31-EDO and 1/4-comma meantone compared

For those interested in 1/4-comma meantone and/or 31-EDO:

I've just made a significant update to the Tuning Dictionary
entry for 1/4-comma meantone:

http://www.ixpres.com/interval/dict/1-4cmt.htm

I've included precise measurements comparing these two
closely-related tunings.

For a more detailed look at the work of Christiaan Huygens,
who first noted the similarity of these two tunings,
download my Microsoft Excel spreadsheet showing all the
calculations and the method used by Huygens:

http://www.ixpres.com/interval/monzo/31edo/Huygens31-qcmt.xls

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

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--- In tuning@y..., "monz" <joemonz@y...> wrote:
> For those interested in 1/4-comma meantone and/or 31-EDO:
>
> I've just made a significant update to the Tuning Dictionary
> entry for 1/4-comma meantone:
>
> http://www.ixpres.com/interval/dict/1-4cmt.htm
>
> I've included precise measurements comparing these two
> closely-related tunings.
>
This statement is not correct:

"Regardless of how many notes the meantone has, comparing the two
notes at either end results in a 'wolf 5th' and a 'wolf 4th', as can
be seen in the following two tables."

Even if you specify that the tuning is an _MOS_ chain of meantone
fifths, there are still two things wrong with this statement. Can you
see what they are?

"The 'wolves' occur between the two '5ths' or '4ths' which bound the
notes at the end of the series of '5ths'."

That's an odd one -- what _two_ '5ths' or '4ths' are you referring to?

> From: Paul Erlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Monday, November 05, 2001 8:03 PM
> Subject: [tuning] Re: 31-EDO and 1/4-comma meantone compared
>

> --- In tuning@y..., "monz" <joemonz@y...> wrote:
> > For those interested in 1/4-comma meantone and/or 31-EDO:
> >
> > I've just made a significant update to the Tuning Dictionary
> > entry for 1/4-comma meantone:
> >
> > http://www.ixpres.com/interval/dict/1-4cmt.htm
> >
> > I've included precise measurements comparing these two
> > closely-related tunings.
> >
>
> [Paul:]
> This statement is not correct:
>
> "Regardless of how many notes the meantone has, comparing the two
> notes at either end results in a 'wolf 5th' and a 'wolf 4th', as can
> be seen in the following two tables."
>
> Even if you specify that the tuning is an _MOS_ chain of meantone
> fifths, there are still two things wrong with this statement. Can you
> see what they are?

Good catch, Paul... my bad. The wolves occur only between any
two pitches which are inclusively 12 steps apart in the
chain of "5ths".

So...

1)
the appearance of the wolves is *not* dependant on the number
of "5ths" in the chain, so that my statement about "regardless
of how many there are" is wrong, and

2)
the wolves do not occur between the pitches at the end of
the chain if it is larger than 12 steps, but rather, always
between those which are 12 steps apart.

> "The 'wolves' occur between the two '5ths' or '4ths' which
> bound the notes at the end of the series of '5ths'."
>
> That's an odd one -- what _two_ '5ths' or '4ths' are you
> referring to?

That is worded badly. By the word "two" I'm referring to
the complementary pair of one wolf "5th" and one wolf "4th".

Thanks for the corrections.

Also, in the Dictionary entry I added a link to the original
letter published by Huygens, which Manuel has on the web at
<http://www.xs4all.nl/~huygensf/doc/lettre.html>. I had meant
to include that link also in my previous Tuning List post, so
that the figures in my Excel spreadsheet at
<http://www.ixpres.com/interval/monzo/31edo/Huygens31-qcmt.xls>
can be compared with Huygens's table.

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

_________________________________________________________
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Get your free @yahoo.com address at http://mail.yahoo.com

--- In tuning@y..., "monz" <joemonz@y...> wrote:

/tuning/topicId_29925.html#29925

> For those interested in 1/4-comma meantone and/or 31-EDO:
>
> I've just made a significant update to the Tuning Dictionary
> entry for 1/4-comma meantone:
>
> http://www.ixpres.com/interval/dict/1-4cmt.htm
>

The link for "vector addition" there gets me a "404 not found..."

Not that I could master that in an afternoon....

JP

Hello, there, Monz and everyone.

To your material on 31-EDO and 1/4-comma meantone, I might add a few
points and possible clarifications.

First, on either tuning as a 31-note circulating system:

"Either 31-EDO or 1/4-comma meantone can serve as a 31-note
circulating system where every step of the cycle has precisely
equivalent (31-EDO) or nearly equivalent (1/4-comma) 5-limit concords
and other intervals available.

"While such a tuning in 31-EDO is by definition precisely symmetrical,
a 31-note cycle of 1/4-comma meantone results in some small
asymmetries. In 31-EDO, a chain of 31 fifths is precisely equivalent
to 18 pure octaves, so that the system has exact mathematical closure.

"In 1/4-comma meantone, 31 tempered fifths would fall short of 18 pure
octaves by about 6.07 cents. To achieve a 'musically closed' 31-note
cycle, we actually tune a chain of 30 fifths or 31 notes within an
octave, then tuning the octave of the starting note to complete the
cycle.

"This results in an 'odd' 31st fifth about 6.07 cents larger than the
others -- as it happens, very close to a pure 3:2, since this
adjustment of 6.07 cents almost exactly counterbalances the usual
temperament of the fifth by 1/4-comma (about 5.38 cents) in the narrow
direction.

"Apart from producing this one near-just fifth, our 31-note cycle of
1/4-comma meantone also produces a few major or minor thirds differing
by about 6.07 cents from their usual sizes, a much smaller variation
than that routinely tolerated in 18th-century well-temperaments.

"This slight disparity between 31-EDO and 1/4-comma meantone means
that the latter has two slightly different sizes for intervals where
the former has a single unvarying size. For example, the neutral third
of 31-EDO is always precisely 9/31 octave, or about 348 cents (in
comparison to a just 11:9 at about 347 cents). In a 31-note cycle of
1/4-comma meantone, however, we find two slightly differing sizes of
neutral thirds at about 345 cents and 351 cents.

"These differences are generally quite minor, so that either system
makes a fine 31-note circulating tuning for music where this variety
of meantone temperament is desired.

"In fact, just how the 31-note meantone cycles used on the instruments
of Nicola Vicentino (1555) and Fabio Colonna (1618) were tempered is
an open question. Both musicians describe a division of the whole-tone
into five dieses, which would be precisely equal in 31-EDO. However,
tuning by ear, a 1/4-comma temperament based on pure 5:4 major thirds
might be easier to judge than the very slight impurity of these thirds
in the wide direction required for a precise 31-EDO. Very possibly the
variations from theoretical interval sizes in practice would have been
greater than the mathematical differences between the two models."

On the theoretical possibility of carrying 1/4-comma meantone beyond
31 notes:

"While 1/4-comma meantone makes a 31-note circulating system very
similar to 31-EDO, one could in principle carry the chain of regular
tempered fifths further. The 31st fifth of such a chain would generate
a step about 6.07 cents lower than the starting note. As one continued
this process, a series of 31-note sets would result, each about 6.07
cents lower than the previous one, generating new steps and interval
sizes."

On the "Wolf fifth/fourth":

"A basic feature of 31-EDO or 1/4-comma meantone, as well as many
other types of tunings, is that notes such as G# and Ab are _not_
equivalent. The difference between such notes in either tuning is
known as an 'enharmonic diesis': 1/31 octave (~38.71 cents) in 31-EDO,
or 128:125 (~41.06 cents) in 1/4-comma.

"An important musical consequence of this difference is that a
diminished sixth such as G#-Eb is _not_ the same interval as a regular
fifth such as G#-D# or Ab-Eb, but rather in usual historical European
timbres a drastically more 'beatful' and dissonant interval a diesis
larger. This beating is said to have resembled the howling of wolves,
thus the term 'Wolf fifth.'

"In 31-EDO, where a usual fifth is 18/31 octave (~696.77 cents), a
Wolf fifth such as G#-Eb is 19/31 octave (~735.48 cents). In
1/4-comma, similarly, a regular fifth is about 696.58 cents, but a
Wolf fifth about 737.65 cents -- a 128:125 diesis larger.

"Likewise an interval such as Eb-G# in either tuning forms a 'Wolf
fourth' a diesis narrower than a usual fourth at 13/31 octave in
31-EDO (~503.23 cents), or about 503.42 cents in 1/4-comma -- 12/31
octave (~464.52 cents) in 31-EDO, or about 462.35 cents in 1/4-comma.

"Generally avoided in historical European music, the Wolf fifth is
nevertheless sometimes used as a 'special effect' where an especially
striking and expressive dissonance is desired.

"Wolf fifths or fourths will occur in any 12-note or larger tuning set
of 31-EDO or 1/4-comma meantone, where they are found between notes at
a distance of 11 fifths or fourths on the chain, e.g. G#-Eb or Eb-G#.

"The 'Wolf problem,' as it is sometimes called, might be taken to
relate not so much to the presence of these intervals as to the
absence of regular fifths or fourths for certain steps of a tuning set
which is not large enough to supply these usual concords.

"Specifically, in a 12-note tuning of 31-EDO or 1/4-comma meantone
such as the chain of fifths from Eb to G# typical of the 16th century,
we find a Wolf fifth or fourth G#-Eb or Eb-G# between the extreme
notes of the chain, but no regular fifth G#-D# or Ab-Eb.

"The problem, if we seek such fifths, is that our tuning set does not
include the note D# or Ab. For most 16th-century keyboard and also
vocal music, where accidentals typically remain in the range of Eb-G#,
this 'problem' is mostly academic.

"However, for more adventurous music, some 16th-18th century meantone
keyboards provide the additional notes for such regular fifths, for
example using "split-key" accidentals with the front portion of a
black key provides G# and the back portion Ab, or likewise Eb and D#.
On such a keyboard, we still find the Wolf fifth G#-Eb, but also now
the regular fifths Ab-Eb and G#-D#.

"To get a circulating version of 31-EDO or 1/4-comma meantone where
_all_ steps have usual fifths or fourths, we need to tune a 31-note
cycle; now these intervals, and also Wolf fifths or fourths, are
available for each step."

Most appreciatively,

Margo Schulter
mschulter@value.net

> From: <jpehrson@rcn.com>
> To: <tuning@yahoogroups.com>
> Sent: Wednesday, November 07, 2001 10:10 AM
> Subject: [tuning] Re: 31-EDO and 1/4-comma meantone compared
>
>
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> /tuning/topicId_29925.html#29925
>
> > For those interested in 1/4-comma meantone and/or 31-EDO:
> >
> > I've just made a significant update to the Tuning Dictionary
> > entry for 1/4-comma meantone:
> >
> > http://www.ixpres.com/interval/dict/1-4cmt.htm
> >
>
> The link for "vector addition" there gets me a "404 not found..."
>
> Not that I could master that in an afternoon....

Thanks, Joe. It's been fixed.

Vector addition really shouldn't take long to understand.
You're simply adding or subtracting the exponents of the
prime-factors.

I find it very useful in a case like this because it gives
an exact measurement of the difference between a meantone
and an ET, with no rounding errors. The rules for addition
and subtraction here simply follow the regular rules for
adding and subtracting fractions.

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

--- In tuning@y..., "monz" <joemonz@y...> wrote:

/tuning/topicId_29925.html#30005

> > From: <jpehrson@r...>
> > To: <tuning@y...>
> > Sent: Wednesday, November 07, 2001 10:10 AM
> > Subject: [tuning] Re: 31-EDO and 1/4-comma meantone compared
> >
> >
> > --- In tuning@y..., "monz" <joemonz@y...> wrote:
> >
> > /tuning/topicId_29925.html#29925
> >
> > > For those interested in 1/4-comma meantone and/or 31-EDO:
> > >
> > > I've just made a significant update to the Tuning Dictionary
> > > entry for 1/4-comma meantone:
> > >
> > > http://www.ixpres.com/interval/dict/1-4cmt.htm
> > >
> >
> > The link for "vector addition" there gets me a "404 not found..."
> >
> > Not that I could master that in an afternoon....
>
>
> Thanks, Joe. It's been fixed.
>
> Vector addition really shouldn't take long to understand.
> You're simply adding or subtracting the exponents of the
> prime-factors.
>
> I find it very useful in a case like this because it gives
> an exact measurement of the difference between a meantone
> and an ET, with no rounding errors. The rules for addition
> and subtraction here simply follow the regular rules for
> adding and subtracting fractions.
>

Thanks, Monz! It's a little disconcerting when the "help file"
crashes, so glad it's operating...

JP

> From: mschulter <MSCHULTER@VALUE.NET>
> To: <tuning@yahoogroups.com>
> Sent: Wednesday, November 07, 2001 10:49 AM
> Subject: [tuning] Re: 31-EDO and 1/4-comma meantone compared
>
>
> Hello, there, Monz and everyone.
>
> To your material on 31-EDO and 1/4-comma meantone, I might add a few
> points and possible clarifications.

Thanks also for your commentary, Margo. I've added a
link to your Tuning List post, to the bottom of the
Dictionary page.

http://www.ixpres.com/interval/dict/1-4cmt.htm

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

_________________________________________________________
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