Yahoo Tuning Groups Ultimate Backup tuning schismatic tuning (was:: new webpage: Ramos's...)

re:
http://sonic-arts.org/monzo/ramos/ramos.htm

> From: "monz" <monz@attglobal.net>
> To: <tuning@yahoogroups.com>
> Sent: Saturday, March 22, 2003 9:56 PM
> Subject: Re: [tuning] Re: new webpage: Ramos's division of the monochord
>
>
> OK, of course you're right about that.
> i'll put something in about schismatic tuning.
> in fact, i should have a page about that in
> the Dictionary.
>
>
>
> -monz

i've added this to the webpage:

>> At some point during the 1400s it became fairly
>> common for keyboards to tuned be to a variant of
>> regular Pythagorean tuning, such that the note
>> whose ratio is prime-factored as 3^-8 represents
>> the JI "major-3rd" whose ratio is 5:4 (e.g., if
>> "C" is the 1:1, "Fb" represents "E"), the difference
>> between those two being only the ~2-cent schisma,
>> which is below the threshold of typical human error
>> in tuning by ear. This is thus known as "schismatic tuning".

i was going to end that paragraph with this:

>> Employing this tuning inevitably led to the occurrence
>> of "wolf" intervals, which in turn led to meantone temperament.

... but decided that i need to find out more about schismatic
tuning. i don't have my copy of Lindley handy ... can
anyone help? was it 12 different notes, or more? what
exactly was the typical range of letter-names-with-accidental?

i've done some calculations on the following 12-note
schismatic tuning; the names of schismatic equivalents
are given in quotation marks:

(as always, on the Yahoo web interface, their stupid
space-saving "feature" will ruin the formatting of my
tables unless you view it with the "Expand messages" option)

- as a Pythagorean chain (actually tuned Bbb to D) :

note 3^x ~cents

"A" -9 882.4049922
"E" -8 384.3599931
"B" -7 1086.314994
"F#" -6 588.2699948
"C#" -5 90.22499567
"G#" -4 792.1799965
Eb -3 294.1349974
Bb -2 996.0899983
F -1 498.0449991
C 0 0
G +1 701.9550009
D +2 203.9100017

- in descending order as a chromatic scale:

note 3^x ~cents

B -7 1086.314994
Bb -2 996.0899983
A -9 882.4049922
G# -4 792.1799965
G +1 701.9550009
F# -6 588.2699948
F -1 498.0449991
E -8 384.3599931
Eb -3 294.1349974
D +2 203.9100017
C# -5 90.22499567
C 0 0

the following is the list all intervals
occuring between pitches of this scale,
broken up into pairs for obvious reasons:

interval size ------- interval function ------
(~cents) Pythagorean schismatic-JI

1109.775004 major-7th diminished-8ve
1086.314994 diminished-8ve major-7th

1019.550009 augmented-6th minor-7th
996.0899983 minor-7th augmented-6th

905.8650026 major-6th diminished-7th
882.4049922 diminished-7th major-6th

815.6400069 augmented-5th minor-6th
792.1799965 minor-6th augmented-5th

701.9550009 perfect-5th

678.4949905 diminished-6th "wolf"-5th
611.7300052 augmented-4th diminished-5th

588.2699948 diminished-5th augmented-4th
521.5050095 augmented-3rd "wolf"-4th

498.0449991 perfect-4th

407.8200035 major-3rd diminished-4th
384.3599931 diminished-4th major-3rd

317.5950078 augmented-2nd minor-3rd
294.1349974 minor-3rd augmented-2nd

203.9100017 major-2nd
180.4499913 diminished-3rd

113.6850061 augmented-prime minor-2nd
90.22499567 minor-2nd augmented-prime

0 unison / prime

("augmented-prime" is usually called "chromatic semitone",
and "minor-2nd" is often called "diatonic semitone".)

any feedback would be appreciated.

-monz

🔗monz <monz@attglobal.net>

> From: "monz" <monz@attglobal.net>
> To: <tuning@yahoogroups.com>
> Sent: Saturday, March 22, 2003 11:08 PM
> Subject: [tuning] schismatic tuning (was:: new webpage: Ramos's...)
> note 3^x ~cents

oops ... my bad. i meant for the table to be
formatted like this:

(... i messed up the spacing between the
"wolf" intervals and the tritones)

- in descending order as a chromatic scale:

note 3^x ~cents

B -7 1086.314994
Bb -2 996.0899983
A -9 882.4049922
G# -4 792.1799965
G +1 701.9550009
F# -6 588.2699948
F -1 498.0449991
E -8 384.3599931
Eb -3 294.1349974
D +2 203.9100017
C# -5 90.22499567
C 0 0

the following is the list all intervals
occuring between pitches of this scale,
broken up into pairs for obvious reasons:

interval size ------- interval function ------
(~cents) Pythagorean schismatic-JI

1109.775004 major-7th diminished-8ve
1086.314994 diminished-8ve major-7th

1019.550009 augmented-6th minor-7th
996.0899983 minor-7th augmented-6th

905.8650026 major-6th diminished-7th
882.4049922 diminished-7th major-6th

815.6400069 augmented-5th minor-6th
792.1799965 minor-6th augmented-5th

701.9550009 perfect-5th

678.4949905 diminished-6th "wolf"-5th

611.7300052 augmented-4th diminished-5th
588.2699948 diminished-5th augmented-4th

521.5050095 augmented-3rd "wolf"-4th

498.0449991 perfect-4th

407.8200035 major-3rd diminished-4th
384.3599931 diminished-4th major-3rd

317.5950078 augmented-2nd minor-3rd
294.1349974 minor-3rd augmented-2nd

203.9100017 major-2nd
180.4499913 diminished-3rd

113.6850061 augmented-prime minor-2nd
90.22499567 minor-2nd augmented-prime

0 unison / prime

("augmented-prime" is usually called "chromatic semitone",
and "minor-2nd" is often called "diatonic semitone".)

any feedback would be appreciated.

-monz

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗monz <monz@attglobal.net>

----- Original Message -----
From: "Gene Ward Smith" <gwsmith@svpal.org>
To: <tuning@yahoogroups.com>
Sent: Sunday, March 23, 2003 12:36 AM
Subject: [tuning] Re: schismatic tuning (was:: new webpage: Ramos's...)

> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > >> Employing this tuning inevitably led to the occurrence
> > >> of "wolf" intervals, which in turn led to meantone temperament.
>
> One thing to notice is that the problem is not wolf fifths, which are
> no worse than meantone, but sharp Pythagorean major thirds. The wolf
> fifth using pure fifths is 678.495 cents, a comma flat but not as bad
> as the howlin' meantone wolf, 737.637 cents for 1/4-comma meantone,
> 35.682 cents sharp. That's pretty sharp!

you're making a relevant point, but have you ever
actually listened to 40/27, the JI "wolf-5th" of
~680 cents? it sounds pretty "wolfish" to me!

> Another thing about schismic temperament is that tuning 17 or 24 notes
> to it works nicely, re a thread we are having now on tuning-math.

right, but the point being made by paul and myself is
that the schismatic tuning used in the 1400s was *not*
a temperament.

... but of course, the other point we're both making
is that the difference between a schismatic temperament
and regular Pythagorean tuning is below the threshold
of normal tuning error, so no-one but theorists and
mathematicians would care about the difference anyway.

-monz

🔗monz <monz@attglobal.net>

> From: "monz" <monz@attglobal.net>
> To: <tuning@yahoogroups.com>
> Sent: Saturday, March 22, 2003 11:08 PM
> Subject: [tuning] schismatic tuning (was:: new webpage: Ramos's...)
>

> re:
> http://sonic-arts.org/monzo/ramos/ramos.htm
>
>
> ... but decided that i need to find out more about schismatic
> tuning. i don't have my copy of Lindley handy ... can
> anyone help? was it 12 different notes, or more? what
> exactly was the typical range of letter-names-with-accidental?

OK, i dug out my copy of Lindley and found
(on page 55) that this was the schismatic tuning
common in keyboards of the 1400s (and Lindley
suggests maybe even the later 1300s) :

note 3^x cents

(Gb)/"F#" -6 588.270
(Db)/"C#" -5 90.225
Ab/"G#" -4 792.180
Eb/"D#" -3 294.135
Bb -2 996.090
F -1 498.045
C 0 0.000
G 1 701.955
D 2 203.910
A 3 905.865
E 4 407.820
B 5 1109.775

the scale was tuned 3^(-6 ... +5), but Lindley says
that Gb and Db were not in use at this time, so that's
why i put them in parentheses.

here it is in descending order of pitch:

note 3^x cents

B 5 1109.775
Bb -2 996.090
A 3 905.865
Ab/"G#" -4 792.180
G 1 701.955
(Gb)/"F#" -6 588.270
F -1 498.045
E 4 407.820
Eb/"D#" -3 294.135
D 2 203.910
(Db)/"C#" -5 90.225
C 0 0.000

the complete set of intervals is the same as that
which i posted before, except that they occur between
different sets of note-names.

-monz

🔗Gene Ward Smith <gwsmith@svpal.org>

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

> you're making a relevant point, but have you ever
> actually listened to 40/27, the JI "wolf-5th" of
> ~680 cents? it sounds pretty "wolfish" to me!

My point was that you don't correct a problem by making it worse, so
this must not have been viewed as the problem.

Let me take the liberty of reposting something from tuning-math; it
shows triads around a circle of fifths for 17-note schismic; and it
looks like a heck of a system. If Western music had gone this way I
think it would have never converted to equal temperament. No one would
have suggest 17-equal as a means of getting rid of one augmented
triad, and even if they did, they would have been laughed to scorn.

Schismic[17]
rms error 0.162 cents

32805/32768
[1, 135/128, 16/15, 9/8, 32/27, 5/4, 81/64, 4/3, 45/32, 64/45, 3/2,
128/81, 8/5, 27/16, 16/9, 15/8, 243/128]
[5/4, 15/8, 45/32, 135/128, 128/81, 32/27, 16/9, 4/3, 1, 3/2, 9/8, 27/
16, 81/64, 243/128, 64/45, 16/15, 8/5]

Circle of fifths
[1, 6/5, 3/2]
[1, 6/5, 3/2]
[1, 6/5, 3/2]
[1, 6/5, 3/2]
[1, 6/5, 3/2]
[1, 6/5, 3/2]
[1, 6/5, 3/2]
[1, 6/5, 3/2]
[1, 5/4, 3/2]
[1, 5/4, 3/2]
[1, 5/4, 3/2]
[1, 5/4, 3/2]
[1, 5/4, 3/2]
[1, 5/4, 3/2]
[1, 5/4, 3/2]
[1, 5/4, 3/2]
[1, 5/4, 25/16]

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> re:
> http://sonic-arts.org/monzo/ramos/ramos.htm
>
> > From: "monz" <monz@a...>
> > To: <tuning@yahoogroups.com>
> > Sent: Saturday, March 22, 2003 9:56 PM
> > Subject: Re: [tuning] Re: new webpage: Ramos's division of the
monochord
> >
> >
> > OK, of course you're right about that.
> > i'll put something in about schismatic tuning.
> > in fact, i should have a page about that in
> > the Dictionary.
> >
> >
> >
> > -monz
>
>
> i've added this to the webpage:
>
>
> >> At some point during the 1400s it became fairly
> >> common for keyboards to tuned be to a variant of
> >> regular Pythagorean tuning, such that the note
> >> whose ratio is prime-factored as 3^-8 represents
> >> the JI "major-3rd" whose ratio is 5:4 (e.g., if
> >> "C" is the 1:1, "Fb" represents "E"), the difference
> >> between those two being only the ~2-cent schisma,
> >> which is below the threshold of typical human error
> >> in tuning by ear. This is thus known as "schismatic tuning".
>
>
>
> i was going to end that paragraph with this:
>
>
> >> Employing this tuning inevitably led to the occurrence
> >> of "wolf" intervals, which in turn led to meantone temperament.
>
>
> ... but decided that i need to find out more about schismatic
> tuning. i don't have my copy of Lindley handy ... can
> anyone help?

you get the same wolf intervals that you get in a JI diatonic scale,
up to a schisma.

> what
> exactly was the typical range of letter-names-with-accidental?

i thought that was explained clearly in margo's post. Gb-B was the
typical *tuning* range, but often flats were *notated* as sharps.

> any feedback would be appreciated.

what exactly were you looking for?

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗jacques dudon <aeh@free.fr>

Sun 23 March, Gene Ward Smith wrote :

> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> > >> Employing this tuning inevitably led to the occurrence
> > >> of "wolf" intervals, which in turn led to meantone temperament.
>
> One thing to notice is that the problem is not wolf fifths, which are
> no worse than meantone, but sharp Pythagorean major thirds. The wolf
> fifth using pure fifths is 678.495 cents, a comma flat but not as bad
> as the howlin' meantone wolf, 737.637 cents for 1/4-comma meantone,
> 35.682 cents sharp. That's pretty sharp!

Using a guitar fretted in what you call "schismic" tuning for about 30 years, I am not certain,
at least for that instrument, that the major thirds are a worse problem than the 40/27.
Not only for it's embarrassing position ! Apart from bending the 40/27 into 3/2,
or within certain forms of minor + 7th chords may be, the 40/27 is rather difficult,
while the various 81/64 can find their place in many minor harmonies.
Remember that 81/64 is an indian sruti, while 40/27 is not ;
also, with a guitar, 81/64 is not far from a 24/19, something that can be easily arranged by a
slight
pressure on the right strings... do you know why I say that ?

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, jacques dudon <aeh@f...> wrote:

> Remember that 81/64 is an indian sruti, while 40/27 is not ;

well, don't confuse interval-ratios with pitch-ratios!

> also, with a guitar, 81/64 is not far from a 24/19, something that
can be easily arranged by a
> slight
> pressure on the right strings... do you know why I say that ?

is it because a high-register 16:19:24 minor triad sounds great on a
distorted guitar, due to the alignment of the difference tones around
the fundamental 4 octaves below the root?

but i think the "embarrassing position" of the wolf fifth in the 12-
tone schismic/schismatic tuning is what gene truly failed to take
note of. a full 5-limit diatonic triadic style was still out of reach
for 15th century keyboardists until meantone tuning came into play.

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> > --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> > wrote:
>
> > Disagreed -- it's far worse -- the wolf fifths are right in the
> > middle of your "good" diatonic scale!!
>
> If the point was not to fix the fifth but to move it, Monzo should
> discuss that, as it is a subtle and interesting point.

right -- that's why i brought up the wolf in the JI diatonic scale to
him.

> > yes -- monz's schismic page already mentions both 24 and 36, but
> > should add the arabic/wilson 17-tone schismic . . .
>
> How did the Arabs uses this scale? Not 5-limit harmony, I presume.

this is medieval, not modern, arabic tuning. the medieval scales
suggest that 5-limit intervals were favored. look 'em up in partch or
helmholtz.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗monz <monz@attglobal.net>

hi paul and Gene,

> From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
> To: <tuning@yahoogroups.com>
> Sent: Monday, March 24, 2003 8:31 PM
> Subject: [tuning] Re: schismatic tuning (was:: new webpage: Ramos's...)
>
>
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> > --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> > <wallyesterpaulrus@y...> wrote:
> > > --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> > > wrote:
> >
> > > Disagreed -- it's far worse -- the wolf fifths are right in the
> > > middle of your "good" diatonic scale!!
> >
> > If the point was not to fix the fifth but to move it, Monzo should
> > discuss that, as it is a subtle and interesting point.

yes, i agree -- that *is* a point which is both
subtle and interesting! i say more about it when
i have time to add it to the webpage.

> right -- that's why i brought up the wolf in the JI diatonic scale to
> him.

uh oh ... you said that to *me*? ... where was that?

-monz

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> hi paul and Gene,
>
> > From: "wallyesterpaulrus" <wallyesterpaulrus@y...>
> > To: <tuning@yahoogroups.com>
> > Sent: Monday, March 24, 2003 8:31 PM
> > Subject: [tuning] Re: schismatic tuning (was:: new webpage:
Ramos's...)
> >
> >
> > --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> > > --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> > > <wallyesterpaulrus@y...> wrote:
> > > > --- In tuning@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> > > > wrote:
> > >
> > > > Disagreed -- it's far worse -- the wolf fifths are right in
the
> > > > middle of your "good" diatonic scale!!
> > >
> > > If the point was not to fix the fifth but to move it, Monzo
should
> > > discuss that, as it is a subtle and interesting point.
>
>
> yes, i agree -- that *is* a point which is both
> subtle and interesting! i say more about it when
> i have time to add it to the webpage.
>
>
>
> > right -- that's why i brought up the wolf in the JI diatonic
scale to
> > him.
>
>
> uh oh ... you said that to *me*? ... where was that?

/tuning/topicId_42967.html#42981

also read all the other responses i wrote to you on this subject . . .

🔗jacques dudon <aeh@free.fr>

wallyesterpaulrus a ï¿½crit :

> --- In tuning@yahoogroups.com, jacques dudon <aeh@f...> wrote:
>
> > Remember that 81/64 is an indian sruti, while 40/27 is not ;
>
> well, don't confuse interval-ratios with pitch-ratios!

That was a bit intentionnal, donï¿½t you think ? normally, when we hear
a major third or a fifth, we place naturally the tonic on the first tone,
thatï¿½s why a reference to modal music makes sense ï¿½ it would have been
different perhaps if we were talking of 10/9, 4/3 or 8/5.
But youï¿½re right, we canï¿½t compare criterions for a non-modal system
with criterions for a modal system, especially with the Ragas where chords
other than between the 22 srutis and SA, or MA, are not supposed to occur.

Besides, to be complete, I should have mentionned that 27/20 IS an indian sruti.
But why 27/20 and not 40/27 ? thatï¿½s an interesting asymmetry.
A strong 6/5 can justify 27/20. What would justify 40/27 ?
40 ï¿½ 27 = 13 is not close to any sruti ;
on the other hand, 81ï¿½ 64 = 17 is quite close to the "commal RE" 2nd sruti.
I only think that to look at a schismic tuning with an ï¿½ indian ï¿½ ear is a good idea,
as much as to look at the srutis with a ï¿½ western ï¿½ ear is a good idea too.
Then two worlds can meet.

> > also, with a guitar, 81/64 is not far from a 24/19, something that
> > can be easily arranged by a slight
> > pressure on the right strings... do you know why I say that ?
>
> is it because a high-register 16:19:24 minor triad sounds great on a
> distorted guitar, due to the alignment of the difference tones around
> the fundamental 4 octaves below the root?

That's a good answer ! I didn't think of the distortion, but I'd go for it !
If your fuzz box is still plugged, you can try also the recurrent series
4 6 8 10 12 15 19 24 30 ... where
12 ï¿½ 10 = 4 / 2
15 ï¿½ 12 = 6 / 2
19 ï¿½ 15 = 8 / 2
24 ï¿½ 19 = 10 / 2
30 ï¿½ 24 = 12 / 2 ... and so on,
resumed in the algorithm x^4 ï¿½ x^3 = 0.5 x^0
(it becomes less "schismic" afterwards ï¿½ sorry for the diversion !)

🔗Mark Rankin <markrankin95511@yahoo.com>

Cher Jacques,

J'etait etonne et heureux a voir ton nom ici. S'il tu
plait, envoyez-moi ton address (direction?) "snail
mail" (Poste normal). J'ai beaucoups des chose a dire
et a montrer. Ca Fait longtemps!

--Mark Rankin

--- jacques dudon <aeh@free.fr> wrote:
> wallyesterpaulrus a �crit :
>
> > --- In tuning@yahoogroups.com, jacques dudon
> <aeh@f...> wrote:
> >
> > > Remember that 81/64 is an indian sruti, while
> 40/27 is not ;
> >
> > well, don't confuse interval-ratios with
> pitch-ratios!
>
> That was a bit intentionnal, don�t you think ?
> normally, when we hear
> a major third or a fifth, we place naturally the
> tonic on the first tone,
> that�s why a reference to modal music makes sense �
> it would have been
> different perhaps if we were talking of 10/9, 4/3 or
> 8/5.
> But you�re right, we can�t compare criterions for a
> non-modal system
> with criterions for a modal system, especially with
> the Ragas where chords
> other than between the 22 srutis and SA, or MA, are
> not supposed to occur.
>
> Besides, to be complete, I should have mentionned
> that 27/20 IS an indian sruti.
> But why 27/20 and not 40/27 ? that�s an interesting
> asymmetry.
> A strong 6/5 can justify 27/20. What would justify
> 40/27 ?
> 40 � 27 = 13 is not close to any sruti ;
> on the other hand, 81� 64 = 17 is quite close to the
> "commal RE" 2nd sruti.
> I only think that to look at a schismic tuning with
> an � indian � ear is a good idea,
> as much as to look at the srutis with a � western �
> ear is a good idea too.
> Then two worlds can meet.
>
> > > also, with a guitar, 81/64 is not far from a
> 24/19, something that
> > > can be easily arranged by a slight
> > > pressure on the right strings... do you know why
> I say that ?
> >
> > is it because a high-register 16:19:24 minor triad
> sounds great on a
> > distorted guitar, due to the alignment of the
> difference tones around
> > the fundamental 4 octaves below the root?
>
> That's a good answer ! I didn't think of the
> distortion, but I'd go for it !
> If your fuzz box is still plugged, you can try also
> the recurrent series
> 4 6 8 10 12 15 19 24 30 ... where
> 12 � 10 = 4 / 2
> 15 � 12 = 6 / 2
> 19 � 15 = 8 / 2
> 24 � 19 = 10 / 2
> 30 � 24 = 12 / 2 ... and so on,
> resumed in the algorithm x^4 � x^3 = 0.5 x^0
> (it becomes less "schismic" afterwards � sorry for
> the diversion !)
>
>

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