Yahoo Tuning Groups Ultimate Backup tuning enharmonics of different pitches (MMM: people's perception of "microtonality")

(i moved this from the MakeMicroMusic list)

--- In MakeMicroMusic@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
>
> --- In MakeMicroMusic@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
> >
> > --- In MakeMicroMusic@yahoogroups.com, "Paul Erlich" <paul@s...>
> wrote:
> >
> > > Performance practice could have been some form of
> > > adaptive JI or adaptive tuning, at least for many
> > > instruments -- chords being sweetly harmonious doesn't
> > > say anything about the horizontal (melodic) intervals . . .
> >
> > But the end result will probably be closer to 31 than 12.
>
> Not if the music makes use of enharmonic equivalence, which
> was very common starting with Beethoven.

You're making the assumption that in the examples you have
in mind here, in which a sequence of two chords has a pair
of enharmonically-equivalent notes tied together which are
spelled differently, the two notes are supposed to be exactly
the same pitch.

I must admit that this probably *was* normally assumed by
anyone who studied music during the "common-practice" period.

But in his response to Fox-Strangways regarding the alleged
impossibility of modulation within a restricted set of
JI pitches:

http://tonalsoft.com/monzo/partch/fs/jimod.htm

in the second paragraph of the section titled
"Getting Down to Cases", Partch demonstrated that the
modulation was "successful in all three instances":

>> "1) sustaining the 10/9 through the third and fourth beats,
>>
>> 2) sustaining the 9/8 through the third and fourth beats,
>>
>> 3) playing the 10/9 on the third beat and 9/8 on the fourth
>> (which is necessary if both chords are in just tuning)."

I know that you, Paul, are bothered by horizonaltal
commatic shifts in pitch, but i'm not. Partch's "instance 3"
exhibits this kind of shift, and i think it's perfectly OK.

In this case (a modulation from C-major to G-major via
a Am7-D progression), the tied notes are two different
pitches a syntonic comma apart, but they are both spelled
as "A".

In a case involving enharmonically-equivalent notes which
are spelled differently, the two notes would be a diesis apart.

In Europe, most keyboards during the first half of
the 1800s would have been in a 12-note well-temperament,
and the two tied notes would have the same pitch.
"Starting with Beethoven", his non-keyboard music would
have been performed in a meantone tuning.

1/4-comma meantone gives a diesis which of ~41.05885841
cents, which is exactly the "standard" 5-limit JI diesis
of ratio 128/125 = 2,3,5-monzo [7 0, -3> .

A meantone in the vicinity of 1/6-comma meantone is
more likely by Beethoven's time, and the diesis of that
tuning is much smaller: ~19.55256881 cents, actually
smaller than both the pythagorean and syntonic commas.

I realize that employing the mathematical term
"equivalence" in "enharmonic equivalence" stipulates
that the two pitches are supposed to be the same,
but i say that your use of "enharmonic equivalence"
can extend to examples like these, where they are
in fact not exactly the same pitch.

I've entered a fair amount of Beethoven string quartet
music into Tonescape using a 15-tone 1/6-comma meantone
which includes 3 pairs of "enharmonically equivalent"
notes, and i think it sounds great. None of these
pieces have examples of tied enharmonic notes, but
hopefully i'll find one. If someone points out some
examples in Beethoven and particularly Schubert, i'll
be happy to render them as Tonescape-produced mp3's
for the tuning community's listening enjoyment. :)

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗monz <monz@tonalsoft.com>

I forgot to close with the point which addresses what
Gene and Paul disagreed about:

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> --- In MakeMicroMusic@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:

> > > [Gene]
> > > But the end result will probably be closer to 31 than 12.
> >
> > Not if the music makes use of enharmonic equivalence, which
> > was very common starting with Beethoven.
>
> > <big snip>
>
> In a case involving enharmonically-equivalent notes which
> are spelled differently, the two notes would be a diesis apart.
>
> In Europe, most keyboards during the first half of
> the 1800s would have been in a 12-note well-temperament,
> and the two tied notes would have the same pitch.
> "Starting with Beethoven", his non-keyboard music would
> have been performed in a meantone tuning.
>
> 1/4-comma meantone gives a diesis which of ~41.05885841
> cents, which is exactly the "standard" 5-limit JI diesis
> of ratio 128/125 = 2,3,5-monzo [7 0, -3> .
>
> A meantone in the vicinity of 1/6-comma meantone is
> more likely by Beethoven's time, and the diesis of that
> tuning is much smaller: ~19.55256881 cents, actually
> smaller than both the pythagorean and syntonic commas.

* 1/4-comma meantone implies ~31-edo as Gene said;

* 1/6-comma meantone implies a subset from ~55-edo
-- and the performance practice of this tuning began
before Beethoven, during Mozart's career, and
continued past 1924 ... fairly common use of tied
enharmonics also began with Mozart, but increased
a lot with Beethoven;

* ~12-edo, of course, is the case where the enharmonic
notes are exactly the same pitch.

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗klaus schmirler <KSchmir@online.de>

monz wrote:

> You're making the assumption that in the examples you have > in mind here, in which a sequence of two chords has a pair
> of enharmonically-equivalent notes tied together which are > spelled differently, the two notes are supposed to be exactly
> the same pitch.
> > I must admit that this probably *was* normally assumed by > anyone who studied music during the "common-practice" period.

Could be, but probably only because there was no reason to think about it much when even false relations were not allowed. (BTW, that's just the old diabolus in musica in new clothes.)

> I've entered a fair amount of Beethoven string quartet
> music into Tonescape using a 15-tone 1/6-comma meantone
> which includes 3 pairs of "enharmonically equivalent"
> notes, and i think it sounds great. None of these
> pieces have examples of tied enharmonic notes, but
> hopefully i'll find one. If someone points out some
> examples in Beethoven and particularly Schubert, i'll
> be happy to render them as Tonescape-produced mp3's
> for the tuning community's listening enjoyment. :)

I can't cite you a piece, but I can corroborate your claim; in Bayreuth they do comma shifts. I was told that by a horn player (may have played there some time between the 1930s and early '60s and a trombone player (late '70s). It's an Eb renotated as D# in one of the lower trombones in some of Wagner's operas. If you can lay your hands on a volume of orchestral studies for trombone, it's sure to be in there. (And I wish I could remember in which direction the shift was supposed to go.)

klaus

🔗Aaron Krister Johnson <aaron@akjmusic.com>

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> (i moved this from the MakeMicroMusic list)
>
>
> --- In MakeMicroMusic@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
> >
> > --- In MakeMicroMusic@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> > wrote:
> > >
> > > --- In MakeMicroMusic@yahoogroups.com, "Paul Erlich" <paul@s...>
> > wrote:
> > >
> > > > Performance practice could have been some form of
> > > > adaptive JI or adaptive tuning, at least for many
> > > > instruments -- chords being sweetly harmonious doesn't
> > > > say anything about the horizontal (melodic) intervals . . .
> > >
> > > But the end result will probably be closer to 31 than 12.
> >
> > Not if the music makes use of enharmonic equivalence, which
> > was very common starting with Beethoven.
>
>
>
> You're making the assumption that in the examples you have
> in mind here, in which a sequence of two chords has a pair
> of enharmonically-equivalent notes tied together which are
> spelled differently, the two notes are supposed to be exactly
> the same pitch.
>
> I must admit that this probably *was* normally assumed by
> anyone who studied music during the "common-practice" period.
>
>
>
> But in his response to Fox-Strangways regarding the alleged
> impossibility of modulation within a restricted set of
> JI pitches:
>
> http://tonalsoft.com/monzo/partch/fs/jimod.htm
>
> in the second paragraph of the section titled
> "Getting Down to Cases", Partch demonstrated that the
> modulation was "successful in all three instances":
>
> >> "1) sustaining the 10/9 through the third and fourth beats,
> >>
> >> 2) sustaining the 9/8 through the third and fourth beats,
> >>
> >> 3) playing the 10/9 on the third beat and 9/8 on the fourth
> >> (which is necessary if both chords are in just tuning)."
>
>
> I know that you, Paul, are bothered by horizonaltal
> commatic shifts in pitch, but i'm not. Partch's "instance 3"
> exhibits this kind of shift, and i think it's perfectly OK.
>
> In this case (a modulation from C-major to G-major via
> a Am7-D progression), the tied notes are two different
> pitches a syntonic comma apart, but they are both spelled
> as "A".
>
> In a case involving enharmonically-equivalent notes which
> are spelled differently, the two notes would be a diesis apart.
>
> In Europe, most keyboards during the first half of
> the 1800s would have been in a 12-note well-temperament,
> and the two tied notes would have the same pitch.
> "Starting with Beethoven", his non-keyboard music would
> have been performed in a meantone tuning.
>
> 1/4-comma meantone gives a diesis which of ~41.05885841
> cents, which is exactly the "standard" 5-limit JI diesis
> of ratio 128/125 = 2,3,5-monzo [7 0, -3> .
>
> A meantone in the vicinity of 1/6-comma meantone is
> more likely by Beethoven's time, and the diesis of that
> tuning is much smaller: ~19.55256881 cents, actually
> smaller than both the pythagorean and syntonic commas.
>
>
> I realize that employing the mathematical term
> "equivalence" in "enharmonic equivalence" stipulates
> that the two pitches are supposed to be the same,
> but i say that your use of "enharmonic equivalence"
> can extend to examples like these, where they are
> in fact not exactly the same pitch.
>
> I've entered a fair amount of Beethoven string quartet
> music into Tonescape using a 15-tone 1/6-comma meantone
> which includes 3 pairs of "enharmonically equivalent"
> notes, and i think it sounds great. None of these
> pieces have examples of tied enharmonic notes, but
> hopefully i'll find one. If someone points out some
> examples in Beethoven and particularly Schubert, i'll
> be happy to render them as Tonescape-produced mp3's
> for the tuning community's listening enjoyment. :)

All due respect Monz, but entering quartets as a listening exercise
into computer software for the sake of demonstration, and making the
claim that non-keyboard Beethoven performers did a shift of a diesis
in mid-performance are two different things. I doubt you will find
references to these type of things in the historical performance
practice. Like always, musicians used their ears in performance to
correct wandering pitch.

Beethoven's contemporaries (Hummel for one) were advocating
quasi-equal tunings just after Beethoven's death. In the absence of
contrary evidence, like the type of diesis shift you claim would have
happened in practice, there is no reason to suppose that by
middle-Beethoven, the trend in tuning was towards well-temperaments
that approach quasi-equal, so I think I'm with Paul here....

OTOH, I think what you're doing is great with these experiments with
the standard rep, and we can learn alot about these issues and make
some very educated guesses, so keep up the good work.

Best,
Aaron.

🔗threesixesinarow <CACCOLA@NET1PLUS.COM>

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> In a case involving enharmonically-equivalent notes which
> are spelled differently, the two notes would be a diesis apart.

"[T]he improvements of Sebastien Erard have made [the pedal harp]
capable of performing any music written for the pianoforte. His double
action harp, perfected in 1808 has a compass of six octaves, from E to
E, with all the semitones, and even quarter tones." ("Harp" The
American cyclopædia D. Appleton and company, 1873-76.)

> In Europe, most keyboards during the first half of
> the 1800s would have been in a 12-note well-temperament,

"A wrong (or at least highly suspect) note in the C. Wheatstone & Co.
publication of Giuglio Regondi's Serenade or English concertina and
piano (1859) should probably be emended with the mean-tone temperament
of the instrument in mind. Until the late 1850s/early 1860s--and
though published in 1859, the Serenade was probably composed in the
1840s--concertina manufacturers used a mean-tone temperament in which
they divided the octave into 14 notes, differentiating between -- and
providing separate buttons for--E/D, on the one hand, and A/G, on the
other, with the flat note of each pair being tuned 41 cents higher
than the sharp note." (Atlas, Allan. "A 41 cent emendation" (abstract)
Early Music. 2005 Vol.33 nr.4 )

Clark

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> * 1/4-comma meantone implies ~31-edo as Gene said;
>
> * 1/6-comma meantone implies a subset from ~55-edo
> -- and the performance practice of this tuning began
> before Beethoven, during Mozart's career, and
> continued past 1924 ... fairly common use of tied
> enharmonics also began with Mozart, but increased
> a lot with Beethoven;

And it can make quite a difference which you use. I made midi versions
of the Hugo Reinhold Impromptu in 12, 103, 55, 43, 31, 81, 50, 69 and
19 equal, put them up in a zip file for comparison purposes. The point
is, this piece brings out the difference in tuning pretty strongly. I
was struck by the fact that even 103, which is very close to 12-equal,
sounds different; less bland, for one thing. If you compare 55 to
flatter systems, I think you'll hear quite a difference. I think
myself the piece is most interesting tuned that way, but clearly less
authentic in sound.

http://66.98.148.43/~xenharmo/scores/impromptus.zip

The 103-et version is in the very mild meantone system of 103 equal,
which wearing other hats also supports miracle and sensi.

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗threesixesinarow <CACCOLA@NET1PLUS.COM>

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In tuning@yahoogroups.com, klaus schmirler <KSchmir@o...> wrote:
>
> > I can't cite you a piece, but I can corroborate your claim; in
> > Bayreuth they do comma shifts.
>
> That's very interesting. It certainly sheds a new light on the
Wagner
> in meantone business.
>
I remembered this after Daniel Wolf wrote about _Tristan Vorspiel_
played on the Oettingen/Schiedmayer harmonium in Berlin.
"When the ventil-mechanism was invented, which gave the horn a
complete scale, nobody was quicker to take exhaustive advantage of it
than Wagner : yet so deeply did he disapprove of the total
disappearance of the old horn technique, that in the preface to the
full score of Tristan he actually says that he would certainly have
had to abandon the advantages of the new instrument if experience had
not shown him that it was possible, by careful training, to induce
players to acquire a technique "almost" as good as the old." Tevey,
Donald. The Pianoforte of Emmanuel Moor. The Aeolian Company, London

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, "threesixesinarow" <CACCOLA@N...> wrote:

> I remembered this after Daniel Wolf wrote about _Tristan Vorspiel_
> played on the Oettingen/Schiedmayer harmonium in Berlin.
> "When the ventil-mechanism was invented, which gave the horn a
> complete scale, nobody was quicker to take exhaustive advantage of it
> than Wagner : yet so deeply did he disapprove of the total
> disappearance of the old horn technique, that in the preface to the
> full score of Tristan he actually says that he would certainly have
> had to abandon the advantages of the new instrument if experience had
> not shown him that it was possible, by careful training, to induce
> players to acquire a technique "almost" as good as the old." Tevey,
> Donald. The Pianoforte of Emmanuel Moor. The Aeolian Company, London

Would the old horn technique have encompassed use of 7th partial tones?
The point being, Tristan, tuned in meantone, sounds septimal. Should it?

🔗threesixesinarow <CACCOLA@NET1PLUS.COM>

🔗monz <monz@tonalsoft.com>

Aaron,

--- In tuning@yahoogroups.com, "Aaron Krister Johnson" <aaron@a...> wrote:

> All due respect Monz, but entering quartets as a listening
> exercise into computer software for the sake of demonstration,
> and making the claim that non-keyboard Beethoven performers
> did a shift of a diesis in mid-performance are two different
> things. I doubt you will find references to these type of
> things in the historical performance practice. Like always,
> musicians used their ears in performance to correct wandering
> pitch.
>
> Beethoven's contemporaries (Hummel for one) were advocating
> quasi-equal tunings just after Beethoven's death. In the
> absence of contrary evidence, like the type of diesis shift
> you claim would have happened in practice, there is no reason
> to suppose that by middle-Beethoven, the trend in tuning was
> towards well-temperaments that approach quasi-equal, so I
> think I'm with Paul here....
>
> OTOH, I think what you're doing is great with these experiments
> with the standard rep, and we can learn alot about these
> issues and make some very educated guesses, so keep up the
> good work.

Thanks for the vote of confidence and encouragement.

Leopold Mozart's popular violin method of 1756 does
have exercises demonstrating how to make the flats a
comma higher than their "enharmonically equivalent" sharps.

If Beethoven tied, say, a D# in a B-major chord to a
D# in what otherwise looks like a C-minor chord, in a
string quartet or other piece not involving a keyboard,
then i would say that he intended both D#'s to be the
same pitch. But if he tied the D# to an Eb in the C-minor
chord -- which is exactly the kind of thing i've seen in
his scores -- then i would say that he intended a diesis
shift.

Of course with Beethoven we're treading on very thin ice
anyway, because for the last 15 years of his life, when
he wrote a lot of his greatest works, he was totally deaf.
So this was all in his mind only, with no possibility to
aurally check the results.

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗monz <monz@tonalsoft.com>

Hi Clark,

--- In tuning@yahoogroups.com, "threesixesinarow" <CACCOLA@N...> wrote:

> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > In a case involving enharmonically-equivalent notes which
> > are spelled differently, the two notes would be a diesis apart.
>
> "[T]he improvements of Sebastien Erard have made [the pedal
> harp] capable of performing any music written for the
> pianoforte. His double action harp, perfected in 1808 has
> a compass of six octaves, from E to E, with all the semitones,
> and even quarter tones." ("Harp" The American cyclopædia
> D. Appleton and company, 1873-76.)

Thanks for that! I never had any idea that the standard harp
could do quarter-tones!

> > In Europe, most keyboards during the first half of
> > the 1800s would have been in a 12-note well-temperament,
>
> "A wrong (or at least highly suspect) note in the C. Wheatstone
> & Co. publication of Giuglio Regondi's Serenade or English
> concertina and piano (1859) should probably be emended with
> the mean-tone temperament of the instrument in mind. Until
> the late 1850s/early 1860s--and though published in 1859,
> the Serenade was probably composed in the 1840s--concertina
> manufacturers used a mean-tone temperament in which they
> divided the octave into 14 notes, differentiating between
> -- and providing separate buttons for--E/D, on the one hand,
> and A/G, on the other, with the flat note of each pair being
> tuned 41 cents higher than the sharp note." (Atlas, Allan.
> "A 41 cent emendation" (abstract) Early Music. 2005 Vol.33
> nr.4 )

Thanks for pointing that out. Yes, the concertina was the
glaring exception to the rule for keyboard instruments.
Meantone did linger on in the concertina longer than on
the piano. I'm sure you mean "Eb/D#" and "Ab/G#".

Meantone also lasted longer on the pipe organ, but
i'm not quite sure when pipe organs began to be tuned
mostly in 12-edo. It was certainly a _fait accompli_ by
the end of the 1800s, because writers then spoke of the
"hellish row" produced by the mixture stops (which played
an entire major triad for each key pressed) on organs
so tuned.

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗monz <monz@tonalsoft.com>

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> Meantone also lasted longer on the pipe organ, but
> i'm not quite sure when pipe organs began to be tuned
> mostly in 12-edo. It was certainly a _fait accompli_ by
> the end of the 1800s, because writers then spoke of the
> "hellish row" produced by the mixture stops (which played
> an entire major triad for each key pressed) on organs
> so tuned.

I should have been more precise about that:

The mixture stop produces a JI major-triad tuned as
a 4:5:6 proportion, for each key pressed.

Thus, if an organist is playing on an organ tuned
nominally in 12-edo, when a major triad is played
on the keyboard with the mixture stop on, the result
is this:

(in ~cents)

6 .... 701.9550009 .. 1101.955001 ... 1401.955001
5 .... 386.3137139 ... 786.3137139 .. 1086.313714
4 ...... 0 ........... 400 ........... 700

........ root ......... 3rd ........... 5th

Note these clashes:

* 400 with ~386
* 700 with both ~702 and ~786
* ~1102 with ~1086 (which, BTW, are both "major-7ths" of 0)

"Hellish row" is a vividly apt description of that!

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗monz <monz@tonalsoft.com>

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > * 1/4-comma meantone implies ~31-edo as Gene said;
> >
> > * 1/6-comma meantone implies a subset from ~55-edo
> > -- and the performance practice of this tuning began
> > before Beethoven, during Mozart's career, and
> > continued past 1924 ... fairly common use of tied
> > enharmonics also began with Mozart, but increased
> > a lot with Beethoven;
>
> And it can make quite a difference which you use.
> I made midi versions of the Hugo Reinhold Impromptu in
> 12, 103, 55, 43, 31, 81, 50, 69 and 19 equal, put them up
> in a zip file for comparison purposes. The point is,
> this piece brings out the difference in tuning pretty
> strongly.

The original discussion here concerned enharmonically
equivalent pairs of notes -- since this is for piano,
did you make both notes of such a pair the same, or different?
(excluding 12-edo, where they're always the same)
-- i.e., are they all 12-note tunings?

> I was struck by the fact that even 103, which is very
> close to 12-equal, sounds different; less bland, for
> one thing.

Yes, i was surprised to hear such a difference too.

> If you compare 55 to flatter systems, I think you'll
> hear quite a difference.

Progressing thru the "softer" meantones in order,
from 43 to 31 to 81 to 50 to 69 to 19, the piece
sounds progressively worse IMO.

> I think myself the piece is most interesting tuned
> that way, but clearly less authentic in sound.
>
> http://66.98.148.43/~xenharmo/scores/impromptus.zip

I also like the 55-edo version the best. Perhaps
Reinhold intended a well-temperament which featured
intervals approximating 1/6-comma meantone? These
were quite popular at one time.

I've been searching but have been able to find no
biographical information about Reinhold.

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> The original discussion here concerned enharmonically
> equivalent pairs of notes -- since this is for piano,
> did you make both notes of such a pair the same, or different?

I didn't have any diesis shifts, if that's what you're asking.

> (excluding 12-edo, where they're always the same)
> -- i.e., are they all 12-note tunings?

No, this is extened meantone; Fx and G, etc.

> Progressing thru the "softer" meantones in order,
> from 43 to 31 to 81 to 50 to 69 to 19, the piece
> sounds progressively worse IMO.

I'd hardly say that. I'd rate it like this:

12: Has a somewhat mechanical, bland flavor.

103: Surprisingly, though it sounds a lot like 12 in some ways, much
livilier and more expressive.

55: Still sounds somewhat like the 12 version, but with clearly
improved triads and expressiveness, and only a little exotic flavoring.

43: The septimal sound becomes more pronounced.

31, 81, 50: These are pretty similar; the triads now have a sweet,
singing sound to them, much superior to the other meantone ranges.
Diminished seventh chords and the like are giving us some distinctly
and exotically septimal sounds, which comes out very clearly.

69: Like above, but slightly more ragged.

19: The sound has now become distinctly harsher.

> I also like the 55-edo version the best. Perhaps
> Reinhold intended a well-temperament which featured
> intervals approximating 1/6-comma meantone? These
> were quite popular at one time.

I haven't a clue, alas.

> I've been searching but have been able to find no
> biographical information about Reinhold.

He's an Austrian composer of little fame, about whom I know nothing
much other than that he wrote this piece, which I think is excellent.

Here's a web page with a little information:

http://hjem.get2net.dk/Brofeldt/Catalogue_r.htm

Hugo Reinhold Austrian composer and pianist

Vienna, 03.03.1854 - Vienna, 04.09.1935

Reinhold started his musical career as a choir boy of court chapel in
Vienna after which he was admitted to the Conservatorium der
Musikfreunde - with the financial support of the Duke of Saxe-Coburg
and Gotha. His most important teachers were Anton Bruckner
(composition), Otto Dessoff and Julius Epstein (who was also Mahler's
piano teacher and fatherly friend).

Reinhold stayed at the conservatory till 1874 and later he became
teacher himself (of piano) at the Akademie der Tonkunst, in Vienna.
He was a prolific composer and his works were performed by the Vienna
Philharmonic Orchestra and the famous Hellmesberger Quartet and were
praised by the Vienna critics.

🔗monz <monz@tonalsoft.com>

Hi Gene,

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> > Progressing thru the "softer" meantones in order,
> > from 43 to 31 to 81 to 50 to 69 to 19, the piece
> > sounds progressively worse IMO.
>
> I'd hardly say that. I'd rate it like this:
>
> <Gene's descriptions snipped>

OK, you went into more detail than i was willing to.
But i think between the two of us there's a consensus
that 55 works best and 19 worst.

> > I've been searching but have been able to find no
> > biographical information about Reinhold.
>
> He's an Austrian composer of little fame, about whom I know nothing
> much other than that he wrote this piece, which I think is excellent.
>
> Here's a web page with a little information:
>
> http://hjem.get2net.dk/Brofeldt/Catalogue_r.htm
>
> Hugo Reinhold Austrian composer and pianist
>
> Vienna, 03.03.1854 - Vienna, 04.09.1935

Thanks for that. Hmm ... i'm surprised i've never
heard of him before, since he's exactly contemporary
with Mahler and from the same mileu - Vienna Conservatory,
student of Bruckner, etc.

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗Ozan Yarman <ozanyarman@ozanyarman.com>

Ouch! Having played the monstrous 1940s church organ at St. Anthony's in
Istanbul and the one in the grand church of Arequipa plaza, Peru (thankfully
not 12-equal in either case... I surmise), I can only try to imagine the
resultant cacophony emerging from the process you describe.

----- Original Message -----
From: "monz" <monz@tonalsoft.com>
To: <tuning@yahoogroups.com>
Sent: 18 Aralï¿½k 2005 Pazar 1:28
Subject: [tuning] mixture stop on organ (was: enharmonics of different
pitches)

> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > Meantone also lasted longer on the pipe organ, but
> > i'm not quite sure when pipe organs began to be tuned
> > mostly in 12-edo. It was certainly a _fait accompli_ by
> > the end of the 1800s, because writers then spoke of the
> > "hellish row" produced by the mixture stops (which played
> > an entire major triad for each key pressed) on organs
> > so tuned.
>
>
>
> I should have been more precise about that:
>
> The mixture stop produces a JI major-triad tuned as
> a 4:5:6 proportion, for each key pressed.
>
> Thus, if an organist is playing on an organ tuned
> nominally in 12-edo, when a major triad is played
> on the keyboard with the mixture stop on, the result
> is this:
>
> (in ~cents)
>
> 6 .... 701.9550009 .. 1101.955001 ... 1401.955001
> 5 .... 386.3137139 ... 786.3137139 .. 1086.313714
> 4 ...... 0 ........... 400 ........... 700
>
> ........ root ......... 3rd ........... 5th
>
>
> Note these clashes:
>
> * 400 with ~386
> * 700 with both ~702 and ~786
> * ~1102 with ~1086 (which, BTW, are both "major-7ths" of 0)
>
>
> "Hellish row" is a vividly apt description of that!
>
>
>
>
> -monz
> http://tonalsoft.com
> Tonescape microtonal music software
>
>
>
>

🔗Ozan Yarman <ozanyarman@ozanyarman.com>

🔗Petr Parízek <p.parizek@chello.cz>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Ozan Yarman <ozanyarman@ozanyarman.com>

Gene, have you tried set nota to e106? There is, normally a 12 cent
difference between neighbouring sharp and flat-tones. It sounds ok to me.

----- Original Message -----
From: "Gene Ward Smith" <gwsmith@svpal.org>
To: <tuning@yahoogroups.com>
Sent: 18 Aralï¿½k 2005 Pazar 11:07
Subject: [tuning] Re: enharmonics of different pitches (MMM: people's
perception of "microtonality

--- In tuning@yahoogroups.com, Petr Parï¿½zek <p.parizek@c...> wrote:
>
> Hi Gene.

> Sorry, are you sure the "103" version really uses the 60/103--octave
as the
> meantone fifth?

It's a Scala bug. Scala is supposed to compute the correct meantone,
using the nearest fifth, even when there is not a "Pythagorean" Pn
type notation to go with it, but in fact it doesn't. It cooks up some
kind of irregular temperament instead. Hence 103 and 69 and possibly
81 are wrong, but not by being the wrong fixed-fifth meantone. The
rest I think are OK, at least they check out on a test "E". 81 does
also but it doesn't have a P81 and so I wouldn't trust it until Manuel
fixes this.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

You're making assumptions about what I had in mind. Your assumptions
are limited and incomplete. I've discussed the problems that arise
with large orchestral scores comprising transposing instruments more
than once before, for one thing. We're talking about long stretches
of out-of-tune unisons in many cases if literal meantone is used.

> "Starting with Beethoven", his non-keyboard music would
> have been performed in a meantone tuning.

Huh? Whose non-keyboard music? And what is the basis for this
seemingly backwards claim?

> If someone points out some
> examples in Beethoven and particularly Schubert, i'll
> be happy to render them as Tonescape-produced mp3's
> for the tuning community's listening enjoyment. :)

Is is Schubert's quartet #14 or #15 that has the perpetually
ascending major third modulations? That would be a wonderful test
case, and I think would show some very interesting differences in
sound under different tuning strategies, but only touches on one of
the potential problems with a failure to observe assumed enharmonic
equivalence.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, "threesixesinarow" <CACCOLA@N...>
wrote:

> > In Europe, most keyboards during the first half of
> > the 1800s would have been in a 12-note well-temperament,
>
> "A wrong (or at least highly suspect) note in the C. Wheatstone &
Co.
> publication of Giuglio Regondi's Serenade or English concertina and
> piano (1859) should probably be emended with the mean-tone
temperament
> of the instrument in mind. Until the late 1850s/early 1860s--and
> though published in 1859, the Serenade was probably composed in the
> 1840s--concertina manufacturers used a mean-tone temperament in
which
> they divided the octave into 14 notes, differentiating between --
and
> providing separate buttons for--E/D, on the one hand, and A/G, on
the
> other, with the flat note of each pair being tuned 41 cents higher
> than the sharp note." (Atlas, Allan. "A 41 cent emendation"
(abstract)
> Early Music. 2005 Vol.33 nr.4 )

England and Spain clung to meantone far longer than the rest of
Europe.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > Meantone also lasted longer on the pipe organ, but
> > i'm not quite sure when pipe organs began to be tuned
> > mostly in 12-edo. It was certainly a _fait accompli_ by
> > the end of the 1800s, because writers then spoke of the
> > "hellish row" produced by the mixture stops (which played
> > an entire major triad for each key pressed) on organs
> > so tuned.
>
>
>
> I should have been more precise about that:
>
> The mixture stop produces a JI major-triad tuned as
> a 4:5:6 proportion, for each key pressed.
>
> Thus, if an organist is playing on an organ tuned
> nominally in 12-edo, when a major triad is played
> on the keyboard with the mixture stop on, the result
> is this:
>
> (in ~cents)
>
> 6 .... 701.9550009 .. 1101.955001 ... 1401.955001
> 5 .... 386.3137139 ... 786.3137139 .. 1086.313714
> 4 ...... 0 ........... 400 ........... 700
>
> ........ root ......... 3rd ........... 5th
>
>
> Note these clashes:
>
> * 400 with ~386
> * 700 with both ~702 and ~786
> * ~1102 with ~1086 (which, BTW, are both "major-7ths" of 0)
>
>
> "Hellish row" is a vividly apt description of that!

Now look at what a minor triad does on such a keyboard. You might
hear voices talking to you!

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗threesixesinarow <CACCOLA@NET1PLUS.COM>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗threesixesinarow <CACCOLA@NET1PLUS.COM>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Tom Dent <stringph@gmail.com>

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In tuning@yahoogroups.com, "threesixesinarow" <CACCOLA@N...> wrote:
>
> > Some French tuning instructions from 1830s here
> >
> > http://geocities.com/threesixesinarow/montalt.htm
>
> This guy wants everything from Fbb to B##, which is the 35 tones of
> Meantone[35]; that beats Mozart's mere 21 tones. No mention is made of
> the possibility of equating Fbb with D##, I see.
>

The piano tuning instructions clearly indicate 12EDO as far as it
could be achieved without beat-counting.

The 'physical scale' of 35 tones employed by non-fixed-intonation
musicmakers appears (from the short quotation which we have) to be
some sort of 1/6 comma, since the author refers to the old subdivision
of the whole tone into 4+5 'commas'. Probably 55EDO.

Why would Fbb be equivalent to D## here?

~~~T~~~

🔗a_sparschuh <a_sparschuh@yahoo.com>

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@g...> wrote:
> >
> > This guy wants everything from Fbb to B##, which is the 35
tones .....
> > No mention is made of
> > the possibility of equating Fbb with D##, I see.
> >
>
> The piano tuning instructions clearly indicate 12EDO as far as it
> could be achieved without beat-counting.
Nobody can tune irrational beat-frequencies exact in practice.

>
> The 'physical scale' of 35 tones employed by non-fixed-intonation
> musicmakers appears (from the short quotation which we have) to be
> some sort of 1/6 comma, since the author refers to the old
> subdivision
> of the whole tone into 4+5 'commas'. Probably 55EDO.
Pilolaos (cit. Böthuis) subdivided the

tonus (t) or 2nd=9:8 into ~9 commata
=
apotome (#) =3^7:2^11=2187:2048 amounting ~5 PC
*
limma (l) =2^8:3^5=256:243 amounting ~4 PC

so that the pythagorean ditonic 7-scale

C-t-D-t-E-l-F-t-G-t-A-t-B-l-C'

C-9-D-9-E-4-F-G-9-A-9-B-4-C'
consists in 53=5*9+2+4 commata steps,
but not in 55.

> Why would Fbb be equivalent to D## here?

if we take C=1 as the root
then

on the one hand
F=4:3,
b=1:#=2^11:3^7=2048:2187
makes
Fbb=2^24:3^15

on the other hand
D=9:8,
#=1:b=3^7:2^11=2187:2048
makes
D##=3^16:2^25

so D##:Fbb=3^31:2^49~=1,09720836200457050324530428042635.......

hence D##-->>Fbb is ~160.6...Cents sharp in the pythagorean
concept, nearly ~7 commata, percieved as an small 2nd.
Remark:
In 31EQ have all 5ths to be ~160.6...Cents/31=~5.2...Cents flat.
like in any other irregular 31 circulating systems on the average
too, as for
Vincentino, Huygens, Sauveur, Suppig &ct.....
Above identification D##=Fbb closes meantonic 31 cycles,
obtained from the SC division into 4 parts
as do
G#=Ab any PC division for coomon dodecaphonic: Schlick, Werckmeister
B#=Cb Gesualdos 19 cycle from SC division into 3 parts
from that on &ct....always in ++12 steps

31 Prätorius,-->> look aboves.. SC/4
43 Harry Partch-->>H.A.Kellner... SC/5
55 Telemann,Valotti,Mozart-->>B.P.Lehman... SC/6
(67 whoever likes...SC/7)...
.
the recent meantonic cul-de-sac of all above dead end woodways
dividing the SC

🔗Brad Lehman <bpl@umich.edu>

> 31 Prätorius,-->> look aboves.. SC/4
> 43 Harry Partch-->>H.A.Kellner... SC/5
> 55 Telemann,Valotti,Mozart-->>B.P.Lehman... SC/6
> (67 whoever likes...SC/7)...
> .
> the recent meantonic cul-de-sac of all above dead end woodways
> dividing the SC

Greetings! But let's clear up a couple of points: my proposed
temperament divides the *Pythagorean* comma, not the SC, in 1/6 and
1/12. My older SC version of it is only a footnote. In practice it's
the same as a 1/13 PC division.

And Kellner's doesn't divide the SC either; it divides the PC into
1/5.

And the 55-division is closer to regular 1/6 PC than to regular 1/6
SC. It's between them. For practical purposes, in tuning acoustic
instruments by ear, there is no audible distinction between
1/6 PC and strict 55-div. But 1/6 SC has cleaner tritones, and its
major 3rds are noticeably wider. Some details:
http://www-personal.umich.edu/~bpl/larips/meantone.html

Bradley Lehman

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, "a_sparschuh" <a_sparschuh@y...> wrote:

> > The 'physical scale' of 35 tones employed by non-fixed-intonation
> > musicmakers appears (from the short quotation which we have) to be
> > some sort of 1/6 comma, since the author refers to the old
> > subdivision
> > of the whole tone into 4+5 'commas'. Probably 55EDO.

> Pilolaos (cit. Böthuis) subdivided the
>
> tonus (t) or 2nd=9:8 into ~9 commata
> =
> apotome (#) =3^7:2^11=2187:2048 amounting ~5 PC
> *
> limma (l) =2^8:3^5=256:243 amounting ~4 PC
>
> so that the pythagorean ditonic 7-scale
>
> C-t-D-t-E-l-F-t-G-t-A-t-B-l-C'
>
> C-9-D-9-E-4-F-G-9-A-9-B-4-C'
> consists in 53=5*9+2+4 commata steps,
> but not in 55.

In 55, the apotome is four steps and the limma is five steps, and the
Pythagorean scale has become the diatonic scale of meantone. 256/243
is equivalent to 16/15, the diatonic semitone, in any meantone system;
and 2187/2048 to 25/24, the chromatic semitone. It is in terms of the
diatonic and chromatic semitones the ideal system is described, and it
seems to be describing 55.

> so D##:Fbb=3^31:2^49~=1,09720836200457050324530428042635.......

And of course in 31-et, where 3^31 and 2^49 are the same, it's a unison.

> 31 Prätorius,-->> look aboves.. SC/4
> 43 Harry Partch-->>H.A.Kellner... SC/5
> 55 Telemann,Valotti,Mozart-->>B.P.Lehman... SC/6
> (67 whoever likes...SC/7)...

What does Partch have to do with 43-et?

> the recent meantonic cul-de-sac of all above dead end woodways
> dividing the SC

Eh?

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, "Brad Lehman" <bpl@u...> wrote:

> But 1/6 SC has cleaner tritones, and its
> major 3rds are noticeably wider. Some details:
> http://www-personal.umich.edu/~bpl/larips/meantone.html

I found the discussion of tuning "Easter eggs", unexpected
higher-limit benefits of a particular meantone tuning, to be very
interesting, because it was based on performance practice.

In the 9-limit, we have the following Easter eggs:

A 7/4 egg near 31-et

A 7/5 egg near 31-et

A 7/6 egg near 31-et, but even closer to 81/et

There is, in other words, a region near 31-et with excellent septimal
harmonies. These are often clearly audible when this tuning is used if
the harmony gets away from the simplest sort of diatonicity; for
instance, when adapting romantic-era music to extended meantone tunings.

We also have a 9/7 egg near 69-et, or the Wilson meantone.

The really interesting thing to me is that Brad has found 11-limit
harmony to be helpful; this confirms my own feeling that 14/11 major
thirds can be "locked on" if you get close enough to them. This
business gets more complicated, however, since there are *two*
versions of 11-limit meantone. We have:

An 11/8 egg near 50-et, and another near 74-et

An 11/7 egg near 81-et, and another near 43-et

An 11/6 egg near 31-et, and another near 43-et

An 11/10 egg near 50-et, and another near 43-et

An 11/9 egg near 31-et

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗threesixesinarow <CACCOLA@NET1PLUS.COM>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗a_sparschuh <a_sparschuh@yahoo.com>

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> > tonus (t) or 2nd=9:8 into ~9 commata
> > =
> > apotome (#) =3^7:2^11=2187:2048 amounting ~5 PC
> > *
> > limma (l) =2^8:3^5=256:243 amounting ~4 PC

> In 55, the apotome is four steps and the limma is five steps, and the
> Pythagorean scale has become the diatonic scale of meantone.

> 256/243
> is equivalent to 16/15,
(16:15):(243:256)=81:80 the SC
that amounts about one step in 55.
Please give me your concrete ratio of yours "apotome" in 55.

> the diatonic semitone, in any meantone system;
> and 2187/2048 to 25/24, the chromatic semitone.
(2187:2048):(25:24)=6561:6400 even sharper than a PC,
already nearly an septimal comma 64:63, more than one step in 55.
What's here an "diatonic-semitone" in ratio?

> It is in terms of the
> diatonic and chromatic semitones the ideal system
Please tell me your personal
criterions for esteeming 55 "ideal"?

> is described, and it
> seems to be describing 55.

> > so D##:Fbb=3^31:2^49~=1,09720836200457050324530428042635.......
that indicates i.m.o. to an 31 division instead an 55 one.
>
> And of course in 31-et, where 3^31 and 2^49 are the same, it's a > >
>unison.
only in deed for 31, but not in 55.

Correction:
43 Harry Partch irregular -->>H.A.Kellner... PC/5 instead SC/5
55 Telemann,Valotti,Mozart SC/5 -->>B.P.Lehman PC/6

> What does Partch have to do with 43-et?
just follow the links to his 43 irregular:
http://www.google.de/search?hl=de&q=harry-partch+43&btnG=Google-Suche&meta=
as far as i understood his theory:
he startet once from irregular
7 first to
19 over
31 to reach final
43 as his own personal limit.
stepwise adding 12 more irr. tones at every stage into the octave.
Questions:
1. Why stopped Harry Partch at 43 irregular?
2. Had he ever considered 55 irregular?
as like Telemann & Mozart had done before him.
>
> > the recent meantonic cul-de-sac of all above dead end woodways
> > dividing the SC or PC into algebraic equal parts however.
that comment refers only to any logarithmic EDOs alone.
Why?
My favourite scala remains the classic 53 Pyth. concept,
reliable since immemorial times, ascribed to
Philolaos (from 9 subdivision of the 2nd, or 4 div. of the limma),
later definitivly (rediscovered again anew?) by Jing Fang.
http://www.google.de/search?hl=de&q=King-fang+comma+mercator&btnG=Suche&meta=
Preferable presented today in terms of Bosanquet and Helmholtz
just for easier handling and labeling:

with 2 different accidentials as sharpening up factors:
Def:
apotome: #:=3^7:2^11=2187:2048
comma: /:=3^12:2^19=531441:528244 the PC

and the according reverse flattening down factors:
Def:
inverse-apotome: b:=1:# the classical "molle"
inverse-comma: \:=1:/

then our antediluvial 53 commta-scale sounds,
as still apporoved and established since
already more than ~2 milleniums unchanged as:

C,C/,C//,Db\,Db,C#,C#/,D\\,D\,
D,D/,D//,Eb\,Eb,D#,D#/,E\\,E\,
E,E/,F\\,F\,
F,F/,F//,Gb\,Gb,F#,F#/,G\\,G\,
G,G/,G//,Ab\,Ab,G#,G#/,A\\,A\,
A,A/,A//,Bb\,Bb,A#,A#/,B\\,B\,
B,B/,c\\,c\
c

Note:
the first column remains stone-aged heptatonic:
C 1
D 9:8
E 81:64
F 4:3
G 3:2
A 27:16,
B 243:128
c 2

note Jing-Fangs comma is called in our
"old europe" also as Mercators comma,
to honour him for the import from china
accordingly

3^53:2^84=E//:F\\ or =B//:c\\

1200*ln(3^53:2^84):ln2 =~3.61504587....Cents

My puristic old fashioned point of view confirms:
The unrivaled 53 Pyth.
remains i.m.o. simply our best tuning,
Barely none competetitor has ever beaten its properties
in 2000 years.
I see no need for the fuzzy 53-edo approximation,
including any other even weaker equal division,
especially the crude 55 meant-nonics.
My reccommendation:
Avoid & get rid of any temperament!

SUMMARY:
Our good old 53 Pyth. can be labeled also too as:

"53 unplugged"

consisting of:
52 bare just pure 5ths + an concluding ~3.6C flat Jing-Fangian one:
cycle closed.

P.S: Nothing new underneath the sun.

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, "a_sparschuh" <a_sparschuh@y...> wrote:

> > 256/243
> > is equivalent to 16/15,
> (16:15):(243:256)=81:80 the SC
> that amounts about one step in 55.

It amounts to very close to one step, but it's zero steps anyway, if
we pick the best tunings for fifths and major thirds, which is
implicit in the discussion.

> Please give me your concrete ratio of yours "apotome" in 55.

It's not a ratio, it's 2^(4/55), and it's equivalent to 25/24 (and
135/128 and 250/243.)

> > It is in terms of the
> > diatonic and chromatic semitones the ideal system
> Please tell me your personal
> criterions for esteeming 55 "ideal"?

I don't assume it is ideal. The text in question assumes it as a
theoretical ideal, which was not uncommon at that time.

> My favourite scala remains the classic 53 Pyth. concept,
> reliable since immemorial times, ascribed to
> Philolaos (from 9 subdivision of the 2nd, or 4 div. of the limma),
> later definitivly (rediscovered again anew?) by Jing Fang.

Does anyone have a cite for Philoas giving second:limma as 9:4?

> I see no need for the fuzzy 53-edo approximation,
> including any other even weaker equal division,
> especially the crude 55 meant-nonics.

What's the big difference? The Mercator comma is 3.615 cents, so it
circulates anyway, with pure fifths or with 53-edo fifths. 55 is a
whole other thing altogether.

> My reccommendation:
> Avoid & get rid of any temperament!

Except if you use your Pythgorean tuning, you are probably going to
end up using C-Fb-G as a major triad, and C-D#-G as a minor triad, and
that's tempering the 5-limit. You might also use C-Fb-G-Cbb as a
utonal tetrad, for 7-limit harmony.

> P.S: Nothing new underneath the sun.

Except that the cycle of 53 pure fifths has been so little used it
qualifies.

🔗Ozan Yarman <ozanyarman@ozanyarman.com>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Ozan Yarman <ozanyarman@ozanyarman.com>

🔗monz <monz@tonalsoft.com>

> Questions:
> 1. Why stopped Harry Partch at 43 irregular?

He based his tuning on the 29-note 11-limit Tonality Diamond,
then filled in the large melodic gaps with other pitches
which were mostly 3/2 and 4/3 ratios above and below those
in the Diamond, until the octave was divided into quasi-equal
steps. The total just happened to be 43.

> 2. Had he ever considered 55 irregular?
> as like Telemann & Mozart had done before him.

Telemann and Mozart advocated subsets of 55-edo,
not "55 irregular".

> My puristic old fashioned point of view confirms:
> The unrivaled 53 Pyth.
> remains i.m.o. simply our best tuning,
> Barely none competetitor has ever beaten its properties
> in 2000 years.
> I see no need for the fuzzy 53-edo approximation,
> including any other even weaker equal division,
> especially the crude 55 meant-nonics.
> My reccommendation:
> Avoid & get rid of any temperament!

55-edo is fundamentally different from 53-edo:

53-edo maps both the syntonic-comma and pythagorean-comma
to 1 degree, and thus preserves both of them.

55-edo makes both of these commas vanish, which is
what makes it a meantone.

Anyway, 53-edo is only slightly different from "53 Pyth."
-- the maximum difference is only +/- ~1.773418726 cents:

("generator" refers to the chain-of-5ths, "deviation"
is the difference in ~cents of 53-edo from "53 Pyth")

generator step . 53-pyth cents . 53-edo cents ... deviation

.... 0 .. 53 .. 1200 ......... 1200 .. ..........0
.. -12 .. 52 .. 1176.53999 ... 1177.358491 .... +0.818500951
.. -24 .. 51 .. 1153.079979 .. 1154.716981 .... +1.637001901
.. +17 .. 50 .. 1133.235015 .. 1132.075472 .... -1.159543013
... +5 .. 49 .. 1109.775004 .. 1109.433962 .... -0.341042063
... -7 .. 48 .. 1086.314994 .. 1086.792453 .... +0.477458888
.. -19 .. 47 .. 1062.854984 .. 1064.150943 .... +1.295959839
.. +22 .. 46 .. 1043.010019 .. 1041.509434 .... -1.500585076
.. +10 .. 45 .. 1019.550009 .. 1018.867925 .... -0.682084126
... -2 .. 44 ... 996.0899983 .. 996.2264151 ... +0.136416825
.. -14 .. 43 ... 972.6299879 .. 973.5849057 ... +0.954917776
.. -26 .. 42 ... 949.1699775 .. 950.9433962 ... +1.773418726
.. +15 .. 41 ... 929.325013 ... 928.3018868 ... -1.023126188
... +3 .. 40 ... 905.8650026 .. 905.6603774 ... -0.204625238
... -9 .. 39 ... 882.4049922 .. 883.0188679 ... +0.613875713
.. -21 .. 38 ... 858.9449818 .. 860.3773585 ... +1.432376664
.. +20 .. 37 ... 839.1000173 .. 837.7358491 ... -1.364168251
... +8 .. 36 ... 815.6400069 .. 815.0943396 ... -0.5456673
... -4 .. 35 ... 792.1799965 .. 792.4528302 ... +0.27283365
.. -16 .. 34 ... 768.7199862 .. 769.8113208 ... +1.091334601
.. +25 .. 33 ... 748.8750216 .. 747.1698113 ... -1.705210314
.. +13 .. 32 ... 725.4150113 .. 724.5283019 ... -0.886709363
... +1 .. 31 ... 701.9550009 .. 701.8867925 ... -0.068208413
.. -11 .. 30 ... 678.4949905 .. 679.245283 .... +0.750292538
.. -23 .. 29 ... 655.0349801 .. 656.6037736 ... +1.568793489
.. +18 .. 28 ... 635.1900156 .. 633.9622642 ... -1.227751426
... +6 .. 27 ... 611.7300052 .. 611.3207547 ... -0.409250475
... -6 .. 26 ... 588.2699948 .. 588.6792453 ... +0.409250475
.. -18 .. 25 ... 564.8099844 .. 566.0377358 ... +1.227751426
.. +23 .. 24 ... 544.9650199 .. 543.3962264 ... -1.568793489
.. +11 .. 23 ... 521.5050095 .. 520.754717 .... -0.750292538
... -1 .. 22 ... 498.0449991 .. 498.1132075 ... +0.068208413
.. -13 .. 21 ... 474.5849887 .. 475.4716981 ... +0.886709363
.. -25 .. 20 ... 451.1249784 .. 452.8301887 ... +1.705210314
.. +16 .. 19 ... 431.2800138 .. 430.1886792 ... -1.091334601
... +4 .. 18 ... 407.8200035 .. 407.5471698 ... -0.27283365
... -8 .. 17 ... 384.3599931 .. 384.9056604 ... +0.5456673
.. -20 .. 16 ... 360.8999827 .. 362.2641509 ... +1.364168251
.. +21 .. 15 ... 341.0550182 .. 339.6226415 ... -1.432376664
... +9 .. 14 ... 317.5950078 .. 316.9811321 ... -0.613875713
... -3 .. 13 ... 294.1349974 .. 294.3396226 ... +0.204625238
.. -15 .. 12 ... 270.674987 ... 271.6981132 ... +1.023126188
.. +26 .. 11 ... 250.8300225 .. 249.0566038 ... -1.773418726
.. +14 .. 10 ... 227.3700121 .. 226.4150943 ... -0.954917776
... +2 ... 9 ... 203.9100017 .. 203.7735849 ... -0.136416825
.. -10 ... 8 ... 180.4499913 .. 181.1320755 ... +0.682084126
.. -22 ... 7 ... 156.989981 ... 158.490566 .... +1.500585076
.. +19 ... 6 ... 137.1450164 .. 135.8490566 ... -1.295959839
... +7 ... 5 ... 113.6850061 .. 113.2075472 ... -0.477458888
... -5 ... 4 .... 90.22499567 .. 90.56603774 ...+0.341042063
.. -17 ... 3 .... 66.76498529 .. 67.9245283 ... +1.159543013
.. +24 ... 2 .... 46.92002077 .. 45.28301887 .. -1.637001901
.. +12 ... 1 .... 23.46001038 .. 22.64150943 .. -0.818500951
.... 0 ... 0 ..... 0 ............ 0 ............ 0

🔗monz <monz@tonalsoft.com>

Hi Gene,

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In tuning@yahoogroups.com, "a_sparschuh" <a_sparschuh@y...> wrote:

> > My favourite scala remains the classic 53 Pyth. concept,
> > reliable since immemorial times, ascribed to
> > Philolaos (from 9 subdivision of the 2nd, or 4 div. of the limma),
> > later definitivly (rediscovered again anew?) by Jing Fang.
>
> Does anyone have a cite for Philoas giving second:limma as 9:4?

http://tonalsoft.com/enc/p/philolaus.aspx

Philolaus did not call the "minor semitone" a "limma",
but a "diesis". But his description clearly shows that
he meant the pythaogrean ratio 256/243, functioning
as a "diatonic semitone":

>> [Boethius's Latin:]
>> "Diesis inquit est spatium quo est maior sesquitertia
>> proportio duobus tonis.""
>>
>> [Bower's English:]
>> "The diesis ... is the interval by which a sesquitertian
>> ratio [4:3] is larger than two tones [(9:8)^2]"

Philolaus clearly states that the whole-tone ("second")
is a comma larger than a 2 "dieses" (_limmata_ ):

>> [Boethius's Latin:]
>> "Comma uero est spatium quo maior est sesquioctaua
>> proportio duabus diesibus. id est duobus semitoniis
>> minoribus."
>>
>> [Bower's English:]
>> "The comma is the interval by which the sesquioctave
>> ratio [9:8] is larger than two dieses [(256:243)^2]
>> -- that is, larger than two minor semitones."

Thus, by implication, the "major semitone" (called
_apotome_ by most later Greek theorists) is a comma
larger than Philolaus's "diesis".

But anyway, the point is that in Philolaus's terminology:

tone = 2 dieses and a comma

So Philolaus's "comma" is our standard pythagorean-comma:
2,3-monzo [-19 12, > = ~23.46001038 cents.

And Philolaus's "diaschisma" is 1/2 "diesis" (1/2 _limma_):

>> [Boethius's Latin:]
>> "Diascissma [Diaschissma corr. supra lin.] uero
>> dimidium dieseos: id est semitonii minoris"
>>
>> [Bower's English:]
>> "The diaschisma is half a diesis -- that is,
>> half a minor semitone. "

So his "diaschisma" has 2,3-monzo [4 -5/2,>
= ~45.11249784 cents.

Thus, in Philolaus's terminology, we have:

diesis (_limma_) = 2 diaschismata
tone = 2 diesis and a comma = 4 diaschismata and a comma

Now it just happens that in this scheme of measurement,
a "comma" is very close to 1/2 of a "diaschisma".

1/2 diaschisma = ~22.55624892 cents
comma = ........ ~23.46001038 cents

Philolaus never equated these two, but if one does
assume that they can be approximately the same, then:

diesis (_limma_) = 2 diaschismata = 4 commas
tone = 4 diaschismata and a comma = 9 commas

This in fact did become "common knowledge" tuning theory
during the medieval period.

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗threesixesinarow <CACCOLA@NET1PLUS.COM>

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > Questions:
> > 1. Why stopped Harry Partch at 43 irregular?
>
>
> He based his tuning on the 29-note 11-limit Tonality Diamond,
> then filled in the large melodic gaps with other pitches
> which were mostly 3/2 and 4/3 ratios above and below those
> in the Diamond, until the octave was divided into quasi-equal
> steps. The total just happened to be 43.

Hi,

Maybe there is a more concrete evidence. At Corporeal Meadows site I
read this:

"Chromelodeon I was his first, and long-lasting, reed organ. He had
another Chromelodeon II that gave out after about 3 years in 1949, and
the one pictured here dates from that time. From the first
Chromelodeon II he salvaged all the reeds and also a separate sub-bass
keyboard, which he subsequently fitted into Chromelodeon I (this can
be seen in the close-up photo as a series of levers above the low end
of the keyboard). The stops bring in different sets of reeds, and on
Chromelodeon II, as heard in the excerpt, harmonies can be played with
one key, in addition to it being one of the more rare reed organs to
span 88 notes on the keyboard."

http://www.corporeal.com/art_inst/incrinst/

The last sentence, I mean.

Clark

🔗Jon Szanto <jszanto@cox.net>

🔗David Beardsley <db@biink.com>

🔗Jon Szanto <jszanto@cox.net>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, "a_sparschuh" <a_sparschuh@y...> wrote:

> > What does Partch have to do with 43-et?
> just follow the links to his 43 irregular:
> http://www.google.de/search?hl=de&q=harry-partch+43&btnG=Google-
Suche&meta=
> as far as i understood his theory:
> he startet once from irregular
> 7 first to
> 19 over
> 31 to reach final
> 43 as his own personal limit.
> stepwise adding 12 more irr. tones at every stage into the octave.

This doesn't describe Partch's theory at all. Where are you geting
this misinformation? Clearly you haven't read Partch's own
writings . . . what did your google search lead you to?

> My puristic old fashioned point of view confirms:
> The unrivaled 53 Pyth.
> remains i.m.o. simply our best tuning,
> Barely none competetitor has ever beaten its properties
> in 2000 years.

Most Western common-practice music sounds wrong to me in this tuning,
no matter what kinds of contortions you go through. So I respect your
opinion, but no thank you.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗Ozan Yarman <ozanyarman@ozanyarman.com>

Very good Paul! This should put a stop to people saying that there can only
be 53 commas per octave.

Cordially,
Ozan

----- Original Message -----
From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: 25 Aralï¿½k 2005 Pazar 3:55
Subject: [tuning] avoid & get rid of any Re: enharmonics of slightly
different pitches by 53 Pyth.

> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > 55-edo is fundamentally different from 53-edo:
> >
> > 53-edo maps both the syntonic-comma and pythagorean-comma
> > to 1 degree, and thus preserves both of them.
> >
> > 55-edo makes both of these commas vanish, which is
> > what makes it a meantone.
>
> Not correct. The pythagorean comma does not vanish in 55-equal. This is
> shown by the fact that "enharmonic equivalents" such as G# and Ab are 1
> degree apart in 55-equal.
>
> Meantone only means that the syntonic comma vanishes, not that the
> pythagorean comma vanishes.
>
> Back to square one, as usual.
>
>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

I don't know if this was a joke, but if you mean it seriously, you
should elaborate -- I don't know what you mean.

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@o...> wrote:
>
> Very good Paul! This should put a stop to people saying that there
can only
> be 53 commas per octave.
>
> Cordially,
> Ozan
>
> ----- Original Message -----
> From: "wallyesterpaulrus" <wallyesterpaulrus@y...>
> To: <tuning@yahoogroups.com>
> Sent: 25 Aralýk 2005 Pazar 3:55
> Subject: [tuning] avoid & get rid of any Re: enharmonics of slightly
> different pitches by 53 Pyth.
>
>
> > --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
> >
> > > 55-edo is fundamentally different from 53-edo:
> > >
> > > 53-edo maps both the syntonic-comma and pythagorean-comma
> > > to 1 degree, and thus preserves both of them.
> > >
> > > 55-edo makes both of these commas vanish, which is
> > > what makes it a meantone.
> >
> > Not correct. The pythagorean comma does not vanish in 55-equal.
This is
> > shown by the fact that "enharmonic equivalents" such as G# and Ab
are 1
> > degree apart in 55-equal.
> >
> > Meantone only means that the syntonic comma vanishes, not that the
> > pythagorean comma vanishes.
> >
> > Back to square one, as usual.
> >
> >
>

🔗Ozan Yarman <ozanyarman@ozanyarman.com>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

And in 72-equal, the Pythagorean comma vanishes while the syntonic
comma doesn't. So? I don't see the relevance.

It doesn't seem that the 'commas' that "people here" (in Turkey?) are
talking about have anything to do with these kinds of commas which
are vectors in the lattice; instead, they're probably just
thinking 'intervals'. Wouldn't you agree?

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@o...> wrote:
>
> Well, people here imagine that an octave cannot contain any more
`commas`
> than 53. 55 equal is a system which, while eliminating the syntonic
comma,
> preserves the `Pythagorean` as you said.
>
> ----- Original Message -----
> From: "wallyesterpaulrus" <wallyesterpaulrus@y...>
> To: <tuning@yahoogroups.com>
> Sent: 25 Aralýk 2005 Pazar 4:29
> Subject: [tuning] avoid & get rid of any Re: enharmonics of slightly
> different pitches by 53 Pyth.
>
>
> I don't know if this was a joke, but if you mean it seriously, you
> should elaborate -- I don't know what you mean.
>

🔗Ozan Yarman <ozanyarman@ozanyarman.com>

----- Original Message -----
From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: 25 Aralï¿½k 2005 Pazar 4:45
Subject: [tuning] avoid & get rid of any Re: enharmonics of slightly
different pitches by 53 Pyth.

[PA]

And in 72-equal, the Pythagorean comma vanishes while the syntonic
comma doesn't. So? I don't see the relevance.

[OZ]

I'm surprised you don't.

[PA]
It doesn't seem that the 'commas' that "people here" (in Turkey?) are
talking about have anything to do with these kinds of commas which
are vectors in the lattice; instead, they're probably just
thinking 'intervals'. Wouldn't you agree?

[OZ]
No, I don't agree. They are specifically referring to the Pythagorean comma
as what remains after a cycle of 12 fifths.

🔗monz <monz@tonalsoft.com>

Hi Paul,

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
>
> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > 55-edo is fundamentally different from 53-edo:
> >
> > 53-edo maps both the syntonic-comma and pythagorean-comma
> > to 1 degree, and thus preserves both of them.
> >
> > 55-edo makes both of these commas vanish, which is
> > what makes it a meantone.
>
> Not correct. The pythagorean comma does not vanish in
> 55-equal. This is shown by the fact that "enharmonic
> equivalents" such as G# and Ab are 1 degree apart in
> 55-equal.
>
> Meantone only means that the syntonic comma vanishes,
> not that the pythagorean comma vanishes.

Duh, my bad. Yes, of course i know that. I muddled this
because in meantones, the pythagorean-comma is the same
as the diesis.

So it's incorrect to say that the pythagorean-comma
vanishes, but it *is* correct to say that it does not
have an "independent" existence -- which *is* true
for 53-edo.

> Back to square one, as usual.

What does that mean?

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Ozan Yarman <ozanyarman@ozanyarman.com>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Ozan Yarman <ozanyarman@ozanyarman.com>

Because, originally:

prime 2 = 1200.000 cents
prime 3 = 1901.955 cents
prime 5 = 2786.314 cents

Best 2 in 1-edo is: 1200 cents
Best approx. of 3 in 1-edo is: 2400 cents (every two octaves/degrees)
Best approx. of 5 in 1-edo is again: 2400 cents

80/81 prime factorized is:

5*2^4 / 3^4

531441/524288 prime factorized is:

3^12 / 2^19

Syntonic comma in 1-edo is:

2 deg + (4* 1 deg) - (4* 2 deg)=
2400+(4* 1200) - (4* 2400)=
7200-9600=every 2400 cents (-2 octaves)

Pythagorean comma in 1-edo is:

(12* 2 deg) - (19* 1 deg)=
(12*2400) - (19*1200)=
28800-22800=every 6000 cents (+5 octaves)

This shall pave the way to a whole new era of deconstructionism in music
theory!

Oz.

----- Original Message -----
From: "Gene Ward Smith" <gwsmith@svpal.org>
To: <tuning@yahoogroups.com>
Sent: 25 Aralï¿½k 2005 Pazar 18:22
Subject: [tuning] avoid & get rid of any Re: enharmonics of slightly
different pitches by 53 Pyth.

> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@o...> wrote:
> >
> > Hahaha, that was a funny one!
>
> The best part of the joke is that it's true. The closest you can get
> to a 3 in terms of otaves is two octaves, or 4, and the same is true
> of 5. This makes 81/80 a 4 also, and the Pythagorean comma a 32, or
> five octaves. These are both positive numbers of octaves, but
> distinct; hence the commas are both preserved and distinguished.
>
>
>

🔗monz <monz@tonalsoft.com>

> --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:
>
> > What is the smallest EDO that preserves and
> > _distinguishes_ both the syntonic comma and
> > the pythagorean comma?

From: "Gene Ward Smith" <gwsmith@...>
>
> 1-edo.
>
>
>

> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@o...> wrote:

> >
> > Hahaha, that was a funny one!
--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

>
> The best part of the joke is that it's true. The closest you can get
> to a 3 in terms of otaves is two octaves, or 4, and the same is true
> of 5. This makes 81/80 a 4 also, and the Pythagorean comma a 32, or
> five octaves. These are both positive numbers of octaves, but
> distinct; hence the commas are both preserved and distinguished.
>

To get the answer you're really looking for, it
depends on exactly what you mean by "preserve".

17-edo is the smallest edo i'm aware of which gives
anything resembling diatonic scales, in which neither
the syntonic-comma nor pythagorean-comma vanish. But
they are both the same size: one degree of the tuning.

22-edo is the smallest edo i know in which they do
not vanish and they also roughly approximate the fact
that the pythagorean-comma is bigger than the syntonic:
p.c.= 2 degrees and s.c. = 1 degree. However, in JI
the p.c. is only ~2 cents higher than the s.c. -- here
it is *twice* as large!

You can see how edos approximate 5-limit JI on my
"bingo-card lattices" webpage -- click on the edo links
on the bottom diagram:

http://tonalsoft.com/enc/b/bingo.aspx

However, you can't see the absolute error values for
the comma sizes ... you'd have to do the math for that
yourself.

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

On Sun, 25 Dec 2005, "Ozan Yarman" wrote:
>
> Because, originally:
>
> prime 2 = 1200.000 cents
> prime 3 = 1901.955 cents
> prime 5 = 2786.314 cents
>
> Best 2 in 1-edo is: 1200 cents
> Best approx. of 3 in 1-edo is: 2400 cents (every two octaves/degrees)
> Best approx. of 5 in 1-edo is again: 2400 cents
>
> 80/81 prime factorized is:
>
> 5*2^4 / 3^4
>
> 531441/524288 prime factorized is:
>
> 3^12 / 2^19
>
> Syntonic comma in 1-edo is:
>
> 2 deg + (4* 1 deg) - (4* 2 deg)=
> 2400+(4* 1200) - (4* 2400)=
> 7200-9600=every 2400 cents (-2 octaves)
>
> Pythagorean comma in 1-edo is:
>
> (12* 2 deg) - (19* 1 deg)=
> (12*2400) - (19*1200)=
> 28800-22800=every 6000 cents (+5 octaves)

Hey, Oz, you're getting quite good at all this mathematical
stuff ... !

> This shall pave the way to a whole new era of deconstructionism
> in music theory!
>
> Oz.

But who will listen to the music anyone writes in it? It
would appear to be singularly lacking in useful musical
resources! However ...

Earlier this evening, I saw an intriguing (and over-hyped)
doco on the National Geographic Channel about the loss
of the musical tradition of "Dabia" - improvised singing
accompanied on a lute - of the Nu tribe of South China.
If anyone else has seen this show, I invite them to share
their thoughts on the tuning used by the "master", both
on his lute and vocally.

Regards,
Yahya

> ----- Original Message -----
> From: "Gene Ward Smith"
>
> > --- In tuning@yahoogroups.com, "Ozan Yarman" wrote:
> > >
> > > Hahaha, that was a funny one!
> >
> > The best part of the joke is that it's true. The closest you can get
> > to a 3 in terms of otaves is two octaves, or 4, and the same is true
> > of 5. This makes 81/80 a 4 also, and the Pythagorean comma a 32, or
> > five octaves. These are both positive numbers of octaves, but
> > distinct; hence the commas are both preserved and distinguished.

--
No virus found in this outgoing message.
Checked by AVG Free Edition.
Version: 7.1.371 / Virus Database: 267.14.7/214 - Release Date: 23/12/05

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

🔗Ozan Yarman <ozanyarman@ozanyarman.com>

Sorry for barging in like this, but I can't help myself...

Al-Javab:

2-edo

ROFLWL
Oz.

----- Original Message -----
From: "Yahya Abdal-Aziz" <yahya@melbpc.org.au>
To: <tuning@yahoogroups.com>
Sent: 27 Aralï¿½k 2005 Salï¿½ 18:20
Subject: [tuning] RE: avoid & get rid of any Re: enharmonics of slightly
different pitches by 53 Pyth.

>
> On Sun, 25 Dec 2005, Gene Ward Smith wrote:
> >
> > --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz"
> > <yahya@m...> wrote:
> >
> > > What is the smallest EDO that preserves and
> > > _distinguishes_ both the syntonic comma and
> > > the pythagorean comma?
> >
> > 1-edo.
>
> Ha! :-)
>
> Alright then, rephrasing and redirecting the question:
>
> Gene,
> What is the smallest nontrivial EDO that preserves
> and _distinguishes_ both the syntonic comma and
> the pythagorean comma?
>
> Regards,
> Yahya
>

🔗Ozan Yarman <ozanyarman@ozanyarman.com>

----- Original Message -----
From: "Yahya Abdal-Aziz" <yahya@melbpc.org.au>
To: <tuning@yahoogroups.com>
Sent: 27 Aralï¿½k 2005 Salï¿½ 18:20
Subject: [tuning] Re: avoid & get rid of any Re: enharmonics of slightly
different pitches by 53 Pyth.

>
> On Sun, 25 Dec 2005, "Ozan Yarman" wrote:
> >
> > Because, originally:
> >
> > prime 2 = 1200.000 cents
> > prime 3 = 1901.955 cents
> > prime 5 = 2786.314 cents
> >
> > Best 2 in 1-edo is: 1200 cents
> > Best approx. of 3 in 1-edo is: 2400 cents (every two octaves/degrees)
> > Best approx. of 5 in 1-edo is again: 2400 cents
> >
> > 80/81 prime factorized is:
> >
> > 5*2^4 / 3^4
> >
> > 531441/524288 prime factorized is:
> >
> > 3^12 / 2^19
> >
> > Syntonic comma in 1-edo is:
> >
> > 2 deg + (4* 1 deg) - (4* 2 deg)=
> > 2400+(4* 1200) - (4* 2400)=
> > 7200-9600=every 2400 cents (-2 octaves)
> >
> > Pythagorean comma in 1-edo is:
> >
> > (12* 2 deg) - (19* 1 deg)=
> > (12*2400) - (19*1200)=
> > 28800-22800=every 6000 cents (+5 octaves)
>
> Hey, Oz, you're getting quite good at all this mathematical
> stuff ... !
>
>

Thanks to my pedant, Paul Erlich. I owe the progress I made mostly to him.
Also, I am very much indebted to George Secor, Gene Ward Smith and others
for their patience and assistance.

> > This shall pave the way to a whole new era of deconstructionism
> > in music theory!
> >
> > Oz.
>
> But who will listen to the music anyone writes in it? It
> would appear to be singularly lacking in useful musical
> resources! However ...
>

So, you are saying that one cannot make music with octaves?

> Earlier this evening, I saw an intriguing (and over-hyped)
> doco on the National Geographic Channel about the loss
> of the musical tradition of "Dabia" - improvised singing
> accompanied on a lute - of the Nu tribe of South China.
> If anyone else has seen this show, I invite them to share
> their thoughts on the tuning used by the "master", both
> on his lute and vocally.
>

And I watched the French Revolution in the History Channel. May Robespierre
be damned, the tyrant sicko.

> Regards,
> Yahya
>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@o...> wrote:
>
>
> ----- Original Message -----
> From: "wallyesterpaulrus" <wallyesterpaulrus@y...>
> To: <tuning@yahoogroups.com>
> Sent: 25 Aralýk 2005 Pazar 4:45
> Subject: [tuning] avoid & get rid of any Re: enharmonics of slightly
> different pitches by 53 Pyth.
>
> [PA]
>
> And in 72-equal, the Pythagorean comma vanishes while the syntonic
> comma doesn't. So? I don't see the relevance.
>
> [OZ]
>
> I'm surprised you don't.
>
> [PA]
> It doesn't seem that the 'commas' that "people here" (in Turkey?)
are
> talking about have anything to do with these kinds of commas which
> are vectors in the lattice; instead, they're probably just
> thinking 'intervals'. Wouldn't you agree?
>
> [OZ]
> No, I don't agree. They are specifically referring to the
Pythagorean comma
> as what remains after a cycle of 12 fifths.

If that is specifically what they are referring to, and no tempering
of these fifths or the octave is allowed, then they are pretty much
correct -- only 51+ pure Pythagorean commas will fit into a pure
octave.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> Hi Paul,
>
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> >
> > --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
> >
> > > 55-edo is fundamentally different from 53-edo:
> > >
> > > 53-edo maps both the syntonic-comma and pythagorean-comma
> > > to 1 degree, and thus preserves both of them.
> > >
> > > 55-edo makes both of these commas vanish, which is
> > > what makes it a meantone.
> >
> > Not correct. The pythagorean comma does not vanish in
> > 55-equal. This is shown by the fact that "enharmonic
> > equivalents" such as G# and Ab are 1 degree apart in
> > 55-equal.
> >
> > Meantone only means that the syntonic comma vanishes,
> > not that the pythagorean comma vanishes.
>
>
> Duh, my bad. Yes, of course i know that. I muddled this
> because in meantones, the pythagorean-comma is the same
> as the diesis.
>
> So it's incorrect to say that the pythagorean-comma
> vanishes, but it *is* correct to say that it does not
> have an "independent" existence -- which *is* true
> for 53-edo.

What is that supposed to mean? Independent existence?

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
>
> > --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...>
wrote:
> >
> > > What is the smallest EDO that preserves and
> > > _distinguishes_ both the syntonic comma and
> > > the pythagorean comma?

> To get the answer you're really looking for, it
> depends on exactly what you mean by "preserve".

Indeed!

> 17-edo is the smallest edo i'm aware of which gives
> anything resembling diatonic scales, in which neither
> the syntonic-comma nor pythagorean-comma vanish. But
> they are both the same size: one degree of the tuning.

I don't agree that the syntonic comma in 17-equal is unambiguously
one degree of the tuning. There are two different mappings of 5-limit
JI to 17-equal that are about equally good (bad), and in one of these
(in fact the slightly better one), the syntonic comma vanishes. If
you're only seeing one of these interpretations of 17-equal, then
your method will be missing out on the best interpretations of many
higher equal temperaments, such as 64-equal.

> 22-edo is the smallest edo i know in which they do
> not vanish and they also roughly approximate the fact
> that the pythagorean-comma is bigger than the syntonic:
> p.c.= 2 degrees and s.c. = 1 degree. However, in JI
> the p.c. is only ~2 cents higher than the s.c. -- here
> it is *twice* as large!

It's easy to find a simpler example. For example, in 15-equal, the
syntonic comma is 1 degree, and the pythagorean comma is 3 degrees!

> You can see how edos approximate 5-limit JI on my
> "bingo-card lattices" webpage -- click on the edo links
> on the bottom diagram:
>
> http://tonalsoft.com/enc/b/bingo.aspx

I don't agree with the assessments you make of many equal
temperaments on this page. 17 is but one example.

🔗Ozan Yarman <ozanyarman@ozanyarman.com>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗Ozan Yarman <ozanyarman@ozanyarman.com>

----- Original Message -----
From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: 28 Aralï¿½k 2005 ï¿½arï¿½amba 0:04
Subject: [tuning] avoid & get rid of any Re: enharmonics of slightly
different pitches by 53 Pyth.

> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@o...> wrote:
> >
> > > [OZ]
> > > No, I don't agree. They are specifically referring to the
> > Pythagorean comma
> > > as what remains after a cycle of 12 fifths.
> >
> > [PA]
> > If that is specifically what they are referring to, and no
> tempering
> > of these fifths or the octave is allowed, then they are pretty much
> > correct -- only 51+ pure Pythagorean commas will fit into a pure
> > octave.
> >
> > [OZ]
> >
> > Exactly.
>
> Then why did you write, "Very good Paul! This should put a stop to
> people saying that there can only be 53 commas per octave."
>
> /tuning/topicId_62918.html#63092
>
> ?
>
>

As you so elegantly put it, only 51.151 untempered `ditonic commas` fit into
one octave.

Newsflash... the current theory of Turkish Music is based on a cycle of pure
fifths alone.

Options:

1. Consider 53-tET and agree to the tempering of every JI or RI interval
known.

2. Stick to the good ol' Pythagorean tuning with 51 commas in one octave as
an abstraction alone.

Take your pick.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@o...> wrote:
>
>
> ----- Original Message -----
> From: "wallyesterpaulrus" <wallyesterpaulrus@y...>
> To: <tuning@yahoogroups.com>
> Sent: 28 Aralýk 2005 Çarþamba 0:04
> Subject: [tuning] avoid & get rid of any Re: enharmonics of slightly
> different pitches by 53 Pyth.
>
>
> > --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@o...>
wrote:
> > >
> > > > [OZ]
> > > > No, I don't agree. They are specifically referring to the
> > > Pythagorean comma
> > > > as what remains after a cycle of 12 fifths.
> > >
> > > [PA]
> > > If that is specifically what they are referring to, and no
> > tempering
> > > of these fifths or the octave is allowed, then they are pretty
much
> > > correct -- only 51+ pure Pythagorean commas will fit into a pure
> > > octave.
> > >
> > > [OZ]
> > >
> > > Exactly.
> >
> > Then why did you write, "Very good Paul! This should put a stop to
> > people saying that there can only be 53 commas per octave."
> >
> > /tuning/topicId_62918.html#63092
> >
> > ?
> >
> >
>
> As you so elegantly put it, only 51.151 untempered `ditonic commas`
fit into
> one octave.
>
> Newsflash... the current theory of Turkish Music is based on a
cycle of pure
> fifths alone.

So?

> Options:
>
> 1. Consider 53-tET and agree to the tempering of every JI or RI
interval
> known.
>
> 2. Stick to the good ol' Pythagorean tuning with 51 commas in one
octave as
> an abstraction alone.
>
> Take your pick.

I don't get it. And I don't agree with this dichotomy. Pythagorean
tuning extended out to 53 notes is more than just an abstraction and
is very close to 53-equal but does not require tempering. The
division of the octave so obtained contains 41 (not 51!) steps each
equal to a Pythagorean comma, and 12 steps of 19.84 cents (Pyth.
comma minus Mercator comma), for 53 comma-sized steps in total.

🔗Ozan Yarman <ozanyarman@ozanyarman.com>

[PA]
I don't get it. And I don't agree with this dichotomy. Pythagorean
tuning extended out to 53 notes is more than just an abstraction and
is very close to 53-equal but does not require tempering. The
division of the octave so obtained contains 41 (not 51!) steps each
equal to a Pythagorean comma, and 12 steps of 19.84 cents (Pyth.
comma minus Mercator comma), for 53 comma-sized steps in total.

[OZ]
Even so, there are never 53 pure ditonic commas in one octave. You prove my
point.

And Mercator diminished Pythagorean commas do not count! There are only
51.151 full sized ditonic commas in an octave, which is only good for
abstraction alone.

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗Ozan Yarman <ozanyarman@ozanyarman.com>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗monz <monz@tonalsoft.com>

Hi Paul,

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
>
> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> > So it's incorrect to say that [in 55-edo] the
> > pythagorean-comma vanishes, but it *is* correct to
> > say that it does not have an "independent" existence
> > -- which *is* true for 53-edo.
>
> What is that supposed to mean? Independent existence?

It's supposed to mean that the pythagorean-comma is
distinguished from the syntonic-comma ... but again,
i must have stayed up too late when i posted that,
because it's wrong. Sorry.

53-edo approximates both the pythagorean and syntonic
commas as one degree of the tuning. So they're not
independent. Again, "duh" ... the schisma is the
difference between those two commas, and 53-edo tempers
out the schisma.

As Gene just pointed out, it would take an edo of
around the cardinality of 612 to distinguish the
schisma, which is what i meant by "independent existence".

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗monz <monz@tonalsoft.com>

Hi Paul,

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
>
> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
> >
> > 22-edo is the smallest edo i know in which they
> > [the pythagorean and syntonic commas] do not vanish
> > and they also roughly approximate the fact that the
> > pythagorean-comma is bigger than the syntonic:
> > p.c.= 2 degrees and s.c. = 1 degree. However, in JI
> > the p.c. is only ~2 cents higher than the s.c. -- here
> > it is *twice* as large!
>
> It's easy to find a simpler example. For example,
> in 15-equal, the syntonic comma is 1 degree, and
> the pythagorean comma is 3 degrees!

Yes, but somewhere in this discussion i qualified my
statements by referring only to edo's which "approximate
diatonic tunings" -- which 15-edo most emphatically
does not.

> > You can see how edos approximate 5-limit JI on my
> > "bingo-card lattices" webpage -- click on the edo links
> > on the bottom diagram:
> >
> > http://tonalsoft.com/enc/b/bingo.aspx
>
> I don't agree with the assessments you make of many equal
> temperaments on this page. 17 is but one example.

I only picked one mapping for each edo, and what you're
saying is that some edo's offer more than one mapping,
which of course i totally understand and agree with.

I guess i need to either say something to that effect
on the webpage, or (better) represent the alternative
mappings on my bingo-card-lattices.

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

On Tue, 27 Dec 2005, Gene Ward Smith wrote:
>
> --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz"
> <yahya@m...> wrote:
>
> > Alright then, rephrasing and redirecting the question:
> >
> > Gene,
> > What is the smallest nontrivial EDO that preserves
> > and _distinguishes_ both the syntonic comma and
> > the pythagorean comma?
>
> To distinguish the two, you need to have 32805/32768, the schisma, set
> to a positive number of steps, which should probably be one step. A
> temperament each of whose steps is very close to a schisma is 612-et,
> but there are smaller ones, of course. It depends on how large you
> want the schisma to be. However, one of the following might be what
> you are looking for: 22, 34, 46, 87, 99, 152, 270, 441, 494 or 612.
> These are all good 5-limit, and often good higher-limit systems. In
> all of them the Pythagorean and Didymus commas are both positive
> numbers of steps, and the schisma is one step.

Thank you, Gene, that's what I wanted to know,
and more.

Choosing any one from this list would give a tuning
with the two commas differing by one step, and I
guess the choice for me would come down to how
close I want the tuning of the scale degrees to be
to JI, which in turn will determine the sizes of
schisma that are acceptable.

Regards,
Yahya

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No virus found in this outgoing message.
Checked by AVG Free Edition.
Version: 7.1.371 / Virus Database: 267.14.7/214 - Release Date: 23/12/05

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

On Tue, 27 Dec 2005, wallyesterpaulrus wrote:
>
[snip]
> > > 53-edo maps both the syntonic-comma and pythagorean-comma
> > > to 1 degree, and thus preserves both of them.
[snip]
> >
> > What is the smallest EDO that preserves and
> > _distinguishes_ both the syntonic comma and
> > the pythagorean comma?
[snip]
>
> Moving right along, how would you define "preserves"
> in this context?

How's this for a definition of my intent:
"A tuning _preserves_ an interval if it maps
two ratios which differ in JI by that interval
to two different steps in the gamut."

Now let's get picky!:
"A tuning preserves an interval _well_ if it
preserves it and if it maps the two JI ratios
to two gamut steps which form an interval
which is not less than half the interval
between the two JI ratios (the JI interval)
and is not more than twice the JI interval.
In symbols, if I is the JI interval and the
tuning is a map t that takes any JI interval
I to an interval of the tuning:
t: I --> t(I)
then the tuning t preserves I well if
I/2 <= t(I) <= 2I"

Clear enough?

> If the direction of the interval is reversed, is the
> interval "preserved"?

Yes, but not well. One might say the tuning
preserves the interval perversely ... :-)

BTW, Gene has given me a full answer to my
question, and monz has also given me some
stuff to mull over. Thank you all for your input!

Regards,
Yahya

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No virus found in this outgoing message.
Checked by AVG Free Edition.
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🔗Gene Ward Smith <gwsmith@svpal.org>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> Hi Paul,
>
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> >
> > --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > > So it's incorrect to say that [in 55-edo] the
> > > pythagorean-comma vanishes, but it *is* correct to
> > > say that it does not have an "independent" existence
> > > -- which *is* true for 53-edo.
> >
> > What is that supposed to mean? Independent existence?
>
>
>
> It's supposed to mean that the pythagorean-comma is
> distinguished from the syntonic-comma ... but again,
> i must have stayed up too late when i posted that,
> because it's wrong. Sorry.
>
> 53-edo approximates both the pythagorean and syntonic
> commas as one degree of the tuning. So they're not
> independent. Again, "duh" ... the schisma is the
> difference between those two commas, and 53-edo tempers
> out the schisma.
>
> As Gene just pointed out, it would take an edo of
> around the cardinality of 612 to distinguish the
> schisma, which is what i meant by "independent existence".

612? You just brought up 22-equal, which does this and is far simpler
than 612-equal. And what about 15-equal? Did you see my post where I
just mentioned that in this context?