Yahoo Tuning Groups Ultimate Backup tuning Tuning Comparison of La messa de Nostre Dame Sanctus by G. Machuat

Dear Chris,

Congratulations on these three files letting us compare
versions of Guillaume de Machaut's _Mass_ in adaptive JI
(which I take to be 5-limit); 22-EDO "superpythagorean"; and
Pythagorean, which I agree is historically most likely and
appropriate.

As I see it from a medievalist perspective, the point is to
convey an impression of the 2:3:4 (or sometimes 2:3:4:6) as
a rich and stable sonority, the most complex restful sound
possible. The _quinta fissa_ or "split fifth" at 64:81:96 or
54:64:81 should be relatively concordant and yet a bit tense
and definitely unstable, a point of motion rather than rest.
A Pythagorean tuning with your timbre nicely succeeds for me
in communicating these things.

One point on which I might question my own hearing: for some
reason, I did not hear the prominent beating of fifths
(e.g. in 2:3:4) that I might have expected in the 22-EDO
version, where they are tuned at 709.09 cents, or 7.14 cents
wide. I'm wondering if that is connected with my aging ears,
or is also a sign of a gentler timbre -- maybe others can
compare their own hearing of this.

But I will confirm that the Pythagorean version was my
favorite, with the caution that I knew in advance which one
it would be, so that my own theoretical preferences could
have influenced my hearing of the examples.

With many thanks, and wishing you a Happy New Year,

Margo

🔗genewardsmith <genewardsmith@...>

🔗Chris Vaisvil <chrisvaisvil@...>

HI Margo,

Thank you for your listens and insightful comments. Since this is an area
of expertise for you I am anxious to know if the minor seconds and major
seconds (for instance the passing tone minor 2nd in measure 21 lower two
voices in the score linked on my blog) would be accurate or a result of
misapplication of musica fictica rules by the transcriber.

I think to answer your question concerning the beating of the fifths I
would have to explore some audio examples with this sound set and see if a
sustained fifth in super-phytagorean indeed does not beat noticeably.

Also, just to check a 2:3:4:6 would be (in C) C G C G ?

For those interested I found this excellent page on 13th century harmony
(so old now it sounds modern - I use, I think, every one of these in a
modernistic context)
http://www.medieval.org/emfaq/harmony/multi.html

When looking up quinta fissa

Best Regards,

Chris
On Mon, Dec 31, 2012 at 12:06 AM, Margo Schulter <mschulter@...>wrote:

> **
>
>
> Dear Chris,
>
> Congratulations on these three files letting us compare
> versions of Guillaume de Machaut's _Mass_ in adaptive JI
> (which I take to be 5-limit); 22-EDO "superpythagorean"; and
> Pythagorean, which I agree is historically most likely and
> appropriate.
>
> As I see it from a medievalist perspective, the point is to
> convey an impression of the 2:3:4 (or sometimes 2:3:4:6) as
> a rich and stable sonority, the most complex restful sound
> possible. The _quinta fissa_ or "split fifth" at 64:81:96 or
> 54:64:81 should be relatively concordant and yet a bit tense
> and definitely unstable, a point of motion rather than rest.
> A Pythagorean tuning with your timbre nicely succeeds for me
> in communicating these things.
>
> One point on which I might question my own hearing: for some
> reason, I did not hear the prominent beating of fifths
> (e.g. in 2:3:4) that I might have expected in the 22-EDO
> version, where they are tuned at 709.09 cents, or 7.14 cents
> wide. I'm wondering if that is connected with my aging ears,
> or is also a sign of a gentler timbre -- maybe others can
> compare their own hearing of this.
>
> But I will confirm that the Pythagorean version was my
> favorite, with the caution that I knew in advance which one
> it would be, so that my own theoretical preferences could
> have influenced my hearing of the examples.
>
> With many thanks, and wishing you a Happy New Year,
>
> Margo
>
>

🔗Graham Breed <gbreed@...>

🔗Chris Vaisvil <chrisvaisvil@...>

🔗Graham Breed <gbreed@...>

🔗Margo Schulter <mschulter@...>

> HI Margo,

> Thank you for your listens and insightful comments. Since this is an
> area of expertise for you I am anxious to know if the minor seconds
> and major seconds (for instance the passing tone minor 2nd in
> measure 21 lower two voices in the score linked on my blog) would be
> accurate or a result of misapplication of musica fictica rules by
> the transcriber.

Hi, Chris, and congratulations again on this comparative study!

As far as measure 21 (measure 20 in Leo Schrade's transcription,
a point on which different modern scores can vary), Schrade has
the two lowest voices as A-C, B-D, C-C, which would make that
vertical minor 2nd a more usual minor 3rd. While bold minor 2nds
can and do occur in 13th-14th century pieces, Schrade's version
as well as my own sense of things agree that you're right to
query that measure in the score that you linked to.

Another thing I can confidently question, based on my own reading
and Schrade's version, is the use of Db where C# would clearly be
the correct accidental at points in the PDF score, maybe an
artifact of going from MIDI to some default notation where 12-EDO
equivalence (C#=Db) is assumed. Remote accidentals like Db can
occur in the 14th century (e.g. an English source with a cadence
of Db-F-Bb to C-G-C), but here it's clearly a mistranscription. Augmented or diminished intervals like C#-F (e.g. in an
ornamental cadence to where the third C#-E contracts to D) or
F-C# (in approaching a cadence of the sixth E-C# to the octave
D-D) can also occur. But here something like Db-F# looks simply
like a 12-EDO equivalence for Schrade's C#-F#.

Beyond that, _musica ficta_ is a matter where performers (or
transcribers) have lots of discretion, part of the fun (and a
complication also!). One thing that can get lost in the process
of making PDF scores is the distinction in a version like
Schrade's between accidentals indicated in a given manuscript,
and others added by the transcriber or editor (typically placed
above the staff, or sometimes in brackets or parentheses, etc.).
What we do know from some 16th-century editions for fretted
instruments where the exact semitones (and thus accidentals) need
to be shown is that sometimes performers would interpret the same
passage a bit differently on repetition, for example. Likely it
was this way in the 14th century also.

> I think to answer your question concerning the beating of the
> fifths I would have to explore some audio examples with this
> sound set and see if a sustained fifth in super-phytagorean
> indeed does not beat noticeably.

That might be one good test.

> Also, just to check a 2:3:4:6 would be (in C) C G C G ?

Exactly! It's a 2:3:4 trine plus a twelfth, which does often
occur in Machaut. In period theory, 2:1 tends to be viewed as the
largest "simple" interval, but intervals beyond 2:1 as more or
less extensions of these simple intervals.

> For those interested I found this excellent page on 13th
> century harmony (so old now it sounds modern - I use, I think,
> every one of these in a modernistic context)
> [57]http://www.medieval.org/emfaq/harmony/multi.html

Great! This page, a project suggested by Todd McComb of
www.medieval.org, was one of the first I became involved in on
the Web. It's a delight to see that you are enjoying the modern
application of these progressions -- and your "so old now it
sounds modern" is my view also!

> When looking up quinta fissa

A very useful term still after seven centuries or so!

> Best Regards,

> Chris

With many thanks, and wishes for a Happy New Year,

Margo

🔗blauschlafer <blauschlafer@...>

Hi Margo and others,

"But I will confirm that the Pythagorean version was my
favorite, with the caution that I knew in advance which one
it would be, so that my own theoretical preferences could
have influenced my hearing of the examples."

This is certainly a plausible explanation, but it's also possible that you've gotten used to Pythagorean thirds as "relatively concordant and yet a bit tense and definitely unstable" as well as 90-cent semitones, but super-Pythagorean 22-EDO thirds and 55-cent steps are still a bridge too far. Ideally, we'd pick a few people to listen for a few months to medieval music tuned in super-Pyth 22-EDO, and see what they think in a blinded trial.

Someone needs to do a developmental psychology PhD and <strike>subject scores of infants to</strike> gift scores of infants with various tunings.

JBSL
--- In tuning@yahoogroups.com, Margo Schulter wrote:
>
> > HI Margo,
> > > Thank you for your listens and insightful comments. Since this is an
> > area of expertise for you I am anxious to know if the minor seconds
> > and major seconds (for instance the passing tone minor 2nd in
> > measure 21 lower two voices in the score linked on my blog) would be
> > accurate or a result of misapplication of musica fictica rules by
> > the transcriber.
> > Hi, Chris, and congratulations again on this comparative study!
> > As far as measure 21 (measure 20 in Leo Schrade's transcription,
> a point on which different modern scores can vary), Schrade has
> the two lowest voices as A-C, B-D, C-C, which would make that
> vertical minor 2nd a more usual minor 3rd. While bold minor 2nds
> can and do occur in 13th-14th century pieces, Schrade's version
> as well as my own sense of things agree that you're right to
> query that measure in the score that you linked to.
> > Another thing I can confidently question, based on my own reading
> and Schrade's version, is the use of Db where C# would clearly be
> the correct accidental at points in the PDF score, maybe an
> artifact of going from MIDI to some default notation where 12-EDO
> equivalence (C#=Db) is assumed. Remote accidentals like Db can
> occur in the 14th century (e.g. an English source with a cadence
> of Db-F-Bb to C-G-C), but here it's clearly a mistranscription. > Augmented or diminished intervals like C#-F (e.g. in an
> ornamental cadence to where the third C#-E contracts to D) or
> F-C# (in approaching a cadence of the sixth E-C# to the octave
> D-D) can also occur. But here something like Db-F# looks simply
> like a 12-EDO equivalence for Schrade's C#-F#.
> > Beyond that, _musica ficta_ is a matter where performers (or
> transcribers) have lots of discretion, part of the fun (and a
> complication also!). One thing that can get lost in the process
> of making PDF scores is the distinction in a version like
> Schrade's between accidentals indicated in a given manuscript,
> and others added by the transcriber or editor (typically placed
> above the staff, or sometimes in brackets or parentheses, etc.).
> What we do know from some 16th-century editions for fretted
> instruments where the exact semitones (and thus accidentals) need
> to be shown is that sometimes performers would interpret the same
> passage a bit differently on repetition, for example. Likely it
> was this way in the 14th century also.
> > > I think to answer your question concerning the beating of the
> > fifths I would have to explore some audio examples with this
> > sound set and see if a sustained fifth in super-phytagorean
> > indeed does not beat noticeably.
> > That might be one good test.
> > > Also, just to check a 2:3:4:6 would be (in C) C G C G ?
> > Exactly! It's a 2:3:4 trine plus a twelfth, which does often
> occur in Machaut. In period theory, 2:1 tends to be viewed as the
> largest "simple" interval, but intervals beyond 2:1 as more or
> less extensions of these simple intervals.
> > > For those interested I found this excellent page on 13th
> > century harmony (so old now it sounds modern - I use, I think,
> > every one of these in a modernistic context)
> > [57]http://www.medieval.org/emfaq/harmony/multi.html
> > Great! This page, a project suggested by Todd McComb of
> www.medieval.org, was one of the first I became involved in on
> the Web. It's a delight to see that you are enjoying the modern
> application of these progressions -- and your "so old now it
> sounds modern" is my view also!
> > > When looking up quinta fissa
> > A very useful term still after seven centuries or so!
> > > Best Regards,
> > > Chris
> > With many thanks, and wishes for a Happy New Year,
> > Margo
>

🔗Margo Schulter <mschulter@...>

> Hi Margo and others,

Hello, JBSL and all.

>> "But I will confirm that the Pythagorean version was my
>> favorite, with the caution that I knew in advance which
>> one it would be, so that my own theoretical preferences
>> could have influenced my hearing of the examples."

> This is certainly a plausible explanation, but it's also
> possible that you've gotten used to Pythagorean thirds as
> "relatively concordant and yet a bit tense and definitely
> unstable" as well as 90-cent semitones, but
> super-Pythagorean 22-EDO thirds and 55-cent steps are
> still a bridge too far. Ideally, we'd pick a few people
> to listen for a few months to medieval music tuned in
> super-Pyth 22-EDO, and see what they think in a blinded
> trial. Someone needs to do a developmental psychology
> PhD and [DEL: subject scores of infants to :DEL] gift
> scores of infants with various tunings.

Thank you for a very logical and fair question, based on my
quoted comment, which gives me the opportunity to clarify my
point of view and inquire a bit into just where my remarks
were coming from in preferring the Pythagorean version.

My conclusion is that 22-EDO could be a pleasant
"modernistic" variation on 14th-century intonation, and
indeed a rather "moderate" express of one facet of the
historical style. The issue, from the viewpoint of a
"classic" or "historically oriented" performance, is not so
much a 436-cent major third or 55-cent semitone step, but
the tempering out of 64/63.

In these terms, a "classic" performance might take one of at
least two approaches:

(1) Simply using regular intervals in Pythagorean
intonation or some temperament not too far from it,
say with fifths within 2 cents or so of just; or

(2) Using regular Pythagorean or near-Pythagorean
intervals through most of a piece, but with
"exuberant" intonation at cadences, especially those
involving sharps, with major thirds and sixths at
around 7:9:12 or even larger, and cadential semitone
or diesis steps around 28:27 or even narrower.

For an example of the second option, based on the theory of
Marchettus of Padua (1318), here is an improvisation:

<http://www.bestII.com/~mschulter/PythEnharImprov01.mp3>

If indeed, as scholars such as Jay Rahn have suggested based
on a monochord interpretation of Marchettus, a cadential
sonority such as E-G#-C# may have been intoned at somewhere
around 37:48:64 (0-451-949 cents), with steps at about 37/36
or 47 cents, then 22-EDO might actually be more of "a bridge
not quite far enough."

In practice, I tend to read Marchettus as leaving the way
open for E-G#-C# or the like at around 7:9:12, with steps
around a typically "tempered" 28/27, which in my practice
means mostly "somewhere around 55-59 cents," or in other
words not at all far from the 55-cent steps of 22-EDO, _at
cadences_, that is!

Here's an example:

<http://www.bestII.com/~mschulter/O_Europae.mid>
<http://www.bestII.com/~mschulter/O_Europae.pdf>

Note the cadences at around 0:24-0:26 (mm. 17-18), 0:33-0:35
(mm. 23-24), 0:42-0:44 (mm. 29-30), and 0:49-0:52
(mm. 34-35) involving steps of 58.6 cents and large major
thirds at 437.1 or 438.3 cents. The second instance involves
~6:7:9, and the others ~7:9:12. These intonations themselves
hardly differ radically from 22-EDO.

Where 22-EDO differs from such "classic" or more
specifically "Marchettan" approaches is in tempering out the
64/63, so that all regular semitones are at 55 cents, for
example. What I'd emphasize is that while this might not be
a classic 14th-century approach, that doesn't mean that it
isn't a good modernistic one!

In fact, I routinely have played tempered Archytan
tetrachords in tunings like 207.4-57.4-232.0 cents, or even
216.8-50.3-224.6 cents. Here I'm not going for some special
effect or intended artistic distortion, but simply accepting
some interval in a given temperament structure as
"equivalent" to 28/27 at 63 cents. So 218.2-54.5-218.2 cents
in 22-EDO hardly seems a bridge further than these examples!

And I would emphasize that while a "historically oriented"
performance of Machaut should probably observe the 64/63 or
whatever if one does choose to use Marchettan inflections,
from a modern perspective 22-EDO is a very reasonable choice
for this kind of style, and one illustrating the qualities
of the tuning as a regular diatonic system. Especially if
the timbres minimize any conspicuous beating of stable
fifths and fourths, as evidently happened in Chris's
example, there's no reason it might not be used, and used
effectively in a (neo)medieval context!

Finally, I would warmly agree that custom and habit can play
a big role in how a given tuning is perceived. About four
decades back, one very justly noted scholar of medieval
European music wrote that early music ensembles were not
using Pythagorean tuning because presumably they wished not
to be criticized for faulty intonation! The experiment you
describe could be a very interesting one.

> JBSL

Best,

Margo

🔗genewardsmith <genewardsmith@...>

🔗Margo Schulter <mschulter@...>

>> For an example of the second option, based on the theory of
>> Marchettus of Padua (1318), here is an improvisation:

> Very striking piece! Can you give an explicit scale for it, and can
> I link it to the Xenwiki?

Please, by all means, Gene! Why don't I give a Scala file and listing,
and also an article by Jay Rahn on Marchettus of Padua proposing an
interpretation very close to this tuning.

While the tuning that follows was designed specifically to yield some
pure sonorities of 14:17:21 and the like, it's more generally intended
for the kind of "Marchettan" intonation heard in this piece. The
cadential diesis steps are 48.230 cents, which happens to be very
close to Jay Rahn's monochord interpretation of Marchettus with these
steps around 37:36 or 47.434 cents. Rahn's reading is based on a
division of a 9/8 or 81/72 tone into 81:79:77:76:74:72, with 74/72 or
37/36 as a possible value for the cadential diesis as "one of the five
parts of a tone."

<http://www.mtosmt.org/issues/mto.98.4.6/mto.98.4.6.rahn.html>

|
0: 1/1 0.000 unison, perfect prime
1: 459/448 41.995
2: 2187/2048 113.685 apotome
3: 1003833/917504 155.680
4: 9/8 203.910 major whole tone
5: 4131/3584 245.905
6: 32/27 294.135 Pythagorean minor third
7: 17/14 336.130 supraminor third
8: 81/64 407.820 Pythagorean major third
9: 37179/28672 449.815
10: 4/3 498.045 perfect fourth
11: 153/112 540.040
12: 729/512 611.730 Pythagorean tritone
13: 334611/229376 653.725
14: 3/2 701.955 perfect fifth
15: 1377/896 743.950
16: 6561/4096 815.640 Pythagorean augmented fifth
17: 3011499/1835008 857.635
18: 27/16 905.865 Pythagorean major sixth
19: 12393/7168 947.860
20: 16/9 996.090 Pythagorean minor seventh
21: 51/28 1038.085
22: 243/128 1109.775 Pythagorean major seventh
23: 111537/57344 1151.770
24: 2/1 1200.000 octave

! xenogj24.scl
!
Neo-Gothic 3/17-flavor JI (keyboards 459:448 apart) 24
!
459/448
2187/2048
1003833/917504
9/8
4131/3584
32/27
17/14
81/64
37179/28672
4/3
153/112
729/512
334611/229376
3/2
1377/896
6561/4096
3011499/1835008
27/16
12393/7168
16/9
51/28
243/128
111537/57344
2/1

Best,

Margo