Re: Draft 2 FAQ: Organs with split keys fewer than 19 per octave

[Ibo Ortgies:]
>Split keys were inserted in those places were there are usually upper
>keys between diatonic notes can be found.
>Usually the split keys in the small and one-stroke octave were "broken"
>to provide additional keys. Occasionally notes in the two-stroke octave
>also were broken, but only eb''/d#'' (more frequent) and g#''/ab''
>(less frequent).

I'm sure I'm VERY ignorant here, but what are "small octave",
"one-stroke octave", and "two-stroke octave"? Could you clarify for
those of us who are not familiar with these terms?

Also (ditto appalling ignorance), how are the keys split? Side to side,
or front to back?

>Special designs were developped...

Spelling quibble: just one 'p'.

JdL

Hi,

Joseph Pehrson wrote:

/tuning/topicId_19545.html#19545

> > resulting practically in

> > ex. 1 B - F#- C#- G#-(Eb- Bb- F - C) ...
> > / \ / \ / \ /
> > ... Eb- Bb- F - C - G - D - A - E

> I know this seems "petty,"

not at all, I welcome your constructive critique!

> but would it be possible to add a little
> text after the first line, like (Eb - Bb - F - C, same chain
> extending downward from below...)

I'd like to keep the diagram "clean" from text. The explaining text
should come befor or below it.

> Otherwise, personally, I was having problems, since I thought the B
> on the top line was a continuation of the fifths from the line
> underneith it... And, of course, that would produce Pythagorean
> thirds, not just ones...

yes, at that point more explanation might be possible or a link to the
resp. FAQ of pythagorean tuning, JI etc.

May be I should add something like:

---------
resulting practically in

ex. 1 B - F#- C#- G#-(Eb- Bb- F - C) ...
/ \ / \ / \ /
... Eb- Bb- F - C - G - D - A - E

All slashes indicate pure intervals in this diagram!
---------

Does this make the point more clear?

> I guess probably we should remember that the FAQ is also intended for
> beginners... and it confused *ME*, already an "intermediate
> beginner..."

Then my writing was not good enough and missed the intended purpose
-thanks for your help in getting this better!

kind regards
Ibo Ortgies

--- In tuning@y..., Ibo Ortgies <ibo.ortgies@m...> wrote:

/tuning/topicId_19553.html#19555

>
> ---------
> resulting practically in
>
> ex. 1 B - F#- C#- G#-(Eb- Bb- F - C) ...
> / \ / \ / \ /
> ... Eb- Bb- F - C - G - D - A - E
>
> All slashes indicate pure intervals in this diagram!
> ---------
>
> Does this make the point more clear?
>

Thanks so much, Ibo, for indulging my "moronic" mind... Would it be
at all possible to include just TWO words and be OK??

resulting practically in

>
> ex. 1 B - F#- C#- G#-(Eb- Bb- F - C)[from below ...]
> / \ / \ / \ /
> ... Eb- Bb- F - C - G - D - A - E

Otherwise, I'm still thinking of the chain going "upward" to the
right from E to B, and I'm not getting the "schismatic" idea...

Thanks!

________ ____ ____ ____
Joseph Pehrson

Dear list members,
answering two mails from
John A. deLaubenfels
Joseph Pehrson

> Date: Wed, 28 Feb 2001 09:41:01 -0700
> From: "John A. deLaubenfels" <jdl@adaptune.com>

> [Ibo Ortgies:]

> >Usually the split keys in the small and one-stroke octave were "broken"
> >to provide additional keys. Occasionally notes in the two-stroke octave
> >also were broken, but only eb''/d#'' (more frequent) and g#''/ab''
> >(less frequent).

> .... what are "small octave", "one-stroke octave", and "two-stroke
> octave"? Could you clarify for those of us who are not familiar
> with these terms?

This refers to the old nomenclature:
C - B Gro�e Oktave great octave
c - b kleine Oktave small octave
c' - b' eingestrichene Oktave one-stroke
c'' - h'' zweigestrichene Oktave two-stroke
c'''- dreigestrichene Oktave three-stroke

How would this have been called in 17th-ct England?

a' is (today) the so-called "Kammerton"

> ... how are the keys split? Side to side, or front to back?

Good question - I forgot to explain that - maybe drawings help more than words:

If you look at the key from above (section)

back back
| | | | |
| | | | |
| | | | |
| | | | |
| | | | |
| | | | |
| | | | d# |
| | | | | | |_____| | ___
| | | | | | | | |
| | | | | | | | |
| | eb | | | | eb | | |
| |_______| | | |_______| | _|_
| | | | | |
| | | | | |
| d | e | | d | e |
|_______|_______| |_______|_______|

The front part of the lower upper key (eb) might be ca. 2 cm long (+/-
2-3 mm) in a few existing organ positives with original keyboard
including split keys by the Manderscheidt-family (mid- 17th century
(area of N�rnberg/Germany and Switzerland) - indicated by the vertical
line to the right.

From the front (section) it will look like

_____
/ \
| d# |
_______ _|_____|
/ \ / | \
| eb | | | eb |
___|_ _ _ _|___ ____| _ _ _|____
| |___|___| | | |_|_____| |
| | | | | |
| d | e | | d | e |
|_______|_______| |________|________|

And in 3-D

/ /
/ /
/ /
/ / /
/ d# / / /
/_____/ / /
| | / / / /
/| | / / / /
/ |_____| / / / /
/ / /| / / / /
/ / eb / |/ / / f# /
/ /______ / / / /______ /
/ | | / / | |
/ | | / / | |
/ |______|/ / |______|
/ / / /
/ / / /
/ d / e / f /
/________/________ /_______ /
| | | |
| | | |
| | | |
|________|________|________|

> >Special designs were developped...

> Spelling quibble: just one 'p'.

Thanks
Ibo
------------------------

> Date: Wed, 28 Feb 2001 18:47:04 -0000
> From: jpehrson@rcn.com
> Subject: Re: Draft 2 FAQ: Organs with split keys fewer than 19 per octave

> --- In tuning@y..., Ibo Ortgies <ibo.ortgies@m...> wrote:

> /tuning/topicId_19553.html#19555

> > ---------
> > resulting practically in

> > ex. 1 B - F#- C#- G#-(Eb- Bb- F - C) ...
> > / \ / \ / \ /
> > ... Eb- Bb- F - C - G - D - A - E

> > All slashes indicate pure intervals in this diagram!

> Thanks so much, Ibo, for indulging my "moronic" mind... Would it be
> at all possible to include just TWO words and be OK??

An explanation below would not be sufficient?
The more brackets and dots etc. the mor eunclear it might be to other
readers - more/other opinions?

> resulting practically in

> > ex. 1 B - F#- C#- G#-(Eb- Bb- F - C)[from below ...]
> > / \ / \ / \ /
> > ... Eb- Bb- F - C - G - D - A - E

> Otherwise, I'm still thinking of the chain going "upward" to the
> right from E to B, and I'm not getting the "schismatic" idea...

There is no line indicating a pure interval (perf. fifth) between E - B

The fifth E - B in this example is a syntonic comma below the pure
fifth, because that E is higher than the pure major third to C -
therefore another E , pure to C, might have been inserted on a keyboard
of a pythagorean tuned organ.

E'
/
C - G - D - A - E

The E in this example is generated by four pure fifths - it is much
higher than pure.
The E' is a pure major third to C.
The difference between those two E-s is called the syntonic comma,
defined by 4 pure fifths minus 1 pure major third.

In my example 1 there is only the first E, therefore a E' might have
been inserted as extra key.

In example 2 there is the E' already there, and another E might have
been added to make the pure chord e-g#-b available

Another
E'
/
Fb- Cb- Gb- Db- Ab- Eb- Bb- F - C - G - D - A - E

The Fb at the left is only 1,9 cents below the E', which serves as pure
major third to C.

The pythagorean chain of perfect fifths

Gb- Db- Ab- Eb- Bb- F - C - G - D - A - E - B

provides with the four notes Gb- Db- Ab- Eb major thirds to some of the
main keys of the modal system. These major thirds are so close to pure,
that they result practically in a cut-out of just intonation
where the diatonic notes (or the notes from the old hexachord-system -
therefore including both B and Bb) get major thirds which are only the
small amount of 1,9 cents , the so called schisma (s. below), lower than
pure.

The scheme
Gb- Db- Ab- Eb- ...
/ \ / \ / \/
...Bb- F - C - G - D - A - E - B

is practically the same as:

F#- C#- G#- D#- ...
/ \ / \ / \/
...Bb- F - C - G - D - A - E - B

(as which it might have originated - adding pure major thirds to the
most used diatonic notes)

The Schisma

The schisma is the difference between the
pythagorean comma (12 pure perfect fifths - 7 octaves) and the
syntonic comma (4 pure perfect fifths - 1 pure major third - 2 octaves)

The schisma is then
(12 pure perfect fifths - 7 octaves)
- ( 4 pure perfect fifths - 2 octaves - 1 pure major third)

Now forget the octaves, because they are just here to keep us in an
"audible range" and we can use the fourths as complementary intervals of course.

Resolving the brackets and without the octaves it is

12 pure perfect fifths
- 4 pure perfect fifths
+ 1 pure major third
--------------------------------------------------------
= 8 pure perfect fifths + 1 pure major third

8 pure perfect fifths + 1 pure major third
is represented by the chain

E'
/
Fb- Cb- Gb- Db- Ab- Eb- Bb- F - C

The difference between the Fb and the E' is the schisma (the E' being
this small amount of 1,955 cents higher than Fb).

Now to the point of departure:

I called those intervalls, actually pythagorean diminuished fourths
(C-Fb in our previous example), which are practically pure major thirds
(very close to C-E') "schismatic thirds"

Does this clarify better? And should this be part of this very FAQ-page?

> Thanks!
> Joseph Pehrson

kind regards
Ibo Ortgies
----------------------------------
http://www.hum.gu.se/goart/ortgies/homepage.htm
http://www.hum.gu.se/goart/ortgies/public.html

GOArt (G�teborg Organ Art center) G�teborgs universitet
http://www.hum.gu.se/goart/w-1.htm

--- In tuning@y..., Ibo Ortgies <ibo.ortgies@m...> wrote:

/tuning/topicId_19553.html#19572

Hello Ibo Ortgies!

> > > resulting practically in
>
> > > ex. 1 B - F#- C#- G#-(Eb- Bb- F - C) ...
> > > / \ / \ / \ /
> > > ... Eb- Bb- F - C - G - D - A - E
>
> > > All slashes indicate pure intervals in this diagram!
>
>
> > Thanks so much, Ibo, for indulging my "moronic" mind... Would it
be at all possible to include just TWO words and be OK??
>
> An explanation below would not be sufficient?

OH! I didn't understand that you were going to include a written
explanation under it! That would, of course, be fine. The problem
with me (well, one of the problems) is that when I look at the chain,
initially, I keep wanting to see the E - B as a continuing link
upward... which would, of course, give a Pythagorean third that would
be quite large... This is just the way the diagram struck me when I
first saw it... I had no idea what you were talking about until
Daniel Wolf explained it to me. (And he did it quite easily in one
short paragraph). Therefore, I really *DO* think some text is
necessary for the beginner... (or "advanced" beginner)

>
> There is no line indicating a pure interval (perf. fifth) between E
- B
>

I know... but without the "explanation" I want to "make" one!

> The fifth E - B in this example is a syntonic comma below the pure
> fifth, because that E is higher than the pure major third to C -
> therefore another E , pure to C, might have been inserted on a
keyboard of a pythagorean tuned organ.
>
> E'
> /
> C - G - D - A - E
>

Yes, of course, this is quite clear...

> The E in this example is generated by four pure fifths - it is much
> higher than pure.
> The E' is a pure major third to C.
> The difference between those two E-s is called the syntonic comma,
> defined by 4 pure fifths minus 1 pure major third.
>
>
> In my example 1 there is only the first E, therefore a E' might have
> been inserted as extra key.
>
> In example 2 there is the E' already there, and another E might have
> been added to make the pure chord e-g#-b available
>
>
> Another
> E'
> /
> Fb- Cb- Gb- Db- Ab- Eb- Bb- F - C - G - D - A - E
>
>
> The Fb at the left is only 1,9 cents below the E', which serves as
pure major third to C.
>

Actually, in Daniel Wolf's post he mentioned the Fb and, with that, I
IMMEDIATELY "got" the idea of the schisma...

>
>
> The Schisma
>
> The schisma is the difference between the
> pythagorean comma (12 pure perfect fifths - 7 octaves) and the
> syntonic comma (4 pure perfect fifths - 1 pure major third - 2
octaves)
>
> The schisma is then
> (12 pure perfect fifths - 7 octaves)
> - ( 4 pure perfect fifths - 2 octaves - 1 pure major third)
>
> Now forget the octaves, because they are just here to keep us in an
> "audible range" and we can use the fourths as complementary
intervals of course.
>
> Resolving the brackets and without the octaves it is
>
> 12 pure perfect fifths
> - 4 pure perfect fifths
> + 1 pure major third
> --------------------------------------------------------
> = 8 pure perfect fifths + 1 pure major third
>
>
> 8 pure perfect fifths + 1 pure major third
> is represented by the chain
>
> E'
> /
> Fb- Cb- Gb- Db- Ab- Eb- Bb- F - C
>
>
> The difference between the Fb and the E' is the schisma (the E'
being this small amount of 1,955 cents higher than Fb).
>

This is quite a fine explanation of this and should DEFINITELY be
included in the FAQ. It might be in the "more info" pages, though,
and presented AFTER the "more basic" concepts like Meantone which, I
believe, is inferred by what Daniel Wolf posted... I think.

Thanks so much for your response...

________ _____ ____ __
Joseph Pehrson

Ibo Ortgies wrote:

> > .... what are "small octave", "one-stroke octave", and
"two-stroke
> > octave"? Could you clarify for those of us who are not familiar
> > with these terms?
>
> This refers to the old nomenclature:
> C - B Große Oktave great octave
> c - b kleine Oktave small octave
> c' - b' eingestrichene Oktave one-stroke
> c'' - h'' zweigestrichene Oktave two-stroke
> c'''- dreigestrichene Oktave three-stroke
>
> How would this have been called in 17th-ct England?
>
> a' is (today) the so-called "Kammerton"

I think numbering octaves is safest for a general audience. That is,
"first octave", "second octave" and so on. Details for organ
specialists would be more appropriate on a website.

> > ... how are the keys split? Side to side, or front to back?
>
> Good question - I forgot to explain that - maybe drawings help more
than words:

Excellent! That makes it crystal clear. It'll make the FAQ bigger,
but I vote it goes in.

> E'
> /
> Fb- Cb- Gb- Db- Ab- Eb- Bb- F - C
>
>
> The difference between the Fb and the E' is the schisma (the E'
being
> this small amount of 1,955 cents higher than Fb).

If cents are to be used, perhaps the sizes of the commas should have
been specified as well.

> Now to the point of departure:
>
> I called those intervalls, actually pythagorean diminuished fourths
> (C-Fb in our previous example), which are practically pure major
thirds
> (very close to C-E') "schismatic thirds"
>
>
> Does this clarify better? And should this be part of this very
FAQ-page?

If you're going to mention schismic temperaments, then the FAQ has to
cover them. I think the whole question of schismic organ tuning
should be moved to that section. It should suffice to say that some
Pythagorean organs had split keys, and which keys were split. But
historical examples would be appropriate under "what is schismic
temperament". I don't think those interested in schismic temperament
would find the idea of splitting organ keys difficult. But coverage
of schismic temperament would be a big diversion for somebody only
interested in split keys.

Note that I cover schismic temperament at
<http://x31eq.com/schismic.htm>. You can copy anything you
like from my site for the FAQ. I don't think I have the historical
overview to write any entries from scratch.

Graham

Hi Graham and all others

I'm rather behind, I try to catch up...

> Date: Thu, 01 Mar 2001 12:24:14 -0000
> From: "Graham Breed" <graham@microtonal.co.uk>
> Subject: Re: Draft 2 FAQ: Organs with split keys fewer than 19 per octave

> Ibo Ortgies wrote:

> > > .... what are "small octave", "one-stroke octave", and
> > > "two-stroke octave"? Could you clarify for those of us
> > > who are not familiar with these terms?

> > This refers to the old nomenclature:
> > C - B Gro�e Oktave great octave
> > c - b kleine Oktave small octave
> > c' - b' eingestrichene Oktave one-stroke
> > c'' - h'' zweigestrichene Oktave two-stroke
> > c'''- dreigestrichene Oktave three-stroke

> I think numbering octaves is safest for a general audience. That is,
> "first octave", "second octave" and so on. Details for organ
> specialists would be more appropriate on a website.

I agree, that it shouldn't be for specialists only but
numbering systems are also not common and need also explanation.
I'd prefer the closest term musical
Is C or b'' not clear to anybody?
How is "Bass octave, tenor octave, middle octave, treble octave" and if
it exceeds (which isn't often the case in the time I'm covering), then
"8ve below bass 8ve" resp. "8ve above treble 8ve"

> > The difference between the Fb and the E' is the schisma (the E'
> > being
> > this small amount of 1,955 cents higher than Fb).

> If cents are to be used, perhaps the sizes of the commas should have
> been specified as well.

yes, maybe it is better. How many figures after the comma/dot. I prefer
actually one: 1,955 -> 2,0, in order not to scare musicians with too
many ciphers.

> > Now to the point of departure:

> > I called those intervalls, actually pythagorean diminuished fourths
> > (C-Fb in our previous example), which are practically pure major
> > thirds (very close to C-E') "schismatic thirds"

> > Does this clarify better? And should this be part of this very
> > FAQ-page?

> If you're going to mention schismic temperaments, then the FAQ has to
> cover them. I think the whole question of schismic organ tuning
> should be moved to that section. It should suffice to say that some
> Pythagorean organs had split keys, and which keys were split. But
> historical examples would be appropriate under "what is schismic
> temperament". I don't think those interested in schismic temperament
> would find the idea of splitting organ keys difficult. But coverage
> of schismic temperament would be a big diversion for somebody only
> interested in split keys.

I'd prefer some redundant information, meaning that the information
might be provided at diefferent places in the FAQ - the less people have
to click around to get here bit of info and there another, seems to me preferable
I'm also voting for a simple e-mail-like ASCII-text in html (no java),
like in my mails, with a clear structured text (hopefully)

Kind regards
Ibo Ortgies

----------------------------------
http://www.hum.gu.se/goart/ortgies/homepage.htm
http://www.hum.gu.se/goart/ortgies/public.html

GOArt (G�teborg Organ Art center) G�teborgs universitet
http://www.hum.gu.se/goart/w-1.htm

Hello, there, Ibo Ortgies and everyone.

First, please let me say what a pleasure it is to see that our FAQ
will have such extensive and scholarly coverage of these split-key
instruments; as I mention in another reply on Lucca, your knowledge of
the actual instruments is something I do not share but can at least
admire. It provides a vital perspective which theory alone, or even
experience with modern equivalents of these schemes, cannot provide.

One small possible change, if you feel it fits, might be to speak of
"Instruments with fewer with 17 notes" rather than "fewer than 19
notes," if the possible split-key instruments of the later Pythagorean
era (1370-1450 or a bit later) are considered an important part of the
draft.

The reason is that while 19 notes (a tuning of Gb-B#) provides a
"Moment of Symmetry" (MOS) for meantone tunings, with the gamut
divided into only two types of adjacent intervals, minor or chromatic
semitones and dieses (e.g. C-C#-Db-D or E-E#-F-F#), 17 notes (Gb-A#)
likewise provide such an MOS in a Pythagorean tuning, with adjacent
limmas (minor or diatonic semitones) and commas (e.g. C-C#-Db-D or
E-F-Gb-F#).

Thus early 15th-century theorists regard a "complete" tuning as having
17 notes, just as 19 notes is common in 16th-century theory and
instrument-building practice.

If, as I read your FAQ draft to say, meantone instruments of 17 or 18
notes are rare, then maybe "fewer than 17 notes" would fit both the
late Pythagorean and meantone eras.

Currently I am planning two related FAQ drafts which I hope can
complement your material:

(1) Pythagorean tunings and keyboard schemes, 13-17 notes (1370-1482);
(2) Keyboard instruments with 19 or more notes (1550-1618).

Here it seems the most common schemes -- with only theoretical schemes
or designs for the Pythagorean era, rather than actual known
instruments, according to my limited and not-so-current sources --
seem to involve regular Pythagorean tunings (within the range of
Gb-A#) or meantone tunings.

However, both Lindley and your FAQ draft mention the possibility of
theoretical schemes or designs, at least, using Pythagorean notes such
as Cb or Fb, and this could be viewed as a kind of transition from
Pythagorean to some form of meantone or syntonic diatonic JI
tuning. The famous 2/3/5-ratio JI monochord of Ramos (1482), for
example, could be viewed as a modified form of a Pythagorean scheme
with the Wolf at G-D.

Lindley takes the very tenable position that schemes of this kind were
theoretical rather than practical designs for keyboards, and certainly
having a Wolf fifth between _musica recta_ notes (diatonic notes and
Bb) would create some of the same complications for a player as later
16th-century keyboards of the kind described by Zarlino and Bottrigari
in the syntonic diatonic.

However, since Zarlino, for example, describes such a syntonic
diatonic keyboard with 16 notes (although noting that it is difficult
to play, and concluding that temperament is much more practical), why
might not some of the transitional Pythagorean schemes also have
built?

One note: while a Pythagorean scheme with a Wolf between diatonic
notes, or at Bb-F, would be complicated to play, something like the
following seems to me quite practical, if one leans toward having as
many schisma thirds as possible (not too unlikely a preference just at
the point where meantone was coming into vogue around 1450-1480).

Here I would emphasize that these are my own schemes, and that I am
not aware of these specific schemes in the 15th century, which is not
to say that they might not have been described or used at that time.

The following three schemes are based on the usual early 15th-century
tuning Gb-B (Wolf at B-Gb, written B-F#), where thirds spelled with
written sharps are played as diminished fourths or augmented seconds
(schisma thirds), with other thirds at their usual Pythagorean sizes.
I use the symbol "@" to show a second version of a _musica recta_ note
lowered by a Pythagorean comma (531441:524288, ~23.46 cents). In my
tuning chain diagrams, I use notes in parentheses to show the written
notes for which certain accidentals might be played:

13 notes: Cb-B (usual Gb-B plus Cb)

(B@-F#-C#-G#)
Cb-Gb-Db-Ab-Eb-Bb-F-C-G-D-A-E-B

Db Eb Gb Ab Bb Cb
C D E F G A B C

(This scheme adds a schisma third G-Cb or G-B@)

* * * *

14 notes: Fb-B (usual Gb-B plus Fb and Cb)

(E@-B@-F#-C#-G#)
Fb-Cb-Gb-Db-Ab-Eb-Bb-F-C-G-D-A-E-B

Db Eb Fb Gb Ab Bb Cb
C D E F G A B C

(This scheme adds another schisma third C-Fb or C-E@)

* * * *

15 notes: Fb-F# (usual Gb-B plus Fb-Cb, F#)

(E@-B@-F#-C#-G#)
Fb-Cb-Gb-Db-Ab-Eb-Bb-F-C-G-D-A-E-B-F#

F#
Db Eb Fb Gb Ab Bb Cb
C D E F G A B C

(This scheme adds a pure fifth B-F#, like a 13-note Gb-F#)

Note that this last scheme is like the usual 15th-century scheme in 13
notes with a split key or the like at Gb/F# to provide a pure fifth at
B-F#, the scheme likely proposed by Ramos's friend Tristan de Silva.

If one has tastes of a kind which may have been common around 1450,
and wants to design a keyboard based on Pythagorean tuning with
schisma thirds available for all diatonic notes, or with a free choice
of either regular Pythagorean or schisma thirds for all these notes,
then a kind of counterpart to the 19-note meantone keyboards of the
16th-17th centuries is also possible. This scheme is based on the
17-note organ (Gb-A#) proposed by Ugolino of Orvieto (c. 1435?), plus
the two added keys Fb and Cb:

19 notes: Fb-A# (usual Gb-A# plus Fb and Cb)

(E@-B@-F#-C#-G#)
Fb-Cb-Gb-Db-Ab-Eb-Bb-F-C-G-D-A-E-B-F#-C#-G#-D#-A#

C# D# F# G# A#
Db Eb Fb Gb Ab Bb Cb
C D E F G A B C

(This scheme adds schisma thirds C-Fb, G-Cb)

Note that all of these schemes have the advantage of not disrupting
the usual _musica recta_ chain of fifths (Bb-E), so that the player
does not have to deal with Wolf fifths between any of these regular
notes of the gamut.

The last scheme is basically the same keyboard layout, but differs in
its tuning, from the 19-note organ described by Ugolino of Orvieto
where the two semitones E-F and B-C are divided by the extra two notes
into equal dieses, as in the ancient enharmonic genus of the Greeks.
Since this interval would be equal to around 45.11 cents (half of
256:243, ~90.22 cents), we might indicate it using something like
Vicentino's later dot to show a note raised by a diesis, or an ASCII
asterisk (*): thus we have Gb-A# plus E* and B*.

Ugolino considers the two extra notes optionally added to his usual
17-note Pythagorean gamut (Gb-A#) as rather an antiquarian nicety,
since he says that they do not occur in modern music. However, they
may be of interest as a 15th-century mention of the enharmonic genus,
which of course became the humanist basis of Vicentino's 31-note
meantone cycle on his archicembalo and arciorgano.

Most appreciatively,

Margo Schulter
mschulter@value.net