A nuanced dialogue -- for Joel and all

Hello, there, Joel Rodrigues.

Thank you very much for reminding me of a post I wrote a bit more than
a year and a half ago which might indeed suggest a new, and I hope
kinder and gentler, "nuance" to some of our discussions here --
including those in which Julia is involved as an accomplished composer
and our valued guest.

First, a link to the post on "nuancing" we're discussing:

/tuning/topicId_16183.html#16183

In that post, I raised the point that some styles of music including
my "neo-Gothic" or "neo-medieval" approaches might fit some aspects
traditionally associated with "JI," but be better viewed as "Rational
Intonation" or RI.

A hallmark of RI is that ratios with diverse degrees of simplicity or
complexity such as 3:2 or 4:3, 7:6 or 9:7 or 7:4, 17:14 or 21:17, and
something like 57344:37179 (about 750.145 cents) are all part of a
continuum.

The concepts of "JI" and "temperament" alike often tend to suggest
tuning either _at_ or some calculated distance _from_ a given "pure"
ratio, suggesting a perceptual distinction between a "just" interval
and a significantly "compromised" one.

Some treasured discussions with Dave Keenan, both almost two years ago
and now, have helped me to realize that much of what I do with RI is
in another category. It is, as you very accurately and relevantly
quote my earlier post, Joel, "nuancing" along a continuum with many
shadings of concord and discord, and often in timbres where almost any
interval can sound rather "concordant" (although stylistically fifths
and fourths define the main stable concords in a neo-medieval
setting).

An aside on Jay Rahn's paper on Marchettus of Padua, Joel, a paper
I've much enjoyed. Curiously, my Pythagorean-based tuning with that
750.145-cent interval could be taken as a very close approximation of
the way Rahn interprets the intonational system described and
recommended for singers in the _Lucidarium_ of Marchettus (1318).

In Rahn's reading, Marchettus is describing the division of the 9:8
whole-tone into "five parts" based on a monochord division, an idea
also discussed by Joe Monzo here and on his excellent Web pages.

Specifically, Marchettus advocates the use of a small melodic step in
cadences he calls a "diesis" equal to "one of the five parts of a
tone." Rahn takes this to refer to a monochord division of the 9:8
tone (measured as 81:72) into five unequal parts, with the cadential
diesis equal to about 74:72, or 37:36, about 47.43 cents. Rahn uses a
value rounded up to 48 cents.

My "Pythagorean enharmonic" tuning has two 12-note chains of pure
fifths at a distance designed to produce some pure sonorities of
14:17:21 (about 0-336.13-701.96 cents). The two chains are therefore
placed at a distance of 459:448, or about 41.99 cents, apart.

As it happens, this method of "adding pitches" divides a regular
Pythagorean diatonic semitone at 256:243 (~90.22 cents) into the
41.99-cent step I've mentioned, plus a larger quartertone-like diesis
at 114688:111537 or about 48.30 cents -- almost identical to Rahn's
suggested size for this 14th-century step, a Gothic "microinterval" in
this reading of Marchettus.

Anyway, in an RI view, I find that all of these intervals -- the pure
fifths and fourths, the "pure" and relatively blending 14:17:21
sonorities from a neo-medieval RI perspective, and the yet more
complex ratios like the ones I've cited above are all part of a
continuum.

Note that I refer to 14:17:21 as "pure" in quotes, because it's an
open question whether a listener taking a perceptual JI viewpoint such
as that often explicated by Dave Keenan would notice anything special
right around 14:17:21 proper, for example a combination-tone effect.

From my point of view, the RI ratio 14:17:21 is a kind of affectionate
name for a region of "supraminor/submajor" thirds somewhere around
330-342 cents and 360-372 cents -- 14:17 (~336.13 cents) and 17:21
(~365.83 cents) are landmarks for this general region, and sometimes
more specifically for its central portion. I tend to associate the
ratio 63:52 (~332.21 cents) with the lower portion of the region of
"supraminor" thirds, and 28:23 (~340.55 cents) with the upper portion.

Curiously, while regarding these ratios as "pure," I do not
necessarily regard other integer-based or irrational intervals in the
region as "impure" or "compromised" -- they are more like fine nuances
on a continuum, with integer-based ratios providing convenient names
for the neighborhood and some of its subregions and gradations.

In fact, this region is one of my favorites not only in RI systems
(pure fifths and fourths), but in systems with fifths typically
tempered gently wide of pure. What I celebrate is the continuum and
all of its infinitesimal "nuances," whether defined by integer-based
or irrational ratios.

Also, from a practical perspective, I suspect as mentioned above that
the timbres I choose often tend to make almost any interval seem
rather "blending," smoothing over some of the contrasts between very
obviously "pure" or "impure" intervals in the strongly harmonic
timbres often associated with JI.

To keep a balance, I should add that neo-medieval RI includes some
very important intervals that _are_ "JI" by lots of traditional
standards: 2:3:4 and 6:8:9 sonorities with pure fifths and fourths,
and also "7-based" sonorities such as 12:14:18:21 and 14:18:21:24.

However, in RI systems and in related temperaments based on irrational
ratios alike, more complex intervals and sonorities are also included
in the continuum of "relative concord," and their inclusion or
"optimal approximation" celebrated.

From a JI perspective, one might ask exactly what it means to
"optimize" a complex ratio which might be perceptually
indistinguishable from "tempered" sizes somewhat to either side.

From an RI perspective, I might reply that including an interval
somewhere in the general vicinity which evokes the "landmark" of the
ratio as an auspicious emblem is one elegant explanation.

Another side of this same pleasure is finding that a tuning system
includes some interval such as an unanticipated and complex integer
ratio, or some minute difference of size between two intervals clearly
equivalent from any practical viewpoint (e.g. differences of around
0.1 cent, with a special delight if they happen to be defined by
superparticular ratios).

Here it may be characteristic that while JI-oriented theory often
speaks of "tuning errors" or "deviations," RI-oriented theory prefers
such terms as "variations" from a given ratio (not to be confused with
statistical measures of "variance").

Anyway, thank you Joel and Dave and Julia for this opportunity to seek
a more "nuanced" discussion. This is already a long enough statement,
so I conclude with an invitation to dialogue.

Most appreciatively,

Margo Schulter
mschulter@value.net

On Thursday, July 11, 2002, at 01:33 , tuning@yahoogroups.com wrote:

> "M. Schulter" <MSCHULTER@VALUE.NET>
>
> Hello, there, Joel Rodrigues.

Hello Margo ! It's always a relief to see a post from you.

> Thank you very much for reminding me of a post I wrote a bit more than
> a year and a half ago which might indeed suggest a new, and I hope
> kinder and gentler, "nuance" to some of our discussions here --
> including those in which Julia is involved as an accomplished composer
> and our valued guest.

Perhaps kinder, gentler, mutually respectful, and in the spirit of friendship ?

> First, a link to the post on "nuancing" we're discussing:
>
> /tuning/topicId_16183.html#16183
>
> In that post, I raised the point that some styles of music including
> my "neo-Gothic" or "neo-medieval" approaches might fit some aspects
> traditionally associated with "JI," but be better viewed as "Rational
> Intonation" or RI.

I think it's perfectly possible and more productive to discuss any aspect of microtonal music without ever mentioning the J word. When I came across your posts about RI, CI, etc. it was a revelation of sorts, it almost immediately got me seeing things in new ways without limitations.

> A hallmark of RI is that ratios with diverse degrees of simplicity or
> complexity such as 3:2 or 4:3, 7:6 or 9:7 or 7:4, 17:14 or 21:17, and
> something like 57344:37179 (about 750.145 cents) are all part of a
> continuum.

> The concepts of "JI" and "temperament" alike often tend to suggest
> tuning either _at_ or some calculated distance _from_ a given "pure"
> ratio, suggesting a perceptual distinction between a "just" interval
> and a significantly "compromised" one.

Indeed, Margo you seem to steal (kidding !) thoughts right out of my head ! The distinction is indeed often lost that a temperament, from the Latin 'TEMPERAMENTUM', meaning right proportion, middle way, mean, moderation, does not necessarily have anything to do with any notion of a 'Just Intonation' or any notion of a 'pure' ratio. I do also get the feeling, which others including Julia possibly share, that the implication is commonly made that anyone not subscribing to 'JI' is 'compromising' or deluding themselves. For example if I'm considering the tonal possibilities of 17ED2:1 as an temperament, it is a temperament to me for many reasons, of which the proximity to any conceived 'pure' ratios is not one.

> Some treasured discussions with Dave Keenan, both almost two years ago
> and now, have helped me to realize that much of what I do with RI is
> in another category. It is, as you very accurately and relevantly
> quote my earlier post, Joel, "nuancing" along a continuum with many
> shadings of concord and discord, and often in timbres where almost any
> interval can sound rather "concordant" (although stylistically fifths
> and fourths define the main stable concords in a neo-medieval
> setting).

> An aside on Jay Rahn's paper on Marchettus of Padua, Joel, a paper
> I've much enjoyed. Curiously, my Pythagorean-based tuning with that
> 750.145-cent interval could be taken as a very close approximation of
> the way Rahn interprets the intonational system described and
> recommended for singers in the _Lucidarium_ of Marchettus (1318).
>
> In Rahn's reading, Marchettus is describing the division of the 9:8
> whole-tone into "five parts" based on a monochord division, an idea
> also discussed by Joe Monzo here and on his excellent Web pages.
>
> Specifically, Marchettus advocates the use of a small melodic step in
> cadences he calls a "diesis" equal to "one of the five parts of a
> tone." Rahn takes this to refer to a monochord division of the 9:8
> tone (measured as 81:72) into five unequal parts, with the cadential
> diesis equal to about 74:72, or 37:36, about 47.43 cents. Rahn uses a
> value rounded up to 48 cents.
>
> My "Pythagorean enharmonic" tuning has two 12-note chains of pure
> fifths at a distance designed to produce some pure sonorities of
> 14:17:21 (about 0-336.13-701.96 cents). The two chains are therefore
> placed at a distance of 459:448, or about 41.99 cents, apart.
>
> As it happens, this method of "adding pitches" divides a regular
> Pythagorean diatonic semitone at 256:243 (~90.22 cents) into the
> 41.99-cent step I've mentioned, plus a larger quartertone-like diesis
> at 114688:111537 or about 48.30 cents -- almost identical to Rahn's
> suggested size for this 14th-century step, a Gothic "microinterval" in
> this reading of Marchettus.
>
> Anyway, in an RI view, I find that all of these intervals -- the pure
> fifths and fourths, the "pure" and relatively blending 14:17:21
> sonorities from a neo-medieval RI perspective, and the yet more
> complex ratios like the ones I've cited above are all part of a
> continuum.
>
> Note that I refer to 14:17:21 as "pure" in quotes, because it's an
> open question whether a listener taking a perceptual JI viewpoint such
> as that often explicated by Dave Keenan would notice anything special
> right around 14:17:21 proper, for example a combination-tone effect.
>
> From my point of view, the RI ratio 14:17:21 is a kind of affectionate
> name for a region of "supraminor/submajor" thirds somewhere around
> 330-342 cents and 360-372 cents -- 14:17 (~336.13 cents) and 17:21
> (~365.83 cents) are landmarks for this general region, and sometimes
> more specifically for its central portion. I tend to associate the
> ratio 63:52 (~332.21 cents) with the lower portion of the region of
> "supraminor" thirds, and 28:23 (~340.55 cents) with the upper portion.
>
> Curiously, while regarding these ratios as "pure," I do not
> necessarily regard other integer-based or irrational intervals in the
> region as "impure" or "compromised" -- they are more like fine nuances
> on a continuum, with integer-based ratios providing convenient names
> for the neighborhood and some of its subregions and gradations.
>
> In fact, this region is one of my favorites not only in RI systems
> (pure fifths and fourths), but in systems with fifths typically
> tempered gently wide of pure. What I celebrate is the continuum and
> all of its infinitesimal "nuances," whether defined by integer-based
> or irrational ratios.

A couple of days ago while I was thinking about similar things, I began using 'fractional intonation' to refer to such irrational ratios. I also tentatively labelled the class of scales with symmetrically repeating intervals, as 'Generated Intonation'.

> Also, from a practical perspective, I suspect as mentioned above that
> the timbres I choose often tend to make almost any interval seem
> rather "blending," smoothing over some of the contrasts between very
> obviously "pure" or "impure" intervals in the strongly harmonic
> timbres often associated with JI.

Which is one of the primary reasons behind my feeling that leaving out the term 'JI' in such a discussion is not only possible, but beneficial. Something like 'anomalous saturated suspensions' <http://www.cix.co.uk/~gbreed/ass.htm> can be better coped within a broader discussion space, like CI.

> To keep a balance, I should add that neo-medieval RI includes some
> very important intervals that _are_ "JI" by lots of traditional
> standards: 2:3:4 and 6:8:9 sonorities with pure fifths and fourths,
> and also "7-based" sonorities such as 12:14:18:21 and 14:18:21:24.

> However, in RI systems and in related temperaments based on irrational
> ratios alike, more complex intervals and sonorities are also included
> in the continuum of "relative concord," and their inclusion or
> "optimal approximation" celebrated.
>
> From a JI perspective, one might ask exactly what it means to
> "optimize" a complex ratio which might be perceptually
> indistinguishable from "tempered" sizes somewhat to either side.
>
> From an RI perspective, I might reply that including an interval
> somewhere in the general vicinity which evokes the "landmark" of the
> ratio as an auspicious emblem is one elegant explanation.
>
> Another side of this same pleasure is finding that a tuning system
> includes some interval such as an unanticipated and complex integer
> ratio, or some minute difference of size between two intervals clearly
> equivalent from any practical viewpoint (e.g. differences of around
> 0.1 cent, with a special delight if they happen to be defined by
> superparticular ratios).
>

> Here it may be characteristic that while JI-oriented theory often
> speaks of "tuning errors" or "deviations," RI-oriented theory prefers
> such terms as "variations" from a given ratio (not to be confused with
> statistical measures of "variance").

Very nicely put !

> Anyway, thank you Joel and Dave and Julia for this opportunity to seek
> a more "nuanced" discussion.

> This is already a long enough statement,

But very welcome. I enjoyed reading all this. Much food for thought, as always ! I hope it sparks similar feeling in others.

> so I conclude with an invitation to dialogue.

> Most appreciatively,
>
> Margo Schulter

Sincerely,
Joel

Joel Rodrigues wrote:

> Indeed, Margo you seem to steal (kidding !) thoughts right out
> of my head ! The distinction is indeed often lost that a
> temperament, from the Latin 'TEMPERAMENTUM', meaning right
> proportion, middle way, mean, moderation, does not necessarily
> have anything to do with any notion of a 'Just Intonation' or
> any notion of a 'pure' ratio. I do also get the feeling, which
> others including Julia possibly share, that the implication is
> commonly made that anyone not subscribing to 'JI' is
> 'compromising' or deluding themselves. For example if I'm
> considering the tonal possibilities of 17ED2:1 as an
> temperament, it is a temperament to me for many reasons, of
> which the proximity to any conceived 'pure' ratios is not one.

Oh no! We don't have to argue about the meaning of "temperament" again do
we? Well, you can start off by finding a dictionary that doesn't have a
notion of pure tuning.

Graham

--- In tuning@y..., graham@m... wrote:
> Joel Rodrigues wrote:
>
> > Indeed, Margo you seem to steal (kidding !) thoughts right out
> > of my head ! The distinction is indeed often lost that a
> > temperament, from the Latin 'TEMPERAMENTUM', meaning right
> > proportion, middle way, mean, moderation, does not necessarily
> > have anything to do with any notion of a 'Just Intonation' or
> > any notion of a 'pure' ratio. I do also get the feeling, which
> > others including Julia possibly share, that the implication is
> > commonly made that anyone not subscribing to 'JI' is
> > 'compromising' or deluding themselves. For example if I'm
> > considering the tonal possibilities of 17ED2:1 as an
> > temperament, it is a temperament to me for many reasons, of
> > which the proximity to any conceived 'pure' ratios is not one.
>
> Oh no! We don't have to argue about the meaning of "temperament"
again do
> we? Well, you can start off by finding a dictionary that doesn't
have a
> notion of pure tuning.

Indeed! Right proportion of what? Middle way between what? Mean of
what?

There is no value judgement here about whether just is better, however
the rather length OED entry (for example) on the musical meaning of
the term, specifically includes the words "... consisting in slight
variations of the pitch of the notes from true or 'just' intonation
...".

There is likewise no value judgement intended by the use of the term
"error" to refer to the difference between a tempered interval and
a nearby just interval. The term is standard in mathematics for any
such deviation from some reference point. But I take Margo's point
that it might be wise to use another term given the feelings on the
list.

On Saturday, July 13, 2002, at 09:46 , tuning@yahoogroups.com wrote:

> "dkeenanuqnetau" <d.keenan@uq.net.au>
>
> --- In tuning@y..., graham@m... wrote:
>> Joel Rodrigues wrote:
>>
>>> Indeed, Margo you seem to steal (kidding !) thoughts right out
>>> of my head ! The distinction is indeed often lost that a
>>> temperament, from the Latin 'TEMPERAMENTUM', meaning right
>>> proportion, middle way, mean, moderation, does not necessarily
>>> have anything to do with any notion of a 'Just Intonation' or
>>> any notion of a 'pure' ratio. I do also get the feeling, which
>>> others including Julia possibly share, that the implication is
>>> commonly made that anyone not subscribing to 'JI' is
>>> 'compromising' or deluding themselves. For example if I'm
>>> considering the tonal possibilities of 17ED2:1 as an
>>> temperament, it is a temperament to me for many reasons, of
>>> which the proximity to any conceived 'pure' ratios is not one.
>>
>> Oh no! We don't have to argue about the meaning of "temperament"
> again do
>> we? Well, you can start off by finding a dictionary that doesn't
> have a
>> notion of pure tuning.
>
> Indeed! Right proportion of what? Middle way between what? Mean of
> what?
>
> There is no value judgement here about whether just is better, however
> the rather length OED entry (for example) on the musical meaning of
> the term, specifically includes the words "... consisting in slight
> variations of the pitch of the notes from true or 'just' intonation
> ...".

Perhaps I can explain with another example. I sometimes examine a scale to look for intervals close to 7ED2:1, nothing more, nothing less.

> There is likewise no value judgement intended by the use of the term
> "error" to refer to the difference between a tempered interval and
> a nearby just interval.

There is an easily made perception that a value judgement is made. Do the words 'true', 'pure', or 'just' ring any bells ?

> The term is standard in mathematics for any
> such deviation from some reference point.

Exactly. That reference point may or may not even be a rational interval, let alone a JI one.

> But I take Margo's point
> that it might be wise to use another term given the feelings on the
> list.

I for one would be happy never to hear the words "Just Intonation' again.

- Joel

Hello Margo,

I could be mistaken, but I found parallels in your ideas
expressed about nuancing & Rational Intonation with sentiments
expressed by Ptolemy and Vincenzo Galilei. To put it another
way, I can see how to use your your ideas to put into practise
their suggestions as gleaned from the quotes cited below. I
think you may be amused by some of this, in particular the last
line in the last paragraph quoted below.

'In Chapter Two, he (Boethius) jumps into the Harmonics'
discussion of the relationship between sense (hearing) and
reason, noting that "the entire judgement is not to be granted
to the sense of hearing; rather, reason must also play a role. Reason should guide and moderate the erring sense, inasmuch as
the sense-tottering and failing-should be supported, as it were,
by a walking stick." In Chapter Three, Boethius discusses
Ptolemy's configuration of the goal of the discipline of
Harmonics: "According to Ptolemy the harmonic scholar directs
his activity in such a way that what the sense estimates, reason
weighs; accordingly, reason searches out ratios to which the
sense expresses no objection."'
<http://humanities.uchicago.edu/classes/zbikowski/week3sum.html>

'Ptolemy begins the Harmonics stating that "the criteria of
harmonia are hearing and reason, not however in the same way."
From the beginning Ptolemy links together perception and reason.
Perception is good at judging and making comparisons, but as
distinctions become smaller and finer, judgement can become
deficient. Reason, working through methods and with appropriate
instruments, is able to accurately detect those things that
perception missed. Ptolemy claims that students' methods should
use reasoning, but also never contradict the perception.'
<http://humanities.uchicago.edu/classes/zbikowski/week2sum.html>

Also see the introduction to Andrew Barker's 'Scientific Method
in Ptolemy’s Harmonics', in PDF at:
<http://books.cambridge.org/0521553725.htm>

'Vincenzo Galilei, although his student, was another thorn in
Zarlino's side. Influenced by Aristoxenus, Galilei challenged
Zarlino's neo-Pythagorean methods, supporting instead a system
of consonances based entirely on "pure sense experience and
artistic usage." All intervals, Galilei argued, are natural, not
simply those mathematically determined. In performance, a 5th
that is a bit off the 3:2 ratio is just as useful as one that is
exactly on the mark. He challenged Zarlino's belief in the
universal harmony of music.

Galilei was also one of the first to impugn the Pythagorean
myth, namely the harmonious qualities of the numbers 6:8:9:12.
Through numerous experiments, Galilei showed that a 2:1 was not
the only ratio for an octave. With two different coins, the
ratio was 3:1, by altering the tension of a string, the ratio
became 4:1, and by changing the volume of an organ pipe, the
ratio was 8:1. This particular experiment was only a part of
Galilei's larger empirical approach. Galilei set up an
experiment to test what tuning system was actually used by
musicians. In the end, neither of the two predetermined systems
were used at all in common practice. The ear, Galilei argued,
had no regard for systems. Only the octave itself could be
critically determined. All other intervals were flexible, part
of a continuum.'
<http://humanities.uchicago.edu/classes/zbikowski/week9sum.html>

Best,
Joel