Early 19th century microtonality

Forgot to say that after recommending that the temperament be as equal
as possible, Schiedmeyersays bemoans the fact that the piano can never
really be properly "in-tune" because, like all keyboard instruments,
it has no separate keys for C#/Db, D#/Eb, E/E#/F, etc.

Ciao,

P

Not too far from Vienna either? - where Valentini and Froberger
probably exercised the emperor's stable of 19-note keyboards in the
mid-17th century.

I wonder what Schubert would do with one of those if they had survived.
~~~T~~~

--- In tuning@yahoogroups.com, "Paul Poletti" <paul@...> wrote:
>
> Forgot to say that after recommending that the temperament be as equal
> as possible, Schiedmeyersays bemoans the fact that the piano can never
> really be properly "in-tune" because, like all keyboard instruments,
> it has no separate keys for C#/Db, D#/Eb, E/E#/F, etc.
>
> Ciao,
>
> P
>

There was a persistance of meantone, or meantone-like, tunings well into the 19th century. Hand horn tutors described different positions for neighboring sharps and flats, with sharps lower than neighboring flats as would be expected in meantone. Berlioz's orchestration manual includes a concertina with a meantone arrangement. Ellis indicates that pianos continued to be tuned in meantone until the middle of the century, probably a geographical fringe phenomena, but one, nevertheless in which a choice had to be made between sharps and flats for each black key on the piano. And, famously, we have Mahler's late testimony, regreting the loss of meantone.

Daniel Wolf

And pipe organs have apparently never ceased to be tuned in meantone.
Of course it is rarer today, but unlike the harpsichord, which died
and was brought back to life, there appears to be an unbroken lineage
with pipe organs. -Carl

--- In tuning@yahoogroups.com, "Daniel Wolf" <djwolf@...> wrote:
>
> There was a persistance of meantone, or meantone-like, tunings well
into
> the 19th century. Hand horn tutors described different positions for
> neighboring sharps and flats, with sharps lower than neighboring
flats as
> would be expected in meantone.

Yes, but I don't consider meantone to have anything to do with
microtonality; it was just the default way to tune a dodecatonic
keyboard for several centuries. It only moves into the MT realm when
you've got subsemitones.

I doubt that Schiedmeyer meant meantone, he was referring to the
natural proportions of the intervals, which of course, would exclude
the rather heavily tempered fifths of 1/4 mean. If he was referring to
anything at all, he probably just meant flexible just intonation, as
implied by Leopold Mozart, Tartinni, Rameau, Muller (1777 London), etc
etc etc. Love that Roger North quote!

Ciao,

P

--- In tuning@yahoogroups.com, "Paul Poletti" <paul@...> wrote:
>
> I doubt that Schiedmeyer meant meantone, he was referring to the
> natural proportions of the intervals, which of course, would exclude
> the rather heavily tempered fifths of 1/4 mean.

from earlier:
'the piano can never really be properly "in-tune" because, like all
keyboard instruments, it has no separate keys for C#/Db, D#/Eb,
E{?}/E#/F, etc.'

The problem in meantone on a standard keyboard *is* specifically these
enharmonics.
But the problem with just intonation (at least the first of many!) on
a keyboard is the need for two D's and/or two A's, a syntonic comma
apart. Forget sharps and flats, even the naturals need extra keys.

If we take Schiedmayer seriously, his lament for lack of in-tuneness
concerns specifically those notes which are 'missing' in a meantone
scheme.
Of course Schiedmayer could have been using the lack of enharmonics as
emblematic of the whole problem of tempering. We don't know how he
would react to a 19-note instrument in meantone.

> If he was referring to
> anything at all, he probably just meant flexible just intonation, as
> implied by Leopold Mozart, Tartinni, Rameau, Muller (1777 London), etc
> etc

- Do you have some reference for 'Muller 1777'? That's a new one on
me. You could add Bremner (also 1777) 'Thoughts on playing concert
music' -
/tuning/topicId_68175.html#68175
~~~T~~~

On 11 Dec 2008, at 6:33 AM, Tom Dent wrote:
>
> from earlier:
> 'the piano can never really be properly "in-tune" because, like all
> keyboard instruments, it has no separate keys for C#/Db, D#/Eb,
> E{?}/E#/F, etc.'
>
> The problem in meantone on a standard keyboard *is* specifically these
> enharmonics.
> But the problem with just intonation (at least the first of many!) on
> a keyboard is the need for two D's and/or two A's, a syntonic comma
> apart. Forget sharps and flats, even the naturals need extra keys.
>
>
What about having 21 keys then:

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

???

Daniel Forro

2008/12/11 Daniel Forró <dan.for@tiscali.cz>:
>
> On 11 Dec 2008, at 6:33 AM, Tom Dent wrote:

>> But the problem with just intonation (at least the first of many!) on
>> a keyboard is the need for two D's and/or two A's, a syntonic comma
>> apart. Forget sharps and flats, even the naturals need extra keys.
>>
>>
> What about having 21 keys then:
>
> C# D# E# F# G# A# B#
> C D E F G A B
> Cb Db Eb Fb Gb Ab Bb
>
> ???

That's meantone, not just intonation. There were accordions (or some
related instruments) made in England in the 19th century that could
play more than 12 notes of meantone. And I followed a link from one
of these lists to an article about some published music that exploited
one of the dieses. However, I'm unable to find that article to link
to again. Not that exciting because it's only one example and only a
brief melodic effect, but still it did happen.

Graham

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:

>
> The problem in meantone on a standard keyboard *is* specifically these
> enharmonics.
> But the problem with just intonation (at least the first of many!) on
> a keyboard is the need for two D's and/or two A's, a syntonic comma
> apart. Forget sharps and flats, even the naturals need extra keys.
>
> If we take Schiedmayer seriously, his lament for lack of in-tuneness
> concerns specifically those notes which are 'missing' in a meantone
> scheme.
> Of course Schiedmayer could have been using the lack of enharmonics as
> emblematic of the whole problem of tempering. We don't know how he
> would react to a 19-note instrument in meantone.

Actually, we do. He says it's too complicated to play an instrument
with lots of keys to the octave.

It's an interesting passage, so here it is in it's entirety:

Die Temperatur der Tonarten soll möglichst gleich seyn. Hat der
Stimmer nicht so viel richtiges Gehör, daß er diese möglichste
Gleicheit auch ohne alle künstliche Regeln herauszubringen weiß, so
ist er zu bedauern und er wird zuverlässig auch nach den besten
Anleitungen zum Stimmen doch keine richtige Stimmung zuwege bringen.
Da es übrigens wegen der allzugroßen Vermehrung der Tasten und der
unüberwindlichen Schwierigkeit, ein solches instrument fertig zu
spielen, nicht möglich ist, die enharmonische Tonleiter auf dem
Claviere zu bilden, und dieses Instrument kein besonderes Ces, Cis,
Des, Dis, Es, Eis, Fes, u.s.w. hat, so kann auf demselben, so wie aur
der orgel und allen andern Tonwerkzeugen, auf welchen die
enhharmonische Tonleiter nicht geblidet werden kann, die Stimmung
überhaupt nie vollkommen seyn, una man muß entweder dem Stimmer
zugestehen, in Einen Theil der Tonarten etwas mehr Reinheit und
befriedigendere Intervalle zu legen, und es in dem andern um so vieles
fehlen zu lassen, oder sich gefallen lassen, daß bei vollkommen
gleicher Temperatur der Mangel, nur in kleinerem Maaße, allen Tonarten
gleich anklebt, indem dann in jeder derselben ein Theil der Intervalle
zu klein, der andere zu groß seyn wird. Dieser Mangel volkommen reiner
Intervalle ist jedoch zo wenig fuhlbar, daß nur ein sehr richtiges,
geübtes musikalisches Ohr ihn bemerkt, und sich vielreicht dadurch
gestört finden kann. Bei den Blas Instrumenten, deren Tonleiter durch
Finger-Applicatur sich bildet, ist die Stimmung noch wiet unter der
Richtigheit, die der Stimmung des Forte-Piano gegeben werden kann,
weswegen man sich durch besondere Klappen für einzelne Töne zu helfen
gesucht hat, wodurch das Uebel theilweise gehoben wird, am
allermeisten leiden aber an diesem Gebrechen die Harfen, Guitarren, u.
dgl.-

I leave it to someone else to provide a translation for non German
readers.

> > If he was referring to
> > anything at all, he probably just meant flexible just intonation, as
> > implied by Leopold Mozart, Tartinni, Rameau, Muller (1777 London), etc
> > etc
>
> - Do you have some reference for 'Muller 1777'? That's a new one on
> me. You could add Bremner (also 1777) 'Thoughts on playing concert
> music' -

You're right, it's Bremner, The Compleat Orchestral Musician, as a
preface to Geminiani sonatas, as I recall.

Ciao,

P

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
>
> .... where Valentini and Froberger
> probably exercised the emperor's stable of 19-note keyboards in the
> mid-17th century....
>
> --- In tuning@yahoogroups.com, "Paul Poletti" <paul@> wrote:
> >....Schiedmeyersays bemoans the fact that the piano can never
> > really be properly "in-tune" because, l
> > ike all keyboard instruments,
> > it has no separate keys for C#/Db, D#/Eb, E/E#/F, etc....

Hi Tom,
for 19th century views
try in the Lesesaal, Signature
LSA, Mus AF 036

Hermann Mendles 1878
"Musicalisches Conversation-Lexicon"
Vol 10, p. 132-138
Article 'Temperatur' on:

12: Marpurg, Chladni, Neidhardt, Werckmeister....
19: Opelt...
31: Galin...
53: Drobisch, Helmholtz...

bye
A.S.

--- In tuning@yahoogroups.com, "Paul Poletti" <paul@...> wrote:

> > Quote: Schiedmayer
> > ...could have been using the lack of enharmonics as
> > emblematic of the whole problem of tempering. We don't know how he
> > would react to a 19-note instrument in meantone.
>
> Actually, we do. He says it's too complicated to play an instrument
> with lots of keys to the octave.
>
> It's an interesting passage, so here it is in it's entirety:
>
> Die Temperatur der Tonarten soll möglichst gleich seyn.
The tempering of all keys should be as equal as possible.

> Hat der
> Stimmer nicht so viel richtiges Gehör, daß er diese möglichste
> Gleicheit auch ohne alle künstliche Regeln herauszubringen weiß, so
> ist er zu bedauern und er wird zuverlässig auch nach den besten
> Anleitungen zum Stimmen doch keine richtige Stimmung zuwege bringen.

If an tuner hasn't enough right sense of heraring,
so that he doesn't know to apply equalness without any artificial
rules, then have pity on him and
however he never will succeed in accomplishing an correct tuning,
even when obeying (strictly) the best tuning instructions.

> Da es übrigens wegen der allzugroßen Vermehrung der Tasten und der
> unüberwindlichen Schwierigkeit, ein solches instrument fertig zu
> spielen, nicht möglich ist, die enharmonische Tonleiter auf dem
> Claviere zu bilden, und dieses Instrument kein besonderes Ces, Cis,
> Des, Dis, Es, Eis, Fes, u.s.w. hat,
Playing enharmonic-instruments is insurmontable difficult to play.
In normal instruments lack special Cb, D#, Eb(sic!), E#, Fb, &.c.t.

> so kann auf demselben, so wie
> auf der orgel und allen andern Tonwerkzeugen, auf welchen die
> enhharmonische Tonleiter nicht geblidet werden kann, die Stimmung
> überhaupt nie vollkommen seyn,
hence on key-instruments without enarmonic-scales,
none tuning can be ideal perfect.

> und man muß entweder dem Stimmer
> zugestehen, in Einen Theil der Tonarten etwas mehr Reinheit und
> befriedigendere Intervalle zu legen,
Consequence:
On the one-hand;
we must allow for the tuner in particular keys
some more pureness and apply there more satifying intervals.

> und es in dem andern um so
> vieles fehlen zu lassen,
But on the other hand:
the more it has to lack there elsewhere in many (aspects)
viceversa.

> oder sich gefallen lassen, daß bei
> vollkommen
> gleicher Temperatur der Mangel, nur in kleinerem Maaße, allen
> Tonarten
> gleich anklebt,
or putting up by passing off the lesser defects of ET,
that adhere unifomly in all keys about the same amount.

> indem dann in jeder derselben ein Theil der Intervalle
> zu klein, der andere zu groß seyn wird.
then generally some (basic-)intervals turn out to be
consistently all to flat or sharp steady-going.

> Dieser Mangel volkommen reiner
> Intervalle ist jedoch zo wenig fuhlbar,
But that deficit of just pure intervals is marginal sensible,
(in his personal view)

> daß nur ein sehr richtiges,
> geübtes musikalisches Ohr ihn bemerkt,
only detectable for the right trained musically ear,
(does that mean, foist ET only on laypersons?)

> und sich vielleicht dadurch
> gestört finden kann.
that maybe disturbed by that (deviations from pure).

Comment:
ET sounds in my ears alike an somehow 'washed-out' Pythagorean.
That won't hold water against JI,
nor holds to a lesser extend the wax in my ears.

> Bei den Blas Instrumenten, deren Tonleiter durch
> Finger-Applicatur sich bildet, ist die Stimmung noch wiet unter der
> Richtigheit,
Wind-instruments turn out to be even worser out of tune,
due to theirs finger-applicature for gaining the (heptatonic) gamut,

> die der Stimmung des Forte-Piano gegeben werden kann,
than the tuning that can given to the piano-forte.

> weswegen man sich durch besondere Klappen für einzelne Töne zu
> helfen
> gesucht hat,
hence they (the wind-instrument builders) tryed to
add some further special throttles for single tones,
in the search for yielding more extra pitches.

> wodurch das Uebel theilweise gehoben wird,
but that resolves the defect only partially.

> am
> allermeisten leiden aber an diesem Gebrechen die Harfen,
> Guitarren, u. dgl.-
The most suffer on that illness are the (worse) afflicted,
harps, guitars &ct.
>
> I leave it to someone else to provide a translation for non German
> readers.
Also it's left up too
to more competent native-speaekers
to improve and complete
my humble attempt in trying that goal.

bye
A.S.

Sort of off topic. Is there a scala file that defines a tuning with “true” accidentials? I would guess in my ignorance that this would be a type of JI. I've read about the fact that g sharp doesn't equal a flat in reality. I'd like to experience it if possible. Thanks. Chris
Sent via BlackBerry from T-Mobile

--- In tuning@yahoogroups.com, chrisvaisvil@... wrote:
>
> Sort of off topic. Is there a scala file that defines a tuning
> with "true" accidentials? I would guess in my ignorance that this
> would be a type of JI. I've read about the fact that g sharp
> doesn't equal a flat in reality. I'd like to experience it if
> possible. Thanks. Chris

Hi Chris,

The bit about flats not equaling sharps isn't to do with JI.
It has to do with the fact that 12-ET came relatively late in
Western music. Before it there was meantone temperament, in
which notes are named according to a chain of fifths.

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

All sharps are to the right, and all flats are to the left.
Any adjacent pair of notes has to be spelled as a true 5th
(five diatonic letter names apart, inclusive).

In meantone, the typical fifth is about 697 cents. That means
that F# will be 4879 cents, or 4 octaves and 79 cents above C.
Db, on the other hand, will be 115 cents above a C three octaves
below where we started. That means Db is pitched above C# in
meantone. In 12-ET they are equal. In any linear scale where
the fifth is greater than 700 cents -- in Pythagorean intonation
for example -- Db will be BELOW C#.

Pythagorean was actually widely used on keyboard instruments
in the medieval period. Meantone hit the scene in the
15th century.

Contrary to what some just intonation proponents will tell
you, 5-limit JI was never widely used in Western music theory
or instrument tuning. Good ensembles perform in adaptive
5-limit JI, but our music notation, keyboard instruments,
guitars, etc. all suppose a linear scale. 5-limit JI is a
2-dimensional affair.

Questions?

-Carl

   Interesting, so, if I have this right...this basically says the "5th" in mean-tone, when multiplied from C to go from C,D,G...adds a different amount of error (vs. 12TET) so when it gets to C# vs. Db, it has different values. 
     In other words, C and Db have 4 notes in between them while C and C# have 6, thus making the resulting notes of C# and Db mathematically different in frequency. 

   On this note...I found 1.5^12 = 129.746 (mean tone) while 1.49831^12 = 128 exactly (12TET).  This would seem to imply an octave away up and down from the middle-C (C4 vs. C6) would not be exactly the same either in mean-tone while they would in 12TET.

-Michael

--- On Thu, 12/18/08, Carl Lumma <carl@...> wrote:

From: Carl Lumma <carl@...>
Subject: [tuning] common-practice accidentals [was Early 19th century microtonality]
To: tuning@yahoogroups.com
Date: Thursday, December 18, 2008, 10:29 AM

--- In tuning@yahoogroups. com, chrisvaisvil@ ... wrote:

>

> Sort of off topic. Is there a scala file that defines a tuning

> with "true" accidentials? I would guess in my ignorance that this

> would be a type of JI. I've read about the fact that g sharp

> doesn't equal a flat in reality. I'd like to experience it if

> possible. Thanks. Chris

Hi Chris,

The bit about flats not equaling sharps isn't to do with JI.

It has to do with the fact that 12-ET came relatively late in

Western music. Before it there was meantone temperament, in

which notes are named according to a chain of fifths.

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

All sharps are to the right, and all flats are to the left.

Any adjacent pair of notes has to be spelled as a true 5th

(five diatonic letter names apart, inclusive).

In meantone, the typical fifth is about 697 cents. That means

that F# will be 4879 cents, or 4 octaves and 79 cents above C.

Db, on the other hand, will be 115 cents above a C three octaves

below where we started. That means Db is pitched above C# in

meantone. In 12-ET they are equal. In any linear scale where

the fifth is greater than 700 cents -- in Pythagorean intonation

for example -- Db will be BELOW C#.

Pythagorean was actually widely used on keyboard instruments

in the medieval period. Meantone hit the scene in the

15th century.

Contrary to what some just intonation proponents will tell

you, 5-limit JI was never widely used in Western music theory

or instrument tuning. Good ensembles perform in adaptive

5-limit JI, but our music notation, keyboard instruments,

guitars, etc. all suppose a linear scale. 5-limit JI is a

2-dimensional affair.

Questions?

-Carl

--- In tuning@yahoogroups.com, djtrancendance@... wrote:
>
>    Interesting, so, if I have this right...this basically says
> the "5th" in mean-tone, when multiplied from C to go from
> C,D,G...adds a different amount of error (vs. 12TET) so when it
> gets to C# vs. Db, it has different values.

Yes. All 5ths will make C# and Db come out different EXCEPT
the 12-ET fifth.

>    On this note...I found 1.5^12 = 129.746 (mean tone)

1.5 isn't a meantone fifth, it's a pure 3:2 fifth.
Meantone is more like 1.4955.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> The bit about flats not equaling sharps isn't to do with JI.
> It has to do with the fact that 12-ET came relatively late in
> Western music. Before it there was meantone temperament, in
> which notes are named according to a chain of fifths.
>
> Db - Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C#

I beg to differ. I think it has everything to do with expanded
flexible JI, which I believe was THE model of intonation for the
majority of the history of western music ensemble performance, as long
as it did not include one of the "imperfect" instruments: keyboards or
fretted strings. That's what Rameau is referring to when he says the
singer will OF COURSE sing the intervals according to their true
proportions, which will require him to deviate from the keyboard.
That's what Roger North is griping about when he says that had the
making of music been left to the voice, which adjusts the interval to
the specifics of every occurrence, there would never have been any
need for all of that clap-trap of commas and limas and diesis and such
forth. That's why the whole lot of violin tutors from the 17th century
tell the player to practice PURE intervals, using Tartinni (resultant)
tones to check the purity. How does that jive with either 1/4 meantone
or the current fad of believing that 1/6 was so sort of default late
Baroque early Classical temperament? Flexible JI STILL is the way a
lot of music is made, including some amazing polyphony in the Corsican
and Sardinian traditions. That is what the function of the "tenor" was
i Medieval music, to "hold" the pitch and keep it from drifting as the
intervals are adjusted. I am not one of those who think that
Pythagorean with its "active thirds" was the blanket application for
Medieval music; I am far more convinced by the performances of
Ensemble Organum, and the idea that one uses a pure third sometimes
and a Pyth. ditone other times, depending on context and voice leading.
>
> All sharps are to the right, and all flats are to the left.
> Any adjacent pair of notes has to be spelled as a true 5th
> (five diatonic letter names apart, inclusive).

This is also true within a system of flexible JI, since sharps and
flats usually result as thirds either above or below an already
established natural. D# above B above G is lower than Eb below G. It
was nothing more than the general awareness and admission of the fact
the major thirds are narrower and minor thirds are wider than any sort
of third which appears in any system of fixed-pitch dodecatonic
division of the octave which satisfies more or less the requirement of
similarity of if not equality of all keys.
>
> Contrary to what some just intonation proponents will tell
> you, 5-limit JI was never widely used in Western music theory
> or instrument tuning.

What's your evidence for that? I know quite a few makers of
reproduction early wind instruments who would bite your head off for
making that statement. I also know quite a few professional "early
music" (mostly classical era) wind instrument players who swear that
flexible JI is "just what appens" when you make music with old
instruments.

> Good ensembles perform in adaptive
> 5-limit JI,

Exactly. Doesn't that contradict your statement?

> but our music notation, keyboard instruments,
> guitars, etc. all suppose a linear scale.

I would give you two out of three. Our music notation is no more than
a mere suggestion of how to realize ANY piece. There are so many
aspects which it cannot convey, and yet do not deny their existence:
ornamentation, rubato, subtle dynamics, timbre, subtle phrasings, the
exact way to use vibrato, portamento, etc etc etc. I fail to see why
we should use the fact that it cannot indicate subtleties of
intonation as proof that JI never played a major role in western music.

> 5-limit JI is a
> 2-dimensional affair.

Only if you insist on restricting it to the straight jacket of
fixed-pitch dodecatonic instruments, which in ancient times, were
called "imperfect". I wonder why? Applying this thinking to music
making in general is what I call "keyboard myopia", a mental disease
which afflicts those who think that Bach conceived of ALL of his music
one or another keyboard compromise (i.e. temperament).

Personally, I agree with Helmholtz: pure intonation is the natural
state of affairs for the uncorrupted ear.
>
> Questions?

Oh so many.

;-)

Ciao,

P

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:

> Pythagorean was actually widely used on keyboard instruments
> in the medieval period.
Historically there had been much more graduations inbetween
http://en.wikipedia.org/wiki/Pythagorean_tuning
http://www.medieval.org/emfaq/harmony/pyth.html
and...
> Meantone hit the scene in the
> 15th century.
http://en.wikipedia.org/wiki/Meantone_temperament
and
http://en.wikipedia.org/wiki/Just_intonation

>
> Contrary to what some just intonation proponents will tell
> you, 5-limit JI was never widely used in Western music theory
> or instrument tuning.
http://www.societymusictheory.org/mto/issues/mto.96.2.6/mto.96.2.6.walker.html
"Intonational Injustice: A Defense of Just Intonation in the
Performance of Renaissance Polyphony"

> Good ensembles perform in adaptive
> 5-limit JI,
The better ones even do prefer
http://en.wikipedia.org/wiki/53_equal_temperament
alike Jutta Stüber's 53-tone
string-quartett adaptions
http://www.orpheus-verlag.de/html/stuber_72.html
for
"Praetorius, Purcell, Tartini, Fasch, Bach, Danzi, Haydn, Beethoven,
Schubert, Glasunow, DvoEák, Sokolow, Ljadow, Puccini, Bizet, Barber"
&ct.
or for singing in 53 :
http://www.bodensee-musikversand.de/product_info.php?products_id=147318
"Es analysiert die gängige Chorliteratur von J. S. Bach bis Hugo Wolf
und legt mit Hilfe von drei leicht erlernbaren Intonationszeichen die
exakte Tonhöhe je-den Tones fest. Der Sänger weiß damit, wie er jeden
Ton zu nehmen hat, viel-leicht etwas tiefer oder höher. Und der
Chorleiter weiß, wo die schwierigen Stellen liegen und wie er sie zu
meistern hat."

'It analyses the usual choir-literature from J.S.Bach to Hugo Wolf,
and fixes the exact pitch with the help of 3 easily to learn
accidnetials. The singer knows by that how to take the tone,
perhaps slightly fattend or sharper. And the choir-master knows,
the position of the difficult positions and how to cope with them.'

http://www.bookfinder.com/author/jutta-stuber/

by the additional "Bosanquet-Helmholtz" signs:

'/' : about an 2^(1/53) comma higer
'\' : about an 2^(1/53) comma lower

http://x31eq.com/notakey.htm
example:
"
C/ D/
C#/ Eb/ F/ G/
C D E/ F#/ GA/ Bb/ C/
C# Eb F G A/ B/
C\ D\ E F# GA Bb C
C#\ Eb\ F\ G\ A B
E\ F#\ GA\ Bb\ C\
A\ B\
"

> but our music notation, keyboard instruments,
> guitars, etc. all suppose a linear scale.
http://www.guitarworld.com/forums/viewtopic.php?f=50&t=61053&start=15

> 5-limit JI is a
> 2-dimensional affair.
yep.

bye
A.S.

Paul wrote:

> > The bit about flats not equaling sharps isn't to do with JI.
> > It has to do with the fact that 12-ET came relatively late in
> > Western music. Before it there was meantone temperament, in
> > which notes are named according to a chain of fifths.
> >
> > Db - Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C#
>
> I beg to differ.

The archives are full of arguments about this. The idea that
meantone is the intellectual and practical heritage of Western
intonation won.

> I am not one of those who think that
> Pythagorean with its "active thirds" was the blanket application
> for Medieval music;

I'm not either, and I didn't say that.

> > All sharps are to the right, and all flats are to the left.
> > Any adjacent pair of notes has to be spelled as a true 5th
> > (five diatonic letter names apart, inclusive).
>
> This is also true within a system of flexible JI, since sharps and
> flats usually result as thirds either above or below an already
> established natural.

Maybe you'd care to notate the 5-limit lattice for us with a
linear notation? I observe the word "flexible" above, which
is potentially a magic word...

> D# above B above G is lower than Eb below G. It
> was nothing more than the general awareness and admission of
> the fact the major thirds are narrower and minor thirds are
> wider than any sort of third which appears in any system of
> fixed-pitch dodecatonic division of the octave which satisfies
> more or less the requirement of similarity of if not equality
> of all keys.

Huh?

> > Contrary to what some just intonation proponents will tell
> > you, 5-limit JI was never widely used in Western music theory
> > or instrument tuning.
>
> What's your evidence for that?

The evidence for linear intonation in Western music theory and
practice could not be more abundant. What's the evidence for
5-limit JI (outside of vague comments about ensemble
performance values)?

> I know quite a few makers of
> reproduction early wind instruments who would bite your head
> off for making that statement.

Yeah? Who are they?

> I also know quite a few professional "early
> music" (mostly classical era) wind instrument players who swear
> that flexible JI is "just what appens" when you make music with
> old instruments.

It's just what happens when you make ensemble music with modern
instruments, too. I addressed this already in my original post.

> > Good ensembles perform in adaptive
> > 5-limit JI,
>
> Exactly. Doesn't that contradict your statement?

No.

> > but our music notation, keyboard instruments,
> > guitars, etc. all suppose a linear scale.
>
> I would give you two out of three. Our music notation is no
> more than a mere suggestion of how to realize ANY piece.

It is much more than that. It is a language for conceiving
music. If you'd care to demonstrate JI renditions of Western
pieces post-1500 I'd like to hear them. Needless to say,
there's been quite a lot of activity in this dept. on this
list over the years, and such renditions start off difficult
and get more difficult as one approaches 1900. It's something
ensemble players intuitively can do, but it is equivocally NOT
something that has ever been widely understood theoretically.
Only in the past decade has adaptive tuning really been sussed
out, and most of this new understanding remains untested in
music. In any case, adaptive JI approximates meantone
intonation, not just intonation. The 'natural perfection' for
diatonic music is meantone, not just intonation.

> I fail to see why
> we should use the fact that it cannot indicate subtleties of
> intonation as proof that JI never played a major role in
> western music.

There's more to JI than vertical intonation. The diatonic
scale doesn't even exist in JI.

> > 5-limit JI is a 2-dimensional affair.
>
> Only if you insist on restricting it to the straight jacket of
> fixed-pitch dodecatonic instruments,

No. 5-limit JI is a rank 2 tuning system.

-Carl

I wrote:

> 1.5 isn't a meantone fifth, it's a pure 3:2 fifth.
> Meantone is more like 1.4955.

The quarter-comma meantone fifth is the 4th root of 5,
or ~ 1.4953487812212205 or ~ 696.578 cents.

-Carl

---I beg to differ. I think it has everything to do with expanded
---flexible JI, which I believe was THE model of intonation for the
---majority of the history of western music ensemble performance

If I have this straight, flexible JI includes
1) JI diatonic (7 tone) as the key of C
2) The exact same scale intervals/ratios, but starting with c#,d,d# as the root note....instead of c....making the "key of c#, key of d..."  I believe Carl described these as "modulations" in a previous e-mail rather than
"transpositions".

   If so, it amazes me just how much fighting goes on about scales that are SO similar.
    All the scales in 1) and 2) coincide very closely to both 12TET and mean-tone...only being a few cents away from each other with each interval (IE 2nd, 3rd, 5th) in the scale being only a tad more/less pure in each different version.
__________________________________
     It all seems to come back to "there are a crapload (excuse my sophisticated "Engrish" :-D) of scales who sole purpose appears to be to estimate "flexible JI" as best as possible. 

    So, you all please tell me...why all the fanaticism about mean-tone vs. 12TET vs. flexible JI vs. ANY scales under any type of TET (IE 31TET) designed to approximate JI diatonic...when they all seem to more or less equal flexible JI-diatonic in the end of the day, minus a few only slightly more sour
intervals?

  I'm not trying to say there are not differences...but rather that a lot of times people seem to exaggerate how different the above group of scales are.

-Michael

--- On Thu, 12/18/08, Paul Poletti <paul@...> wrote:

From: Paul Poletti <paul@...>
Subject: [tuning] Re: common-practice accidentals [was Early 19th century microtonality]
To: tuning@yahoogroups.com
Date: Thursday, December 18, 2008, 12:00 PM

--- In tuning@yahoogroups. com, "Carl Lumma" <carl@...> wrote:

>

> The bit about flats not equaling sharps isn't to do with JI.

> It has to do with the fact that 12-ET came relatively late in

> Western music. Before it there was meantone temperament, in

> which notes are named according to a chain of fifths.

>

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

I beg to differ. I think it has everything to do with expanded

flexible JI, which I believe was THE model of intonation for the

majority of the history of western music ensemble performance, as long

as it did not include one of the "imperfect" instruments: keyboards or

fretted strings. That's what Rameau is referring to when he says the

singer will OF COURSE sing the intervals according to their true

proportions, which will require him to deviate from the keyboard.

That's what Roger North is griping about when he says that had the

making of music been left to the voice, which adjusts the interval to

the specifics of every occurrence, there would never have been any

need for all of that clap-trap of commas and limas and diesis and such

forth. That's why the whole lot of violin tutors from the 17th century

tell the player to practice PURE intervals, using Tartinni (resultant)

tones to check the purity. How does that jive with either 1/4 meantone

or the current fad of believing that 1/6 was so sort of default late

Baroque early Classical temperament? Flexible JI STILL is the way a

lot of music is made, including some amazing polyphony in the Corsican

and Sardinian traditions. That is what the function of the "tenor" was

i Medieval music, to "hold" the pitch and keep it from drifting as the

intervals are adjusted. I am not one of those who think that

Pythagorean with its "active thirds" was the blanket application for

Medieval music; I am far more convinced by the performances of

Ensemble Organum, and the idea that one uses a pure third sometimes

and a Pyth. ditone other times, depending on context and voice leading.

>

> All sharps are to the right, and all flats are to the left.

> Any adjacent pair of notes has to be spelled as a true 5th

> (five diatonic letter names apart, inclusive).

This is also true within a system of flexible JI, since sharps and

flats usually result as thirds either above or below an already

established natural. D# above B above G is lower than Eb below G. It

was nothing more than the general awareness and admission of the fact

the major thirds are narrower and minor thirds are wider than any sort

of third which appears in any system of fixed-pitch dodecatonic

division of the octave which satisfies more or less the requirement of

similarity of if not equality of all keys.

>

> Contrary to what some just intonation proponents will tell

> you, 5-limit JI was never widely used in Western music theory

> or instrument tuning.

What's your evidence for that? I know quite a few makers of

reproduction early wind instruments who would bite your head off for

making that statement. I also know quite a few professional "early

music" (mostly classical era) wind instrument players who swear that

flexible JI is "just what appens" when you make music with old

instruments.

> Good ensembles perform in adaptive

> 5-limit JI,

Exactly. Doesn't that contradict your statement?

> but our music notation, keyboard instruments,

> guitars, etc. all suppose a linear scale.

I would give you two out of three. Our music notation is no more than

a mere suggestion of how to realize ANY piece. There are so many

aspects which it cannot convey, and yet do not deny their existence:

ornamentation, rubato, subtle dynamics, timbre, subtle phrasings, the

exact way to use vibrato, portamento, etc etc etc. I fail to see why

we should use the fact that it cannot indicate subtleties of

intonation as proof that JI never played a major role in western music.

> 5-limit JI is a

> 2-dimensional affair.

Only if you insist on restricting it to the straight jacket of

fixed-pitch dodecatonic instruments, which in ancient times, were

called "imperfect". I wonder why? Applying this thinking to music

making in general is what I call "keyboard myopia", a mental disease

which afflicts those who think that Bach conceived of ALL of his music

one or another keyboard compromise (i.e. temperament) .

Personally, I agree with Helmholtz: pure intonation is the natural

state of affairs for the uncorrupted ear.

>

> Questions?

Oh so many.

;-)

Ciao,

P

[ Attachment content not displayed ]

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     I meant, so far and how many fifths you have to circle around in the circle of 5ths before you get to the desired note in the circle of fifths (which is important since mean-tone is based on the circle of fifths).

--- On Thu, 12/18/08, Chris Vaisvil <chrisvaisvil@...> wrote:

From: Chris Vaisvil <chrisvaisvil@...>
Subject: Re: [tuning] common-practice accidentals [was Early 19th century microtonality]
To: tuning@yahoogroups.com
Date: Thursday, December 18, 2008, 1:40 PM

I think this
  In other words, C and Db have 4 notes in between them while C and C# have 6

is incorrect due to enharmonic spelling. Someone correct me if I'm wrong.

On Thu, Dec 18, 2008 at 1:53 PM, <djtrancendance@ yahoo.com> wrote:

   Interesting, so, if I have this right...this basically says the "5th" in mean-tone, when multiplied from C to go from C,D,G...adds a different amount of error (vs. 12TET) so when it gets to C# vs. Db, it has different values. 

     In other words, C and Db have 4 notes in between them while C and C# have 6, thus making the resulting notes of C# and Db mathematically different in frequency. 

   On this note...I found 1.5^12 = 129.746 (mean tone) while 1.49831^12 = 128 exactly (12TET).  This would seem to imply an octave away up and down from the middle-C (C4 vs. C6) would not be exactly the same either in mean-tone while they would in 12TET.

-Michael

--- On Thu, 12/18/08, Carl Lumma <carl@...> wrote:

From: Carl Lumma <carl@...>
Subject: [tuning] common-practice accidentals [was Early 19th century microtonality]
To: tuning@yahoogroups. com

Date: Thursday, December 18, 2008, 10:29 AM

--- In tuning@yahoogroups. com, chrisvaisvil@ ... wrote:

>

> Sort of off topic. Is there a scala file that defines a tuning

> with "true" accidentials? I would guess in my ignorance that this

> would be a type of JI. I've read about the fact that g sharp

> doesn't equal a flat in reality. I'd like to experience it if

> possible. Thanks. Chris

Hi Chris,

The bit about flats not equaling sharps isn't to do with JI.

It has to do with the fact that 12-ET came relatively late in

Western music. Before it there was meantone temperament, in

which notes are named according to a chain of fifths.

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

All sharps are to the right, and all flats are to the left.

Any adjacent pair of notes has to be spelled as a true 5th

(five diatonic letter names apart, inclusive).

In meantone, the typical fifth is about 697 cents. That means

that F# will be 4879 cents, or 4 octaves and 79 cents above C.

Db, on the other hand, will be 115 cents above a C three octaves

below where we started. That means Db is pitched above C# in

meantone. In 12-ET they are equal. In any linear scale where

the fifth is greater than 700 cents -- in Pythagorean intonation

for example -- Db will be BELOW C#.

Pythagorean was actually widely used on keyboard instruments

in the medieval period. Meantone hit the scene in the

15th century.

Contrary to what some just intonation proponents will tell

you, 5-limit JI was never widely used in Western music theory

or instrument tuning. Good ensembles perform in adaptive

5-limit JI, but our music notation, keyboard instruments,

guitars, etc. all suppose a linear scale. 5-limit JI is a

2-dimensional affair.

Questions?

-Carl

[ Attachment content not displayed ]

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:

> >
> > I beg to differ.
>
> The archives are full of arguments about this. The idea that
> meantone is the intellectual and practical heritage of Western
> intonation won.

I will be the first to admit that I have not cruised the archives to
read these arguments. However, I in all honesty, I have to say that in
the year or so that I have been participating here, I would have to
say that this forum seems to be populated to large extent by those who
enjoy constructing elegant complex intellectual constructs to deal
with the issue of intonation and rather short on those who actually do
it in real music making with acoustic instruments, especially early
instruments. that's the world I've been haning out in for the last 35
years, and I don't know anybody who would agree that meantone was
anything other than the default KEYBOARD tuning for about 200 years.
You would be very hard pressed indeed to find any practicing musician
specialized playing Baroque music who thinks that the excessively
narrow fifths of 1/4 meantone is any sort of thing would should adopt
in general. Pure thirds, yes, but these bad fifths, no.
>
> > I am not one of those who think that
> > Pythagorean with its "active thirds" was the blanket application
> > for Medieval music;
>
> I'm not either, and I didn't say that.

Granted, you didn't, but there are many who do.
>
> > > All sharps are to the right, and all flats are to the left.
> > > Any adjacent pair of notes has to be spelled as a true 5th
> > > (five diatonic letter names apart, inclusive).
> >
> > This is also true within a system of flexible JI, since sharps and
> > flats usually result as thirds either above or below an already
> > established natural.
>
> Maybe you'd care to notate the 5-limit lattice for us with a
> linear notation? I observe the word "flexible" above, which
> is potentially a magic word...

No, I wouldn't, precisely because what I am talking about is not any
sort of lattice or matrix or scale any sort of fixed system. This is
PRECISELY the point, what is meant by flexible JI. Yeah, I know all
the arguments, and all the pitfalls of linear intonation. But when you
hear Corsican ensembles realizing some fairly complex polyphony and
doing it dead pure and not drifting up or down, your ears are opened,
and hopefully your mind follows.
>
> > D# above B above G is lower than Eb below G. It
> > was nothing more than the general awareness and admission of
> > the fact the major thirds are narrower and minor thirds are
> > wider than any sort of third which appears in any system of
> > fixed-pitch dodecatonic division of the octave which satisfies
> > more or less the requirement of similarity of if not equality
> > of all keys.
>
> Huh?

Don't play dumb, you know what I mean. Sharps are generally lower than
flats because as you move around within a system of generally pure
intonation, any time you define a major third upward it is low and a
major third downward it's high. Now moving upward by a major third
from a natural usually implies a sharp, and moving downward by a major
third from a natural usually implies a flat. This is like intorduction
to Baroque music making 1A, regardless of what sort "lattice" you want
to corset your mind into.
>
> > > Contrary to what some just intonation proponents will tell
> > > you, 5-limit JI was never widely used in Western music theory
> > > or instrument tuning.
> >
> > What's your evidence for that?
>
> The evidence for linear intonation in Western music theory and
> practice could not be more abundant. What's the evidence for
> 5-limit JI (outside of vague comments about ensemble
> performance values)?

What IS the evidence for linear intonation? Do we have reocrdings of
ancinet music? I assume you mean that once a given note has been
established, it is then used as a reference point for other notes, and
one proceeds through the piece in a sort of leap-frog manner. Did you
every consider the option that notes once defined can be changed when
the harmonic environment changes, even whilst they continue to sound,
perhaps as a tied note - like Quantz tells us to do? Did you every
consider the idea that two notes can be changed simultaneously in
order to meet the harmonic context, like the Corsicans do?
>
> > I know quite a few makers of
> > reproduction early wind instruments who would bite your head
> > off for making that statement.
>
> Yeah? Who are they?

Alfredo Bernardini (maker + player), Robin Howell (maker + player),
Stanley King (maker + player), Peter de Konigh, Leslie Ross, Bob
Marvin, just to name a few who come immediately to mind. I used to
schlep my pianos to early music exhibitions all over Europe, and the
states when I lived there, and I always spent a lot of time talking to
the wind instrument makers about tuning and temperament. I would be
hard pressed to make any sort of concrete summary of all these
discussions, but I can tell you the idea that meantone is the norm
just ain't so. I'm saying this not to win points, just to let you know
that there is a huge world out there beyond the rather narrow
intellectual straight jacket of so much of what appears on this list
that doesn't jive with the conclusions reached here in the hopes that
it will help you to open your mind to other possibilities.

>
> > > Good ensembles perform in adaptive
> > > 5-limit JI,
> >
> > Exactly. Doesn't that contradict your statement?
>
> No.

Ah, ensemble playing has never been important in western music, then,
I guess... just casting about for some way to understand how it can be
the basis of god ensemble playing and yet not be an important part of
the history of performance practice in western music.

>
> It is much more than that. It is a language for conceiving
> music. If you'd care to demonstrate JI renditions of Western
> pieces post-1500 I'd like to hear them.

Aye, there's the rub indeed. I think the early music movement has
largely failed in this respect. Part of the problem is the lack of
good singers, I fear. Too much modern technique, no ears for pure
tuning, no interest. Only the wind players seem good at it.

But yeah, it's a very thorny issue indeed. Yet there persists
historical commentary, not only Rameau, but things like Burney
describing the Italian oboist who subtly changed the intonation of a
single sustained note according to the harmonic function. While I
don't have time to dig through all my notes and post all the quotes, I
think the sum total represent more of an indication of performance
practice that is worthy of a dismissal as "vague comments".
>
> There's more to JI than vertical intonation. The diatonic
> scale doesn't even exist in JI.

That's putting the cart before the horse. tur enough, but it doesn't
mean that a series of notes which we would identify as a diatonic
scale cannot be intoned by a series of relationships governed by JI,
flexible or fixed. Let's face it, intonation at this fine a level
really only becomes important in a harmonic context. Except maybe for
those working in the maqam traditions, whose hearing is far superior
to most western musicians, but we are not talking about that.
>
> > > 5-limit JI is a 2-dimensional affair.
> >
> > Only if you insist on restricting it to the straight jacket of
> > fixed-pitch dodecatonic instruments,
>
> No. 5-limit JI is a rank 2 tuning system.

Maybe we are talking about two different things. I am not talking
about any tuning SYSTEM, just how to make harmony in a free
environment. Just pick up the axes and play them in tune. Not always
easy, but if EVERYBODY in the ensemble has the same goal, and if they
have been practicing pure intonation their whole life long
(practicing, as Leopold and Tartinni and Bremner tell the student to
do - not necessarily always "putting into practice"),once you get used
to it, it's not as hard at the lattice weavers would have one believe.
Part of the problem is that our sense of intonation has been so
polluted by ET that it is hard for us to imagine how music sounded to
those who largely had never heard it. If purity is your point of
departure, you push and bend to get around without tain wrecks, but
you always tend back toward purity. I honestly don't know if we ever
can reconstruct that. One must experience it to believe it is
possible. Again, the best I can say is go to Corsica. Or go study
singing in the Byzantine tradition. Or get a recording of The Voice Squad.

Ciao,

p

--- In tuning@yahoogroups.com, djtrancendance@... wrote:

> If I have this straight, flexible JI includes

Paul hasn't told us what flexible JI is, but adaptive JI
can require hundreds of pitches, depending on the piece
of music.

>      It all seems to come back to "there are a crapload (excuse
> my sophisticated "Engrish" :-D) of scales who sole purpose
> appears to be to estimate "flexible JI" as best as possible.

Most of the scales that get analyzed around here do try to
provide exact or approximate JI harmonies. That goes for
temperaments like meantone. But temperaments like meantone
go further than that -- they simplify melodic relationships.
One can argue that this simplification of melodic
relationships (e.g. that a single pitch "D" can be used
in three triads -- GMaj, Dmin, and FMaj) constitutes a
unique musical effect that lends itself to the development
musical languages.

>     So, you all please tell me...why all the fanaticism about
> mean-tone vs. 12TET vs. flexible JI vs. ANY scales under any
> type of TET (IE 31TET) designed to approximate JI diatonic...
> when they all seem to more or less equal flexible JI-diatonic
> in the end of the day, minus a few only slightly more sour
> intervals?

Probably the fuss isn't warranted. :)

-Carl

Paul wrote:

> > Maybe you'd care to notate the 5-limit lattice for us with a
> > linear notation? I observe the word "flexible" above, which
> > is potentially a magic word...
>
> No, I wouldn't, precisely because what I am talking about is not
> any sort of lattice or matrix or scale any sort of fixed system.
> This is PRECISELY the point, what is meant by flexible JI. Yeah,
> I know all the arguments, and all the pitfalls of linear
> intonation.

Er, 5-limit JI is not "linear intonation".

> But when you hear Corsican ensembles realizing some fairly
> complex polyphony and doing it dead pure and not drifting up or
> down, your ears are opened, and hopefully your mind follows.

Indeed, but what these ensembles do is not trivial. Try
some elegant intellectual constructs sometime... they can open
the mind, and hopefully the ear follows. :P

> > > D# above B above G is lower than Eb below G. It
> > > was nothing more than the general awareness and admission of
> > > the fact the major thirds are narrower and minor thirds are
> > > wider than any sort of third which appears in any system of
> > > fixed-pitch dodecatonic division of the octave which satisfies
> > > more or less the requirement of similarity of if not equality
> > > of all keys.
> >
> > Huh?
>
> Don't play dumb, you know what I mean.

I don't. If you actually try to notate a piece like this I
think you'd find that what you're saying doesn't make sense...

> any time you define a major third upward it is low and a
> major third downward it's high.

Um, yes, but these commas are not the source of the accidentals
in our notation system. They will instead result in pitch
shift or drift (two distinct things in the lingo here) which
will be invisible in the notation.

> > The evidence for linear intonation in Western music theory and
> > practice could not be more abundant. What's the evidence for
> > 5-limit JI (outside of vague comments about ensemble
> > performance values)?
>
> What IS the evidence for linear intonation?

As mentioned, Western notation is a linear notation and
thus implies linear intonation, and all (with a handful of
exceptions in 500 years) fixed-pitch instruments are tuned
linearly. Theory books generally corroborate the linearity.
Where theorists have tried to explain Western music in terms
of JI, they inevitably get confused and make errors. That's
true from Zarlino to the present day. One exception to this
seems to be the whole Willaert school. Or so we suppose;
there's very little extant on it. But Vicentino, at least,
seemed to nail it. But then all is quiet until Groven or so,
in the 1930s and '40s. But anyway...

> Did you
> every consider the option that notes once defined can be changed
> when the harmonic environment changes, even whilst they continue
> to sound, perhaps as a tied note - like Quantz tells us to do?

You bet.

> > It is much more than that. It is a language for conceiving
> > music. If you'd care to demonstrate JI renditions of Western
> > pieces post-1500 I'd like to hear them.
>
> Aye, there's the rub indeed. I think the early music movement has
> largely failed in this respect. Part of the problem is the lack of
> good singers, I fear. Too much modern technique, no ears for pure
> tuning, no interest. Only the wind players seem good at it.

No, no, there's plenty of that, and I bet I have more recordings
than you do. :P I mean tried to do it where you know what you're
doing, i.e. could notate it.

> > There's more to JI than vertical intonation. The diatonic
> > scale doesn't even exist in JI.
>
//
> Let's face it, intonation at this fine a level really only
> becomes important in a harmonic context.

I don't agree. Again, if you try it, you'll run into
melodic artifacts some people don't like. Unless you follow
Vicentino's suggestion, and it's not clear he even understood
how clever his suggestion was.

> > Except maybe for those working in the maqam traditions,
> > whose hearing is far superior to most western musicians, but
> > we are not talking about that.

I don't buy the blanket superiority argument. They are
no doubt better at hearing maqamat. Anyway.

-Carl

Paul & all concerned,

It occurred to me that we may simply have a lingo or
point of view disconnect.

It seems to me that you (and probably many in the early
music community) take the diatonic / 5-limit basis of
music as a given. Then "JI" or "temperament" is just a
question of fine tuning it.

But if you're like me (and many others here), you do
NOT take this language as given. Then, you're off looking
for ways to explain and classify it (it is only one of
many valid systems). The result of this classification
is inevitably calling it a diatonic, 5-limit sort of
affair, with 81/80 in the kernel since about 1400 and
the diminished and augmented commas stirred between
1800-1900. "JI" isn't a question of tuning, it's a
question of fundamental construction.

I think my key point (made previously) is that fine
Western ensemble intonation (which is vertically JI but
horizontally all over the map) *approximates* meantone
intonation. This means that, as a question of the
fundamantal classification of Western music, it is a
meantone universe.

There was a lot of confusion on this list about this in
the early days. I started out saying "JI" to mean
ensemble intonation, just like I think you're doing.
But others took it to mean JI, that is, a system of
pitches completely interconnected by rational intervals.
Western ensemble intonation moves pitches all the heck
around in an effort to maintain melodic sensibility
(which winds up being: approximating a linear scale)
while keeping harmonies pure.

To resolve the confusion, we decided to always call the
thing that ensembles do "adaptive JI" rather than just
"JI".

Maybe this helps.

-Carl

2008/12/19 Carl Lumma <carl@...>:
> Paul wrote:

>> > 5-limit JI is a 2-dimensional affair.
>>
>> Only if you insist on restricting it to the straight jacket of
>> fixed-pitch dodecatonic instruments,
>
> No. 5-limit JI is a rank 2 tuning system.

Rank 3, unless "tuning system" has some special meaning.

Graham

2008/12/19 Carl Lumma <carl@...>:

> As mentioned, Western notation is a linear notation and
> thus implies linear intonation, and all (with a handful of
> exceptions in 500 years) fixed-pitch instruments are tuned
> linearly. Theory books generally corroborate the linearity.
> Where theorists have tried to explain Western music in terms
> of JI, they inevitably get confused and make errors. That's
> true from Zarlino to the present day. One exception to this
> seems to be the whole Willaert school. Or so we suppose;
> there's very little extant on it. But Vicentino, at least,
> seemed to nail it. But then all is quiet until Groven or so,
> in the 1930s and '40s. But anyway...

Vicentino certainly got confused and made errors.

This paper analyzes Willaert's just intonation:

http://www.societymusictheory.org/mto/issues/mto.04.10.1/mto.04.10.1.wibberley1_frames.html

Note that "syntonic tuning" is 5-limit just intonation. What's called
"solmization" and "hexachord theory" is essentially "xenharmonic
moving windows" or a fixed JI scale that you can move to different
scale degrees. This article rejects that interpretation although I'm
sure I saw one that explained the same piece in terms of hexachords.

"Willaert has therefore devised a means of notating comma inflection.
While it is true that the exact notation of Syntonic comma inflection
cannot be attained graphically, Willaert has realized that the
notation of Pythagorean comma inflection certainly can. . . . What
Willaert has done is to use a traditional singer's Pythagoreanism
against him by constructing a melodic matrix in such a way that its
purely Pythagorean execution will actually cause the resulting harmony
to be Ptolemaic."

Graham

--- In tuning@yahoogroups.com, "Graham Breed" <gbreed@...> wrote:
>
> 2008/12/19 Carl Lumma <carl@...>:
> > Paul wrote:
>
> >> > 5-limit JI is a 2-dimensional affair.
> >>
> >> Only if you insist on restricting it to the straight jacket of
> >> fixed-pitch dodecatonic instruments,
> >
> > No. 5-limit JI is a rank 2 tuning system.
>
> Rank 3, unless "tuning system" has some special meaning.
>
> Graham

Rank 3 if you count octaves as I mentioned elsewhere. -Carl

Dear Carl,
On Dec 19, 2008, at 2:22 AM, Carl Lumma wrote:
> > > There's more to JI than vertical intonation. The diatonic
> > > scale doesn't even exist in JI.
> >
> //
> > Let's face it, intonation at this fine a level really only
> > becomes important in a harmonic context.
>
> I don't agree. Again, if you try it, you'll run into
> melodic artifacts some people don't like. Unless you follow
> Vicentino's suggestion, and it's not clear he even understood
> how clever his suggestion was.

Could you perhaps summarise what Vicentino's suggestion was you are talking about here?

Thanks!

Best
Torsten

--
Torsten Anders
Interdisciplinary Centre for Computer Music Research
University of Plymouth
Office: +44-1752-586219
Private: +44-1752-558917
http://strasheela.sourceforge.net
http://www.torsten-anders.de

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> The quarter-comma meantone fifth is the 4th root of 5,
> or ~ 1.4953487812212205 or ~ 696.578 cents.

that's only valid for
http://en.wikipedia.org/wiki/Thomas_Young_(scientist)
's averaged "regular" 1/4-Syntonic-Comma.

Historically before Young,
the most theoreticans divided the SC into:
81/80 = (324/323)(323/322)(322/321)(321/320)

with 4 different seizes of 5ths:

1: (1 200 * ln(1.5 / (324 / 323))) / ln(2) = ~696.603423...Cents
2: (1 200 * ln(1.5 / (323 / 322))) / ln(2) = ~696.586829...Cents
3: (1 200 * ln(1.5 / (322 / 321))) / ln(2) = ~696.570132...Cents
4: (1 200 * ln(1.5 / (321 / 320))) / ln(2) = ~696.55333....Cents

alike Aron, Zarlino, Prätorius & Werckmeister did also.

Mark Lindley wrote an that gernrally:
New-Grove Ed.2001, Vol.16, 205+206, article 'Mean-tone'
and more en detail in his
'Stimmung und Temperatur'

Quest:
Does anybody here in that group know about an usage of

(1200 * ln(1.5 / ((81 / 80)^(1 / 4)))) / ln(2) = ~696.578428...

before Young?

bye
A.S.

> Could you perhaps summarise what Vicentino's suggestion was you are
> talking about here?
>
> Thanks!
>
> Best
> Torsten

He made two suggestions, actually. It's the 2nd I was
referring to:

http://en.wikipedia.org/wiki/Archicembalo

See also this excellent web page:
http://tonalsoft.com/monzo/vicentino/vicentino.aspx

-Carl

--- In tuning@yahoogroups.com, "Paul Poletti" <paul@...> wrote:
>[...]in all honesty, I have to say that in
> the year or so that I have been participating here, I would have
> to say that this forum seems to be populated to large extent by
> those who enjoy constructing elegant complex intellectual
> constructs to deal with the issue of intonation and rather
> short on those who actually do it in real music making with
> acoustic instruments, especially early instruments.
> That's the world I've been hanging out in for the last 35
> years...

Right, this is an obvious and practical split between two different
sets of practices, goals and expectations. It seems be at least
partly, perhaps mostly, a division on the lines of modern electronic/
digital instruments vs. traditional/acoustic instruments. It might
even show a division between "eye" and "ear" approaches to tuning
issues, because all this math and digitizing appears to be a very
different kind of activity from that of listening to and tweaking
one's intonation in real time performance on an acoustic instrument
without reference to a set of numbers in six decimals.

Curious that great minds like Newton and Mercator digitized
everything back in the 17th century but didn't produce any memorable
music, while a great mind like Bach just tweaked his tuning until it
worked for him and got on with composing.

Although I've learned a lot in my short time on this list, and will
continue to learn, from the lists of digitized temperaments, it's
also true that the guys who are tuning historical keyboards have a
lot to say that seems directly relevant to my own guitar-playing
habits and issues.

For me personally this split implies two possible paths: (not that I
can't have both, but...) (1) buy a virtual strobe tuner and go
digital, or (2) keep listening and tweaking, and learn to listen
better.

(Paul wrote)
"Yeah, I know all the arguments, and all the pitfalls of linear
intonation. But when you hear Corsican ensembles realizing some
fairly complex polyphony and doing it dead pure and not drifting up
or down, your ears are opened, and hopefully your mind follows."

I'm wondering how far I'm going to be able to develop my ears, now
that I'm finally aware of the challenge! (I've been listening to some
of the guitar-playing microtonalists lately just for listening
practice.) The digital approach might be a way of defining the
territory to some degree, but an approach through pure listening (and
thinking about listening) is another way of thinking, isn't it? I'm
interested in some comments here... ?

(Paul wrote)
">[...]I am not talking about any tuning SYSTEM, just how to make
harmony in a free environment. Just pick up the axes and play them in
tune. Not always easy, but if EVERYBODY in the ensemble has the same
goal, and if they have been practicing pure intonation their whole
life long ... it's not as hard at the lattice weavers would have one
believe.
Part of the problem is that our sense of intonation has been so
polluted by ET that it is hard for us to imagine how music sounded to
those who largely had never heard it. ... I honestly don't know if we
ever can reconstruct that. One must experience it to believe it is
possible. Again, the best I can say is go to Corsica. Or go study
singing in the Byzantine tradition. Or get a recording of The Voice
Squad."

(Hear, hear.)

Dear Carl,

Thanks for your reply.

On 19 Dec 2008, at 16:02, Carl Lumma wrote:
> > Could you perhaps summarise what Vicentino's suggestion was you are
> > talking about here?
> >
> > Thanks!
> >
> > Best
> > Torsten
>
> He made two suggestions, actually. It's the 2nd I was
> referring to:
>
> http://en.wikipedia.org/wiki/Archicembalo
>
> See also this excellent web page:
> http://tonalsoft.com/monzo/vicentino/vicentino.aspx
Ah, you are referring to his tuning of the Archicembalo with 31 pitches per octave (approximating?) 31 ET.

BTW: in case anyone is interested in "hands-on" experience with 31 ET, I recently whipped up an a tuning for the Tonal Plexus (yes, I meanwhile bought one :) which arranges 31 ET in a generalized keyboard fashion. Only later I found out that the keyboard layout of Adriaan Fokker's 31-tone organ is basically the same (only that his keyboard is not tilted as the TPX, and the sharp and flat keys are swapped). Anyway, the tuning file is available here

http://www.h-pi.com/tpxfiles/31ET-regions.tpx

If you do not have a Tonal Plexus, you can still play around with it using the software version which comes with the software TPXE

http://www.h-pi.com/TPXEsoftware.html

I plan to extend this tuning later to some "quasi adaptive tuning" by complementing the Meantone fifths with pure fifths (nice to have so many keys on the Tonal Plexus :).

Best
Torsten

--- In tuning@yahoogroups.com, Torsten Anders <torsten.anders@...> wrote:

> > He made two suggestions, actually. It's the 2nd I was
> > referring to:
> >
> > http://en.wikipedia.org/wiki/Archicembalo
> >
> > See also this excellent web page:
> > http://tonalsoft.com/monzo/vicentino/vicentino.aspx
>
> Ah, you are referring to his tuning of the Archicembalo with 31
> pitches per octave (approximating?) 31 ET.

Bzzz.

-Carl

On Dec 19, 2008, at 7:20 PM, Carl Lumma wrote:
> --- In tuning@yahoogroups.com, Torsten Anders <torsten.anders@...> > wrote:
> > > He made two suggestions, actually. It's the 2nd I was
> > > referring to:
> > >
> > > http://en.wikipedia.org/wiki/Archicembalo
> > >
> > > See also this excellent web page:
> > > http://tonalsoft.com/monzo/vicentino/vicentino.aspx
> >
> > Ah, you are referring to his tuning of the Archicembalo with 31
> > pitches per octave (approximating?) 31 ET.
>
> Bzzz.
>

?? Sorry, I am no native English speaker. Is that a confirmation?

Torsten

>
>
> -Carl
>
>
>

--- In tuning@yahoogroups.com, "Jack" <gvr.jack@...> wrote:

> Curious that great minds like Newton and Mercator digitized
> everything back in the 17th century but didn't produce any memorable
> music, while a great mind like Bach just tweaked his tuning until it
> worked for him and got on with composing.
>
http://www.amazon.de/Johann-Sebastian-Bach-Musician-Paperback/dp/product-description/0393322564
"Christoph Wolff cogently compares the German composer to English
scientist Isaac Newton. Both men "brought about fundamental changes
and established new principles" in their chosen fields, he argues;
both sought to reveal God's harmonious ordering of their world.....
.....Wolff devotes a great deal of space to examining how Bach was
viewed by his contemporaries, to whom, of course, the idea of a
musician as an artist--as opposed to a sort of scientist of sound
(there are valuable comparisons of Bach's achievement to that of his
contemporary, Isaac Newton)--was quite foreign."

www.bachnetwork.co.uk/ub2/wolff.pdf

Bach's cousin, a Werckmeister pupil:
http://www.answers.com/topic/johann-gottfried-walther
http://www.bach-cantatas.com/Lib/Walther-Johann-Gottfried.htm
The guy that digitized music with ruler and compass
http://en.wikipedia.org/wiki/File:Walther-Johann-Gottfried-01.jpg
defines the credo of the Bach-family tradition:

"Musica Poetica or musical composition is a mathematical science
through which an agreeable and correct harmony of the notes is brought
to paper in order that it might later be sung or played, thereby
appropriately moving the listeners to Godly devotion as well as to
please and delight both mind and soul…. It is so called because the
composer must not only understand language as does the poet in order
not to violate the meter of the text but because he also writes
poetry, namely a melody, thus deserving the title Melopoeta or
Melopoeus." (22)

http://www.bachnetwork.co.uk/ub2/tatlow.pdf

http://imslp.org/index.php?title=Category:Walther,_Johann_Gottfried&
http://web.auth.gr/cim08/cim08_abstracts/079_CIM08_abstracts.pdf
http://web.auth.gr/cim08/cim08_papers/Kikou-Karagiannis/Kikou-Karagiannis.pdf

A member of:
http://www.mizlersociety.org/
http://en.wikipedia.org/wiki/Lorenz_Christoph_Mizler
"In intellect and study he was a polymath, his interests encompassing
music, mathematics, philosophy, theology, law, and the natural
sciences in great detail."

http://www.naxosdirect.com/WALTHER-Organ-Works-Vol--2/title/8554317/
"The following autumn he went to Magdeburg and made a particularly
significant visit to Halberstadt, where he met Andreas Werckmeister,
one of the most distinguished names in German music at that time, an
organist and a noted writer of major works on music theory.
Werckmeister was very sympathetic to the young Walther, presented him
with a gift, and subsequently corresponded regularly and sent him
music including the keyboard works of Buxtehude. "

bye
A:S.

bye
A.S.

> > > > He made two suggestions, actually. It's the 2nd I was
> > > > referring to:
> > > >
> > > > http://en.wikipedia.org/wiki/Archicembalo
> > > >
> > > > See also this excellent web page:
> > > > http://tonalsoft.com/monzo/vicentino/vicentino.aspx
> > >
> > > Ah, you are referring to his tuning of the Archicembalo with 31
> > > pitches per octave (approximating?) 31 ET.
> >
> > Bzzz.
> >
>
> ?? Sorry, I am no native English speaker. Is that a confirmation?
>
> Torsten

Hi Torsten, sorry for being cryptic. It's a case of
onomatopoeia
http://en.wikipedia.org/wiki/Onomatopoeia
taken from the sound of a buzzer on TV game shows (I think)
when incorrect answers are given.

Anyway, the Archicembalo had 36 tones per octave.
As explained on Wikipedia, Vicentino's first suggestion
was to tune it in 31-ET with 5 spare notes. The second
suggestion was to tune the manuals each in meantone, a
quarter-comma apart.

You can hear the result of such a procedure on Monz's
website (link above).

-Carl

Dear Carl,

On Dec 19, 2008, at 10:10 PM, Carl Lumma wrote:
> > > > > He made two suggestions, actually. It's the 2nd I was
> > > > > referring to:
> > > > >
> > > > > http://en.wikipedia.org/wiki/Archicembalo
> > > > >
> > > > > See also this excellent web page:
> > > > > http://tonalsoft.com/monzo/vicentino/vicentino.aspx
> > > >
> > > > Ah, you are referring to his tuning of the Archicembalo with 31
> > > > pitches per octave (approximating?) 31 ET.
>
> Anyway, the Archicembalo had 36 tones per octave.
> As explained on Wikipedia, Vicentino's first suggestion
> was to tune it in 31-ET with 5 spare notes. The second
> suggestion was to tune the manuals each in meantone, a
> quarter-comma apart.
>
Ah. I just realised that the Tonal Plexus tuning I mentioned before is a superset of both these Vicentino tunings. For each pitch of 31 ET I have 2 variants +/- 5.19 cent which gives me meantone plus all 31 pure fifths/fourth. It also gives a virtually just minor third. It goes back to a suggestion of George Secor in a private mail from 12 May, of which a CC went to you. In total there are now 3 * 31 pitches (could be extended to 5 + 31 pitches).

I just did the Total Plexus tuning and I am trying it out right now. I will upload it later to the same page as the other one.

Best
Torsten

Dear Carl,

On Dec 19, 2008, at 10:10 PM, Carl Lumma wrote:
> > > > He made two suggestions, actually. It's the 2nd I was
> > > > > referring to:
> > > > >
> > > > > http://en.wikipedia.org/wiki/Archicembalo
> > > > >
> > > > > See also this excellent web page:
> > > > > http://tonalsoft.com/monzo/vicentino/vicentino.aspx
> > > >
> > > > Ah, you are referring to his tuning of the Archicembalo with 31
> > > > pitches per octave (approximating?) 31 ET.
> > >
> > > Bzzz.
> > >
> >
> > ?? Sorry, I am no native English speaker. Is that a confirmation?
> >
> > Torsten
>
> Hi Torsten, sorry for being cryptic. It's a case of
> onomatopoeia
> http://en.wikipedia.org/wiki/Onomatopoeia
> taken from the sound of a buzzer on TV game shows (I think)
> when incorrect answers are given.
>
> Anyway, the Archicembalo had 36 tones per octave.
> As explained on Wikipedia, Vicentino's first suggestion
> was to tune it in 31-ET with 5 spare notes. The second
> suggestion was to tune the manuals each in meantone, a
> quarter-comma apart.
>
> You can hear the result of such a procedure on Monz's
> website (link above).
>
I had another look at these examples. I understood, in a nutshell Vicentino's idea was that you get virtually JI but without any comma pumps if you intonate chord *roots* always in meantone (not JI), play many remaining tones in meantone as well, but correct pitches if necessary (e.g., fifth, minor third) by 1/4 comma compared with their meantone counterpart. And you where suggesting to Paul that with this approach musicians actually play just chords (which was the most important aspect for Paul I understood), but at the same time melodic artifacts are avoided because the music is essentially in meantone with only a 1/4 comma correction here and there, and these 5.2 cent do not introduce any melodic artifacts.

Did I get that right? Thank you!

Best
Torsten

> I had another look at these examples. I understood, in a nutshell
> Vicentino's idea was that you get virtually JI but without any
> comma pumps if you intonate chord *roots* always in meantone (not
> JI), play many remaining tones in meantone as well, but correct
> pitches if necessary (e.g., fifth, minor third) by 1/4 comma
> compared with their meantone counterpart. And you where
> suggesting to Paul that with this approach musicians actually
> play just chords (which was the most important aspect for Paul I
> understood), but at the same time melodic artifacts are avoided
> because the music is essentially in meantone with only a 1/4 comma
> correction here and there, and these 5.2 cents do not introduce
> any melodic artifacts.
>
> Did I get that right? Thank you!
>
> Best
> Torsten

That's correct. 1/4-syntonic comma is just below the melodic
JND of ~ 6 cents. This approach works for music until about
1800, when 128/125 and 648/625 start to be used. The same
approach can be applied then, by putting melodic intervals into
the augmented and diminished linear temperaments, respectively.
Unfortunately I think the max melodic shifts are then 13.7 and
15.6 cents, respectively. That last number is the same as the
max 5-limit error in 12-ET. So Hermode tuning can only be
improved by detecting in a score when 648/625 is not needed.
That however might be tricky, especially in realtime.

There is one possible drawback to any of these adaptive
tuning schemes: "magic chords" also called "supersaturated
suspensions". These are structures such as the dim7 chord in
12-ET which have no single analog in just intonation, and in
fact may be more consonant than any of their JI analogs.
1/1-4/3-7/4 in 22-ET is another example I mentioned recently
on the MMM list. It is composed of two perfect fourths in
22-ET, rather than 4/3 and the dissonant 21/16 as in JI.

-Carl

Carl, are you saying one cannot obtain 4 stacked pure minor 3rds in JI? Presumably because it belongs to too many scales simultaneously?
Sent via BlackBerry from T-Mobile

-----Original Message-----
From: "Carl Lumma" <carl@lumma.org>

Date: Sun, 21 Dec 2008 01:29:02
To: <tuning@yahoogroups.com>
Subject: [tuning] Re: common-practice accidentals [was Early 19th century microtonality]

> I had another look at these examples. I understood, in a nutshell
> Vicentino's idea was that you get virtually JI but without any
> comma pumps if you intonate chord *roots* always in meantone (not
> JI), play many remaining tones in meantone as well, but correct
> pitches if necessary (e.g., fifth, minor third) by 1/4 comma
> compared with their meantone counterpart. And you where
> suggesting to Paul that with this approach musicians actually
> play just chords (which was the most important aspect for Paul I
> understood), but at the same time melodic artifacts are avoided
> because the music is essentially in meantone with only a 1/4 comma
> correction here and there, and these 5.2 cents do not introduce
> any melodic artifacts.
>
> Did I get that right? Thank you!
>
> Best
> Torsten

That's correct. 1/4-syntonic comma is just below the melodic
JND of ~ 6 cents. This approach works for music until about
1800, when 128/125 and 648/625 start to be used. The same
approach can be applied then, by putting melodic intervals into
the augmented and diminished linear temperaments, respectively.
Unfortunately I think the max melodic shifts are then 13.7 and
15.6 cents, respectively. That last number is the same as the
max 5-limit error in 12-ET. So Hermode tuning can only be
improved by detecting in a score when 648/625 is not needed.
That however might be tricky, especially in realtime.

There is one possible drawback to any of these adaptive
tuning schemes: "magic chords" also called "supersaturated
suspensions". These are structures such as the dim7 chord in
12-ET which have no single analog in just intonation, and in
fact may be more consonant than any of their JI analogs.
1/1-4/3-7/4 in 22-ET is another example I mentioned recently
on the MMM list. It is composed of two perfect fourths in
22-ET, rather than 4/3 and the dissonant 21/16 as in JI.

-Carl

--- In tuning@yahoogroups.com, chrisvaisvil@... wrote:
>
> Carl, are you saying one cannot obtain 4 stacked pure minor
> 3rds in JI? Presumably because it belongs to too many scales
> simultaneously?

One can stack 6/5 minor thirds alright. But compare 216/125
to 900 cents (900 cents approximates 5/3). Then compare
1296/625 to 1200 cents... which is more consonant?

Same sort of thing happens when stacking 7/6. Not to say
these JI stacks don't sound good, but some people do find them
less consonant than the 12-ET dim7 chord.

Probably the most consonant dim7 JI tetrad is 10:12:15:17.
But it changes its sound quite a bit as it's inverted. In
12-ET, of course, there's a neat effect where you can
endlessly invert the dim7.

-Carl

2008/12/19 Andreas Sparschuh <a_sparschuh@...>:
> Quest:
> Does anybody here in that group know about an usage of
>
> (1200 * ln(1.5 / ((81 / 80)^(1 / 4)))) / ln(2) = ~696.578428...
>
> before Young?

I checked Dr. Robert's book, dated 1754, OCR-d here:

http://www.archive.org/details/harmonicsorphilo00smit

He certainly knows it isn't a rational number. Maybe there's some
mathematical notation that got garbled -- I didn't check the
facsimile. But I found the number 2.99069756 which implies a fifth of
696.5784269 cents.

Close enough for you?

Graham

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:

> There is one possible drawback to any of these adaptive
> tuning schemes: "magic chords" also called "supersaturated
> suspensions". These are structures such as the dim7 chord in
> 12-ET which have no single analog in just intonation, and in
> fact may be more consonant than any of their JI analogs.
> 1/1-4/3-7/4 in 22-ET is another example I mentioned recently
> on the MMM list. It is composed of two perfect fourths in
> 22-ET, rather than 4/3 and the dissonant 21/16 as in JI.

Hi Carl,

there are at least two ways to treat these chords in adaptive schemes:

1) let them be tempered, not necessarily in that tuning where the
tuning scheme is centered (for example 22-equal) as a more consonant
version may exist. This works when the chords are thought as not in
need of a resolution, for example some quartal harmonies.

2) as tuning these in JI will produce dissonant intervals and as these
kind of chords are often treated as in need of a resolution one could
choose a JI analog so that those dissonant intervals are resolved in
the next chord. This also removes the ambiguity from these chords.
This would not work in real-time systems though.

Kalle

Hmm, ironically I missed all of this exchange due to rehearsals and
(two) performances of Monteverdi Vespers over the last days.

Lots of 'justness' in evidence there, but how or whether it fitted
into any linear scheme is extremely debatable, except for the obvious
point that the only keyboard (a small chamber organ) probably had some
sort of meantone-like tuning.

It might be reasonable to talk about tweaked meantone as a good
approximation, except that large stretches of the music are
harmonically simple enough to not even have comma pumps. (As in the
closing section of Tallis Lamentations part 1, discussed by Duffin...)
In those parts, JI is a perfectly good standard, although as always at
the mercy of singers and instrumentalists.

I will give a passage from Smith (1749) with my emphasis...

p.225 Section X

Of occasional temperaments used in concerts well performed upon
perfect instruments.

By a perfect instrument I mean a voice, violin
or violoncello, &c, with which a good musician
can perfectly express any sound which his
ear requires.

PROPOSITION XXII.

The several Parts of a concert well performed upon perfect
instruments, do not move exactly by the given intervals of any one
system whatever, but only pretty nearly, and so as to make perfect
harmony as near as possible.

For instance, If the base be supposed to move
by the best system of perfect intervals (d), the
other part or parts cannot constantly move by it
too, without making some of the concords imperfect
by a comma (e), which would grievously
offend the musicians (f). Consequently if they
are pleased, those intervals are occasionally tem-

p.226
-pered by the upper part or parts, which therefore
do not move by the same intervals which
the base is supposed to move by.
Likewise if the base be supposed to move by
the system of mean tones and limmas (g), the
other part or parts cannot constantly do so too,
without making about two thirds of all the concords
imperfect by a quarter of a comma (h).

--->
But whenever concords are held out by good <---
musicians, they seem to me to be always perfect. <---
--->

And if so, the upper part or parts cannot
move by the system of mean tones, which the
base is supposed to move by. (...)

Coroll. 2. The proposition holds true though
one of the instruments be imperfect, as when
the thorough base is played upon an organ or
harpsichord : because the performers of the upper
parts are more attentive to make perfect harmony
with the base notes, than with the chords
to them. Consequently those parts do not move
by the tempered system of the thorough base.

Coroll. 3. Coeteris paribus the same piece of
music well performed upon perfect instruments,
is more agreeable than it would be if it were as
well performed upon imperfect ones, as an organ,
&c.
For nothing gives greater offence to the hearer,
though ignorant of the cause of it, than those rapid
rattling beats of high and loud sounds, which
make imperfect consonances with one another.
And yet a few slow beats, like the slow undulations
of a close shake now and then introduced,
are far from being disagreeable.

... End of excerpt.

It seems to me that 'Western' teachings, as recorded in most textbooks
for non-keyboard musicians, have remained hopelessly schizophrenic and
sloppy ever since the Renaissance started to put a premium on 5-limit
intervals. There was no very good and widely-known theory of tuning,
nevertheless musicians got through anyway, by sensitive adjustments,
to the point of producing 'good' tuning in places where the difference
is audible.

The notation is based on a diatonic scale and its transpositions,
which is fine if all you're doing is melody or Medieval harmonies with
a chain of pure fifths. But Renaissance theorists and many following
them were kind enough to tell their readers that there was a major
tone and a minor one, although notation doesn't give even an inkling
of the difference. Even in the English translation of Tosi (mid
18th-century) there is a nice little diagram of two diatonic scales,
with their different placements of 'tone major' and 'tone minor'
indicated. (All useless of course, as you can't learn to sing in tune
with anything else just by learning major and minor tones.)

And a lot of the monochords, diatonic or chromatic, described for (one
assumes) the benefit of singers and instrumentalists were laid out
precisely with 5-limit intervals, apparently without the least thought
of commas or temperament.

So, as a matter of historical record, for some hundreds of years, the
diatonic scale *was* modelled via a small collection or selection of
JI intervals. In this sense JI was a theoretical reality. You may
object that it was not a very good or satisfactory theory, but it is
the theory that was taught.
Except for the occasional comma-pump-free passages, it cannot have
been a practical reality. But JI, though a bad theory in theory,
turned out to be a good theory in practice, from the point of view of
musicians who used aural knowledge of just intervals to adjust their
tuning without much caring if the result made any exact ratio to the
previous note. Whether this adjustment which tempered the theory's
just sets of pitches to the reality of musical performance was
something like a linear temperament, or rather less orderly and
controlled, is not something we can know.

From the modern point of view, of course this is 'adaptive JI' with
the caveat that the 'adaptation' may be a lot more variable than we
think. For the non-keyboard performer, it is simply using pure
intervals to the best of one's ability, adjusting where necessary.
~~~T~~~

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Paul & all concerned,
>
> It occurred to me that we may simply have a lingo or
> point of view disconnect.
>
> It seems to me that you (and probably many in the early
> music community) take the diatonic / 5-limit basis of
> music as a given. Then "JI" or "temperament" is just a
> question of fine tuning it.
>
> But if you're like me (and many others here), you do
> NOT take this language as given. Then, you're off looking
> for ways to explain and classify it (it is only one of
> many valid systems). The result of this classification
> is inevitably calling it a diatonic, 5-limit sort of
> affair, with 81/80 in the kernel since about 1400 and
> the diminished and augmented commas stirred between
> 1800-1900. "JI" isn't a question of tuning, it's a
> question of fundamental construction.
>
> I think my key point (made previously) is that fine
> Western ensemble intonation (which is vertically JI but
> horizontally all over the map) *approximates* meantone
> intonation. This means that, as a question of the
> fundamantal classification of Western music, it is a
> meantone universe.
>
> There was a lot of confusion on this list about this in
> the early days. I started out saying "JI" to mean
> ensemble intonation, just like I think you're doing.
> But others took it to mean JI, that is, a system of
> pitches completely interconnected by rational intervals.
> Western ensemble intonation moves pitches all the heck
> around in an effort to maintain melodic sensibility
> (which winds up being: approximating a linear scale)
> while keeping harmonies pure.
>
> To resolve the confusion, we decided to always call the
> thing that ensembles do "adaptive JI" rather than just
> "JI".
>
> Maybe this helps.
>
> -Carl

--- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@...> wrote:

> Hi Carl,
> there are at least two ways to treat these chords in adaptive
> schemes:
> 1) let them be tempered, not necessarily in that tuning where
> the tuning scheme is centered (for example 22-equal) as a more
> consonant version may exist. This works when the chords are
> thought as not in need of a resolution, for example some quartal
> harmonies.
> 2) as tuning these in JI will produce dissonant intervals and
> as these kind of chords are often treated as in need of a
> resolution one could choose a JI analog so that those dissonant
> intervals are resolved in the next chord. This also removes the
> ambiguity from these chords. This would not work in real-time
> systems though.

There's another point I keep forgetting to mention, which I'll
bring up tangentially now: A lion's share of the sense of
melodic pitch height seems to come from the very early phase of
musical sounds (the "attack"), whereas the sense of harmony seems
to come from all phases, or even moreso from later phases
(the "sustain"). Thus, an adaptive tuning scheme can minimize
the perception of melodic shifts by attacking notes at the
pitches desired for them melodically, and then bending them pure.
This is no doubt the general pattern followed by musicians, who
'aim' melodically but then refine their intonation with feedback
as notes are sounding. Manuel once cited a study of this here.
But to my knowledge, it has never been employed in an adaptive
tuning algorithm.

-Carl

A little bit OT:
As far as I know, similar adaptive principle was used in some Guitar To MIDI converters (Yamaha, Blue Axon especially, maybe other branches as well). They try to analyze incoming pitch roughly in attack phase, generate the nearest MIDI notes and then after few cycles when pitch can be exactly determined glide very quickly to the right frequency by Pitch Bend. This is possible thanks to hexaphonic divided pickup and sending MIDI data on six adjacent channels. Using this small trick a delay between start of incoming note to generating MIDI note can be less then 30 msec even on low strings. Despite my using of Yamaha G10 guitar controller plus G10C converter (which was quite special revolutionary concept, unfortunately not successful on the market), and G50 converter with pickup on one of my six string custom made bassguitars I'm not specialist in this, I just have written some reviews for music magazines years ago. I think some material can be found on internet if somebody is interested.

Daniel Forro

On 22 Dec 2008, at 10:59 AM, Carl Lumma wrote:

> --- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@...> wrote:
>
> > Hi Carl,
> > there are at least two ways to treat these chords in adaptive
> > schemes:
> > 1) let them be tempered, not necessarily in that tuning where
> > the tuning scheme is centered (for example 22-equal) as a more
> > consonant version may exist. This works when the chords are
> > thought as not in need of a resolution, for example some quartal
> > harmonies.
> > 2) as tuning these in JI will produce dissonant intervals and
> > as these kind of chords are often treated as in need of a
> > resolution one could choose a JI analog so that those dissonant
> > intervals are resolved in the next chord. This also removes the
> > ambiguity from these chords. This would not work in real-time
> > systems though.
>
> There's another point I keep forgetting to mention, which I'll
> bring up tangentially now: A lion's share of the sense of
> melodic pitch height seems to come from the very early phase of
> musical sounds (the "attack"), whereas the sense of harmony seems
> to come from all phases, or even moreso from later phases
> (the "sustain"). Thus, an adaptive tuning scheme can minimize
> the perception of melodic shifts by attacking notes at the
> pitches desired for them melodically, and then bending them pure.
> This is no doubt the general pattern followed by musicians, who
> 'aim' melodically but then refine their intonation with feedback
> as notes are sounding. Manuel once cited a study of this here.
> But to my knowledge, it has never been employed in an adaptive
> tuning algorithm.
>
> -Carl
>
>
>

--- In tuning@yahoogroups.com, "Graham Breed" <gbreed@...> wrote:
>
> I checked Dr. Robert's book, dated 1754, OCR-d here:
>
> http://www.archive.org/details/harmonicsorphilo00smit
>
> He certainly knows it isn't a rational number. Maybe there's some
> mathematical notation that got garbled -- I didn't check the
> facsimile. But I found the number 2.99069756 which implies a fifth of
> 696.5784269 cents.
>
> Close enough for you?
in deed,
rechecking Smith's calculation delivers his accurate result,
of the geometric mean precisely:

1200Cents * ln(~2.99069756... / 2) / ln(2) = ~696.578427...Cents

But personally i do prefer the older arithmetically mean of the SC
81:80 = 324:320 = 324:323:322:321:320
because the calculation in that subdivision remains fully rational.

many thanks for yours advise
A.S.

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@> wrote:
>
> > Hi Carl,
> > there are at least two ways to treat these chords in adaptive
> > schemes:
> > 1) let them be tempered, not necessarily in that tuning where
> > the tuning scheme is centered (for example 22-equal) as a more
> > consonant version may exist. This works when the chords are
> > thought as not in need of a resolution, for example some quartal
> > harmonies.
> > 2) as tuning these in JI will produce dissonant intervals and
> > as these kind of chords are often treated as in need of a
> > resolution one could choose a JI analog so that those dissonant
> > intervals are resolved in the next chord. This also removes the
> > ambiguity from these chords. This would not work in real-time
> > systems though.
>
> There's another point I keep forgetting to mention, which I'll
> bring up tangentially now: A lion's share of the sense of
> melodic pitch height seems to come from the very early phase of
> musical sounds (the "attack"), whereas the sense of harmony seems
> to come from all phases, or even moreso from later phases
> (the "sustain"). Thus, an adaptive tuning scheme can minimize
> the perception of melodic shifts by attacking notes at the
> pitches desired for them melodically, and then bending them pure.
> This is no doubt the general pattern followed by musicians, who
> 'aim' melodically but then refine their intonation with feedback
> as notes are sounding. Manuel once cited a study of this here.
> But to my knowledge, it has never been employed in an adaptive
> tuning algorithm.

Hi Carl,

I wonder what determines where the melodically desired pitches are?

--- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@...> wrote:
>
> > There's another point I keep forgetting to mention, which I'll
> > bring up tangentially now: A lion's share of the sense of
> > melodic pitch height seems to come from the very early phase of
> > musical sounds (the "attack"), whereas the sense of harmony seems
> > to come from all phases, or even moreso from later phases
> > (the "sustain"). Thus, an adaptive tuning scheme can minimize
> > the perception of melodic shifts by attacking notes at the
> > pitches desired for them melodically, and then bending them pure.
> > This is no doubt the general pattern followed by musicians, who
> > 'aim' melodically but then refine their intonation with feedback
> > as notes are sounding. Manuel once cited a study of this here.
> > But to my knowledge, it has never been employed in an adaptive
> > tuning algorithm.
>
> Hi Carl,
>
> I wonder what determines where the melodically desired pitches are?

One presumably has some way of doing that. :):)

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@> wrote:
> >
> > > There's another point I keep forgetting to mention, which I'll
> > > bring up tangentially now: A lion's share of the sense of
> > > melodic pitch height seems to come from the very early phase of
> > > musical sounds (the "attack"), whereas the sense of harmony seems
> > > to come from all phases, or even moreso from later phases
> > > (the "sustain"). Thus, an adaptive tuning scheme can minimize
> > > the perception of melodic shifts by attacking notes at the
> > > pitches desired for them melodically, and then bending them pure.
> > > This is no doubt the general pattern followed by musicians, who
> > > 'aim' melodically but then refine their intonation with feedback
> > > as notes are sounding. Manuel once cited a study of this here.
> > > But to my knowledge, it has never been employed in an adaptive
> > > tuning algorithm.
> >
> > Hi Carl,
> >
> > I wonder what determines where the melodically desired pitches are?
>
> One presumably has some way of doing that. :):)

Yes, but why in a given situation is one pitch preferred melodically
for a certain note rather than another?

I suspect that in addition to cultural preferences this is mostly or
completely determined by the other versions of that note heard before
while listening to a piece of music. In addition to harmonic purity
and small local melodic shifts in the already playing notes the
listener desires that there is as little global variance in the pitch
of a given note as possible.

If this is so there is little reason to choose this or that tuning as
the melodic center in non-real time adaptive schemes! The center
tuning could be one of the things calculated for the piece. I don't
know if such non-centered adaptive schemes have been proposed before.

The simplest (at least definitionally if not computationally) adaptive
scheme I can think of is one where in addition to choosing the tunings
of the vertical simultaneities (which would mostly be justly intoned)
the tuner would have to specify which pitches are the same note.
Merely octave equivalent notes would be treated as different notes.
Then the computer program would search for a solution where the pitch
variations of notes are minimized. One could for example minimize the
maximal pitch difference occurring in the piece which is simple to
define but hard to calculate. Or one could
sum the squares of the distances of the notes from their average
pitches and minimize that (which is in principle solvable by linear
algebra) and so on.

This basic scheme could be supplemented with other things like
minimizing the melodic shifts in sustaining notes.

There are potential problems though: in some pieces of music there
might be outliers, that is pitches that are not harmonically connected
to other pitches. Their tuning would have to be somehow determined too.

And a piece might for example begin with a single melodic line with no
harmony. But there is an easy solution to this: use the melodic center
the program calculates from the vertical simultaneities.

Kalle

---Yes, but why in a given situation is one pitch preferred melodically
---for a certain note rather than another?
I never quite got this either.
    So, in theory (as I understand it), the mind wants to hear pure intervals in the sustained harmony (which I already figured it did)...but then different ones in the attack/melody? 
  This is all new to me...I did not even know that, melodically, the mind prefers something other than pure intervals for anything at all. 

     >>>So then..what schema/intervals does the mind prefer melodically?<<<  Does it by chance have anything to do with which note is played first in a series of quickly occurring notes (IE all notes coming shortly after must be in "tone" with the first
note)?

   And, from what I understand of how a computer would "morph"/slide between ideal melodic and harmonic adaptive tunings...the best way is to find the shortest path between the two schema (and hope there are no points between the schema which simply don't match) and follow that path. 

   But I still have doubts...how much grinding the brain has to do to hear a slide/warp between schema. 
     I remember one of Sethare's old experiments on morphing between simple individual sounds (with many overtones).  He ran into huge problems with fading/sliding two tones in a linear fashion (the slides sounded grossly robotic and grating) so he decided to change the amplitude as he slid so the consonance between notes slid between the two schema smoothly during the side.   He also used cross fading in the case a point on one schema did not match with another.  Hopefully those
concepts could spawn some new ideas so far as adaptive JI is concerned.

-Michael

--- On Thu, 1/8/09, Kalle Aho <kalleaho@...sinki.fi> wrote:

From: Kalle Aho <kalleaho@...>
Subject: [tuning] Melodic Preferences In Adaptive Schemes (was: common-practice accidentals)
To: tuning@yahoogroups.com
Date: Thursday, January 8, 2009, 2:20 AM

--- In tuning@yahoogroups. com, "Carl Lumma" <carl@...> wrote:

>

> --- In tuning@yahoogroups. com, "Kalle Aho" <kalleaho@> wrote:

> >

> > > There's another point I keep forgetting to mention, which I'll

> > > bring up tangentially now: A lion's share of the sense of

> > > melodic pitch height seems to come from the very early phase of

> > > musical sounds (the "attack"), whereas the sense of harmony seems

> > > to come from all phases, or even moreso from later phases

> > > (the "sustain"). Thus, an adaptive tuning scheme can minimize

> > > the perception of melodic shifts by attacking notes at the

> > > pitches desired for them melodically, and then bending them pure.

> > > This is no doubt the general pattern followed by musicians, who

> > > 'aim' melodically but then refine their intonation with feedback

> > > as notes are sounding. Manuel once cited a study of this here.

> > > But to my knowledge, it has never been employed in an adaptive

> > > tuning algorithm.

> >

> > Hi Carl,

> >

> > I wonder what determines where the melodically desired pitches are?

>

> One presumably has some way of doing that. :):)

Yes, but why in a given situation is one pitch preferred melodically

for a certain note rather than another?

I suspect that in addition to cultural preferences this is mostly or

completely determined by the other versions of that note heard before

while listening to a piece of music. In addition to harmonic purity

and small local melodic shifts in the already playing notes the

listener desires that there is as little global variance in the pitch

of a given note as possible.

If this is so there is little reason to choose this or that tuning as

the melodic center in non-real time adaptive schemes! The center

tuning could be one of the things calculated for the piece. I don't

know if such non-centered adaptive schemes have been proposed before.

The simplest (at least definitionally if not computationally) adaptive

scheme I can think of is one where in addition to choosing the tunings

of the vertical simultaneities (which would mostly be justly intoned)

the tuner would have to specify which pitches are the same note.

Merely octave equivalent notes would be treated as different notes.

Then the computer program would search for a solution where the pitch

variations of notes are minimized. One could for example minimize the

maximal pitch difference occurring in the piece which is simple to

define but hard to calculate. Or one could

sum the squares of the distances of the notes from their average

pitches and minimize that (which is in principle solvable by linear

algebra) and so on.

This basic scheme could be supplemented with other things like

minimizing the melodic shifts in sustaining notes.

There are potential problems though: in some pieces of music there

might be outliers, that is pitches that are not harmonically connected

to other pitches. Their tuning would have to be somehow determined too.

And a piece might for example begin with a single melodic line with no

harmony. But there is an easy solution to this: use the melodic center

the program calculates from the vertical simultaneities.

Kalle

Of relevance here I think is an scientific american article that argued that some thoughts, such as a popular song or idealolgy or joke, was similar to a computer virus. This it said explain why that song you hate gets stuck in your head. Also let me point out that world music demonstrates that consonnance and dissonance is in part cultural. Not all melodic intervals in all cultures are universally consonant. My point is a catchy melody "fits" the brain and is not necessarily rational - it just needs to be viral. Excuse my black berry typing. Chris
Sent via BlackBerry from T-Mobile

--- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@...> wrote:

> why in a given situation is one pitch preferred melodically
> for a certain note rather than another?
>
> I suspect that in addition to cultural preferences this is mostly
> or completely determined by the other versions of that note heard
> before while listening to a piece of music. In addition to
> harmonic purity and small local melodic shifts in the already
> playing notes the listener desires that there is as little global
> variance in the pitch of a given note as possible.

It's notoriously hard to pin down. I'm not sure this last
is a general principle. I outline what I think are two
general principles here:

/tuning/topicId_79656.html#79659

There are also Rothenberg's propriety and efficiency to
consider...

> Then the computer program would search for a solution where the
> pitch variations of notes are minimized. One could for example
> minimize the maximal pitch difference occurring in the piece
> which is simple to define but hard to calculate. Or one could
> sum the squares of the distances of the notes from their average
> pitches and minimize that (which is in principle solvable by
> linear algebra) and so on.

Not sure exactly what you mean, but John deLaubenfels' system
first calculated an optimum well temperament for a piece, and
then minimized the sum of the notes' squared deviations from it
over the course of the adaptive tuning.

My own adaptive tuning scheme (as yet unimplemented) tries to
eliminate short term shift but allows long-term drift.

> There are potential problems though: in some pieces of music
> there might be outliers, that is pitches that are not
> harmonically connected to other pitches. Their tuning would
> have to be somehow determined too.

My scheme is designed around 12-ET being the input, and for
most configurations, it'll have several output intervals
for each 12-ET interval, so isolated pitches are not possible.
It knows about chords and uses a greedy algorithm to extract
as many chords as it can, largest first. Then it connects
them with dyads.

> And a piece might for example begin with a single melodic line
> with no harmony. But there is an easy solution to this: use the
> melodic center the program calculates from the vertical
> simultaneities.

Again not sure what you mean, but would certainly be interested
to read more about your idea.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@> wrote:
>
> > why in a given situation is one pitch preferred melodically
> > for a certain note rather than another?
> >
> > I suspect that in addition to cultural preferences this is mostly
> > or completely determined by the other versions of that note heard
> > before while listening to a piece of music. In addition to
> > harmonic purity and small local melodic shifts in the already
> > playing notes the listener desires that there is as little global
> > variance in the pitch of a given note as possible.
>
> It's notoriously hard to pin down. I'm not sure this last
> is a general principle. I outline what I think are two
> general principles here:
>
> /tuning/topicId_79656.html#79659
>
> There are also Rothenberg's propriety and efficiency to
> consider...

I don't this answers my question which was not about scale structure
at all. Let me try again, you wrote:

> Thus, an adaptive tuning scheme can minimize
> the perception of melodic shifts by attacking notes at the
> pitches desired for them melodically, and then bending them pure.

Why should musicians attack a certain note at this exact pitch rather
than another pitch?

> > Then the computer program would search for a solution where the
> > pitch variations of notes are minimized. One could for example
> > minimize the maximal pitch difference occurring in the piece
> > which is simple to define but hard to calculate. Or one could
> > sum the squares of the distances of the notes from their average
> > pitches and minimize that (which is in principle solvable by
> > linear algebra) and so on.
>
> Not sure exactly what you mean, but John deLaubenfels' system
> first calculated an optimum well temperament for a piece, and
> then minimized the sum of the notes' squared deviations from it
> over the course of the adaptive tuning.

This is similar to my proposal but mine is more flexible as the
temperament also is determined from the piece. If the piece doesn't
require the vanishing of any commas and the simultaneities are JI then
the result will be pure JI also melodically.

> My own adaptive tuning scheme (as yet unimplemented) tries to
> eliminate short term shift but allows long-term drift.

If you just eliminate (or minimize) short-term shifts and don't care
about long-term drift, don't you just end up with comma pumps?

> > There are potential problems though: in some pieces of music
> > there might be outliers, that is pitches that are not
> > harmonically connected to other pitches. Their tuning would
> > have to be somehow determined too.
>
> My scheme is designed around 12-ET being the input, and for
> most configurations, it'll have several output intervals
> for each 12-ET interval, so isolated pitches are not possible.
> It knows about chords and uses a greedy algorithm to extract
> as many chords as it can, largest first. Then it connects
> them with dyads.

In my scheme the input is the desired tunings for the vertical
simultaneities with the specification which notes (types, not tokens)
are the same across all simultaneities, for example which ones are
supposed to be c1:s, f#1:s, e2:s and so on.

> > And a piece might for example begin with a single melodic line
> > with no harmony. But there is an easy solution to this: use the
> > melodic center the program calculates from the vertical
> > simultaneities.
>
> Again not sure what you mean, but would certainly be interested
> to read more about your idea.

My scheme requires that all notes (again types, not tokens) are
harmonically connected to at least one other note so that we get a
connected web of notes. Here is an example (view with fixed width font)

G11 A11 A12 B11 B12 B13 C2
E11 F11 F12 G12 G13 G13 G14
C11 C12 D11 D12 E12 (F1) E13
C01 F0 D0 G01 G02 G02 C02

The columns are the simultaneities.
The first number in the note name is the octave register, and the
second one is the number of the note token, that is a particular pitch
of a note. The desired tunings for the simultaneities in this example
are standard 5-limit JI ones except for the lonely F1 in brackets
which is not harmonically connected to the other notes in its
simultaneity. It is simply calculated from the average of F11 and F12.
Here are the relevant pitches:

C01 C02
C11 C12
E11 E12 E13
G11 G12 G13 G14
F11 F12
A11 A12
D11 D12
G01 G02
B11 B12 B13

some are left out from the calculation because they can be directly
determined from the result, for example C2 is just C02 + 4800 cents.

The program might for example search through different pitch choices
for the bass notes:

C01 F0 D0 G01 G02 G02 C02

Now find a sort of minimax tuning: calculate the pitch range (the
difference for the lowest and highest tokens of a note) for all notes.
Minimize the maximum pitch range. That's it!

I hope it is clearer now!

Kalle

--- In tuning@yahoogroups.com, djtrancendance@... wrote:
>     So, in theory (as I understand it), the mind wants to hear
> pure intervals in the sustained harmony (which I already
> figured it did)...but then different ones in the attack/melody?

That's right. The idea is, the ear is better at hearing
the quality of intervals the longer it listens. But it
prefers to get the pitch height right away. When you hear
a sound in your environment, you label it right away, and
then monitor it for its status going forward. If the pitch
changes slightly while you're in monitoring mode (as with
a tied note between two chords), it is less noticeable than
when there is no tie -- the new attack gets your attention
and gets you to give it a new label... it's a new event.
This is an auditory scene analysis way of looking at it.

>   This is all new to me...I did not even know that,
> melodically, the mind prefers something other than pure
> intervals for anything at all. 

Well sure, it's the comma problem. In Western music it's
an issue, anyway. One wants melodic continuity and pure
harmonies, but the pure harmony requires little melodic
comma shifts that throw off the melody.

-Carl

[ Attachment content not displayed ]

--- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@...> wrote:
>
> > /tuning/topicId_79656.html#79659
> >
> > There are also Rothenberg's propriety and efficiency to
> > consider...
>
> I don't this answers my question which was not about scale
> structure at all.

I understood you to say that other than a desire for harmonic
purity and a lack of comma shifts, there are no guiding principles
in melody other than cultural preference. I think that's false.

> > Thus, an adaptive tuning scheme can minimize
> > the perception of melodic shifts by attacking notes at the
> > pitches desired for them melodically, and then bending them pure.
>
> Why should musicians attack a certain note at this exact pitch
> rather than another pitch?

Because they have the melody in their mind. Speaking as a
trumpet player, I would 'aim' based on the melody of the
part I was playing (even if it was not _the_ melody), and
then refine the result once I heard my instrument in the
ensemble.

> > Not sure exactly what you mean, but John deLaubenfels' system
> > first calculated an optimum well temperament for a piece, and
> > then minimized the sum of the notes' squared deviations from it
> > over the course of the adaptive tuning.
>
> This is similar to my proposal but mine is more flexible as the
> temperament also is determined from the piece. If the piece
> doesn't require the vanishing of any commas and the simultaneities
> are JI then the result will be pure JI also melodically.

If the piece doesn't require any commas vanish, then John's
COFT (calculated optimum fixed temperament) will be JI, or
very close to it.

> > My own adaptive tuning scheme (as yet unimplemented) tries to
> > eliminate short term shift but allows long-term drift.
>
> If you just eliminate (or minimize) short-term shifts and don't
> care about long-term drift, don't you just end up with
> comma pumps?

Yes, but "just having pumps" is not trivial. For many
progressions there is more than one way to do it. For
example: tritone substitution. In the 19-limit, there
are multiple ways to do any progression. 12-ET isn't
even unique with respect to 9-limit tetrads.

> > > There are potential problems though: in some pieces of music
> > > there might be outliers, that is pitches that are not
> > > harmonically connected to other pitches. Their tuning would
> > > have to be somehow determined too.
> >
> > My scheme is designed around 12-ET being the input, and for
> > most configurations, it'll have several output intervals
> > for each 12-ET interval, so isolated pitches are not possible.
> > It knows about chords and uses a greedy algorithm to extract
> > as many chords as it can, largest first. Then it connects
> > them with dyads.
>
> In my scheme the input is the desired tunings for the vertical
> simultaneities with the specification which notes (types, not
> tokens) are the same across all simultaneities, for example
> which ones are supposed to be c1:s, f#1:s, e2:s and so on.

What's a C1? Presumably this means 12-ET.

> > > And a piece might for example begin with a single melodic line
> > > with no harmony. But there is an easy solution to this: use the
> > > melodic center the program calculates from the vertical
> > > simultaneities.
> >
> > Again not sure what you mean, but would certainly be interested
> > to read more about your idea.
>
> My scheme requires that all notes (again types, not tokens) are
> harmonically connected to at least one other note so that we get
> a connected web of notes. Here is an example (view with fixed
> width font)
>
> G11 A11 A12 B11 B12 B13 C2
> E11 F11 F12 G12 G13 G13 G14
> C11 C12 D11 D12 E12 (F1) E13
> C01 F0 D0 G01 G02 G02 C02
>
> The columns are the simultaneities.
> The first number in the note name is the octave register, and the
> second one is the number of the note token, that is a particular
> pitch of a note. The desired tunings for the simultaneities in
> this example are standard 5-limit JI ones except for the lonely
> F1 in brackets which is not harmonically connected to the other
> notes in its simultaneity. It is simply calculated from the
> average of F11 and F12.

Is it not better to connect it to G0 by means of a 7:4, or
16:9, etc.?

> Now find a sort of minimax tuning: calculate the pitch range
> (the difference for the lowest and highest tokens of a note) for
> all notes. Minimize the maximum pitch range. That's it!
>
> I hope it is clearer now!

I think it's very close to deLaubenfels. He minimized the
mean squared error from COFT, if I'm not mistaken.

-Carl

Chris wrote:

>> Well sure, it's the comma problem. In Western music it's
>> an issue, anyway. One wants melodic continuity and pure
>> harmonies, but the pure harmony requires little melodic
>> comma shifts that throw off the melody.
>
> Isn't this handled in choral music, at least older music,
> without upset?

Good choirs and a capella ensembles can do it, yes.
Nobody knows exactly how, but we have some ideas:

1. They sing the vertical intervals (harmony) in JI,
but the horizontal intervals (melodic intervals between
roots of chords, for example) in a temperament. This
reduces the size of the comma shifts. Recently discussed
here regarding Vicentino's 1555 proposal of this kind.

2. They attack notes closer to a temperament and then
bend chords pure. Currently being discussed.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@> wrote:
> >
> > > /tuning/topicId_79656.html#79659
> > >
> > > There are also Rothenberg's propriety and efficiency to
> > > consider...
> >
> > I don't this answers my question which was not about scale
> > structure at all.
>
> I understood you to say that other than a desire for harmonic
> purity and a lack of comma shifts, there are no guiding principles
> in melody other than cultural preference. I think that's false.

That depends what is included in the cultural preferences. But I agree
that scale structures are important. But these structures are already
given to musicians in the score, even the desire for no comma shifts
is specified in the score. I was asking about the exact tuning of
those scale structures. So once these scale structures are given in
the score, what else determines what pitches to play in the attacks of
the notes other than the desire to keep the pitch ranges of notes at a
minimum both in the short and long term?

> > > Thus, an adaptive tuning scheme can minimize
> > > the perception of melodic shifts by attacking notes at the
> > > pitches desired for them melodically, and then bending them pure.
> >
> > Why should musicians attack a certain note at this exact pitch
> > rather than another pitch?
>
> Because they have the melody in their mind. Speaking as a
> trumpet player, I would 'aim' based on the melody of the
> part I was playing (even if it was not _the_ melody), and
> then refine the result once I heard my instrument in the
> ensemble.

Of course you hear the melody of your part in your mind, but is it in
some exact tuning? If it is, why is it in this tuning rather than
another?

> > > Not sure exactly what you mean, but John deLaubenfels' system
> > > first calculated an optimum well temperament for a piece, and
> > > then minimized the sum of the notes' squared deviations from it
> > > over the course of the adaptive tuning.
> >
> > This is similar to my proposal but mine is more flexible as the
> > temperament also is determined from the piece. If the piece
> > doesn't require the vanishing of any commas and the simultaneities
> > are JI then the result will be pure JI also melodically.
>
> If the piece doesn't require any commas vanish, then John's
> COFT (calculated optimum fixed temperament) will be JI, or
> very close to it.

But you said his system calculates an optimum well temperament for a
piece. How could any result then be in horizontal JI?

By the way, does JdL:s COFT assume (melodic) octave equivalence? (That
is, is the tuning the "same" in different octaves?)

Mine doesn't.

> > > My own adaptive tuning scheme (as yet unimplemented) tries to
> > > eliminate short term shift but allows long-term drift.
> >
> > If you just eliminate (or minimize) short-term shifts and don't
> > care about long-term drift, don't you just end up with
> > comma pumps?
>
> Yes, but "just having pumps" is not trivial.

But how does eliminating (or even minimizing) short-term shifts differ
from free-style JI if you allow long-term drift?

> For many
> progressions there is more than one way to do it. For
> example: tritone substitution. In the 19-limit, there
> are multiple ways to do any progression. 12-ET isn't
> even unique with respect to 9-limit tetrads.

In my proposed scheme one has to do additional work in personally
choosing the tunings for all the simultaneities. So if the retuned
piece was originally composed in 12-equal there is a lot of choosing
to do. But of course some of this work could be automated.

> > > > There are potential problems though: in some pieces of music
> > > > there might be outliers, that is pitches that are not
> > > > harmonically connected to other pitches. Their tuning would
> > > > have to be somehow determined too.
> > >
> > > My scheme is designed around 12-ET being the input, and for
> > > most configurations, it'll have several output intervals
> > > for each 12-ET interval, so isolated pitches are not possible.
> > > It knows about chords and uses a greedy algorithm to extract
> > > as many chords as it can, largest first. Then it connects
> > > them with dyads.
> >
> > In my scheme the input is the desired tunings for the vertical
> > simultaneities with the specification which notes (types, not
> > tokens) are the same across all simultaneities, for example
> > which ones are supposed to be c1:s, f#1:s, e2:s and so on.
>
> What's a C1? Presumably this means 12-ET.

No, it doesn't. C1 is just a label. It could be any symbol.
Now, obviously the piece has to be *composed* in some temperament.
This then determines what notes are the same across the
simultaneities. If the piece is composed in Pajara this job is
different from what it would be if the piece was composed in Meantone.
But no exact tuning for Pajara (or Meantone) is assumed beforehand
when the piece is entered into the program. If the piece was composed
in meantone but it happened to avoid comma pumps the result would be
horizontal and vertical JI (assuming that the simultaneities could be
tuned to JI).

> > > > And a piece might for example begin with a single melodic line
> > > > with no harmony. But there is an easy solution to this: use the
> > > > melodic center the program calculates from the vertical
> > > > simultaneities.
> > >
> > > Again not sure what you mean, but would certainly be interested
> > > to read more about your idea.
> >
> > My scheme requires that all notes (again types, not tokens) are
> > harmonically connected to at least one other note so that we get
> > a connected web of notes. Here is an example (view with fixed
> > width font)
> >
> > G11 A11 A12 B11 B12 B13 C2
> > E11 F11 F12 G12 G13 G13 G14
> > C11 C12 D11 D12 E12 (F1) E13
> > C01 F0 D0 G01 G02 G02 C02
> >
> > The columns are the simultaneities.
> > The first number in the note name is the octave register, and the
> > second one is the number of the note token, that is a particular
> > pitch of a note. The desired tunings for the simultaneities in
> > this example are standard 5-limit JI ones except for the lonely
> > F1 in brackets which is not harmonically connected to the other
> > notes in its simultaneity. It is simply calculated from the
> > average of F11 and F12.
>
> Is it not better to connect it to G0 by means of a 7:4, or
> 16:9, etc.?

Yep, you could do that and I might prefer to tune it to 16:9 myself
based on (surprise!) melodic preferences. But the example is 5-limit
adaptive JI, so the intervals G0-F1, F1-G1 and F1-B1 are supposed to
be dissonances. The example just shows one way to tune something that
is not determined by the odd-limit of the piece.

> > Now find a sort of minimax tuning: calculate the pitch range
> > (the difference for the lowest and highest tokens of a note) for
> > all notes. Minimize the maximum pitch range. That's it!
> >
> > I hope it is clearer now!
>
> I think it's very close to deLaubenfels. He minimized the
> mean squared error from COFT, if I'm not mistaken.

OK!

>1. They sing the vertical intervals (harmony) in JI,

>but the horizontal intervals (melodic intervals between

>roots of chords, for example) in a temperament. This

>reduces the size of the comma shifts. Recently discussed

>here regarding Vicentino's 1555 proposal of this kind.

>2. They attack notes closer to a temperament and then
>bend chords pure. Currently being discussed.

    Hmm...I common pattern between both methods and things like MOS scale is that apparently uniform step size increases the ease of melodic sense...while JI intervals increase harmonic sense.  It seems again the sudden attack IE first second or so of a note counts as melody while the rest counts as harmony and that's where the shifting from ET to JI takes place.

--- On Thu, 1/8/09, Carl Lumma <carl@...> wrote:

From: Carl Lumma <carl@...>
Subject: [tuning] Re: Melodic Preferences In Adaptive Schemes (was: common-practice accidentals)
To: tuning@yahoogroups.com
Date: Thursday, January 8, 2009, 10:54 PM

Chris wrote:

>> Well sure, it's the comma problem. In Western music it's

>> an issue, anyway. One wants melodic continuity and pure

>> harmonies, but the pure harmony requires little melodic

>> comma shifts that throw off the melody.

>

> Isn't this handled in choral music, at least older music,

> without upset?

Good choirs and a capella ensembles can do it, yes.

Nobody knows exactly how, but we have some ideas:

1. They sing the vertical intervals (harmony) in JI,

but the horizontal intervals (melodic intervals between

roots of chords, for example) in a temperament. This

reduces the size of the comma shifts. Recently discussed

here regarding Vicentino's 1555 proposal of this kind.

2. They attack notes closer to a temperament and then

bend chords pure. Currently being discussed.

-Carl

Hi Kalle,

>> I understood you to say that other than a desire for harmonic
>> purity and a lack of comma shifts, there are no guiding
>> principles in melody other than cultural preference. I think
>> that's false.
>
> That depends what is included in the cultural preferences.

I think there are distinctly non-cultural factors at work.
Limitations of short-term memory, for example, which come
from our physiology.

> But these structures are already given to musicians in
> the score, even the desire for no comma shifts
> is specified in the score. I was asking about the exact
> tuning of those scale structures. So once these scale
> structures are given in the score, what else determines
> what pitches to play in the attacks of the notes other
> than the desire to keep the pitch ranges of notes at a
> minimum both in the short and long term?

As pertains the attack/sustain suggestion, the extreme
example is to put all attacks completely within the source
scale (say 12-ET) and then bend sustains to JI. If the
portamenti are too wide as a result, yes, one can allow
the attacks to wander from the source scale, and then one
simply tries to minimize the wandering.

That's, as you say, assuming the source scale perfectly
captures the melodic ideal. But just as our scores do not
fully capture the harmonic ideal, they may not capture the
melodic. Suppose there is a fundamental desire for two
different major 7ths, depending on context. Then it would
be a mistake to try to minimize the range of pitches
assigned to "B". That's just an example (though many have
argued its truth), but if what I say about physiology-
derived melodic factors is true, there may be a way to
improve upon both harmony and melody in an adaptive tuning.

>> Because they have the melody in their mind. Speaking as a
>> trumpet player, I would 'aim' based on the melody of the
>> part I was playing (even if it was not _the_ melody), and
>> then refine the result once I heard my instrument in the
>> ensemble.
>
> Of course you hear the melody of your part in your mind, but
> is it in some exact tuning?

Absolutely. Human melodic pitch discrimination is almost as
good as harmonic. Actually I would guess it's better if you
give normal timbres melodically and sine waves harmonically.

> If it is, why is it in this tuning rather than another?

That's the hard question. I know of a few general principles
I believe are true, but at the moment I have no suggestions
for how to add them to an adaptive tuning algorithm that
would be a significant improvement over 'narrow pitch classes'
for Western music. But if you have a musician perform a
lone melody and measure it, you'll get more than the notated
number of pitches, that's for sure. That's true for voice or
any orchestral instrument, and it's true for maqam music as
well (they do not stick to the notation). In fact you'll
find it's nontrivial even to assign discrete pitches to
violin and voice performances.

>> If the piece doesn't require any commas vanish, then John's
>> COFT (calculated optimum fixed temperament) will be JI, or
>> very close to it.
>
> But you said his system calculates an optimum well temperament
> for a piece. How could any result then be in horizontal JI?

The "optimum well temperament" will be JI if there's no
modulation.

>> By the way, does JdL:s COFT assume (melodic) octave
>> equivalence? (That is, is the tuning the "same" in
>> different octaves?)
>
> Mine doesn't.

Yes, his does, and I would say this is a bona fide advantage
of your approach.

>>> If you just eliminate (or minimize) short-term shifts and
>>> don't care about long-term drift, don't you just end up
>>> with comma pumps?
>>
>> Yes, but "just having pumps" is not trivial.
>
> But how does eliminating (or even minimizing) short-term shifts
> differ from free-style JI if you allow long-term drift?

My system is essential automatic free-style JI. Except
technically, since it allows irrational horizontal intervals,
it isn't (according to the Lou Harrison definition).

>> For many
>> progressions there is more than one way to do it. For
>> example: tritone substitution. In the 19-limit, there
>> are multiple ways to do any progression. 12-ET isn't
>> even unique with respect to 9-limit tetrads.
>
> In my proposed scheme one has to do additional work in
> personally choosing the tunings for all the simultaneities.
> So if the retuned piece was originally composed in 12-equal
> there is a lot of choosing to do. But of course some of this
> work could be automated.

My thing chooses them based on achieving as many common tones
as possible from the notated progression, using the generalized
Tenney height of the chords (a*b*c) to resolve ties.

>> I think it's very close to deLaubenfels. He minimized the
>> mean squared error from COFT, if I'm not mistaken.
>
> OK!

The octave-specific nature of your suggestion seems to be the
main difference.

-Carl

Good choirs and a capella ensembles can do it, yes.Nobody knows exactly how, but we have some ideas:

Ok - I would think the place to start is at the beginning. And that would be gregorian chant. I would assume monks didn't sing melodies in 12 TET. But early chant was sung in parallel 5ths and 4ths. So does not the comma problem occur in this simple harmonization?
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Chris wrote:

> Ok - I would think the place to start is at the beginning. And
> that would be gregorian chant. I would assume monks didn't sing
> melodies in 12 TET. But early chant was sung in parallel 5ths
> and 4ths. So does not the comma problem occur in this simple
> harmonization?

No, there's no comma problem in gregorian chant.

-Carl

[ Attachment content not displayed ]

[ Attachment content not displayed ]

Early chant was sung in unison, organum came later.

Daniel Forro

On 10 Jan 2009, at 8:34 AM, chrisvaisvil@... wrote:

> Good choirs and a capella ensembles can do it, yes.Nobody knows > exactly how, but we have some ideas:
>
> Ok - I would think the place to start is at the beginning. And that > would be gregorian chant. I would assume monks didn't sing melodies > in 12 TET. But early chant was sung in parallel 5ths and 4ths. So > does not the comma problem occur in this simple harmonization?

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Gregorian Chant was exclusively monophonic for a long time. They
thought pretty exclusively in a diatonic modal sense - there are no
chromatic notes or anything. So since this was sung with no
accompaniment, I can't see how comma drifts would be much of a
conscious problem or thought at all. More likely the tricky part was
getting the whole choir to stay in one key for the entire chant and
not drift upwards or downwards tonally, and for reasons besides comma
drifts :-P

-Mike

On Fri, Jan 9, 2009 at 9:30 PM, Chris Vaisvil <chrisvaisvil@...> wrote:
> true but I don't think I understand the meaning.
>
> On Fri, Jan 9, 2009 at 9:05 PM, Daniel Forro <dan.for@...> wrote:
>>
>> Early chant was sung in unison, organum came later.
>>
>> Daniel Forro
>>
>> On 10 Jan 2009, at 8:34 AM, chrisvaisvil@... wrote:
>>
>> > Good choirs and a capella ensembles can do it, yes.Nobody knows
>> > exactly how, but we have some ideas:
>> >
>> > Ok - I would think the place to start is at the beginning. And that
>> > would be gregorian chant. I would assume monks didn't sing melodies
>> > in 12 TET. But early chant was sung in parallel 5ths and 4ths. So
>> > does not the comma problem occur in this simple harmonization?
>
>

[ Attachment content not displayed ]

They harmonized by a pure perfect fifth every time. There were no wolf
fifths, and generally comma differences as well as diatonic
differences were ignored.

-Mike

On Fri, Jan 9, 2009 at 10:43 PM, Chris Vaisvil <chrisvaisvil@...> wrote:
> ok, please excuse my terminology. I meant organum and use of comma is
> totally incorrect.
>
> Obviously monophonic music doesn't make sense in my example.
> But diatonic does not mean 12 TET - and therefore when they started to sing
> in parallel 5ths and 4ths a melody line A may not have been the same as the
> A sung in harmony to a melodic D.
>
> Perhaps I have misunderstood but I thought the challenge was singing and
> harmonizing in pure intervals simultaneously and the compromises and
> techniques involved to do it. My point is that this problem is not limited
> to triads, tetrads or tridadic harmony at all.
>
> On Fri, Jan 9, 2009 at 10:29 PM, Mike Battaglia <battaglia01@...>
> wrote:
>>
>> Gregorian Chant was exclusively monophonic for a long time. They
>> thought pretty exclusively in a diatonic modal sense - there are no
>> chromatic notes or anything. So since this was sung with no
>> accompaniment, I can't see how comma drifts would be much of a
>> conscious problem or thought at all. More likely the tricky part was
>> getting the whole choir to stay in one key for the entire chant and
>> not drift upwards or downwards tonally, and for reasons besides comma
>> drifts :-P
>>
>> -Mike
>>
>> On Fri, Jan 9, 2009 at 9:30 PM, Chris Vaisvil <chrisvaisvil@...>
>> wrote:
>> > true but I don't think I understand the meaning.
>> >
>> > On Fri, Jan 9, 2009 at 9:05 PM, Daniel Forro <dan.for@...> wrote:
>> >>
>> >> Early chant was sung in unison, organum came later.
>> >>
>> >> Daniel Forro
>> >>
>> >> On 10 Jan 2009, at 8:34 AM, chrisvaisvil@... wrote:
>> >>
>> >> > Good choirs and a capella ensembles can do it, yes.Nobody knows
>> >> > exactly how, but we have some ideas:
>> >> >
>> >> > Ok - I would think the place to start is at the beginning. And that
>> >> > would be gregorian chant. I would assume monks didn't sing melodies
>> >> > in 12 TET. But early chant was sung in parallel 5ths and 4ths. So
>> >> > does not the comma problem occur in this simple harmonization?
>> >
>> >
>
>

--- In tuning@yahoogroups.com, "Chris Vaisvil" <chrisvaisvil@...> wrote:
>
> Why would you say that? Are they singing melodically in 12-tet?
>

The music contains no comma pumps that I know of.

http://tonalsoft.com/enc/c/comma-pump.aspx

-Carl

--- In tuning@yahoogroups.com, "Chris Vaisvil" <chrisvaisvil@...> wrote:
>
> for instance if the distance between E and F is not the same
> as the distance between B and C I think the problem exists.
> perhaps it is incorrectly term by me as comma - nonetheless
> the interval has to be adjusted unless the distances are
> exactly the same.

Generally speaking, any small interval is a "comma". But
there is only a "comma problem" when a piece of music requires
a comma change between two adjacent notes that are named the
same way in the score. That won't happen in gregorian chant.

-Carl

--- In tuning@yahoogroups.com, Daniel Forro <dan.for@...> wrote:
>
> Early chant was sung in unison, organum came later.
>
> Daniel Forro

Thanks Daniel, that's right; we're actually talking
about organum.

-Carl

Ok so you and Daniel are saying the top half of a diatonic scale exactly mirrors the bottm half in melodic interval size? I thought organum started to necessitate the use of accidentals, especially in some modal scales.
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-----Original Message-----
From: "Carl Lumma" <carl@lumma.org>

Date: Sat, 10 Jan 2009 08:24:40
To: <tuning@yahoogroups.com>
Subject: [tuning] Re: Melodic Preferences In Adaptive Schemes (was: common-practice accidentals)

--- In tuning@yahoogroups.com, "Chris Vaisvil" <chrisvaisvil@...> wrote:
>
> for instance if the distance between E and F is not the same
> as the distance between B and C I think the problem exists.
> perhaps it is incorrectly term by me as comma - nonetheless
> the interval has to be adjusted unless the distances are
> exactly the same.

Generally speaking, any small interval is a "comma". But
there is only a "comma problem" when a piece of music requires
a comma change between two adjacent notes that are named the
same way in the score. That won't happen in gregorian chant.

-Carl

I didn't say this, but OK, I can answer this.

I suppose you mean tetrachords by that "half of the scale", and standard 7-tone scale, as there are also diatonic scales with less or more tones.

If you are talking about "mirror", that would create a scale (modus) for example C D E F - G Ab Bb C, which didn't exist in Gregorian.

Or if you have meant not "mirroring", but parallel following, then result will be for example C D E F - G A B C, which is Ionian (later called major). Also this scale was not used in Gregorian chant.

Parallel following PLUS mirroring is possible with Dorian: D E F G - A B C D. Still no accidentals are necessary for fifth organum.

Parallel following exists in Phrygian: E F G A - B C D E. Still no accidentals are necessary for fifth organum.

For parallel following in Lydian we need accidental: F G A B - C D E F# (F), but it fights with F which is 8th note of the scale. I'm not sure now if this was allowed those times, probably yes. Of course F and F# never meet, but ...

Same with Mixolydian, F# against F as 7th note of the scale: G A B C - D E F# (F) G

Aeolian and Locrian were not used in Gregorian chant (but there are very rare examples of their using).

For sure here are experts on Middle Age music who can answer better. As i know, the first accidental was Bb, then F# came. These too were enough to help in very slow process of disappearing of the original 4 modes (8 with HYPO- variants) and establishing Major/Minor system used since 17th century. Look:

Bb changed Dorian to Aeolian:
D E F G A B C D became D E F G A Bb C D

and Lydian to Ionian:
F G A B C D E F became F G A Bb C D E F

F# changed Phrygian to Aeolian:
E F G A B C D E became E F# G A B C D E

and Mixolydian to Ionian:
G A B C D E F G became G A B C D E F# G

Many early composers used only those few accidentals. Of course there were crazy exceptions, like Solage's Fumeaux fume or Gesualdo da Venosa...

Daniel Forro

On 10 Jan 2009, at 11:42 PM, chrisvaisvil@... wrote:

> Ok so you and Daniel are saying the top half of a diatonic scale > exactly mirrors the bottm half in melodic interval size? I thought > organum started to necessitate the use of accidentals, especially > in some modal scales.
>
> Sent via BlackBerry from T-Mobile
>
>
> From: "Carl Lumma"
> Date: Sat, 10 Jan 2009 08:24:40 -0000
> To: <tuning@yahoogroups.com>
> Subject: [tuning] Re: Melodic Preferences In Adaptive Schemes (was: > common-practice accidentals)
>
> --- In tuning@yahoogroups.com, "Chris Vaisvil" <chrisvaisvil@...> > wrote:
> >
> > for instance if the distance between E and F is not the same
> > as the distance between B and C I think the problem exists.
> > perhaps it is incorrectly term by me as comma - nonetheless
> > the interval has to be adjusted unless the distances are
> > exactly the same.
>
> Generally speaking, any small interval is a "comma". But
> there is only a "comma problem" when a piece of music requires
> a comma change between two adjacent notes that are named the
> same way in the score. That won't happen in gregorian chant.
>
> -Carl
>
>
>
>

Thanks Daniel. I think I'm misunderstanding the tuning implications. So if you have perfectly in tune C major (JI?) you can play anything w/o accidentals and it will all be spot on tuning wise - all thirds are pure etc.
Sent via BlackBerry from T-Mobile

I haven't Cent table at my hand now, but I don't think all thirds will be pure.

And I suppose we are talking about fifth parallels used in organum. I don't think even all fifths will be pure. I can check in the morning, or somebody else will answer.

Unfortunately somehow we have no audio records from Middle Age :-), so it's difficult to judge in which tuning they sang those parallel organum... Maybe there are some theories about it.

Daniel Forro

On 11 Jan 2009, at 1:18 AM, chrisvaisvil@... wrote:

> Thanks Daniel. I think I'm misunderstanding the tuning > implications. So if you have perfectly in tune C major (JI?) you > can play anything w/o accidentals and it will all be spot on tuning > wise - all thirds are pure etc.

Ok, that was my point. It seems singers naturally sing in perfect intervals. They adjust which is then adaptive tuning and organum is the simplest example we have of that practice. And probably then organum is the easist to study since the question here was the best method to implement adaptive tuning.
Sent via BlackBerry from T-Mobile

-----Original Message-----
From: Daniel Forro <dan.for@tiscali.cz>

Date: Sun, 11 Jan 2009 01:48:58
To: <tuning@yahoogroups.com>
Subject: Re: [tuning] Re: Melodic Preferences In Adaptive Schemes (was: common-practice accidentals)

I haven't Cent table at my hand now, but I don't think all thirds
will be pure.

And I suppose we are talking about fifth parallels used in organum. I
don't think even all fifths will be pure. I can check in the morning,
or somebody else will answer.

Unfortunately somehow we have no audio records from Middle Age :-),
so it's difficult to judge in which tuning they sang those parallel
organum... Maybe there are some theories about it.

Daniel Forro

On 11 Jan 2009, at 1:18 AM, chrisvaisvil@gmail.com wrote:

> Thanks Daniel. I think I'm misunderstanding the tuning
> implications. So if you have perfectly in tune C major (JI?) you
> can play anything w/o accidentals and it will all be spot on tuning
> wise - all thirds are pure etc.

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Hi Kalle,
>
> >> I understood you to say that other than a desire for harmonic
> >> purity and a lack of comma shifts, there are no guiding
> >> principles in melody other than cultural preference. I think
> >> that's false.
> >
> > That depends what is included in the cultural preferences.
>
> I think there are distinctly non-cultural factors at work.
> Limitations of short-term memory, for example, which come
> from our physiology.

I understand that these psychological and physiological limitations
constrain for example the number of distinct notes in a comprehensible
scale and so on but these are considerations that are important when
we construct musical scales. Again, what I'm talking about is the
exact *tuning* of these scales when actual music is played. Once the
musical scale is already in existence, what determines its exact
preferred melodic tuning in actual performance other than the already
mentioned factors of cultural preferences and narrow pitch ranges for
distinct notes?

> As pertains the attack/sustain suggestion, the extreme
> example is to put all attacks completely within the source
> scale (say 12-ET) and then bend sustains to JI. If the
> portamenti are too wide as a result, yes, one can allow
> the attacks to wander from the source scale, and then one
> simply tries to minimize the wandering.
>
> That's, as you say, assuming the source scale perfectly
> captures the melodic ideal. But just as our scores do not
> fully capture the harmonic ideal, they may not capture the
> melodic. Suppose there is a fundamental desire for two
> different major 7ths, depending on context. Then it would
> be a mistake to try to minimize the range of pitches
> assigned to "B". That's just an example (though many have
> argued its truth), but if what I say about physiology-
> derived melodic factors is true, there may be a way to
> improve upon both harmony and melody in an adaptive tuning.

Wouldn't it be better then to treat them as different notes at least
when entering them into the adaptive tuning program?

> > Of course you hear the melody of your part in your mind, but
> > is it in some exact tuning?
>
> Absolutely. Human melodic pitch discrimination is almost as
> good as harmonic. Actually I would guess it's better if you
> give normal timbres melodically and sine waves harmonically.
>
> > If it is, why is it in this tuning rather than another?
>
> That's the hard question. I know of a few general principles
> I believe are true, but at the moment I have no suggestions
> for how to add them to an adaptive tuning algorithm that
> would be a significant improvement over 'narrow pitch classes'
> for Western music.

Bear in mind that pitch classes are octave-equivalent. I prefer narrow
pitch ranges for notes understood as octave-specific.

> But if you have a musician perform a
> lone melody and measure it, you'll get more than the notated
> number of pitches, that's for sure.

Pitches yes but the question is the number of distinct notes
perceived. Notes allow a lot of pitch variation without becoming
different notes perceptually.

> That's true for voice or
> any orchestral instrument, and it's true for maqam music as
> well (they do not stick to the notation). In fact you'll
> find it's nontrivial even to assign discrete pitches to
> violin and voice performances.
>
> >> If the piece doesn't require any commas vanish, then John's
> >> COFT (calculated optimum fixed temperament) will be JI, or
> >> very close to it.
> >
> > But you said his system calculates an optimum well temperament
> > for a piece. How could any result then be in horizontal JI?
>
> The "optimum well temperament" will be JI if there's no
> modulation.

But isn't it limited to only 12 notes then?

> >> By the way, does JdL:s COFT assume (melodic) octave
> >> equivalence? (That is, is the tuning the "same" in
> >> different octaves?)
> >
> > Mine doesn't.
>
> Yes, his does, and I would say this is a bona fide advantage
> of your approach.

Along with the fact that it is not limited to 12 notes (if there is
such a limitation in JdL:s COFT, that is).

> > But how does eliminating (or even minimizing) short-term shifts
> > differ from free-style JI if you allow long-term drift?
>
> My system is essential automatic free-style JI. Except
> technically, since it allows irrational horizontal intervals,
> it isn't (according to the Lou Harrison definition).

Is there any documentation of your system? I would like to take a look.

> >> For many
> >> progressions there is more than one way to do it. For
> >> example: tritone substitution. In the 19-limit, there
> >> are multiple ways to do any progression. 12-ET isn't
> >> even unique with respect to 9-limit tetrads.
> >
> > In my proposed scheme one has to do additional work in
> > personally choosing the tunings for all the simultaneities.
> > So if the retuned piece was originally composed in 12-equal
> > there is a lot of choosing to do. But of course some of this
> > work could be automated.
>
> My thing chooses them based on achieving as many common tones
> as possible from the notated progression, using the generalized
> Tenney height of the chords (a*b*c) to resolve ties.

This is a bit cryptic to me I'm afraid. :)

> >> I think it's very close to deLaubenfels. He minimized the
> >> mean squared error from COFT, if I'm not mistaken.
> >
> > OK!
>
> The octave-specific nature of your suggestion seems to be the
> main difference.

Ok, that was my point. It seems singers naturally sing in perfect intervals. They adjust which is then adaptive tuning and organum is the simplest example we have of that practice. And probably then organum is the easist to study since the question here was the best method to implement adaptive tuning.
Sent via BlackBerry from T-Mobile

-----Original Message-----
From: Daniel Forro <dan.for@tiscali.cz>

Date: Sun, 11 Jan 2009 01:48:58
To: <tuning@yahoogroups.com>
Subject: Re: [tuning] Re: Melodic Preferences In Adaptive Schemes (was: common-practice accidentals)

I haven't Cent table at my hand now, but I don't think all thirds
will be pure.

And I suppose we are talking about fifth parallels used in organum. I
don't think even all fifths will be pure. I can check in the morning,
or somebody else will answer.

Unfortunately somehow we have no audio records from Middle Age :-),
so it's difficult to judge in which tuning they sang those parallel
organum... Maybe there are some theories about it.

Daniel Forro

On 11 Jan 2009, at 1:18 AM, chrisvaisvil@gmail.com wrote:

> Thanks Daniel. I think I'm misunderstanding the tuning
> implications. So if you have perfectly in tune C major (JI?) you
> can play anything w/o accidentals and it will all be spot on tuning
> wise - all thirds are pure etc.

--- In tuning@yahoogroups.com, chrisvaisvil@... wrote:
>
> Ok so you and Daniel are saying the top half of a diatonic scale
> exactly mirrors the bottm half in melodic interval size? I
> thought organum started to necessitate the use of accidentals,
> especially in some modal scales.

Did you study the link I provided on comma pumps? I'd be
happy to answer any questions you have about it. -Carl

--- In tuning@yahoogroups.com, chrisvaisvil@... wrote:
>
> Thanks Daniel. I think I'm misunderstanding the tuning
> implications. So if you have perfectly in tune C major (JI?)
> you can play anything w/o accidentals and it will all be spot
> on tuning wise - all thirds are pure etc.

No. The syntonic comma problem arises in the 5-limit diatonic
scale with no accidentals at all. -Carl

--- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@...> wrote:

> > I think there are distinctly non-cultural factors at work.
> > Limitations of short-term memory, for example, which come
> > from our physiology.
>
> I understand that these psychological and physiological
> limitations constrain for example the number of distinct notes
> in a comprehensible scale and so on but these are considerations
> that are important when we construct musical scales.
> Again, what I'm talking about is the exact *tuning* of these
> scales when actual music is played.

I'm not sure the distinction is valid. If I give you two bags
of pitches, can you tell me whether they are two tunings of the
same "scale"?

> Once the musical scale is already in existence, what determines
> its exact preferred melodic tuning in actual performance other
> than the already mentioned factors of cultural preferences and
> narrow pitch ranges for distinct notes?

I've already mentioned memory contraints. Boomsliter & Creel
suggest there is a preference for melodic JI, and Paul Erlich's
omnitetrachordality is similar. I also already mentioned
the 'leading tone rule'. Some people think anchor/pivot tones
(in a Schenkerian sense) are the only tones whose melodic
intonation matters, and others (e.g. passing tones) can be tuned
anyhow...

> > That's the hard question. I know of a few general principles
> > I believe are true, but at the moment I have no suggestions
> > for how to add them to an adaptive tuning algorithm that
> > would be a significant improvement over 'narrow pitch classes'
> > for Western music.
>
> Bear in mind that pitch classes are octave-equivalent. I prefer
> narrow pitch ranges for notes understood as octave-specific.

Eck- right; sorry. And again, I think it's a very good idea.

> Pitches yes but the question is the number of distinct notes
> perceived. Notes allow a lot of pitch variation without becoming
> different notes perceptually.

How do you quantify this?

> > >> If the piece doesn't require any commas vanish, then John's
> > >> COFT (calculated optimum fixed temperament) will be JI, or
> > >> very close to it.
> > >
> > > But you said his system calculates an optimum well temperament
> > > for a piece. How could any result then be in horizontal JI?
> >
> > The "optimum well temperament" will be JI if there's no
> > modulation.
>
> But isn't it limited to only 12 notes then?

Yes, but that comes out of the setup here -- you're saying
there's an input piece, which we've agree has at most 12 notes,
and it can be performed in JI without comma problems. So
there's no reason to use > 12 pitches for it, is there?

-Carl

>
> > >> By the way, does JdL:s COFT assume (melodic) octave
> > >> equivalence? (That is, is the tuning the "same" in
> > >> different octaves?)
> > >
> > > Mine doesn't.
> >
> > Yes, his does, and I would say this is a bona fide advantage
> > of your approach.
>
> Along with the fact that it is not limited to 12 notes (if there is
> such a limitation in JdL:s COFT, that is).
>
> > > But how does eliminating (or even minimizing) short-term shifts
> > > differ from free-style JI if you allow long-term drift?
> >
> > My system is essential automatic free-style JI. Except
> > technically, since it allows irrational horizontal intervals,
> > it isn't (according to the Lou Harrison definition).
>
> Is there any documentation of your system? I would like to take a look.
>
> > >> For many
> > >> progressions there is more than one way to do it. For
> > >> example: tritone substitution. In the 19-limit, there
> > >> are multiple ways to do any progression. 12-ET isn't
> > >> even unique with respect to 9-limit tetrads.
> > >
> > > In my proposed scheme one has to do additional work in
> > > personally choosing the tunings for all the simultaneities.
> > > So if the retuned piece was originally composed in 12-equal
> > > there is a lot of choosing to do. But of course some of this
> > > work could be automated.
> >
> > My thing chooses them based on achieving as many common tones
> > as possible from the notated progression, using the generalized
> > Tenney height of the chords (a*b*c) to resolve ties.
>
> This is a bit cryptic to me I'm afraid. :)
>
> > >> I think it's very close to deLaubenfels. He minimized the
> > >> mean squared error from COFT, if I'm not mistaken.
> > >
> > > OK!
> >
> > The octave-specific nature of your suggestion seems to be the
> > main difference.
>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@> wrote:
>
> > > I think there are distinctly non-cultural factors at work.
> > > Limitations of short-term memory, for example, which come
> > > from our physiology.
> >
> > I understand that these psychological and physiological
> > limitations constrain for example the number of distinct notes
> > in a comprehensible scale and so on but these are considerations
> > that are important when we construct musical scales.
> > Again, what I'm talking about is the exact *tuning* of these
> > scales when actual music is played.
>
> I'm not sure the distinction is valid.

Well, I jolly well consider 7-note diatonic collections of 1/4-, 1/5-
and 1/6-comma meantone tunings of the same scale.

> If I give you two bags
> of pitches, can you tell me whether they are two tunings of the
> same "scale"?

I guess it depends on the actual contents of the bags. What's the
relevance of this?

> > Once the musical scale is already in existence, what determines
> > its exact preferred melodic tuning in actual performance other
> > than the already mentioned factors of cultural preferences and
> > narrow pitch ranges for distinct notes?
>
> I've already mentioned memory contraints. Boomsliter & Creel
> suggest there is a preference for melodic JI, and Paul Erlich's
> omnitetrachordality is similar. I also already mentioned
> the 'leading tone rule'. Some people think anchor/pivot tones
> (in a Schenkerian sense) are the only tones whose melodic
> intonation matters, and others (e.g. passing tones) can be tuned
> anyhow...

Now we are getting somewhere! Except I don't think omnitetrachordality
has much to do with exact tuning: I think well-tempered diatonic
collection can be considered omnitetrachordal even though its not
exactly (tuning-wise) so.

> > > That's the hard question. I know of a few general principles
> > > I believe are true, but at the moment I have no suggestions
> > > for how to add them to an adaptive tuning algorithm that
> > > would be a significant improvement over 'narrow pitch classes'
> > > for Western music.
> >
> > Bear in mind that pitch classes are octave-equivalent. I prefer
> > narrow pitch ranges for notes understood as octave-specific.
>
> Eck- right; sorry. And again, I think it's a very good idea.

Thanks!

> > Pitches yes but the question is the number of distinct notes
> > perceived. Notes allow a lot of pitch variation without becoming
> > different notes perceptually.
>
> How do you quantify this?

I don't know about that but isn't the whole adaptive tuning business
based on the assumption that close enough pitches are heard as
versions of the same note?

> > > >> If the piece doesn't require any commas vanish, then John's
> > > >> COFT (calculated optimum fixed temperament) will be JI, or
> > > >> very close to it.
> > > >
> > > > But you said his system calculates an optimum well temperament
> > > > for a piece. How could any result then be in horizontal JI?
> > >
> > > The "optimum well temperament" will be JI if there's no
> > > modulation.
> >
> > But isn't it limited to only 12 notes then?
>
> Yes, but that comes out of the setup here -- you're saying
> there's an input piece, which we've agree has at most 12 notes,
> and it can be performed in JI without comma problems. So
> there's no reason to use > 12 pitches for it, is there?

Aha! Actually I haven't agreed on that. Where did I say anything like
that? The system I'm proposing is not limited to 12 notes. There can
be as many notes as the score specifies. If only 12 distinct notes are
used in the score then the situation is like you said.

Kalle

--- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@...> wrote:

> > > Again, what I'm talking about is the exact *tuning* of these
> > > scales when actual music is played.
//
> > If I give you two bags
> > of pitches, can you tell me whether they are two tunings of
> > the same "scale"?
>
> I guess it depends on the actual contents of the bags. What's the
> relevance of this?

You keep making a distinction between "exact tuning"s and
"scales", saying that what I'm talking about only applies
to one or the other. I'm wondering if you have a way to
tell them apart.

Rothenberg has the only sensical method for doing this that
I'm aware of, and even it has a problem with small pitch
differences (two Bs 2 cents apart probably won't be perceived
as different, but Rothenberg doesn't account for this).

> > > Pitches yes but the question is the number of distinct notes
> > > perceived. Notes allow a lot of pitch variation without becoming
> > > different notes perceptually.
> >
> > How do you quantify this?
>
> I don't know about that but isn't the whole adaptive tuning business
> based on the assumption that close enough pitches are heard as
> versions of the same note?

Yes, but as I say, maybe that's not desirable. Maybe we want
to draw out the two different Bs inherent in the score.
I'm sure you can ignore such stuff and get excellent results,
I'm just pointing it out...

> > Yes, but that comes out of the setup here -- you're saying
> > there's an input piece, which we've agree has at most 12 notes,
> > and it can be performed in JI without comma problems. So
> > there's no reason to use > 12 pitches for it, is there?
>
> Aha! Actually I haven't agreed on that. Where did I say anything
> like that? The system I'm proposing is not limited to 12 notes.
> There can be as many notes as the score specifies. If only 12
> distinct notes are used in the score then the situation is like
> you said.

John's system is real software that converts MIDI files, which
are in 12-ET by default, into 5- and 7-limit adaptive tunings.
Only in that sense is his system limited to 12 notes on the
input. I keep asking you about input and you keep dodging the
question. Are you saying that whatever pajara notation I throw
at you, you'll support it? It seems that for every input
scale, you'll need a mapping to JI...

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@> wrote:
>
> > > > Again, what I'm talking about is the exact *tuning* of these
> > > > scales when actual music is played.
> //
> > > If I give you two bags
> > > of pitches, can you tell me whether they are two tunings of
> > > the same "scale"?
> >
> > I guess it depends on the actual contents of the bags. What's the
> > relevance of this?
>
> You keep making a distinction between "exact tuning"s and
> "scales", saying that what I'm talking about only applies
> to one or the other. I'm wondering if you have a way to
> tell them apart.
>
> Rothenberg has the only sensical method for doing this that
> I'm aware of, and even it has a problem with small pitch
> differences (two Bs 2 cents apart probably won't be perceived
> as different, but Rothenberg doesn't account for this).

Okay, I get it now.

I do believe there's not always a fact of the matter whether two
tunings are tunings of the same scale or not. But for many scales and
temperaments like Meantone or Pajara I can tell what tunings are
"acceptable". If the piece is composed in meantone what determines the
preferred pitches in the attacks of notes? Why even assume that there
is a single preferred choice rather than an acceptable range of
pitches to choose from?

> > I don't know about that but isn't the whole adaptive tuning business
> > based on the assumption that close enough pitches are heard as
> > versions of the same note?
>
> Yes, but as I say, maybe that's not desirable. Maybe we want
> to draw out the two different Bs inherent in the score.
> I'm sure you can ignore such stuff and get excellent results,
> I'm just pointing it out...

That's an excellent point. I think that if such a situation should
arise the two Bs can be treated as if they were different notes. It
seems to me the system I'm proposing is flexible enough to allow this.
But in general I suppose the whole business of keeping the pitches at
narrow ranges makes the music easier to comprehend as it is intended
in the score. Anyway we can't be sure if we don't try these things out.

> > > Yes, but that comes out of the setup here -- you're saying
> > > there's an input piece, which we've agree has at most 12 notes,
> > > and it can be performed in JI without comma problems. So
> > > there's no reason to use > 12 pitches for it, is there?
> >
> > Aha! Actually I haven't agreed on that. Where did I say anything
> > like that? The system I'm proposing is not limited to 12 notes.
> > There can be as many notes as the score specifies. If only 12
> > distinct notes are used in the score then the situation is like
> > you said.
>
> John's system is real software that converts MIDI files, which
> are in 12-ET by default, into 5- and 7-limit adaptive tunings.
> Only in that sense is his system limited to 12 notes on the
> input. I keep asking you about input and you keep dodging the
> question. Are you saying that whatever pajara notation I throw
> at you, you'll support it? It seems that for every input
> scale, you'll need a mapping to JI...

Yes, exactly. I'm sorry that I haven't been more clear about this. The
proposed system is not restricted to any specific temperament or
gamut. The input is the desired tunings for the simultaneities along
with the labellings of the notes across the simultaneities. These must
be entered by hand or automated by entirely separate algorithm.
Personally I would prefer to enter the tunings of the simultaneities
myself but surely this work could be automated for MIDI files for
example. So my proposal is less specific in details than JdL:s or
yours. It only concerns the latter part of the process, that is
finding a solution.

I initially thought that the most innovative thing about it was that
no center tuning is assumed beforehand but I should have known JdL had
thought about that! Now it seems that the octave-specificity of the
notes is the meat of it.

Kalle

[ Attachment content not displayed ]

I have found early using of quartal harmony in Satie's works from 80ies of 19th century. He could find it by chance, or by trying to do something different than the others those times (as he mostly did), or by deriving from pentatonics.

Daniel Forro

On 11 Jan 2009, at 9:04 AM, Chris Vaisvil wrote:

> I would think organum wouldn't have comma pumps by definition since > stacked quartal or quintal harmony was not practiced until the 20th > century (barring a 2nd inversion suspended 4th) And the suggestion > was to avoid triadic harmony.
>
> This I think does not preclude adjustments of pitch to square > melodic and harmonic use.
>

--- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@...> wrote:

> > Rothenberg has the only sensical method for doing this that
> > I'm aware of, and even it has a problem with small pitch
> > differences (two Bs 2 cents apart probably won't be perceived
> > as different, but Rothenberg doesn't account for this).
>
> Okay, I get it now.
> I do believe there's not always a fact of the matter whether two
> tunings are tunings of the same scale or not.

Rothenberg's suggestion is dangerously seductive by the way.
Are you interested to hear about it?

> If the piece is composed in meantone what determines the
> preferred pitches in the attacks of notes? Why even assume that
> there is a single preferred choice rather than an acceptable
> range of pitches to choose from?

There may be an acceptable range, but whatever it is, if pure
harmony is ever at odds with it, then the attack/sustain
technique is supposed to help you satisfy both anyway.

> > Yes, but as I say, maybe that's not desirable. Maybe we want
> > to draw out the two different Bs inherent in the score.
> > I'm sure you can ignore such stuff and get excellent results,
> > I'm just pointing it out...
>
> That's an excellent point. I think that if such a situation should
> arise the two Bs can be treated as if they were different notes.
> It seems to me the system I'm proposing is flexible enough to
> allow this.

Sounds like it is. But you'd have to have a way to identify
when to allow two output ranges (e.g. B' and B'') for a given
input note (e.g. B).

> But in general I suppose the whole business of keeping the
> pitches at narrow ranges makes the music easier to comprehend
> as it is intended in the score. Anyway we can't be sure if we
> don't try these things out.

Yes. I think to start, just map input pitches to output
pitch ranges 1:1.

> > John's system is real software that converts MIDI files, which
> > are in 12-ET by default, into 5- and 7-limit adaptive tunings.
> > Only in that sense is his system limited to 12 notes on the
> > input. I keep asking you about input and you keep dodging the
> > question. Are you saying that whatever pajara notation I throw
> > at you, you'll support it? It seems that for every input
> > scale, you'll need a mapping to JI...
>
> Yes, exactly. I'm sorry that I haven't been more clear about
> this. The proposed system is not restricted to any specific
> temperament or gamut.

It's not restricted in any way from working in principle, but
that's a far cry from saying it works with them already! There
are devils in details.

Even with the mappings from input scales to JI in hand, there
are performance considerations with the pitch range minimization.
John's approach was to model a physical system, excite it, and
let it relax into the solution (a box of springs, one spring for
each note... and now, I'm even wondering if I had it right that
there were only 12 springs... I'll have to look that up...).

-Carl

[ Attachment content not displayed ]

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@> wrote:
>
> > > Rothenberg has the only sensical method for doing this that
> > > I'm aware of, and even it has a problem with small pitch
> > > differences (two Bs 2 cents apart probably won't be perceived
> > > as different, but Rothenberg doesn't account for this).
> >
> > Okay, I get it now.
> > I do believe there's not always a fact of the matter whether two
> > tunings are tunings of the same scale or not.
>
> Rothenberg's suggestion is dangerously seductive by the way.
> Are you interested to hear about it?

Yes!

> > If the piece is composed in meantone what determines the
> > preferred pitches in the attacks of notes? Why even assume that
> > there is a single preferred choice rather than an acceptable
> > range of pitches to choose from?
>
> There may be an acceptable range, but whatever it is, if pure
> harmony is ever at odds with it, then the attack/sustain
> technique is supposed to help you satisfy both anyway.

Right.

> > > Yes, but as I say, maybe that's not desirable. Maybe we want
> > > to draw out the two different Bs inherent in the score.
> > > I'm sure you can ignore such stuff and get excellent results,
> > > I'm just pointing it out...
> >
> > That's an excellent point. I think that if such a situation should
> > arise the two Bs can be treated as if they were different notes.
> > It seems to me the system I'm proposing is flexible enough to
> > allow this.
>
> Sounds like it is. But you'd have to have a way to identify
> when to allow two output ranges (e.g. B' and B'') for a given
> input note (e.g. B).
>
> > But in general I suppose the whole business of keeping the
> > pitches at narrow ranges makes the music easier to comprehend
> > as it is intended in the score. Anyway we can't be sure if we
> > don't try these things out.
>
> Yes. I think to start, just map input pitches to output
> pitch ranges 1:1.

Yep!

> > > John's system is real software that converts MIDI files, which
> > > are in 12-ET by default, into 5- and 7-limit adaptive tunings.
> > > Only in that sense is his system limited to 12 notes on the
> > > input. I keep asking you about input and you keep dodging the
> > > question. Are you saying that whatever pajara notation I throw
> > > at you, you'll support it? It seems that for every input
> > > scale, you'll need a mapping to JI...
> >
> > Yes, exactly. I'm sorry that I haven't been more clear about
> > this. The proposed system is not restricted to any specific
> > temperament or gamut.
>
> It's not restricted in any way from working in principle, but
> that's a far cry from saying it works with them already! There
> are devils in details.

Quite right. :)

> Even with the mappings from input scales to JI in hand, there
> are performance considerations with the pitch range minimization.
> John's approach was to model a physical system, excite it, and
> let it relax into the solution (a box of springs, one spring for
> each note... and now, I'm even wondering if I had it right that
> there were only 12 springs... I'll have to look that up...).

I think JdL's approach and a version of my system where the sum of the
squared distances from the averages is minimized could in theory find
a solution by linear algebra manipulations of matrices. But for a
typical piece of even moderate length the matrix would be enormous.
Some kind of iterative Monte Carlo-method might be more practical
(from your description it seems JdL might use something like this). I
don't know if some linear programming method could be used to find a
minimax solution but surely the Monte Carlo-method would work too.

I'm not much of a programmer but I have a short Pajara progression in
mind which I intend to use as an example. I'll write a dedicated
program (in FreeBasic) just to find a solution for this progression
and probably start fantasizing about a general program after that.

:)

Kalle

I wrote:

> and now, I'm even wondering if I had it right that
> there were only 12 springs... I'll have to look that up...).

Yup, I was right:

/tuning/topicId_7890.html#7890
for background and
/tuning-math/message/127
for details.

-Carl

Kalle wrote:

> > Rothenberg's suggestion is dangerously seductive by the way.
> > Are you interested to hear about it?
>
> Yes!

So he starts with the notion that melodies are not (or are
not merely) series of pitches (e.g. E3, D3 etc), but rather
series of intervals. And not specific intervals (5:4 9:8 etc)
but rather scalar intervals (3rd, 2nd etc). The scale is the
language of the melody, in other words. Evidence for this
notion includes the fact that diatonically transposing a
melody to, say, the relative minor, preserves a great deal
of its essence.

He further assumes that listeners do not carry out something
like: "I'm hearing a 5:4. I know all 5:4s are 3rds, therefore
I am hearing a 3rd." Instead, it is assumed that they do
something like: "This one is smaller than the previous one.
Therefore if the previous one was a 3rd, this one must be
and 2nd". **

Now, for any scale, we can show all its intervals via a
tonality-diamond-like matrix with modes on the y axis and
scale degrees on the x axis. Here we can indeed see that
for the diatonic scale, all 5:4s are in the 3rds column.
But since only relative sizes matter in the above, we can
replace each interval listed with its size rank among all
intervals in the matrix. You can do this in Scala by
choosing View > Interval Ranking Matrix.

These "rank-order matrices" as Rothenberg calls them are
thus like scale classes. His claim is that if two scales
have the same rank-order matrix, they are in a sense
different tunings of the same scale.

Very clever and I think there's a lot of truth to it. But
the problem of small deviations needs to be addressed
before it can be practically applied to arbitrary scales.
Rothenberg used it for ET subsets (e.g. all 5-tone subsets
of 31-ET). There it works, because the 31-ET step is still
fairly large.

** R. claims that in order for this to work, the scale has
to have the property that no 2nd is larger than any 3rd, and
so on. Such scales are called "proper". Improper scales are
predicted to be unsuitable for diatonic-like music, and
instead must be used against a drone. Again I think he's
onto something, but there is likewise the problem with tiny
deviations. I believe the problem can be solved in this case
by measuring, in fractions of an octave, the amount of overlap
between scale degrees. This is implemented in Scala under
show data > Lumma impropriety factor.

> But for a typical piece of even moderate length the matrix
> would be enormous.

Yup.

> Some kind of iterative Monte Carlo-method might be more practical
> (from your description it seems JdL might use something like this).

Yup.

> I'm not much of a programmer but I have a short Pajara
> progression in mind which I intend to use as an example. I'll
> write a dedicated program (in FreeBasic) just to find a solution
> for this progression and probably start fantasizing about a
> general program after that.

Sweet!

-Carl

Yes, definitely he was a very original composer, some of his works are rather primitive from the point of classical music theory (probably that's the main reason why he was so many years almost in oblivion, ignored by official music world, and his works were considered unmusical), as he wrote them without deeper knowledge of music theory (and therefore he showed some new ways and attitudes how music can be done in far future), in some of them he tried to follow theory rules when he later studied music, but results are not so interesting, in some pieces he reached or even crossed limits of tonality... He was the first composer who wrote music without bar lines. One of the first who did motivic montage rather then motivic evolution. First one writing athematic music. And I could continue. His innovations are numerous.

His quartal chords are used as single chords or in rows, both was exploited later by Debussy and others (Schonberg, Scriabine, Janacek...).

Pentatonic scales started to be used intentionally in European art music since Mussorgskij. Some composers derived them from English, Irish and Scottish folk music, some other were inspired by African, Chinese, Japanese and Bali music (especially after Paris Exhibition Universelle 1889), Dvorak learned it in America from English, Afro-american and Native American music, Bartok and Kodaly find pentatonic base in real Hungarian folk music, other composers in folk music of some Baltic nations.

When you order for example anhemitonic pentatonic scale on black keys of piano as chain of fourths, you will get nice quartal five tone chord:

Bb Eb Ab Db Gb

From this is not too far to those typical Debussy progressions:

Ab Db Gb - Bb Eb Ab - Db Gb Bb - Eb Ab Db - Gb Bb Eb

Daniel Forro

On 11 Jan 2009, at 10:16 AM, Chris Vaisvil wrote:

> Satie is becoming more interesting the more I hear of his work. > Amazingly we devoted very little time on him in music theory. Is > this quartal harmony a single chord in a progression or a string of > quartal harmonies?
>
> From pentatonic scale... that is interesting too.
>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:

> These "rank-order matrices" as Rothenberg calls them are
> thus like scale classes. His claim is that if two scales
> have the same rank-order matrix, they are in a sense
> different tunings of the same scale.

Thanks for the info, Carl!

I think we should add that the periods of the scales must be
approximately the same. For example all the modes of the 13-tone equal
division of 3:1 have corresponding modes in 13-EDO which have the same
rank-order matrix. It's pretty hard to think of them as tunings of the
same scales!

Kalle

> Thanks for the info, Carl!
>
> I think we should add that the periods of the scales must be
> approximately the same. For example all the modes of the 13-tone equal
> division of 3:1 have corresponding modes in 13-EDO which have the same
> rank-order matrix. It's pretty hard to think of them as tunings of the
> same scales!
>
> Kalle

Indeed. I think Rothenberg considered only octave-based
scales.

-Carl

2009/1/11 Carl Lumma <carl@...>:

> Rothenberg has the only sensical method for doing this that
> I'm aware of, and even it has a problem with small pitch
> differences (two Bs 2 cents apart probably won't be perceived
> as different, but Rothenberg doesn't account for this).

Rothenberg did handle that. He said intervals have to be subjectively
ordered. If you hear two disjunct intervals and you can't tell which
is larger, they're equal. If these two Bs are perceived as the same,
they're the same.

Graham

--- In tuning@yahoogroups.com, "Graham Breed" <gbreed@...> wrote:
>
> 2009/1/11 Carl Lumma <carl@...>:
>
> > Rothenberg has the only sensical method for doing this that
> > I'm aware of, and even it has a problem with small pitch
> > differences (two Bs 2 cents apart probably won't be perceived
> > as different, but Rothenberg doesn't account for this).
>
> Rothenberg did handle that. He said intervals have to be
> subjectively ordered. If you hear two disjunct intervals and
> you can't tell which is larger, they're equal. If these two
> Bs are perceived as the same, they're the same.
>
> Graham

OK, but how do you take an arbitrary scale and compute its
perceived rank-order matrix. I don't recall him offering a
method for that.

-Carl