bring you up to speed on this -- for now, try to
absorb the 'gentle introduction' as best you can . . .

> > An attraction of 12 is that it is an MOS,

> OK, now the mysterious "tritone", which is curiously absent from
> the music books I have (or is given one or two sentences). I did
> read the full article in Joe's dictionary, but I still don't
> see the significance of the tritone...

it's only significant if you happen to be using the diatonic scale.

> Very basic question: in C major, the tritone is the interval
> C-Gb and/or C-F# (assuming a tuning which distinguishes Gb/F#)?

actually, in C major, the tritone is B-F!

> Now, I really want to understand the defining properties of
> major/minor tonality and what turns an arbitrary bunch of pitches
> into a "Key". I suspect it is the ability to form a cadence
> like G^7 ---> C major, but then where is the tritone?

the dyad B-F is present in the G7 chord. the diatonic scale only has
one tritone in it so you immediately know 'where you are' in the
scale once you hear it. and as if that weren't enough, the dissonant
tritone *resolves*, both voices moving a half-step, in contrary
motion, to the C-E dyad in the C major triad. this is psychologically
powerful stuff!

> A related question is what properties of the Ionian mode and
> Aeolian mode made them special? What can they do that they
> others can't?

only in those modes can the native tritone resolve in this
particular, powerful way, to notes of the tonic triad. in all other
modes, one (or both) of the notes of the tritone are already
*members* of the tonic triad -- no opportunity for a powerful
resolution!

i'd be happy to restate this if that wasn't clear.

> Is the Ionian mode the same thing as the modern
> standard major scale?

pretty much. though really there wasn't much music a western scholar
would today consider to be in 'major tonality' but was created back
in the 'modal' days when Ionian was used. 'tonality' as the west
knows it today didn't fully become a feature of our music until
around 1670 (says margo). tonality means much more than just a set of
frequencies.

--- In tuning@y..., "M. Schulter" <MSCHULTER@V...> wrote:

/tuning/topicId_35909.html#36190

>
> For me, the 14th century is another such era; thank you for remarks
> which both enrich and stimulate this kind of discourse both on the
> specifics of intonation and the general use of evolutionary concepts
> in music history.
>
> Please let me comment on a few points which may further illustrate
> these themes, while providing a link to your complete article:
>
> /tuning/message/
>

***Hello Margo!

I believe the link above is not correct. Maybe you mean something
else on www.medieval.org.. (??)

> Such systems could have evolved in a 15th-16th century European
milieu with different musical tastes, or in an early 20th-century
xenharmonic setting with more of a "neo-medieval" inclination. Why
this should actually happen around the end of the 20th century, I'm
not sure.
>

***Well, I guess a simple answer, which you've probably already
though of, is the fact that modern technology both facilitates
research into the past as well as provides opportunity to *easily*
try alternate tunings.

BTW, I *really* enjoyed your essay with the fascinating implications
that Gesualdo's music could have, and probably was, performed on a 19-
tone meantone keyboard...

Joe Pehrson

Paul, thanks for the info. Now things are really getting
interesting...!

--- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:
> there is a huge academic literature on this very topic -- clough
> might be a good name to begin with, or clampitt . . . i myself long
> had little appreciation for the musical attractiveness of such
> structures . . . until i realized that they (or at least the many
> specific ones that theorists of 'our school' ["start with nature"]
> have proposed for the last millenium) could all be understood as:
>
>
> *****fokker periodicity blocks with all but one of the unison
vectors
> tempered out*****
>

> i'd love to help bring you up to speed on this -- for now, try to
> absorb the 'gentle introduction' as best you can . . .

I've read parts 1 and 2, and it doesn't seem too difficult so far.
I'm also going to read the discussion that Dave Benson gives in
Math and Music. My main question so far is: What are some big
dragons that we can slay with this formalism?

> > but I still don't
> > see the significance of the tritone...
>
> it's only significant if you happen to be using the diatonic scale.

So then the existence of the tritone is an important defining
property of the diatonic scale?

>
> > Very basic question: in C major, the tritone is the interval
> > C-Gb and/or C-F# (assuming a tuning which distinguishes Gb/F#)?
>
> actually, in C major, the tritone is B-F!

Well, in 12tet I can look for the place where there are 3
consecutive whole tones and that would be F-B. So far so good.
But then people say things like the tritone is the "center of
the octave" or an "augmented 4th" and that would seem to
necessarily mean C-F#???

> > Now, I really want to understand the defining properties of
> > major/minor tonality and what turns an arbitrary bunch of pitches
> > into a "Key". I suspect it is the ability to form a cadence
> > like G^7 ---> C major, but then where is the tritone?
>
> the dyad B-F is present in the G7 chord. the diatonic scale only
has
> one tritone in it so you immediately know 'where you are' in the
> scale once you hear it. and as if that weren't enough, the
dissonant
> tritone *resolves*, both voices moving a half-step, in contrary
> motion, to the C-E dyad in the C major triad. this is
psychologically powerful stuff!

Yes, this is what I want to understand! My plan is to
extend my sensory dissonance applet to compute the net dissonance
for chords and then try to cook up a mathematical function that
will quantify how much a given chord "wants" to resolve to
some other chord. At the moment, this is all vague. I'm doing
some reading on tonal harmony to try to firm things up.

Yes, your point is well taken that sensory dissonance may only
explain part of the phenomeneon of dissonance. Ideally, the
applet will be sufficiently modular that it would be possible
to just plug in a harmonic entropy function, say, and then
crank out different numbers with that.

> tonality means much more than just a set of frequencies.

So we need a precise list of what it does mean, or at least,
a first guess (to be refined ) at such a list...

Jeff

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

> > i'd love to help bring you up to speed on this -- for now, try to
> > absorb the 'gentle introduction' as best you can . . .
>
> I've read parts 1 and 2, and it doesn't seem too difficult so far.
> I'm also going to read the discussion that Dave Benson gives in
> Math and Music. My main question so far is: What are some big
> dragons that we can slay with this formalism?

for that, you should check out the tuning-math list -- our resident
dragon slayer gene ward smith has smitten some fearsome beasts!

> > > but I still don't
> > > see the significance of the tritone...
> >
> > it's only significant if you happen to be using the diatonic
scale.
>
> So then the existence of the tritone is an important defining
> property of the diatonic scale?

i wouldn't put it that way. but if you're using the diatonic scale,
the tritone that's found within it is an interval with great
grammatical importance.

> > > Very basic question: in C major, the tritone is the interval
> > > C-Gb and/or C-F# (assuming a tuning which distinguishes Gb/F#)?
> >
> > actually, in C major, the tritone is B-F!
>
> Well, in 12tet I can look for the place where there are 3
> consecutive whole tones and that would be F-B. So far so good.
> But then people say things like the tritone is the "center of
> the octave" or an "augmented 4th" and that would seem to
> necessarily mean C-F#???

an augmented fourth up from F is B: F, G, A, B. F-B is the naturally
occuring tritone in C major and A natural minor. C-F# doesn't come up
unless you're talking about G major or E natural minor.

> > > Now, I really want to understand the defining properties of
> > > major/minor tonality and what turns an arbitrary bunch of
pitches
> > > into a "Key". I suspect it is the ability to form a cadence
> > > like G^7 ---> C major, but then where is the tritone?
> >
> > the dyad B-F is present in the G7 chord. the diatonic scale only
> has
> > one tritone in it so you immediately know 'where you are' in the
> > scale once you hear it. and as if that weren't enough, the
> dissonant
> > tritone *resolves*, both voices moving a half-step, in contrary
> > motion, to the C-E dyad in the C major triad. this is
> psychologically powerful stuff!
>
> Yes, this is what I want to understand! My plan is to
> extend my sensory dissonance applet to compute the net dissonance
> for chords and then try to cook up a mathematical function that
> will quantify how much a given chord "wants" to resolve to
> some other chord.

well, psychoacoutical musical dissonance is not solely a function of
sensory dissonance, and desire to resolve is not solely a function of
psychoacoustical musical dissonance, but sounds like a fun project
anyway.

i also want to amplify the statements of mine you quoted above that
if you try to make music in a mode of the diatonic scale *other* than
ionian or aeolian, and play the entire scale over the tonic chord,
you'll produce a vertical (harmonic) tritone somewhere, giving an
unstable sound (at least to most western ears between 1670 and 1950).
this is another way of quickly "explaining" why the other modes fell
out of favor.

> At the moment, this is all vague. I'm doing
> some reading on tonal harmony to try to firm things up.
>
> Yes, your point is well taken that sensory dissonance may only
> explain part of the phenomeneon of dissonance. Ideally, the
> applet will be sufficiently modular that it would be possible
> to just plug in a harmonic entropy function, say, and then
> crank out different numbers with that.

well i wish harmonic entropy were more well-developed. i haven't even
calculated harmonic entropy for triads yet, let alone larger chords,
but now that i have a new faster computer (750 or 833 MHz or so), i
may have to get back to this project!

> > tonality means much more than just a set of frequencies.
>
> So we need a precise list of what it does mean, or at least,
> a first guess (to be refined ) at such a list...

my 22-tone paper tries to give some rules that make a scale, and
certain of its modes, conducive to tonality . . .

I've started discussing these on xenharmony.org if anyone wants to
check it out.