Mailing list post
From: mschulter (2001-08-10)
Subject: Re: question on Zarlino scale
Hello, there, Joseph Pehrson and everyone.
Indeed Zarlino took a great interest in _both_ just intonation
(5-limit) and meantone temperaments for keyboard, as well as the
standard temperament with equal semitones for the lute.
While regarding JI as the "natural" basis for vocal intonation, by the
way, he noted in both early and late writings that voices seems to
avoid the "small steps" or commas which appear on keyboard instruments
in Ptolemy's syntonic diatonic. In other words, he suggests that the
pure ratios of this scheme are realized by something which might now
be called "adaptive JI."
For keyboards, he generally finds temperament much more practical,
describing his own 2/7-comma scheme in 1558, and adding in 1571
descriptions of 1/4-comma and 1/3-comma also.
Note that while terms such as "temperament" or _participatio_ are
common in the 16th century, "meantone" or "mesotonic" is as far as I
know a later term. Strictly speaking, as has been discussed, only a
1/4-comma temperament with pure major thirds fits the definition of a
"mean-tone" with the whole-tone equal to the precise mean of the 9:8
and 10:9 steps of the syntonic diatonic, although often the term is
used more generally for other temperaments such as 2/7-comma,
1/3-comma, etc.
Likewise, Zarlino's contemporary Francisco Salinas takes an interest
in both JI and meantone keyboards: he favors 1/3-comma, but considers
it less sonorous than alternatives with less heavy tempering of the
fifth (and major third). The possibility of a circular system in only
19 notes may have been a special attraction.
Here's Zarlino's 16-note JI system implemented on a keyboard, which he
presents as interesting but difficult to play in practice -- an
experimental solution we might say, in contrast to the usual approach
of temperament. I'll show the octave 3C-4C (octave numbers first, 4C
as middle C), which I map to two 12-note keyboards, although Zarlino's
diagram shows the usual 16th-century approach of split keys for an
instrument of this tuning size. Signed numbers show notes raised or
lowered by a syntonic comma with reference to the chain of fifths
F0-C0-G0:
3Eb0 3F#-2 3Bb0
32:27 25:18 16:9
294.13 568.72 996.09
....... ...... .....
3C#-2 3Eb+1 3F#-1 3G#-2 3Bb+1
25:24 6:5 45:32 25:16 9:5
70.67 315.64 590.22 772.63 1017.60
3D-1
10:9
182.40
......
3C0 3D0 3E-1 3F0 3G0 3A-1 3B-1 4C0
1:1 9:8 5:4 4:3 3:2 5:3 15:8 2:1
0 203.91 386.31 498.04 701.96 884.36 1088.27 1200
Picture the "D" key split into two parts, and likewise the "Eb," "F#,"
and "Bb" keys.
Here's a Scala file of this 16-note tuning:
! zarlin16.scl
!
Zarlino's 16-note JI scale implemented on an instrument with split keys
16
!
25/24
10/9
9/8
32/27
6/5
5/4
4/3
25/18
45/32
3/2
25/16
5/3
16/9
9/5
15/8
2/1
Or, if a downloadable version of this file might be more convenient for
some people:
<http://value.net/~mschulter/zarlin16.scl>
Here's Scala's information on ratios and sizes in cents, and let's
hope that I've entered the data correctly:
Zarlino's 16-note JI scale implemented on an instrument with split keys
0: 1/1 0.000000 unison, perfect prime
1: 25/24 70.67245 classic chromatic semitone
2: 10/9 182.4038 minor whole tone
3: 9/8 203.9100 major whole tone
4: 32/27 294.1351 Pythagorean minor third
5: 6/5 315.6414 minor third
6: 5/4 386.3139 major third
7: 4/3 498.0452 perfect fourth
8: 25/18 568.7176 classic augmented fourth
9: 45/32 590.2239 tritone
10: 3/2 701.9553 perfect fifth
11: 25/16 772.6278 classic augmented fifth
12: 5/3 884.3591 major sixth, BP sixth
13: 16/9 996.0905 Pythagorean minor seventh
14: 9/5 1017.596 just minor seventh, BP seventh
15: 15/8 1088.269 classic major seventh
16: 2/1 1200.000 octave
Incidentally, I tend to follow Zarlino in taking rational and tempered
tunings as each valid and important approaches.
Curiously, this applies for me with neo-Gothic tunings and
temperaments, where I might lean now toward a system with pure ratios
such as 14:18:21:24, now to the complex ratios of a usual Pythagorean
tuning, and now to some tempered system.
Most appreciatively,
Margo Schulter
[email protected]
Full thread (6 messages)
From: [email protected] (2001-08-10)
Subject: question on Zarlino scale
This is an easy one... (probably)
In Kyle Gann's excellent website on just intonation, he mentions
Zarlino's scale of 1588. Margo had also mentioned that Zarlino was
the first person to mathematically define temperament.
The Zarlino scale shown on the Gann page is clearly *not* a
temperament, but is a just intonation scale.
I'm assuming, then, that Zarlino worked with *both* tempered and just
scales?? Kind of looks that way, yes??
The Zarlino scale cited has two D's: D- at 10/9 and D at 9/8
as we have been discussing...
but also has other "enharmonic" pitches:
Eb's: Eb- at 32/27 and Eb at 6/5
and
F#'s: F#- at 25/18 and F# at 45/32
Bb's: Bb- at 16/9 and Bb at 9/5
NOW, my question is... are the differences between these pitches ALL
SYNTONIC COMMAS as in the case of the D-, D? (I think D- is my grade
here...)
??
And, if so, are they *all* "unison vectors??" That's a lot of them,
yes??
Any help would be GREATLY appreciated!
Thanks!
__________ _________ _______
Joseph Pehrson
From: Paul Erlich (2001-08-10)
Subject: Re: question on Zarlino scale
--- In tuning@y..., jpehrson@r... wrote:
> This is an easy one... (probably)
>
> In Kyle Gann's excellent website on just intonation, he mentions
> Zarlino's scale of 1588. Margo had also mentioned that Zarlino was
> the first person to mathematically define temperament.
>
> The Zarlino scale shown on the Gann page is clearly *not* a
> temperament, but is a just intonation scale.
>
> I'm assuming, then, that Zarlino worked with *both* tempered and
just
> scales?? Kind of looks that way, yes??
Margo and I had quite a lot of discussion about this some time
ago . . . Zarlino advocated 2/7-comma meantone temperament in
practice, but the scholarly tradition of his day, being so obsessed
with ancient Greek and Roman civilization, made it almost impossible
for anyone to write a treatise on tuning without laying down some
scales in ratios as the "ideal" or "theoretical" basis for their
work. The ratios of Zarlino's just scale do a pretty good job of
conveying what 2/7-comma meantone can actually do, though the
meantone does it better, with fewer notes, no commatic shifts, and no
commatic drifts.
>
> The Zarlino scale cited has two D's: D- at 10/9 and D at 9/8
>
> as we have been discussing...
>
> but also has other "enharmonic" pitches:
>
> Eb's: Eb- at 32/27 and Eb at 6/5
>
> and
>
> F#'s: F#- at 25/18 and F# at 45/32
>
> Bb's: Bb- at 16/9 and Bb at 9/5
>
I see Ben Johnston's notaion is being used here . . .
>
> NOW, my question is... are the differences between these pitches
ALL
> SYNTONIC COMMAS as in the case of the D-, D?
You betcha! And this is to be expected, as the syntonic comma is just
the interval that meantone temperament does away with.
> And, if so, are they *all* "unison vectors??" That's a lot of
them,
> yes??
Not sure what you're getting at . . . I guess, given Zarlino's
ultimate adoption of meantone temperament, the syntonic comma must be
viewed as a unison vector . . . but that's just _one_ unison
vector . . . a vector is just an arrow with a certain length and
direction, regardless of where it appears in the lattice (between D-
and D, between Eb- and Eb, etc.).
From: mschulter (2001-08-10)
Subject: Re: question on Zarlino scale
Hello, there, Joseph Pehrson and everyone.
Indeed Zarlino took a great interest in _both_ just intonation
(5-limit) and meantone temperaments for keyboard, as well as the
standard temperament with equal semitones for the lute.
While regarding JI as the "natural" basis for vocal intonation, by the
way, he noted in both early and late writings that voices seems to
avoid the "small steps" or commas which appear on keyboard instruments
in Ptolemy's syntonic diatonic. In other words, he suggests that the
pure ratios of this scheme are realized by something which might now
be called "adaptive JI."
For keyboards, he generally finds temperament much more practical,
describing his own 2/7-comma scheme in 1558, and adding in 1571
descriptions of 1/4-comma and 1/3-comma also.
Note that while terms such as "temperament" or _participatio_ are
common in the 16th century, "meantone" or "mesotonic" is as far as I
know a later term. Strictly speaking, as has been discussed, only a
1/4-comma temperament with pure major thirds fits the definition of a
"mean-tone" with the whole-tone equal to the precise mean of the 9:8
and 10:9 steps of the syntonic diatonic, although often the term is
used more generally for other temperaments such as 2/7-comma,
1/3-comma, etc.
Likewise, Zarlino's contemporary Francisco Salinas takes an interest
in both JI and meantone keyboards: he favors 1/3-comma, but considers
it less sonorous than alternatives with less heavy tempering of the
fifth (and major third). The possibility of a circular system in only
19 notes may have been a special attraction.
Here's Zarlino's 16-note JI system implemented on a keyboard, which he
presents as interesting but difficult to play in practice -- an
experimental solution we might say, in contrast to the usual approach
of temperament. I'll show the octave 3C-4C (octave numbers first, 4C
as middle C), which I map to two 12-note keyboards, although Zarlino's
diagram shows the usual 16th-century approach of split keys for an
instrument of this tuning size. Signed numbers show notes raised or
lowered by a syntonic comma with reference to the chain of fifths
F0-C0-G0:
3Eb0 3F#-2 3Bb0
32:27 25:18 16:9
294.13 568.72 996.09
....... ...... .....
3C#-2 3Eb+1 3F#-1 3G#-2 3Bb+1
25:24 6:5 45:32 25:16 9:5
70.67 315.64 590.22 772.63 1017.60
3D-1
10:9
182.40
......
3C0 3D0 3E-1 3F0 3G0 3A-1 3B-1 4C0
1:1 9:8 5:4 4:3 3:2 5:3 15:8 2:1
0 203.91 386.31 498.04 701.96 884.36 1088.27 1200
Picture the "D" key split into two parts, and likewise the "Eb," "F#,"
and "Bb" keys.
Here's a Scala file of this 16-note tuning:
! zarlin16.scl
!
Zarlino's 16-note JI scale implemented on an instrument with split keys
16
!
25/24
10/9
9/8
32/27
6/5
5/4
4/3
25/18
45/32
3/2
25/16
5/3
16/9
9/5
15/8
2/1
Or, if a downloadable version of this file might be more convenient for
some people:
<http://value.net/~mschulter/zarlin16.scl>
Here's Scala's information on ratios and sizes in cents, and let's
hope that I've entered the data correctly:
Zarlino's 16-note JI scale implemented on an instrument with split keys
0: 1/1 0.000000 unison, perfect prime
1: 25/24 70.67245 classic chromatic semitone
2: 10/9 182.4038 minor whole tone
3: 9/8 203.9100 major whole tone
4: 32/27 294.1351 Pythagorean minor third
5: 6/5 315.6414 minor third
6: 5/4 386.3139 major third
7: 4/3 498.0452 perfect fourth
8: 25/18 568.7176 classic augmented fourth
9: 45/32 590.2239 tritone
10: 3/2 701.9553 perfect fifth
11: 25/16 772.6278 classic augmented fifth
12: 5/3 884.3591 major sixth, BP sixth
13: 16/9 996.0905 Pythagorean minor seventh
14: 9/5 1017.596 just minor seventh, BP seventh
15: 15/8 1088.269 classic major seventh
16: 2/1 1200.000 octave
Incidentally, I tend to follow Zarlino in taking rational and tempered
tunings as each valid and important approaches.
Curiously, this applies for me with neo-Gothic tunings and
temperaments, where I might lean now toward a system with pure ratios
such as 14:18:21:24, now to the complex ratios of a usual Pythagorean
tuning, and now to some tempered system.
Most appreciatively,
Margo Schulter
[email protected]
From: [email protected] (2001-08-13)
Subject: Re: question on Zarlino scale
--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
http://groups.yahoo.com/group/tuning/message/26900
> --- In tuning@y..., jpehrson@r... wrote:
> > This is an easy one... (probably)
> >
> > In Kyle Gann's excellent website on just intonation, he mentions
> > Zarlino's scale of 1588. Margo had also mentioned that Zarlino
was
> > the first person to mathematically define temperament.
> >
> > The Zarlino scale shown on the Gann page is clearly *not* a
> > temperament, but is a just intonation scale.
> >
> > I'm assuming, then, that Zarlino worked with *both* tempered and
> just
> > scales?? Kind of looks that way, yes??
>
> Margo and I had quite a lot of discussion about this some time
> ago . . . Zarlino advocated 2/7-comma meantone temperament in
> practice, but the scholarly tradition of his day, being so obsessed
> with ancient Greek and Roman civilization, made it almost
impossible
> for anyone to write a treatise on tuning without laying down some
> scales in ratios as the "ideal" or "theoretical" basis for their
> work. The ratios of Zarlino's just scale do a pretty good job of
> conveying what 2/7-comma meantone can actually do, though the
> meantone does it better, with fewer notes, no commatic shifts, and
no commatic drifts.
Got it... Thanks, Paul...
> > And, if so, are they *all* "unison vectors??" That's a lot of
> them, yes??
>
> Not sure what you're getting at . . . I guess, given Zarlino's
> ultimate adoption of meantone temperament, the syntonic comma must
be viewed as a unison vector . . . but that's just _one_ unison
> vector . . . a vector is just an arrow with a certain length and
> direction, regardless of where it appears in the lattice (between D-
and D, between Eb- and Eb, etc.).
We had been discussing some time ago, with regard to your paper _The
Forms of Tonality_ the concept of eliminating unison vectors...
I don't have the paper right here... and this is a little fuzzy, but
it seemed that eliminating *one* unison vector created a certain kind
of scale, and eliminating *two* created another...
Could you please refresh my "swiss cheese" recall on this item??
I'm assuming in the above, all the "enharmonics" of the Zarlino scale
refer to only ONE vector, as you seem to indicate...
_______ _________ ______
Joseph Pehrson
From: [email protected] (2001-08-13)
Subject: Re: question on Zarlino scale
--- In tuning@y..., mschulter <MSCHULTER@V...> wrote:
http://groups.yahoo.com/group/tuning/message/26909
> Hello, there, Joseph Pehrson and everyone.
>
> Indeed Zarlino took a great interest in _both_ just intonation
> (5-limit) and meantone temperaments for keyboard, as well as the
> standard temperament with equal semitones for the lute.
>
Thank you, Margo, for your clear explanation of this system... which
I printed out right away...
_______ _______ ________
Joseph Pehrson
From: Paul Erlich (2001-08-14)
Subject: Re: question on Zarlino scale
--- In tuning@y..., jpehrson@r... wrote:
>
> We had been discussing some time ago, with regard to your paper
_The
> Forms of Tonality_ the concept of eliminating unison vectors...
>
> I don't have the paper right here... and this is a little fuzzy,
but
> it seemed that eliminating *one* unison vector created a certain
kind
> of scale, and eliminating *two* created another...
Yes . . . in a 2-dimensional system (say a 5-limit lattice),
eliminating one unison vector creates a linear temperament, still
infinite, while eliminating two results in a _finite_ scale.
> I'm assuming in the above, all the "enharmonics" of the Zarlino
scale
> refer to only ONE vector, as you seem to indicate...
Yes. Thus Zarlino was in all likelihood using the scale to convey an
open-ended meantone system.