Re: 7-limit thinking -- for Bob Wendell

Hello, there, Bob Wendell and Paul Erlich and everyone.

Bob, you asked a very fair and provocative question: how do we know
what kinds of vocal intonations were used for different intervals and
sonorities in the medieval era in Europe.

One brief and obvious answer, which as an enthusiast for Pythagorean
tuning I feel especially important to acknowledge, is that since we
don't have tapes or CD's of performances from that era, whatever we
reconstruct or advocate as "period intonation" is an educated guess,
however informed.

Such a guess will be informed not only by the writings of that era,
but by our own musical experiences with compositions and styles from
this era, and maybe also related improvisations or compositions in
styles inspired by 13th-14th century practice.

At the same time, in approaching this question, I'm aware that not
everything I happen personally to like very much in the intonation of
this music was necessarily practice during the era, although a
theorist like Marchettus of Padua may give valuable clues about
flexible pitch ensembles in some localities and epochs.

To say that flexible-pitch ensembles may generally have followed a
"Pythagorean" type of intonation is obviously not to say that the
elementary intervals of the system were tuned precisely at 2:1, 3:2,
4:3, and 9:8, nor that more complex intervals such as major and minor
thirds were always tuned precisely at 81:64 and 32:27.

The variability of human vocal intonation, not to speak of possible
contextual factors pulling a note or sonority in one direction or
another, would preclude such an unlikely interpretation.

However, the argument for a generally Pythagorean-like intonation
follows three basic lines:

(1) Pythagorean is an excellent tuning for the various
modes or octave species of medieval monophony and
polyphony, with its generous 9:8 whole-tones around
204 cents, and its compact diatonic semitones at
256:243, around 90 cents;

(2) Pythagorean nicely fits the vertical parameters of
style, with pure concords of 2:3:4 or 3:4:6, and
also relatively concordant sonorities at 4:6:9 or
6:8:9, for example; rather complex thirds and sixths
nicely resolving cadentially to stable concords; and
a rich spectrum of concord/discord fitting the subtle
theory of the period;

(3) Pythagorean, while providing a regular tuning structure,
neatly fits the 14th-century ideal of "closest approach"
resolutions (e.g. m3-1, M3-5, M6-8, m7-5, M2-4) where
the unstable interval resolves by contrary motion as
efficiently as possible to its stable goal -- the total
motion being equal to a 32:27 minor third, or about
294 cents.

We can cite various kinds of medieval statements on these points, for
example the use of the _minor_ semitone at 256:243 as the "singable"
one -- in contrast to Renaissance theorists who tell us that the
_major_ semitone is the singable (or more usually sung) one.

We can also cite the graduated scales of concord/discord in the 13th
century, the "closest approach" doctrine of the 14th and early 15th
centuries, and the taste expressed by theorists for "fully perfected"
major thirds and sixths at 81:64 (~408 cents) and 27:16 (~906 cents),
which expand most efficiently to stable fifths and octaves
respectively.

However, this is theory -- at best, an approximate model, given the
kind of variability that singers and flexible pitch instrumentalists
find at once unavoidable and often expressive.

For example, I have seen a study of string ensemble intonation dating
to around 1948 showing that the surveyed players tend to come very
close to Pythagorean -- but with some variations, of course.

Now we come to my own opinion, speaking as a keyboard player from a
vertical perspective, and realizing that singers may be influenced by
intuitive melodic considerations to which I might be much less
sensitive.

From a vertical point of view, in a 14th-century kind of style I often
like semitones _narrower_ than Pythagorean, and major thirds and
sixths _wider_, approaching ratios of 9:7 and 12:7 rather than 5:4 and
5:3 (the latter, of course, fitting Renaissance music).

The attraction of tuning a cadential sonority like G3-B3-E4 before
F3-C4-F4 at 7:9:12 (around 0-435-933 cents) is that we get "smoother"
thirds and sixths at "simpler" ratios, and the same time make the
cadence yet more efficient than in Pythagorean, with semitones of
28:27 or about 63 cents. One could argue that this is "optimizing"
both vertical and melodic dimensions at the same time.

However, the usual Pythagorean intonation with its active and "bouncy"
major thirds and sixths is also very attractive -- in a keyboard
improvisation, I might use both forms as free variations.

Possibly some remarks about vocal intonation by Marchettus of Padua in
1318, pointing to cadential semitones narrower than Pythagorean so
that major thirds and sixths can "more closely approach" the stable
fifths and octaves toward which they strive, provide a kind of
historical sanction for such variations.

Taken literally, Marchettus seems to go yet further, suggesting a
tuning of something like A3-C#4-F#4 before G3-D4-G4 -- note the
element of accidental inflection outside the regular gamut of Bb-B --
at around 30:39:52, or 0-454-952 cents, with a cadential diesis only
about half the size of a usual diatonic semitone. He says that the
cadential major sixth is equally distant from a 2:1 octave or a 3:2
fifth, suggesting a size of about 951 cents, if we take this as a
geometric mean.

If one takes a "sweet spot" approach of leaning toward simple ratios,
then 7:9:12 would be very attractive in providing such ratios which
nicely fit the ideal of closest approach.

Another question one could raise from this kind of viewpoint is that
of the minor seventh, a prominent interval in some pieces by Perotin
and Machaut and others. Could this lean toward 7:4 or around 969
cents, possibly favored along with a 7:6 minor third for sonorities
combining these two intervals?

If one follows the recorded theory, or performs melodic regularity,
then the usual Pythagorean 16:9 (around 996 cents) and 32:27 (around
294 cents) are fine, and I would consider them musically effective as
well as "theoretically correct."

For example, there's a piece of English provenance from around the
late 14th century with some sonorities like D3-F3-C4 which have for me
a rather "floating" quality, suggesting what Jacobus of Liege calls
the "imperfect concord" of the minor seventh. The quality comes
through for me in a usual Pythagorean tuning.

However, especially in directed cadential progressions, a tuning of
12:14:21 (0-267-969 cents) can be very appealing, again at once
"simple" or "streamlined" vertically, and very efficient in terms of
closest approach, e.g. D3-F3-C4 before E3-B3, or E3-G3-D4 before
F3-C4.

Here I'm inclined to guess that 14th-century practice may generally
have stayed rather close to Pythagorean, but that either this tuning
or a "7-flavor" tuning at or near 12:14:21 is fitting for new
21st-century music in this kind of style.

In my own music of this kind, I'm very enthusiastic about using a full
four-voice sonority of 14:18:21:24 or 0-435-702-933 cents,
e.g. G3-B3-D4-E4 before F3-C4-F4; or 12:14:18:21 or 0-267-702-969,
e.g. G3-Bb3-D4-F4 before A3-E4.

However, this is a new kind of interpretation rather than an attempt
at following period practice -- although Marchettus might give a kind
of problematic license to guess that some ensembles _might_ have used
this kind of intonation at certain cadences.

Incidentally, one might argue that Pythagorean intonation closely
approximates a "sweet spot" for the _quinta fissa_ or "split fifth"
sonority of Jacobus -- outer fifth "split" into major and minor third
by a middle voice -- in its form with the minor third below,
e.g. A3-C4-E4.

In Pythagorean tuning, this will be 54:64:81 or about 0-294-702 cents,
not too far from 16:19:24 or 0-297-702 cents. One could argue that the
16:19:24-like qualities of this Pythagorean sonority would tend to
accentuate its relatively concordant feeling, "sweet" although
unstable.

Curiously, at least one English piece around 1300 and also, as I
recall, a couple of pieces from the Apt manuscript on the Continent of
around 1400, have concluding sonorities of this type; here
Pythagorean, possibly shaded by singers toward 16:19:24, might
actually be more conclusive than a 5-limit tuning at 10:12:15
(around 0-316-702 cents).

To Paul's comment about the "style of the Western Isles" influencing a
shift toward 5-limit in the era of Dunstable and Dufay, I would agree,
while adding that the apparent tendency toward some prominent schisma
third sonorities in the instrumental music of around 1400 in Italy
(Faenza Codex) might suggest a potential leaning in this direction
also.

Dufay's early style, combining as it does traditional French elements
of the 14th-century Ars Nova with Italian elements of the new century
and the "English countenance" of Dunstable, may draw on both sources
for a new kind of intonational approach.

From my own JI perspective, I would say that Pythagorean tuning
optimizes the stable 2:3:4 sonority of Gothic music, analogous to
4:5:6 in the Renaissance or 4:5:6:7 in your tetradic/decatonic system,
Paul.

It also nicely fits the scale of concord/discord, the melodic
structure of the music, and the ideal of "closest approach"
progressions.

However, if one wants to adjust unstable sonorities to find "sweet
spots" -- or harmonic entropy "valleys" -- then we have such choices
open as 6:7:9, 7:9:12, or 12:14:21, or 16:19:24, as well as the
subsequent Renaissance preferences for 4:5:6 or 10:12:15, etc.

Without tapes or CD's, we can only guess what ensembles may have done;
I find Pythagorean a fine model in theory and practice, but not ruling
out various experiments and less conventional offshoots, some of which
I have discussed here.

Most appreciatively,

Margo Schulter
mschulter@value.net

--- In tuning@y..., mschulter <MSCHULTER@V...> wrote:

> From a vertical point of view, in a 14th-century kind of style I
often
> like semitones _narrower_ than Pythagorean, and major thirds and
> sixths _wider_, approaching ratios of 9:7 and 12:7 rather than 5:4
and
> 5:3 (the latter, of course, fitting Renaissance music).

Have you tried playing 14th century music on a 22-tone keybaord? It
has sharp major tones, splitting the difference from 9/8 to 8/7, and
nearly pure 9/7s when we take two major tones in sequence. It might
be interesting to hear some late Medieval examples tuned in various
systems, including the 22-tone system.

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., mschulter <MSCHULTER@V...> wrote:
>
> > From a vertical point of view, in a 14th-century kind of style I
> often
> > like semitones _narrower_ than Pythagorean, and major thirds and
> > sixths _wider_, approaching ratios of 9:7 and 12:7 rather than
5:4
> and
> > 5:3 (the latter, of course, fitting Renaissance music).
>
> Have you tried playing 14th century music on a 22-tone keybaord?

Margo has discussed just this is quite a bit of detail . . . try
searching the archives . . .

Hi, Margo! I'm sure I speak for us all here when I express my deep
appreciation for yours, Paul's, Monz's and a few others' erudition
with respect to intonational history! Thank you for your many
wonderful and detailed contributions. Your knowledge is a blessing
for me.

To clarify my thinking a bit, I do not question the role of melodic
habit in infuencing intonation. In those who do not specifically pay
close attention to tuning harmonies (which represents a the vast
majority of singers, for example, in the common practice music of
today), melodic habit is essentially all there is. I am simply
suggesting that the EVOLUTION of intonational preferences is more
strongly influenced by the attractive power of just ratios than some
here seem to allow, if I understand them correctly.

The microtonal commas, schismas, etc. that were clearly recognized
and discussed historically in medieval times, as well as before and
after, indicate a sensitivity to fine harmonic distinctions and
considerations that is generally lacking in today's world of common
practice music. Such sensitivity implies to me more than the ability
to tune keyboards according to accepted practice. I feel it also
implies a sense of both harmonic and melodic discrimination that was
quite highly developed.

Given these "truths", it seems likely that the attractive power of
just intervals other than 2:3 would not have been forever lost on
musicians as they evolved ever richer polyphonic textures. This must
have been especially true of vocalists and other pitch-flexible souls
whose sensitized ears were capable of picking up on fairly subtle
musical acoustic phenomena.

Someone stated recently in one of these threads that Pythagorean
tuning endured for "many centuries" in spite of the attractive
potential of 4:5 thirds. If we look at the evolutionary long view of
European church music, this tuning in fact endured relatively briefly
vis-a-vis its previous long history in a MELODIC environment once the
textures got really sophisticated polyphonically (implying
harmonically).

I believe it endured as long as it did only because of the treatment
of thirds as dissonances (tuning dissonances) requiring resolution to
fifths, fourths, and octaves in the polyphonic styles of the periods
in question. This was a self-reinforcing cycle of "thirds avoidance"
as a stable consonance that delayed the ultimate triumph of the JI
4:5 attractive power.

I would like to also refer you, Margo, to some of the other postings
I have made today that are also relevant to the issues you discuss
here.

Gratefully,

Bob

--- In tuning@y..., mschulter <MSCHULTER@V...> wrote:
> Hello, there, Bob Wendell and Paul Erlich and everyone.
>
> Bob, you asked a very fair and provocative question: how do we know
> what kinds of vocal intonations were used for different intervals
and
> sonorities in the medieval era in Europe.
>
> One brief and obvious answer, which as an enthusiast for Pythagorean
> tuning I feel especially important to acknowledge, is that since we
> don't have tapes or CD's of performances from that era, whatever we
> reconstruct or advocate as "period intonation" is an educated guess,
> however informed.
>
> Such a guess will be informed not only by the writings of that era,
> but by our own musical experiences with compositions and styles from
> this era, and maybe also related improvisations or compositions in
> styles inspired by 13th-14th century practice.
>
> At the same time, in approaching this question, I'm aware that not
> everything I happen personally to like very much in the intonation
of
> this music was necessarily practice during the era, although a
> theorist like Marchettus of Padua may give valuable clues about
> flexible pitch ensembles in some localities and epochs.
>
> To say that flexible-pitch ensembles may generally have followed a
> "Pythagorean" type of intonation is obviously not to say that the
> elementary intervals of the system were tuned precisely at 2:1, 3:2,
> 4:3, and 9:8, nor that more complex intervals such as major and
minor
> thirds were always tuned precisely at 81:64 and 32:27.
>
> The variability of human vocal intonation, not to speak of possible
> contextual factors pulling a note or sonority in one direction or
> another, would preclude such an unlikely interpretation.
>
> However, the argument for a generally Pythagorean-like intonation
> follows three basic lines:
>
> (1) Pythagorean is an excellent tuning for the various
> modes or octave species of medieval monophony and
> polyphony, with its generous 9:8 whole-tones around
> 204 cents, and its compact diatonic semitones at
> 256:243, around 90 cents;
>
> (2) Pythagorean nicely fits the vertical parameters of
> style, with pure concords of 2:3:4 or 3:4:6, and
> also relatively concordant sonorities at 4:6:9 or
> 6:8:9, for example; rather complex thirds and sixths
> nicely resolving cadentially to stable concords; and
> a rich spectrum of concord/discord fitting the subtle
> theory of the period;
>
> (3) Pythagorean, while providing a regular tuning
structure,
> neatly fits the 14th-century ideal of "closest
approach"
> resolutions (e.g. m3-1, M3-5, M6-8, m7-5, M2-4) where
> the unstable interval resolves by contrary motion as
> efficiently as possible to its stable goal -- the
total
> motion being equal to a 32:27 minor third, or about
> 294 cents.
>
> We can cite various kinds of medieval statements on these points,
for
> example the use of the _minor_ semitone at 256:243 as the "singable"
> one -- in contrast to Renaissance theorists who tell us that the
> _major_ semitone is the singable (or more usually sung) one.
>
> We can also cite the graduated scales of concord/discord in the 13th
> century, the "closest approach" doctrine of the 14th and early 15th
> centuries, and the taste expressed by theorists for "fully
perfected"
> major thirds and sixths at 81:64 (~408 cents) and 27:16 (~906
cents),
> which expand most efficiently to stable fifths and octaves
> respectively.
>
> However, this is theory -- at best, an approximate model, given the
> kind of variability that singers and flexible pitch instrumentalists
> find at once unavoidable and often expressive.
>
> For example, I have seen a study of string ensemble intonation
dating
> to around 1948 showing that the surveyed players tend to come very
> close to Pythagorean -- but with some variations, of course.
>
> Now we come to my own opinion, speaking as a keyboard player from a
> vertical perspective, and realizing that singers may be influenced
by
> intuitive melodic considerations to which I might be much less
> sensitive.
>
> From a vertical point of view, in a 14th-century kind of style I
often
> like semitones _narrower_ than Pythagorean, and major thirds and
> sixths _wider_, approaching ratios of 9:7 and 12:7 rather than 5:4
and
> 5:3 (the latter, of course, fitting Renaissance music).
>
> The attraction of tuning a cadential sonority like G3-B3-E4 before
> F3-C4-F4 at 7:9:12 (around 0-435-933 cents) is that we get
"smoother"
> thirds and sixths at "simpler" ratios, and the same time make the
> cadence yet more efficient than in Pythagorean, with semitones of
> 28:27 or about 63 cents. One could argue that this is "optimizing"
> both vertical and melodic dimensions at the same time.
>
> However, the usual Pythagorean intonation with its active and
"bouncy"
> major thirds and sixths is also very attractive -- in a keyboard
> improvisation, I might use both forms as free variations.
>
> Possibly some remarks about vocal intonation by Marchettus of Padua
in
> 1318, pointing to cadential semitones narrower than Pythagorean so
> that major thirds and sixths can "more closely approach" the stable
> fifths and octaves toward which they strive, provide a kind of
> historical sanction for such variations.
>
> Taken literally, Marchettus seems to go yet further, suggesting a
> tuning of something like A3-C#4-F#4 before G3-D4-G4 -- note the
> element of accidental inflection outside the regular gamut of Bb-B
--
> at around 30:39:52, or 0-454-952 cents, with a cadential diesis only
> about half the size of a usual diatonic semitone. He says that the
> cadential major sixth is equally distant from a 2:1 octave or a 3:2
> fifth, suggesting a size of about 951 cents, if we take this as a
> geometric mean.
>
> If one takes a "sweet spot" approach of leaning toward simple
ratios,
> then 7:9:12 would be very attractive in providing such ratios which
> nicely fit the ideal of closest approach.
>
> Another question one could raise from this kind of viewpoint is that
> of the minor seventh, a prominent interval in some pieces by Perotin
> and Machaut and others. Could this lean toward 7:4 or around 969
> cents, possibly favored along with a 7:6 minor third for sonorities
> combining these two intervals?
>
> If one follows the recorded theory, or performs melodic regularity,
> then the usual Pythagorean 16:9 (around 996 cents) and 32:27 (around
> 294 cents) are fine, and I would consider them musically effective
as
> well as "theoretically correct."
>
> For example, there's a piece of English provenance from around the
> late 14th century with some sonorities like D3-F3-C4 which have for
me
> a rather "floating" quality, suggesting what Jacobus of Liege calls
> the "imperfect concord" of the minor seventh. The quality comes
> through for me in a usual Pythagorean tuning.
>
> However, especially in directed cadential progressions, a tuning of
> 12:14:21 (0-267-969 cents) can be very appealing, again at once
> "simple" or "streamlined" vertically, and very efficient in terms of
> closest approach, e.g. D3-F3-C4 before E3-B3, or E3-G3-D4 before
> F3-C4.
>
> Here I'm inclined to guess that 14th-century practice may generally
> have stayed rather close to Pythagorean, but that either this tuning
> or a "7-flavor" tuning at or near 12:14:21 is fitting for new
> 21st-century music in this kind of style.
>
> In my own music of this kind, I'm very enthusiastic about using a
full
> four-voice sonority of 14:18:21:24 or 0-435-702-933 cents,
> e.g. G3-B3-D4-E4 before F3-C4-F4; or 12:14:18:21 or 0-267-702-969,
> e.g. G3-Bb3-D4-F4 before A3-E4.
>
> However, this is a new kind of interpretation rather than an attempt
> at following period practice -- although Marchettus might give a
kind
> of problematic license to guess that some ensembles _might_ have
used
> this kind of intonation at certain cadences.
>
> Incidentally, one might argue that Pythagorean intonation closely
> approximates a "sweet spot" for the _quinta fissa_ or "split fifth"
> sonority of Jacobus -- outer fifth "split" into major and minor
third
> by a middle voice -- in its form with the minor third below,
> e.g. A3-C4-E4.
>
> In Pythagorean tuning, this will be 54:64:81 or about 0-294-702
cents,
> not too far from 16:19:24 or 0-297-702 cents. One could argue that
the
> 16:19:24-like qualities of this Pythagorean sonority would tend to
> accentuate its relatively concordant feeling, "sweet" although
> unstable.
>
> Curiously, at least one English piece around 1300 and also, as I
> recall, a couple of pieces from the Apt manuscript on the Continent
of
> around 1400, have concluding sonorities of this type; here
> Pythagorean, possibly shaded by singers toward 16:19:24, might
> actually be more conclusive than a 5-limit tuning at 10:12:15
> (around 0-316-702 cents).
>
> To Paul's comment about the "style of the Western Isles"
influencing a
> shift toward 5-limit in the era of Dunstable and Dufay, I would
agree,
> while adding that the apparent tendency toward some prominent
schisma
> third sonorities in the instrumental music of around 1400 in Italy
> (Faenza Codex) might suggest a potential leaning in this direction
> also.
>
> Dufay's early style, combining as it does traditional French
elements
> of the 14th-century Ars Nova with Italian elements of the new
century
> and the "English countenance" of Dunstable, may draw on both sources
> for a new kind of intonational approach.
>
> From my own JI perspective, I would say that Pythagorean tuning
> optimizes the stable 2:3:4 sonority of Gothic music, analogous to
> 4:5:6 in the Renaissance or 4:5:6:7 in your tetradic/decatonic
system,
> Paul.
>
> It also nicely fits the scale of concord/discord, the melodic
> structure of the music, and the ideal of "closest approach"
> progressions.
>
> However, if one wants to adjust unstable sonorities to find "sweet
> spots" -- or harmonic entropy "valleys" -- then we have such choices
> open as 6:7:9, 7:9:12, or 12:14:21, or 16:19:24, as well as the
> subsequent Renaissance preferences for 4:5:6 or 10:12:15, etc.
>
> Without tapes or CD's, we can only guess what ensembles may have
done;
> I find Pythagorean a fine model in theory and practice, but not
ruling
> out various experiments and less conventional offshoots, some of
which
> I have discussed here.
>
> Most appreciatively,
>
> Margo Schulter
> mschulter@v...

--- In tuning@y..., BobWendell@t... wrote:

> I believe it endured as long as it did only because of the
treatment
> of thirds as dissonances (tuning dissonances) requiring resolution
to
> fifths, fourths, and octaves in the polyphonic styles of the
periods
> in question. This was a self-reinforcing cycle of "thirds
avoidance"
> as a stable consonance that delayed the ultimate triumph of the JI
> 4:5 attractive power.

If you are willing to accept this, then why wouldn't you accept that,
for a long period in the 5-limit era, sonorities such as dominant
seventh chords, which were treated as dissonances requiring strict
resolution of the parts, were not in fact performed in a 7-limit
tuning?

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., BobWendell@t... wrote:
>
> > I believe it endured as long as it did only because of the
> treatment
> > of thirds as dissonances (tuning dissonances) requiring
resolution
> to
> > fifths, fourths, and octaves in the polyphonic styles of the
> periods
> > in question. This was a self-reinforcing cycle of "thirds
> avoidance"
> > as a stable consonance that delayed the ultimate triumph of the
JI
> > 4:5 attractive power.
>
> If you are willing to accept this, then why wouldn't you accept
that,
> for a long period in the 5-limit era, sonorities such as dominant
> seventh chords, which were treated as dissonances requiring strict
> resolution of the parts, were not in fact performed in a 7-limit
> tuning?

Bob:
See my post just previous to this, in which I explain what I mean by
drive toward resolution owing to a tuning dissonance (e.g.,
Pythagorean thirds) and drive toward resolution owing to intrinsic
harmonic dissonance (e.g., JI tritones).

--- In tuning@y..., BobWendell@t... wrote:

/tuning/topicId_28495.html#28555
>
> The microtonal commas, schismas, etc. that were clearly recognized
> and discussed historically in medieval times, as well as before and
> after, indicate a sensitivity to fine harmonic distinctions and
> considerations that is generally lacking in today's world of common
> practice music. Such sensitivity implies to me more than the
ability to tune keyboards according to accepted practice. I feel it
also implies a sense of both harmonic and melodic discrimination that
was quite highly developed.
>

Hello Bob!

Well, the Pythagorean third is quite a bit different from the just
third, so I'm not so sure the discrimination had to be so refined...??

What captivates me is the fact that possibly the composers of that
time got hold of a *new* sound, a *new* third, schizmatic, that
didn't seem to need to resolve so much, and which could be held over
as a sustained harmony, with melodic lines going through it...

This would probably be a very "modern" concept at that time and since
the composers were conceivably hearing other people use the 5:4, or
at least parallel minor 6ths (isn't that Dunstable?) they might
consider this long, sustained, practically "minimalistic" sonority
(by *our* standards) since it was already present in the Pythagorean
tuning....

The "birth" of the textural 5:4...

_______ _______ __________
Joseph Pehrson

Joseph ! :)

What do you discute here ?
A just third ?
Do you know what's one just third ?
Look in the harmonics the place of ...
And look how OCTAVES also !
Why you spend lot of time for nothing ?
The third is too fare in the spectre(harmonic or
other) to respect so hardly :)
I ask me -when we will speak really about any usefull
tning ?

Dimitrov

--- jpehrson@rcn.com a �crit�: > --- In tuning@y...,
BobWendell@t... wrote:
>
> /tuning/topicId_28495.html#28555
> >
> > The microtonal commas, schismas, etc. that were
> clearly recognized
> > and discussed historically in medieval times, as
> well as before and
> > after, indicate a sensitivity to fine harmonic
> distinctions and
> > considerations that is generally lacking in
> today's world of common
> > practice music. Such sensitivity implies to me
> more than the
> ability to tune keyboards according to accepted
> practice. I feel it
> also implies a sense of both harmonic and melodic
> discrimination that
> was quite highly developed.
> >
>
> Hello Bob!
>
> Well, the Pythagorean third is quite a bit different
> from the just
> third, so I'm not so sure the discrimination had to
> be so refined...??
>
> What captivates me is the fact that possibly the
> composers of that
> time got hold of a *new* sound, a *new* third,
> schizmatic, that
> didn't seem to need to resolve so much, and which
> could be held over
> as a sustained harmony, with melodic lines going
> through it...
>
> This would probably be a very "modern" concept at
> that time and since
> the composers were conceivably hearing other people
> use the 5:4, or
> at least parallel minor 6ths (isn't that Dunstable?)
> they might
> consider this long, sustained, practically
> "minimalistic" sonority
> (by *our* standards) since it was already present in
> the Pythagorean
> tuning....
>
> The "birth" of the textural 5:4...
>
> _______ _______ __________
> Joseph Pehrson
>
>
>

___________________________________________________________
Do You Yahoo!? -- Un e-mail gratuit @yahoo.fr !
Yahoo! Courrier : http://fr.mail.yahoo.com

--- In tuning@y..., Latchezar Dimitrov <latchezar_d@y...> wrote:

/tuning/topicId_28495.html#28592

> Joseph ! :)
>
> What do you discute here ?
> A just third ?
> Do you know what's one just third ?
> Look in the harmonics the place of ...
> And look how OCTAVES also !
> Why you spend lot of time for nothing ?
> The third is too fare in the spectre(harmonic or
> other) to respect so hardly :)
> I ask me -when we will speak really about any usefull
> tning ?
>
> Dimitrov
>

Hello Latch!

Basically we were discussing Margo Schulter's post here:

/tuning/topicId_28294.html#28367

And it involves the two kinds of "major" thirds that are present in
Pythagorean tuning: a "schismatic" (not "schizoid" but "schismoid")
third (say from A to Db) and a regular Pythagorean one (say from G to
B) generated from a chain of Pythagorean thirds from Gb to B.

This is a theory of Medieval/Renaissance harmonic "evolution" from
the 3-limit to 5-limit thirds as advanced by scholar Mark Lindley.

The English is kind of difficult, though, so probably you'd better
get our your dictionary... and good luck!

________ ________ _______
Joseph Pehrson

No problem, Joseph :))

When will do you respond me what's for you one just
third ?
If no, ok -so sorry but my electronic english-french
card dont work actually :)
Dont profit,pls :P

Latchezar

--- jpehrson@rcn.com a �crit�: > --- In tuning@y...,
Latchezar Dimitrov
> <latchezar_d@y...> wrote:
>
> /tuning/topicId_28495.html#28592
>
> > Joseph ! :)
> >
> > What do you discute here ?
> > A just third ?
> > Do you know what's one just third ?
> > Look in the harmonics the place of ...
> > And look how OCTAVES also !
> > Why you spend lot of time for nothing ?
> > The third is too fare in the spectre(harmonic or
> > other) to respect so hardly :)
> > I ask me -when we will speak really about any
> usefull
> > tning ?
> >
> > Dimitrov
> >
>
> Hello Latch!
>
> Basically we were discussing Margo Schulter's post
> here:
>
> /tuning/topicId_28294.html#28367
>
>
> And it involves the two kinds of "major" thirds that
> are present in
> Pythagorean tuning: a "schismatic" (not "schizoid"
> but "schismoid")
> third (say from A to Db) and a regular Pythagorean
> one (say from G to
> B) generated from a chain of Pythagorean thirds from
> Gb to B.
>
>
> This is a theory of Medieval/Renaissance harmonic
> "evolution" from
> the 3-limit to 5-limit thirds as advanced by scholar
> Mark Lindley.
>
>
> The English is kind of difficult, though, so
> probably you'd better
> get our your dictionary... and good luck!
>
> ________ ________ _______
> Joseph Pehrson
>
>
>
>

___________________________________________________________
Do You Yahoo!? -- Un e-mail gratuit @yahoo.fr !
Yahoo! Courrier : http://fr.mail.yahoo.com

--- In tuning@y..., Latchezar Dimitrov <latchezar_d@y...> wrote:

/tuning/topicId_28495.html#28596

> No problem, Joseph :))
>
> When will do you respond me what's for you one just
> third ?
> If no, ok -so sorry but my electronic english-french
> card dont work actually :)
> Dont profit,pls :P
>
> Latchezar
>

I'm not following you, Latch... A "just third" has to be a 5:4 ratio
at 386 cents...?? At least, it was last time *I* checked on it...

_______ _______ ______
Joseph Pehrson

Hi, Joseph! Yes, I'm quite sure this interesting mathematical tuning
coincidence was a significant contributor to the evolution toward the
use of thirds as consonant and the advent of meantone predominance.
However, it should be obvious that the attractive power of 4:5 was
behind even the interest that this schisma held for composers and
musicians of the time.

It is interesting to ask a chicken and egg question. Was the schisma
reinforcing the attractive power of 4:5 or vice versa? Hard to answer
that one without going back there in a time machine, but who knows?
Maybe Margo has some interesting contribution to make on this point.

Also has anyone pointed out to all interested readers that the
schisma thirds are a result of the happy mathematical proximity of
the synontic and Pythagorean commas, which differ only by 1.96 cents?

If you go up four just fifths, you will be a syntonic comma sharp on
the third you find at the top. Then if you back down 12 fifths for
the full Pythagorean cycle, you will slightly more than cancel out
the syntonic comma with the negative Pythagorean, ending up with a
major third only two cents flat to a perfect 4:5! Mighty pure, folks!

--- In tuning@y..., jpehrson@r... wrote:
> --- In tuning@y..., BobWendell@t... wrote:
>
> /tuning/topicId_28495.html#28555
> >
> > The microtonal commas, schismas, etc. that were clearly
recognized
> > and discussed historically in medieval times, as well as before
and
> > after, indicate a sensitivity to fine harmonic distinctions and
> > considerations that is generally lacking in today's world of
common
> > practice music. Such sensitivity implies to me more than the
> ability to tune keyboards according to accepted practice. I feel it
> also implies a sense of both harmonic and melodic discrimination
that
> was quite highly developed.
> >
>
> Hello Bob!
>
> Well, the Pythagorean third is quite a bit different from the just
> third, so I'm not so sure the discrimination had to be so
refined...??
>
> What captivates me is the fact that possibly the composers of that
> time got hold of a *new* sound, a *new* third, schizmatic, that
> didn't seem to need to resolve so much, and which could be held
over
> as a sustained harmony, with melodic lines going through it...
>
> This would probably be a very "modern" concept at that time and
since
> the composers were conceivably hearing other people use the 5:4, or
> at least parallel minor 6ths (isn't that Dunstable?) they might
> consider this long, sustained, practically "minimalistic" sonority
> (by *our* standards) since it was already present in the
Pythagorean
> tuning....
>
> The "birth" of the textural 5:4...
>
> _______ _______ __________
> Joseph Pehrson

--- In tuning@y..., BobWendell@t... wrote:
> Hi, Joseph! Yes, I'm quite sure this interesting mathematical
tuning
> coincidence was a significant contributor to the evolution toward
the
> use of thirds as consonant and the advent of meantone predominance.
> However, it should be obvious that the attractive power of 4:5 was
> behind even the interest that this schisma held for composers and
> musicians of the time.

But of course.
>
> It is interesting to ask a chicken and egg question. Was the
schisma
> reinforcing the attractive power of 4:5 or vice versa? Hard to
answer
> that one without going back there in a time machine, but who knows?
> Maybe Margo has some interesting contribution to make on this
point.

Hmm . . . there were a lot of "unusual" intervals in the 12-tone
Pythagorean system. For example the wolf fifth, which was normally
avoided. However, the near-4:5 schismatic thirds were clearly desired
for their own sake, as evidenced in the 15th-century practice of
tuning the 12-tone Pythagorean chain from Gb to B instead of the
traditional Eb to G#. This had the effect of providing near-just
major triads on D, A, and E, which were much more useful pitches to
base them on than the B, F#, and C# roots in the traditional tuning.
The main disadvantage of this practice was to introduce the wolf
between B and F# . . . and evidence is that this interval was avoided
by composers using this tuning . . . Margo has of course explained
all this much better . . .

> Also has anyone pointed out to all interested readers that the
> schisma thirds are a result of the happy mathematical proximity of
> the synontic and Pythagorean commas, which differ only by 1.96
cents?
>
> If you go up four just fifths, you will be a syntonic comma sharp
on
> the third you find at the top. Then if you back down 12 fifths for
> the full Pythagorean cycle, you will slightly more than cancel out
> the syntonic comma with the negative Pythagorean, ending up with a
> major third only two cents flat to a perfect 4:5! Mighty pure,
folks!

Well, yes, this is exactly what we've just been saying (though you're
probably saying it more clearly, and I encourage repeating it in any
case!).

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

/tuning/topicId_28495.html#28621

> Hmm . . . there were a lot of "unusual" intervals in the 12-tone
> Pythagorean system. For example the wolf fifth, which was normally
> avoided. However, the near-4:5 schismatic thirds were clearly
desired for their own sake, as evidenced in the 15th-century practice
of
> tuning the 12-tone Pythagorean chain from Gb to B instead of the
> traditional Eb to G#. This had the effect of providing near-just
> major triads on D, A, and E, which were much more useful pitches to
> base them on than the B, F#, and C# roots in the traditional
tuning.
> The main disadvantage of this practice was to introduce the wolf
> between B and F# . . . and evidence is that this interval was
avoided by composers using this tuning . . . Margo has of course
explained all this much better . . .
>
> > Also has anyone pointed out to all interested readers that the
> > schisma thirds are a result of the happy mathematical proximity
of the synontic and Pythagorean commas, which differ only by 1.96
> cents?
> >
> > If you go up four just fifths, you will be a syntonic comma sharp
> on the third you find at the top. Then if you back down 12 fifths
for the full Pythagorean cycle, you will slightly more than cancel
out the syntonic comma with the negative Pythagorean, ending up with
a major third only two cents flat to a perfect 4:5! Mighty pure,
> folks!
>
> Well, yes, this is exactly what we've just been saying (though
you're probably saying it more clearly, and I encourage repeating it
in any case!).

Actually, *both* you guys are helping clarify this situation in my
mind... Thanks, Paul, for the clear description of the difference in
the start of the Pythagorean chain and the possible schismatic
intervals resulting. That part wasn't in the recent Schulter
article, although it may have been in some before that which I
missed....

Thanks again!

________ __________ ______
Joseph Pehrson

Hello, there, Bob and Paul and everyone, and I'd like to attempt a
response on some questions about the stable or unstable quality in
context of intervals such as 5:4 major thirds, or 7:5 diminished
fifths, or 7:6 minor thirds.

First, I might offer a possible point of semantics which may suggest
some of the stylistic assumptions and orientations coming into play
here, and also in much of the literature of European music history,
xenharmonic or otherwise.

To speak of "third avoidance" in Perotin or Machaut, or "seventh
avoidance" in Monteverdi or Bach, would suggest a frame of analysis in
which intervals such as major thirds or minor sevenths are assumed to
be "inherently" stable or conclusive, "all things equal, but for some
reason this stable treatment isn't taking place."

If applied to Monteverdi or Bach, such a concept would produce the
curious result that these composers would be noted for "seventh
avoidance," since they "avoid the seventh" in closing or other stable
sonorities.

Of course, in usual histories, it is quite the contrary. As early as
1600, when Giovanni Maria Artusi wrote his famous dialogue against
"modern music," Monteverdi had gained a reputation for his bold use of
the minor seventh, not for his "avoidance" of it at places where
stability was desired -- the latter going without saying for either
Monteverdi or his less favorable critics.

To explain this bold use of the seventh in Monteverdi, and also its
noninclusion in stable sonorities, we might point to the basic
"grammar" of the music, a grammar which might be termed "triadic" or
"5-limit," in which the complete 4:5:6 is the most complex stable
sonority.

At the same time, we might observe that the regular minor seventh in a
meantone tuning such as 1/4-comma will have a relatively complex
nature, with a ratio somewhere roughly midway between 16:9 and 9:5 --
the precise mean, in 1/4-comma, or around 1007 cents. In comparison,
the simplest ratio for a minor seventh is 7:4, at around 969 cents.

Here we might say that the grammar and the phonology of the music seem
nicely to fit together: a minor seventh is treated as decidedly tense,
and is realized as a rather complex ratio, one with a relatively high
degree of "harmonic entropy" vis-a-vis the simplest ratio of 7:4.

However, these two aspects of music -- grammatical expectations, and
intonational or "phonological" realization -- can combine in different
ways.

Let us consider another kind of "seventh avoidance," to use that term
again in a rather curious way: the practice of barbershop or similar
ensembles who lean toward a tuning of the dominant seventh in a
major/minor style of tonality at 4:5:6:7, but who follow a musical
grammar still calling for a resolution of the 17th-19th century type
with the 7:5 contracting to a 5:4.

Is it the "nature" of the 7:5 that somehow requires resolution to a
simpler interval, or is this a matter of style -- much like the
resolution of a major third to a fifth in 14th-century style?

With Paul Erlich, I would say style is here a main factor, since we
can use the same 4:5:6:7 as a stable and conclusive sonority in a
tetradic style such as that based on a decatonic scale in 22-tET.

Here one might say that the two variables of grammar and phonology
interact in various ways. With Monteverdi or Erlich, the two factors
seem to "match": a relatively complex type of ratio treated as
decidedly unstable, or a simpler type treated as stable and
conclusive.

Similarly, comparing Perotin or Machaut to Palestrina or Lasso, we
have a relatively complex type of major third around the Pythagorean
size of 81:64 used in an active and unstable role, in contrast with a
major third at or near the simplest ratio of 5:4 used in a role of
rest and stability.

However, the fact that some people still feel a need for or tendency
toward "seventh avoidance" (otherwise known as a "resolution") in a
setting with prominent 4:5:6:7 sonorities suggests that in a medieval
type of setting, a sonority such as 4:5:6 may likewise call for "third
avoidance" -- or "resolving the third to stability."

Playing some standard medieval progressions in 1/4-comma meantone
using the regular meantone intervals, I find that the usual
"grammatical" expectations still hold, although their intonational
expression is muted a bit in both directions both by the purity of the
5:4 major thirds and the tempering of the fifths. Categorical
perception -- and grammatical orientation -- still make the standard
Gothic resolutions recognizable and enjoyable.

One "real-world" experiment of just about this time in 1968 sticks
with me. Newly arrived at college, I was excited about the musical
activity to be found on campus, and had written a simple three-voice
conductus in the style of around 1200 that I hoped to have performed.

Three singers very generously offered to sight-read on a totally
impromptu basis, and in a rather distracting setting.

When they reached the end of the first phrase, where I expected to
hear a resolution to a stable fifth, one of the singers, likely
through habit and custom, inserted a third -- and to me it sounded
quite "wrong," a frustration of the cadential intent.

Presumably the singers, who were reading unaccompanied, were tuning
such a third at a smooth and concordant ratio -- but it was the wrong
interval, at least from my point of view as a listener and composer,
however tuned.

Such things may suggest how, for example, early meantone organs around
the late 15th and early 16th centuries could be used in a "backward
compatible" manner for music conceived in a late Pythagorean setting.

Here I would like to suggest, possibly agreeing with both of you, Paul
and Bob, that the factors of musical grammar or style and phonology or
intonation can sometimes act either to reinforce each other within a
given setting, or to promote change.

Thus in a Gothic setting on the Continent, the active and "partially
concordant" quality of thirds and sixths is nicely expressed by
Pythagorean tuning; and the tuning reinforces this tendency of the
intervals to follow the patterns of the musical grammar.

(Incidentally, while theorists in French and related traditions often
speak of thirds as "imperfect" or "intermediate" concords, Italian
writers such as Marchettus of Padua speak of "tolerable dissonances,"
so both kind of descriptions have some historical warrant.)

In England, however, during this same era, we have evidence through
actual composed music and through accounts of listeners and theorists
that thirds are often used as "the best concords," sometimes in
closing sonorities, and that they are taken as approximations of the
simple ratios 5:4 and 6:5.

During the early and middle 15th century, Continental style moves both
toward the "English countenance," and toward the simpler ratios --
first through schisma thirds in a Pythagorean system, then through a
shift to meantone.

Here, however, we may note that "evolution" could have moved in other
directions not realized until much later, at least in composed music,
as far as we know.

For example, Jacobus of Liege in 1325 describes the minor seventh as
an "imperfect concord" -- rather tense, but somewhat "compatible" --
and compares its Pythagorean ratio of 16:9 with a simpler 7-based
ratio, actually the minor-seventh-plus-double-octave at 64:9 with a
ratio of 63:9 or 7:1, thus demonstrating what we should call the
septimal comma at 64:63.

In 21st-century neo-Gothic music, the resolution of a pure 7:4 or
close approximation to a stable fifth is one favorite progression, and
we have standard four-voice progressions like this, with a "^" showing
a note a septimal comma higher than usual Pythagorean:

D4 C4
B^3 C4
G3 F3
E^3 F3

Here the first sonority has a simple ratio of 12:14:18:21 (around
0-267-702-969 cents), resolving to a 3:2 fifth -- with all these
intervals pure, in a neo-Gothic kind of JI setting based on ratios of
2-3-7-9 odd, or 2-3-7 prime.

It will be seen that from a "neo-medieval" point of view, the narrow
28:27 semitones E^3-F3 and B^3-C4 nicely carry further the Pythagorean
aesthetic of narrow semitones and efficient resolutions; at the same
time, E^3-G3-B^3-D4 is at a "valley" of harmonic entropy, or "sweet
spot" of acoustical simplicity and concoord, making the resolution
"smooth" and "streamlined."

Why should this progression have apparently come into vogue around
2001 rather than around 1401? It is easy enough to derive from
progressions in Perotin or Machaut with minor thirds and sevenths --
if one is inclined in this direction.

Why should the role of minor sevenths, prominently used by various
composers during the era of 1200-1400, have been rather consistently
restricted during the era of roughly 1420-1590, albeit with some
notable nuances (Josquin's treatment is freer than Palestrina's)?

I am inclined to answer: acoustics supplies the materials, but taste
and fashion largely decide how it is woven.

The resolution I cite above, of 12:14:18:21 to a 3:2 fifth, might
illustrate this point from another perspective also.

One might very reasonably ask: why must a pure or near-pure
12:14:18:21 resolve at all?

Here I would answer that only the "grammar" of a typical neo-Gothic
style sets up expectations for such a resolution -- the convention
that fifths and fourths are the most complex stable intervals.

In various 20th-century kinds of settings, a 12:14:18:21 sonority
could be described as a kind of "minor seventh chord" -- or, to use
Dave Keenan's fitting terminology, a "subminor seventh chord."

While this sonority does not exhibit what I might term your "strict"
form of "rootedness," Paul, with the lowest note as an octave of the
fundamental (as in 2:3;4, 4:5:6, 4:5:6:7, or 16:19:24), nevertheless
both the "anchoring" or "semirooted" effect of the 3:2 fifth above
this note, and also the 7:4 minor seventh above, may impart an
acoustical quality tending to reinforce a stylistic treatment of
stability.

Thus it appears that sonorities such as either 4:5:6:7 or 12:14:18:21
may be heard in the early 21st century as inviting directed
resolutions, or serving as places of restful concord, depending on
style.

In approaching the question of stylistic change, I would like to
suggest that a general principle may apply which can interact with
intonational factors: the tendency for sonorities used in a
"dissonant" or "incidental" way often to take on a more "blending" or
"prominent" quality, and even eventually to be regarded as stable and
potentially conclusive.

For example, a given interval or sonority might arise as a result of a
melodic ornament or a figure promoting more conjunct motion between
two stable concords. Then the moment of tension is appreciated for its
deliberate contrast to stability, inviting treatment as a bold and
directed instance of cadential instability. The unstable cadential
sonority may gradually come to be heard as "indeed unstable, but not
so tense," inviting coloristic treatments also. Finally, the sonority
may come to be heard as representing a new standard of saturated
concord or "complete harmony."

This may be a process driven by internal "grammatical" factors, with
or without any obvious or dramatic intonational change. For example,
in 12-tET, a sonority of 0-300-700-1000 cents or 0-400-700-900 cents
may be either definitely unstable or quite stable depending on whether
the milieu is one of 18th-century Classicism or 20th-century harmony.

If we focus specifically on multi-voice styles of European and related
composition, then this process appears often to be a gradual one, at
least if we focus on the criterion of stability.

Thus in moving from the use of thirds as "partial concords" in the
complex polyphony of Perotin and his colleagues around 1200, to the
increasingly common acceptance of thirds as elements of closing
sonorities by Josquin and his colleagues around 1500, we cover about
three centuries of richly diverse styles. If we take the young Dufay
(around 1420-1430) or the likely advent of meantone (around 1450) as a
landmark of increasingly "tertian" verticality, then we are still
dealing with some 200-250 years of history.

Similarly, if we take a conventional historical perspective, we might
start with the bold seventh sonorities of Monteverdi and his
colleagues around 1600, and find that about 300 years of history take
place before sonorities with minor sevenths or major seconds are
treated as fully concordant or stable -- or, as one might have said
around 1900, as not requiring a resolution. Here, also, one could look
to the "floating" seventh chords of the 19th century as an earlier
landmark of transformation, inviting some would say a 4:5:6:7 tuning.

From a somewhat less conventional perspective, we might also note the
bold use of major seconds and minor sevenths by Perotin and his
colleagues, and ask why it required from around 1200 to 1900 for
sonorities such as 6:8:9 or 4:6:9 or 9:12:16 to be treated as stable
in European composition. They often seem to play a relatively
concordant role in Gothic style, but are restricted in the
Renaissance-Romantic eras, coming into their own in the era of Debussy
around 1900.

(Note that I choose these three-voice sonorities because they are
already optimized in a medieval Pythagorean intonation.)

What I would like to suggest is that each period or style has its own
standard of vertical saturation, and its sets of grammatically favored
progressions.

Finally, I would note that one kind of aesthetic preference might be
termed "mellow instability": the use of a sonority such as 4:5:6:7 or
12:14:18:21 in a setting where the stylistic or grammatical
expectations favor some kind of "usual" resolution, but where the
smoothness or "mellowness" of the contextually unstable sonority is
much esteemed and relished.

Most appreciatively,

Margo Schulter
mschulter@value.net

--- In tuning@y..., BobWendell@t... wrote:

> If you go up four just fifths, you will be a syntonic comma sharp
on
> the third you find at the top. Then if you back down 12 fifths for
> the full Pythagorean cycle, you will slightly more than cancel out
> the syntonic comma with the negative Pythagorean, ending up with a
> major third only two cents flat to a perfect 4:5! Mighty pure,
folks!

Today I learned that the shisma and the enharmonic diesis are
both "ABC good"--not to mention the ragisma and the breedsma. It's
been a very educational day since before today I had never even heard
the names "ragisma" or "breedsma", familiar though the numbers
themselves were to me. I also found out that the interval of 23/9,
repeated five times, is 109. No, make that 109, sharp by 1/1857th of
a cent. Mighty pure, folks, and mighty useless too!

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., BobWendell@t... wrote:
>
> > If you go up four just fifths, you will be a syntonic comma sharp
> on
> > the third you find at the top. Then if you back down 12 fifths
for
> > the full Pythagorean cycle, you will slightly more than cancel
out
> > the syntonic comma with the negative Pythagorean, ending up with
a
> > major third only two cents flat to a perfect 4:5! Mighty pure,
> folks!
>
> Today I learned that the shisma and the enharmonic diesis are
> both "ABC good"--not to mention the ragisma and the breedsma. It's
> been a very educational day since before today I had never even
heard
> the names "ragisma" or "breedsma", familiar though the numbers
> themselves were to me. I also found out that the interval of 23/9,
> repeated five times, is 109. No, make that 109, sharp by 1/1857th
of
> a cent. Mighty pure, folks, and mighty useless too!

Bob:
Ha-ha-ha! Well, I still don't know what the "ragisma" or "breedsma"
are, but the closeness of the syntonic and Pythagorean commas is
highly significant, contributing substantially through schisma thirds
toward the tonal evolution in western music from the Greek 3-limit to
5-limit JI in the Renaissance as noted in other threads. The advent
of 5-limit tonality was the breeding ground for 1/4-comma meantone
and all its variants, etc.

Also 53-tET has 22.6 cent steps, which is extremely close (+0.1
cents) to the average of 23.5 (Pythagorean) and 21.5 (syntonic).
That's why 53-tET is such a very accurate ET for 5-limit JI
approximation, even though you can arrive at it using continued
fractions to find close rational approximations for the base two
logarithmic portion of the octave represented by the JI perfect fifth
(7/12, 31/53, etc.)

--- In tuning@y..., BobWendell@t... wrote:

> Bob:
> Ha-ha-ha! Well, I still don't know what the "ragisma"

4375:4375 = 2^(-1) * 3^(-7) * 5^4 * 7^1 = (-1,-7,4,1)

> or "breedsma"

2401:2400 = 2^(-5) * 3^(-1) * 5^(-2) * 7^4 = (-5,-1,-2,4)
This is named after Graham Breed's chord progression in Blackjack
which absorbs said interval.

> are