Vicentino's adaptive-JI tuning

i've just made a new webpage analyzing Vicentino's
adaptive-JI tuning scheme, his "2nd tuning of 1555",
as paul erlich always calls it.

http://sonic-arts.org/monzo/vicentino/vicentino.htm

it started out as a posting to this list, which
is quoted below. but once i got started i just
couldn't stop ...

-----------------

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Vicentino's "2nd tuning of 1555" is composed of
two chains of 1/4-comma meantone, the first one
a 19-tone chain from Gb to B#, which can be closely
approximated by a 19-tone subset of 31edo, and the
second a 17-tone chain from Gb to A#, 1/4-comma
higher than the first chain.

here's a rectangular lattice of Vicentino's
"2nd tuning of 1555"; the 3-axis is horizontal,
the 5-axis is vertical and the pitches indicate
exponents which increase by increments of 1/4.
notes a "5th" apart (3:2 ratio) which have integer
exponents of 5 are connected by dashed lines.

B# 269
E# 773-----
A# 76
D# 579 A# 81
G# 1083 D# 585
C# 386-----G# 1088
F# 890 C# 392
B 193 F# 895
E 697 B 199
A 0-----E 702
D 503 A 5
G 1007 D 509
C 310 G 1012
F 814-----C 316
Bb 117 F 819
Eb 621 Bb 122
Ab 1124 Eb 626
Db 427-----Ab 1129
Gb 931 Db 433
Gb 936

the 19-tone "first chain" is the column on the
left-hand side of this lattice.

the generator "5th" in 1/4-comma meantone can
be factored as 5^(1/4), thus 4 steps of the generator
(i.e., 4 "5ths") is exactly 5^1, which represents
the 5:4 ratio; thus, 4 "5ths" = a "major-3rd",
the basic equation of meantone, which is approximated
by other variants belonging to the meantone family
of temperaments, but is exact here in 1/4-comma MT.
thus, on the lattice, a chain of 1/4-comma meantone
goes straight up and down the 5-axis.

Vicentino's second chain, 1/4-comma higher than
the first chain, thus ends up being in effect a
3:2 higher than the first chain, as the lattice shows
in the right-hand column.

here's an attempt at redrawing the same lattice
in triangular format:

B# 269
E# 773 -----
A# 76
D# 579 A# 81
G# 1083 D# 585
C# 386 ----- G# 1088
F# 890 C# 392
B 193 F# 895
E 697 B 199
A 0 ----- E 702
D 503 A 5
G 1007 D 509
C 310 G 1012
F 814 ----- C 316
Bb 117 F 819
Eb 621 Bb 122
Ab 1124 Eb 626
Db 427 ----- Ab 1129
Gb 931 Db 433
Gb 936

the lattices show that the following triads are
exactly in tune to low-integer proportions:

Major = 4:5:6
Gb, Db, Ab, Eb, Bb, F, C, G, D, A, E, B, F#, C#, G#

minor = 1/(6:5:4)
eb, bb, f, c, g, d, a, e, b, f#, c#, g#, d#

here is the whole scale. i'd have to do some
reading to find out Vicentino's notation, so
i'll just use a plus sign to designate the
1/4-comma-higher note of each like-named pair.

note ~cents
Ab+ 1129
Ab 1124
G#+ 1088
G# 1083
G+ 1012
G 1007
Gb+ 936
Gb 931
F#+ 895
F# 890
F+ 819
F 814
E# 773
E+ 702
E 697
Eb+ 626
Eb 621
D#+ 585
D# 579
D+ 509
D 503
Db+ 433
Db 427
C#+ 392
C# 386
C+ 316
C 310
B# 269
B+ 199
B 193
Bb+ 122
Bb 117
A#+ 81
A# 76
A+ 5
A 0

i made an interval matrix of this tuning, and
just out of curiosity i analyzed what intervals
might approximate the 7:4 ratio. here's the list:

~971 cents:

Gb : E+
F : D#+
Eb : C#+
Db : B+
C : A#+
Bb : G#+
Ab : F#+

~966 cents:

G : E#
Gb+ : E+
Gb : E
F+ : D#+
F : D#
Eb+ : C#+
Eb : C#
D : B#
Db+ : B+
Db : B
C+ : A#+
C : A#
Bb : G#
Bb+ : G#+
Ab+ : F#+
Ab : F#

-monz
"all roads lead to n^0"

--- In tuning@y..., "monz" <monz@a...> wrote:
> i've just made a new webpage analyzing Vicentino's
> adaptive-JI tuning scheme, his "2nd tuning of 1555",
> as paul erlich always calls it.
>
> http://sonic-arts.org/monzo/vicentino/vicentino.htm

hi monz,

nice work! really, it's the meantone chain itself that should be
latticed -- the 19-tone chain is a 5-limit periodicity block with
81:80 tempered out, of course . . . the slight difference between the
chains allows one to obtain pure JI sonorities, but doesn't affect
the basic structure or function of the scale.

i like what you're saying about 217-equal, but see below.

perhaps a more impressive coincidence is that 1/4-comma meantone, if
carried out beyond 31 notes, actually repeats itself about 6 cents
away -- close to 1/4-comma! along these lines, margo schulter has
proposed a 62-tone 1/4-comma meantone chain as a marvelous tuning
system, since it practically encompasses vicentino's tuning while
allowing one to circulate around the cycle of 31 meantone fifths (or
thirds or whatever). this is an example of an NMOS (currently being
discussed on tuning-math), since 62 = 2 * the 31-tone MOS of
meantone . . . other NMOSs are helmholtz 24 and groven 36, multiples
of the 12-tone MOS of schismic temperament . . .

. . . if you take a look, i'm sure you'll find that 205-equal
provides a much better approximation of vicentino's tuning than 217-
equal does. 205-equal also gives a better approximation, than 217-
equal, of schulter 62, but 174-equal positively excels in the latter
regard.

btw, who's aaron hunt?

monz -- i'm amazed at how quickly you included this information at

http://sonic-arts.org/monzo/vicentino/vicentino.htm

!!!!!!!!!!!!!! wow !!!!!!!!!!!!!!

. . . in the bibliography, i don't think the alves article is
relevant, as it concerns a different tuning altogether . . . instead,
expanding the other reference would tell you where margo, and hence
the rest of us, got her information on the topic:

Vicentino, Nicola. L'antica musica ridotta alla moderna prattica.
Antonio Barre, Rome, 1555, 1557. English translation Ancient music
adapted to modern practice by Maria Rika Maniates (ed.), Claude V.
Palisca (ed.), Yale University Press, New Haven CT, 1996, p. xlix.

--- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...>
wrote:
> --- In tuning@y..., "monz" <monz@a...> wrote:
> > i've just made a new webpage analyzing Vicentino's
> > adaptive-JI tuning scheme, his "2nd tuning of 1555",
> > as paul erlich always calls it.
> >
> > http://sonic-arts.org/monzo/vicentino/vicentino.htm
>
> hi monz,
>
> nice work! really, it's the meantone chain itself that should be
> latticed -- the 19-tone chain is a 5-limit periodicity block with
> 81:80 tempered out, of course . . . the slight difference between
the
> chains allows one to obtain pure JI sonorities, but doesn't affect
> the basic structure or function of the scale.
>
> i like what you're saying about 217-equal, but see below.
>
> perhaps a more impressive coincidence is that 1/4-comma meantone,
if
> carried out beyond 31 notes, actually repeats itself about 6 cents
> away -- close to 1/4-comma! along these lines, margo schulter has
> proposed a 62-tone 1/4-comma meantone chain as a marvelous tuning
> system, since it practically encompasses vicentino's tuning while
> allowing one to circulate around the cycle of 31 meantone fifths
(or
> thirds or whatever). this is an example of an NMOS (currently being
> discussed on tuning-math), since 62 = 2 * the 31-tone MOS of
> meantone . . . other NMOSs are helmholtz 24 and groven 36,
multiples
> of the 12-tone MOS of schismic temperament . . .
>
> . . . if you take a look, i'm sure you'll find that 205-equal
> provides a much better approximation of vicentino's tuning than 217-
> equal does. 205-equal also gives a better approximation, than 217-
> equal, of schulter 62, but 174-equal positively excels in the
latter
> regard.
>
> btw, who's aaron hunt?

--- In tuning@y..., "monz" <monz@a...> wrote:
> i've just made a new webpage analyzing Vicentino's
> adaptive-JI tuning scheme, his "2nd tuning of 1555",
> as paul erlich always calls it.
>
> http://sonic-arts.org/monzo/vicentino/vicentino.htm
>
>
> it started out as a posting to this list, which
> is quoted below. but once i got started i just
> couldn't stop ...

Good! It's an interesting topic.

Hello, there, everyone, and I'd like quickly to clarify a slight
distinction between my 62-note proposal of 2001

</tuning/topicId_26884.html#26884>

and Paul Erlich's ingenious proposal for a single 62-note chain of
1/4-comma meantone.

My proposal has two 31-note chains of 1/4-comma meantone at 1/4-comma
apart (~5.38 cents), and thus represents a kind of superset including in
essence both of Vicentino's archicembalo tunings.

Vicentino's first tuning is a circulating 31-note meantone "loop" (to use
Joe Pehrson's term) plus a few extra "comma keys" which in one
interpretation add a few just fifths to the most common diatonic notes of
the tuning.

The second tuning, discussed in this thread and in the new Monz web page,
consists of a basic 19-note tuning (Gb-B#) on the main manual, with the
second manual (ideally 19 notes, but 17 notes in Vicentino's practical
instrument) providing "just fifths" to all or most of these notes. This
arrangement is generally synonymous with two meantone chains at 1/4-comma
apart.

In my 62-note tuning, each 31-note set is a complete cycle or loop (as in
Vicentino's first tuning), with the two sets 1/4-comma apart (as in his
second tuning). The advantage of this arrangement is that one can play
enharmonic compositions involving fifthtone steps (about 1/5-tone, and
actually slightly unequal at 128:125 or ~41.06 cents for the larger
fifthtone and ~34.99 cents for the smaller fifthtone) in adaptive JI.

In the 62-note system which you describe, Paul, we have the slightly
different although musically almost equivalent arrangement of a single
62-note meantone chain, with the second 31-note set thus at a distance of
about 6.07 cents from the first set -- very close to the ~5.38 cents
required for pure vertical 5-limit JI (4:5:6 or 10:12:15).

What you give is an excellent of where one can both "have one's cake and
eat it too" with a circulating loop. Each 31-note set could serve as a
complete meantone cycle; but the 62-note chain makes available virtually
pure JI sonorities which would not be available in a single 31-note
tuning.

Similarly, while 53 notes makes a fine circulating Pythagorean loop,
carrying the chain to 106 notes makes available virtual JI intervals such
as the near-7:6 minor third formed from 68 fourths up -- thanks to Monz
for pointing this out!

With Paul's 62-note meantone chain, the near-equivalence between the
6.07-cent difference of 31 meantone fifths vs. 18 pure octaves and the
adaptive-JI adjustment of 1/4-comma (~5.38 cents) makes such a result
possible. With the Pythagorean example, it is the close resemblance
between the comma of Mercator (53 pure fifths vs. 31 pure octaves), about
3.62 cents, and the 3-7 schisma at about 3.80 cents, differing only by a
nanisma.

Anyway, while my original 62-note proposal followed the apparent
historical method of Vicentino's "just fifths," Paul's described
alterative of a single 62-note meantone chain nicely shows the potential
of "almost-closed" circulating loops.

Most appreciatively,

Margo Schulter
mschulter@value.net

thanks for this, Margo. i've added this to the
Vicentino page as well.

http://sonic-arts.org/monzo/vicentino/vicentino.htm

-monz
"all roads lead to n^0"

----- Original Message -----
From: "M. Schulter" <MSCHULTER@VALUE.NET>
To: <tuning@yahoogroups.com>
Sent: Thursday, October 31, 2002 4:57 PM
Subject: [tuning] Re: Vicentino's adaptive-JI tuning

> Hello, there, everyone, and I'd like quickly to clarify a slight
> distinction between my 62-note proposal of 2001
>
> </tuning/topicId_26884.html#26884>
>
> and Paul Erlich's ingenious proposal for a single 62-note chain of
> 1/4-comma meantone.
>
> <etc.>

--- In tuning@y..., "M. Schulter" <MSCHULTER@V...> wrote:

> Hello, there, everyone, and I'd like quickly to clarify a slight
> distinction between my 62-note proposal of 2001
>
> </tuning/topicId_26884.html#26884>
>
> and Paul Erlich's ingenious proposal for a single 62-note chain of
> 1/4-comma meantone.

whoops -- sorry i confused the two! monz, i hope you will update your
page to correct my error.