Stretched octaves on acoustic pianos

Hi there,

still wondering why acoustic pianos are tuned with stretched octaves. Should someone get beating octaves, why aren't they tuned with, for example, shrunk octaves ... Any idea?

Petr

The explanation that I've gotten is that this is done because an
acoustic piano's partials are stretched and inharmonic due to the
thickness of the strings. The partials are not in a strict harmonic
series. This inharmonicity varies from piano to piano - generally, the
longer the strings, the less this effect will be. Stretching the
octaves on a piano means that there is actually less beating, as the
partials line up more accurately in this case. It's sort of a
real-world application of using altered timbres to get more consonant
sounds out of, say, 11-tet or the like.

-Mike

On Tue, Jun 17, 2008 at 12:54 PM, Petr Pařízek <p.parizek@chello.cz> wrote:
> Hi there,
>
> still wondering why acoustic pianos are tuned with stretched octaves. Should
> someone get beating octaves, why aren't they tuned with, for example, shrunk
> octaves ... Any idea?
>
> Petr
>
>
>
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--- In tuning@yahoogroups.com, Petr Paøízek <p.parizek@...> wrote:
>
> Hi there,
>
> still wondering why acoustic pianos are tuned with stretched
> octaves. Should someone get beating octaves, why aren't they
> tuned with, for example, shrunk octaves ... Any idea?
>
> Petr

One theory is to achieve concord with the partials of the
instrument, which are stretched. Another (not mutually
exclusive) theory is that the human ear prefers some
stretch -- even for perfectly harmonic timbres -- over
large distances (e.g. if you double a melody 6 octaves
away it will sound flat if you use exact 2/1s). A final
explanation might be that if you stretch the octaves, you
can get pure 3:1 or even pure 3:2 intervals in all keys.

As far as tuning them shrunk, I'm the only person in the
world I know of doing that. The instrument loses its
'crisp' sound and some people balk at first... but the
3rds and 6ths definitely do benefit.

-Carl

--- In tuning@yahoogroups.com, "Mike Battaglia" <battaglia01@...> wrote:
>
> The explanation that I've gotten is that this is done because an
> acoustic piano's partials are stretched and inharmonic due to the
> thickness of the strings.

Stiffness of the strings, yes.

> The partials are not in a strict harmonic
> series. This inharmonicity varies from piano to piano - generally,
> the longer the strings, the less this effect will be. Stretching
> the octaves on a piano means that there is actually less beating,
> as the partials line up more accurately in this case.

It's been claimed that tunings are often stretched to a greater
degree than the inharmonicity of the instrument would explain.
Concert grands have less inharmonicity than regular pianos, but
according to Easley Blackwood are often stretched more.

-Carl

On Tue, Jun 17, 2008 at 3:02 PM, Carl Lumma <carl@...> wrote:
> --- In tuning@yahoogroups.com, "Mike Battaglia" <battaglia01@...> wrote:
>>
>> The explanation that I've gotten is that this is done because an
>> acoustic piano's partials are stretched and inharmonic due to the
>> thickness of the strings.
>
> Stiffness of the strings, yes.

See http://falstad.com/barwaves/j2/ for an enlightening simulation of
this phenomenon. (Select "stiff string, pinned" from the "Setup"
menu.)

Keenan

Carl wrote:
> One theory is to achieve concord with the partials of the
> instrument, which are stretched. Another (not mutually
> exclusive) theory is that the human ear prefers some
> stretch -- even for perfectly harmonic timbres -- over
> large distances (e.g. if you double a melody 6 octaves
> away it will sound flat if you use exact 2/1s). A final
> explanation might be that if you stretch the octaves, you
> can get pure 3:1 or even pure 3:2 intervals in all keys.

I've heard this, but does the standard MIDI GM synth for example do
this? Because with an 6-octave range the pitch seems to stay pretty
much the same to me, except in the extreme low register which
consistently sounds sharp. However, I think that is a separate
phenomenon.

On Tue, Jun 17, 2008 at 5:02 PM, Carl Lumma <carl@...> wrote:
> --- In tuning@yahoogroups.com, "Mike Battaglia" <battaglia01@...> wrote:
>>
>> The explanation that I've gotten is that this is done because an
>> acoustic piano's partials are stretched and inharmonic due to the
>> thickness of the strings.
>
> Stiffness of the strings, yes.
>
>> The partials are not in a strict harmonic
>> series. This inharmonicity varies from piano to piano - generally,
>> the longer the strings, the less this effect will be. Stretching
>> the octaves on a piano means that there is actually less beating,
>> as the partials line up more accurately in this case.
>
> It's been claimed that tunings are often stretched to a greater
> degree than the inharmonicity of the instrument would explain.
> Concert grands have less inharmonicity than regular pianos, but
> according to Easley Blackwood are often stretched more.

I would probably agree with this. It has to do with the same principle
that we were discussing earlier about 34-tet's stretched harmonics. To
me, at least, the stretching of an interval "accentuates" or
"brightens" it slightly, whereas the slight flattening of it
"diminishes" its presence or makes it "darker." Whether this is a
psychological thing or whether it has something to do with the brain
is beyond me, but I tend to think it's most likely psychological. It's
sort of like increasing the "contrast" knob on the interval slightly.

-Mike

Carl wrote:

> Another (not mutually
> exclusive) theory is that the human ear prefers some
> stretch -- even for perfectly harmonic timbres -- over
> large distances (e.g. if you double a melody 6 octaves
> away it will sound flat if you use exact 2/1s).

Maybe that's because the "natural" sounds usually have overtones with larger than pure harmonics, so the human hearing is "adjusted" in a certain way. It might be an interesting question whether this kind of "correlation" was an intentional result of some development or not -- similarly to the frequent statement that we hear frequencies in the range of about 2-4kHz louder than the others as a result of our ears being gradually adapted to the need of efficiently "evaluating" the formant frequencies in the human speech.

Petr

Mike wrote:
> > One theory is to achieve concord with the partials of the
> > instrument, which are stretched. Another (not mutually
> > exclusive) theory is that the human ear prefers some
> > stretch -- even for perfectly harmonic timbres -- over
> > large distances (e.g. if you double a melody 6 octaves
> > away it will sound flat if you use exact 2/1s). A final
> > explanation might be that if you stretch the octaves, you
> > can get pure 3:1 or even pure 3:2 intervals in all keys.
>
> I've heard this, but does the standard MIDI GM synth for example do
> this? Because with an 6-octave range the pitch seems to stay pretty
> much the same to me, except in the extreme low register which
> consistently sounds sharp. However, I think that is a separate
> phenomenon.

"General MIDI" doesn't really say much about a synth.
Ivory and Pianoteq do it, and any pure sampler with
full-keyboard samples will do it too if the sampled
instrument was stretched (which is very likely). Cheap
sampled pianos of the kind often called "General MIDI"
with only 6 samples for the whole keyboard may even
retain some of it, who knows.

> > It's been claimed that tunings are often stretched to a greater
> > degree than the inharmonicity of the instrument would explain.
> > Concert grands have less inharmonicity than regular pianos, but
> > according to Easley Blackwood are often stretched more.
>
> I would probably agree with this. It has to do with the same
> principle that we were discussing earlier about 34-tet's
> stretched harmonics. To me, at least, the stretching of an
> interval "accentuates" or "brightens" it slightly, whereas the
> slight flattening of it "diminishes" its presence or makes it
> "darker." Whether this is a psychological thing or whether
> it has something to do with the brain is beyond me, but I
> tend to think it's most likely psychological. It's sort of
> like increasing the "contrast" knob on the interval slightly.

Terhardt has a whole thing about "pitch shifts".

http://www.mmk.e-technik.tu-muenchen.de/persons/ter/top/pshifts.html

-Carl

Sorry for the clipboard snafu (I'm deleting it from the archives
so don't worry if you don't know what I'm talking about).
Reposting:

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@> wrote:
> > Another (not mutually
> > exclusive) theory is that the human ear prefers some
> > stretch -- even for perfectly harmonic timbres -- over
> > large distances (e.g. if you double a melody 6 octaves
> > away it will sound flat if you use exact 2/1s).
>
> Maybe that's because the "natural" sounds usually have
> overtones with larger than pure harmonics, so the human
> hearing is "adjusted" in a certain way.

According to Paul Erlich this is a fallacy. Most naturally-
occurring pitched sounds have perfectly harmonic overtones.

>It might be an interesting question whether this kind of
>"correlation" was an intentional result of some development
>or not -- similarly to the frequent statement that we hear
>frequencies in the range of about 2-4kHz louder than the
>others as a result of our ears being gradually adapted to
>the need of efficiently "evaluating" the formant frequencies
>in the human speech.

Maybe we should all have a look at that Terhardt page
http://www.mmk.e-technik.tu-muenchen.de/persons/ter/top/pshifts.html
I'm rusty myself.

-Carl

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

> >
> > Maybe that's because the "natural" sounds usually have
> > overtones with larger than pure harmonics, so the human
> > hearing is "adjusted" in a certain way.
>
> According to Paul Erlich this is a fallacy. Most naturally-
> occurring pitched sounds have perfectly harmonic overtones.

Right. Took the words right out of my mouth. Speaking of which, THE
most important "natural" sound is of course the human voice, which
even when speaking, is harmonic.
>
> >It might be an interesting question whether this kind of
> >"correlation" was an intentional result of some development
> >or not -- similarly to the frequent statement that we hear
> >frequencies in the range of about 2-4kHz louder than the
> >others as a result of our ears being gradually adapted to
> >the need of efficiently "evaluating" the formant frequencies
> >in the human speech.
>
> Maybe we should all have a look at that Terhardt page
> http://www.mmk.e-technik.tu-muenchen.de/persons/ter/top/pshifts.html
> I'm rusty myself.

Or just Google "formants". For instance:

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

Just one of the many on-line charts that will show that in fact most
of the formant frequencies lie well below 2 kHz, and of the higher
ones, very few are located in the 2-4 kHz range. Thus the slightly
increased sensitivity we have in that range (Google "phons" and look
for the dip) is NOT due to anything having to do with speech, a fact
which can easily be proven by low-pass filtering any recording of
speech. Chopping everything above about 1,5 kHz reduces the
intelligibility only ever so slightly, and the greatest loss is in
consonants, not vowels.

Far more important for human survival are auditory cues of approaching
prey: snapping twigs, respiration, snarls, flapping wings, things of
this nature, even things as subtle as acoustic shading of ambient
white noise caused by large approaching sound absorbing bodies, like
lions, tigers, and such. These sorts of life-saving cues DO tend to be
composed primarily of higher frequencies, which might well explain the
dip in the phons curve. I'm not sure about the exact frequencies, but
our outer ear is shaped in order to turn it into a highly sensitive
directional locater at higher frequencies.

Regarding octave stretch. Whenever that comes up someone always chimes
in with the same old arguments about how we have a preference for wide
octaves. But as Terhard quite rightly points out, this is only
melodically. When we hear octaves harmonically, we are spot on 2:1, of
4:1, or 8:1, etc.

Pianos are tuned with stretch primarily in order to compensate for the
inharmonicity of the strings, which is not constant over the entire
instrument; it tends to be more pronounced in the lowest and highest
octaves. A good tuner simply tunes "pure" (no beats), which in the
case of a slightly inharmonic tone is by definition an inharmonic
octave. It SOUNDS pure, but if measured, it is not 2:1. In the case of
more complex inharmonic tones, like bells, the perception of "pitch"
becomes highly complicated, and can be higher or lower than the
"fundamental", which in any event in the case of bells is usually the
second partial, the "hum tone" being to weak to be taken as the
fundamental.

Once again, read Sethares. A truly landmark work (excepting the
chapter on historical temperaments, which is fundamentally flawed in
concept - oh well, nobody's perfect!).

That said, a number of piano tuners don't know what they are doing and
stretch too much or in regions which don't need it. This is especially
true of those who have learned the craft with electronic devices. They
simply enter a certain "stretch curve" and go! Throw in the ubiquitous
lack of discrimination between melodic and harmonic perception theory
and you've got a recipe for general confusion and a large body of ugly
sounding pianos. Until you get used to it, and then it sounds
"normal", and a properly tuned piano sounds out-of-tune. Who really
listens, anyway?

Ciao,

P

For Carl and Paul P.:

Carl wrote:

> According to Paul Erlich this is a fallacy. Most naturally-
> occurring pitched sounds have perfectly harmonic overtones.

Hmmm, if that should be true, then I'm not sure what's meant by "naturally-occurring". Wonder where Paul E. got this claim. Whatever they get stretched or shrunk, I don't think they can get totally pure. For at least two reasons:

- 1. Pick a mid-pitched string (like the high E on a guitar, for example) and try to tune it terribly low (like two octaves lower or something similar) and pluck it. You'll immediately HEAR (you don't need to measure anything) that the intervals in the overtone series are now much wider than when the string was tight. After trying to reach a meaningful conclusion for years, I gave it up and asked prof. Joe Wolfe of the Australian UNSW what he thought about it. He himself has told me that to get the overtones 100% harmonic, the string would have to be "infinitely tight", which is impossible as this would stop it vibrating. This could explain why some musicians (especially somewhere around Wienna, AFAIK) tend to tune A4 to 444 or 445Hz.

- 2. If I make a hole into a hollow pipe and put something like a piece of cork very far from it, then, when blowing into the hole, I'll not only hear the higher overtones softer but also all the intervals will sound shrunk compared to pure harmonics. The closer I put the piece of cork to the hole, the closer the overtones get to pure harmonics.

And maybe there could be even more examples of stretching and shrinking overtones, I don't know. But anyway, why should human voice be an exception breaking all of these rules? Or is there perhaps something wrong with my hearing and with prof. Wolfe's statements?

Petr

Hiya Paul,

> Far more important for human survival are auditory cues of
> approaching prey: snapping twigs, respiration, snarls, flapping
> wings, things of this nature,

I strongly disagree. The most important thing for a social
animal like a human is to be able to understand the vocalizations
of other humans. If you're twig-snap deaf you might just stay
home telling stories or knitting. Human audition has plenty
of stuff for detecting sharp noises, like many other animals.
But the bulk of the hardware -- and the part unique to our
species* -- is the stuff for processing human speech.

* Other species with highly refined vocalizations do have
specialized hardware for them -- birds for example.

-Carl

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
>
> For Carl and Paul P.:
>
>
>
> Carl wrote:
>
>
>
> > According to Paul Erlich this is a fallacy. Most naturally-
> > occurring pitched sounds have perfectly harmonic overtones.
>
>
>
> Hmmm, if that should be true, then I'm not sure what's meant by
"naturally-occurring". Wonder where Paul E. got this claim. Whatever
they get stretched or shrunk, I don't think they can get totally pure.
For at least two reasons:
>
> - 1. Pick a mid-pitched string (like the high E on a guitar, for
example) and try to tune it terribly low (like two octaves lower or
something similar) and pluck it. You'll immediately HEAR (you don't
need to measure anything) that the intervals in the overtone series
are now much wider than when the string was tight. After trying to
reach a meaningful conclusion for years, I gave it up and asked prof.
Joe Wolfe of the Australian UNSW what he thought about it. He himself
has told me that to get the overtones 100% harmonic, the string would
have to be "infinitely tight", which is impossible as this would stop
it vibrating.

Yes, this is correct. But don't turn a molehill into a mountain. Most
string instruments with relatively thin strings have overtones which
are very very close to being harmonic, for all practical purposes,
harmonic. Only at very low tensions do they become noticably inharmonic.

It's really rather simple. The normal restoring force of a stretched
string is tension. Rigidity creates an additional restoring force, the
more rigid, the more extra force. This force increases the
acceleration of the returning string during each cycle, and is more
significant at higher modes. Thus higher overtones are shifted upward
in frequency.

When you tune a string way down, you have reduced the normal restoring
force drastically, but rigidty is the same, meaning it is much more
signficant. Thus inharmonicity goes up. So what? This proves nothing.

> This could explain why some musicians (especially somewhere around
Wienna, AFAIK) tend to tune A4 to 444 or 445Hz.

Say WHAT? There's nothing Holy or universal about ANY reference pitch.
As far as I know the idea does not apply to any body of folk music -
not even to traditions like western classical.
>
> - 2. If I make a hole into a hollow pipe and put something like a
piece of cork very far from it, then, when blowing into the hole, I'll
not only hear the higher overtones softer but also all the intervals
will sound shrunk compared to pure harmonics. The closer I put the
piece of cork to the hole, the closer the overtones get to pure harmonics.

Your explanation is little unclear, so I don't know what you mean. I
DO know that all wind instruments produce harmonic tones. They can be
very INharmonic in register shifts, but that has nothing to do with
the structure of each individual tone. So if you are hearing
inharmonicity in a single tone, get thee to a lab, FAST! You having
discovered something which previously has not existed.
>
> And maybe there could be even more examples of stretching and
shrinking overtones, I don't know. But anyway, why should human voice
be an exception breaking all of these rules? Or is there perhaps
something wrong with my hearing and with prof. Wolfe's statements?

I vote for the latter. Or a version of it: something wrong with your
hearing and not understanding what Wolfe is saying.

Ciao,

p

--- In tuning@yahoogroups.com, "Paul Poletti" <paul@...> wrote:
>
>
> Pianos are tuned with stretch primarily in order to compensate for the
> inharmonicity of the strings, which is not constant over the entire
> instrument; it tends to be more pronounced in the lowest and highest
> octaves. A good tuner simply tunes "pure" (no beats), which in the
> case of a slightly inharmonic tone is by definition an inharmonic
> octave. It SOUNDS pure, but if measured, it is not 2:1.
>
Here some references on that observation:

http://en.wikipedia.org/wiki/Piano_acoustics
http://en.wikipedia.org/wiki/Inharmonicity
http://www.postpiano.com/support/updates/tech/Tuning.htm
http://dafx04.na.infn.it/WebProc/Proc/P_212.pdf
http://www.afn.org/~afn49304/youngnew.htm
http://www.acoustics.auckland.ac.nz/research/research_files/keane_nzas04.pdf
http://www.pianoworld.com/ubb/cgi-bin/ultimatebb.cgi?ubb=get_topic;f=3;t=002745;p=0
http://www.speech.kth.se/music/5_lectures/introd/introd.html
http://www.acoustics.org/press/134th/galembo.htm
http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JASMAN0000760000S1000S22000004&idtype=cvips&gifs=yes
http://adsabs.harvard.edu/abs/1994ASAJ...95.2887Z
http://lib.tkk.fi/Diss/2007/isbn9789512290666/isbn9789512290666.pdf
http://www.wellesley.edu/Physics/brown/pubs/freqRatV99P1210-P1218.pdf
http://www.amarilli.co.uk/piano/theory/paradigm.asp
"
Provided it is understood that "beating" and "beats" are somewhat
generic terms, it is reasonable to say that (aural) piano tuners tune,
in part, by "listening to beats". However, rather than using the
paradigm of tuners applying beat rates to "aurally estimate" some
theoretically "worked out" tuning, perhaps a more accurate paradigm
would be:
The finest piano tuning is an art, carried out by tuners drawing on a
deep empirical knowledge of piano tone and tuning behaviour. All
current theoretical models for piano tuning are just models. Even if
we improved the model to account for the data we now have for piano
tone behaviour, in the words of Samuel Karlin, The purpose of models
is not to fit the data, but to sharpen the questions.
(Karlin, S, 11th RA Fischer Memorial Lecture, Royal Society,
20/4/1983, cited in Buchanan, Mark, Ubiquity, Phoenix, 2000)
"

http://www.acoustics.hut.fi/~bbank/demo.html
http://iwk.mdw.ac.at/instrumentenkunde/literatur/klavier.htm
http://www.ibs.it/book/9781905209262/guillaume-philippe/music-and-acoustics.html
http://www.citeulike.org/user/wmppaul/article/2763561
http://www.britannica.com/EBchecked/topic/288184/inharmonicity
http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JASMAN000094000001000046000001&idtype=cvips&gifs=yes
http://www.acoustics.org/press/146th/Wogram.htm
http://www.amarilli-books.co.uk/ecom/index1.html
http://www.ingentaconnect.com/content/dav/aaua/2004/00000090/00000003/art00013
http://www.ncbi.nlm.nih.gov/pubmed/11572374http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?tp=&arnumber=784113&isnumber=17013
http://ieeexplore.ieee.org/iel5/89/17013/00784113.pdf?arnumber=784113
http://www.speech.kth.se/prod/publications/files/qpsr/1994/1994_35_1_135-144.pdf
http://www.time2tune.com/Ear_or_Machine.html
http://www.economicexpert.com/a/Piano:acoustics.htm

in German:
http://de.wikipedia.org/wiki/Inharmonizit%C3%A4t
http://musikforskning.se/stm/STM1973/STM1973Sirker.pdf
http://www.troostmuziek.nl/gehoor/auf_die_ohren.htm
http://www.ptb.de/de/org/1/17/173/pdf/klavierfluegel.pdf

A diagramm says more than 1000 words:
http://www.petersontuners.com/oldweb/images/bank/Rhodes-stretchW.jpg

hope that at least a few of them can help you

Yours Sincerly
A.S.

Petr wrote:
> > According to Paul Erlich this is a fallacy. Most naturally-
> > occurring pitched sounds have perfectly harmonic overtones.
>
> Hmmm, if that should be true, then I'm not sure what's meant by
> "naturally-occurring".

Human voice, all reed and brass winds, and bowed strings.

> Wonder where Paul E. got this claim.

His substantial understanding of physics.

> - 1. Pick a mid-pitched string (like the high E on a guitar,

Plucked and struck strings are an exception. But even
with plucked strings, the inharmonicity is usually pretty
well confined to the "A" part of the ASDR envelope.

> for example) and try to tune it terribly low (like two octaves
> lower or something similar) and pluck it. You'll immediately
> HEAR (you don't need to measure anything) that the intervals
> in the overtone series are now much wider

Yes, loose strings are also an exception. But the quality
of the *pitch* will also deteriorate (noticed I said "pitched"
sounds in my previous message above). Likewise metal bars
are inherently inharmonic. A number of subtle tricks is also
required to elicit a strong pitch from them for use in
instruments like xylophones.

> He himself has told me that to get the overtones 100% harmonic,
> the string would have to be "infinitely tight", which is
> impossible as this would stop it vibrating. This could explain
> why some musicians (especially somewhere around Wienna, AFAIK)
> tend to tune A4 to 444 or 445Hz.

That sounds right, except it doesn't apply to bowed strings.
Also in practice a good harpsichord or guitar will have
harmonic spectra (after the attack) within the limits of
the ear (through the upper-middle of their range, anyway).

> And maybe there could be even more examples of stretching and
>shrinking overtones, I don't know. But anyway, why should human
>voice be an exception breaking all of these rules?

Because the vocal folds exhibit SHM:
http://en.wikipedia.org/wiki/Simple_harmonic_motion

-Carl

Paul Poletti wrote:

> Yes, this is correct. But don't turn a molehill into a mountain. Most
> string instruments with relatively thin strings have overtones which
> are very very close to being harmonic, for all practical purposes,
> harmonic. Only at very low tensions do they become noticably inharmonic.

I know. And I don't say that I disagree. I was just trying to find out if there is some kind of rule according to which it's possible to tell something like an "average amount" of overtone detuning, for example, for a particular string thickness of X (something) and tension of Y (something else).

> Your explanation is little unclear, so I don't know what you mean. I
> DO know that all wind instruments produce harmonic tones. They can be
> very INharmonic in register shifts, but that has nothing to do with
> the structure of each individual tone. So if you are hearing
> inharmonicity in a single tone, get thee to a lab, FAST! You having
> discovered something which previously has not existed.

That's exageration.

I don't think I need to explain how an overtone flute works --- briefly, it's the one with no finger holes (you change the tones just by changing the speed and thickness of the air stream or by opening/closing the opposite end of the flute with your hand). A hollow pipe with a hole near one end can serve as such a flute if you put a piece of cork (or simply something which prevents the air from getting to places you don't want) near the blowing hole. But if the cork is not close enough to the blowing hole, than the overtone series is rather mistuned as compared to regular harmonics -- to be more precise, the intervals are "audibly" smaller. OTOH, if the cork is too close to the hole, then the overtones themselves are essentially harmonic but this could stop the sound echoing. So it's important to find some position "in-between" where the sound is echoing well in the pipe and at the same time the overtones sound more or less harmonic. Concerning normal non-overtone flutes, care should be also taken while overblowing to higher registers because if the overtones were harmonic and you blowed stronger than when playing in the low register, this could actually make octaves significantly wider. Prof. Wolfe mentions this on his webpage about Boehm flutes: www.phys.unsw.edu.au/jw/fluteacoustics.html#cork

Petr

An Acoustician at Sydney university recently told me the same.

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Carl Lumma wrote:
>
> -
>
> According to Paul Erlich this is a fallacy. Most naturally-
> occurring pitched sounds have perfectly harmonic overtones.
>
>
>

Funny it was Wolfe who told me two weeks ago that they were basically harmonic. and i questioned him about it further

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere:
North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Petr Par�zek wrote:
>
> For Carl and Paul P.:
>
> Carl wrote:
>
> > According to Paul Erlich this is a fallacy. Most naturally-
> > occurring pitched sounds have perfectly harmonic overtones.
>
> Hmmm, if that should be true, then I�m not sure what�s meant by > �naturally-occurring�. Wonder where Paul E. got this claim. Whatever > they get stretched or shrunk, I don�t think they can get totally pure. > For at least two reasons:
>
> - 1. Pick a mid-pitched string (like the high E on a guitar, for > example) and try to tune it terribly low (like two octaves lower or > something similar) and pluck it. You�ll immediately HEAR (you don�t > need to measure anything) that the intervals in the overtone series > are now much wider than when the string was tight. After trying to > reach a meaningful conclusion for years, I gave it up and asked prof. > Joe Wolfe of the Australian UNSW what he thought about it. He himself > has told me that to get the overtones 100% harmonic, the string would > have to be �infinitely tight�, which is impossible as this would stop > it vibrating. This could explain why some musicians (especially > somewhere around Wienna, AFAIK) tend to tune A4 to 444 or 445Hz.
>
> - 2. If I make a hole into a hollow pipe and put something like a > piece of cork very far from it, then, when blowing into the hole, I�ll > not only hear the higher overtones softer but also all the intervals > will sound shrunk compared to pure harmonics. The closer I put the > piece of cork to the hole, the closer the overtones get to pure harmonics.
>
> And maybe there could be even more examples of stretching and > shrinking overtones, I don�t know. But anyway, why should human voice > be an exception breaking all of these rules? Or is there perhaps > something wrong with my hearing and with prof. Wolfe�s statements?
>
> Petr
>
>

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
> I know. And I don't say that I disagree. I was just trying to find
out if there is some kind of rule according to which it's possible to
tell something like an "average amount" of overtone detuning, for
example, for a particular string thickness of X (something) and
tension of Y (something else).
>

Hi Petr,

One simple rule is: If the system is being continuously driven, i.e.
energy is being put into the system continuously, as opposed to being
put in in a single impulse, then the partials will be strictly
harmonic. It doesn't matter whether the natural resonances of the
system are harmonic or not.

If the natural resonances are inharmonic, as in the flute you
described, these inharmonic resonances will only be heard when the
input stops and the system decays.

Examples of continuously driven systems are: voice, wind instruments,
brass instruments, bowed strings.

Struck or plucked instruments may or may not be inharmonic depending
on stiffness and mass distribution and nonlinearities in same. There
will be some formulae for simple cases. I'm sorry I can't direct you
to them.

> > DO know that all wind instruments produce harmonic tones. They can be
> > very INharmonic in register shifts, but that has nothing to do with
> > the structure of each individual tone. So if you are hearing
> > inharmonicity in a single tone, get thee to a lab, FAST! You having
> > discovered something which previously has not existed.
>
> That's exageration.
>
> I don't think I need to explain how an overtone flute works ---
briefly, it's the one with no finger holes (you change the tones just
by changing the speed and thickness of the air stream or by
opening/closing the opposite end of the flute with your hand). A
hollow pipe with a hole near one end can serve as such a flute if you
put a piece of cork (or simply something which prevents the air from
getting to places you don't want) near the blowing hole. But if the
cork is not close enough to the blowing hole, than the overtone series
is rather mistuned as compared to regular harmonics -- to be more
precise, the intervals are "audibly" smaller. OTOH, if the cork is too
close to the hole, then the overtones themselves are essentially
harmonic but this could stop the sound echoing. So it's important to
find some position "in-between" where the sound is echoing well in the
pipe and at the same time the overtones sound more or less harmonic.
Concerning normal non-overtone flutes, care should be also taken while
overblowing to higher registers because if the overtones were harmonic
and you blowed stronger than when playing in the low register, this
could actually make octaves significantly wider. Prof. Wolfe mentions
this on his webpage about Boehm flutes:
www.phys.unsw.edu.au/jw/fluteacoustics.html#cork
>

You seem not to have understood Paul Poletti's point. The resonances
of the flute may be very inharmonic, leading to an inharmonic series
of tones, but each tone will still have harmonic partials (except
during the short decay after blowing is stopped).

I believe Joe Wolfe's paper agrees with this, although I can see how
it could easily be misunderstood.

-- Dave Keenan

Dave Keenan wrote:

> One simple rule is: If the system is being continuously driven, i.e.
> energy is being put into the system continuously, as opposed to being
> put in in a single impulse, then the partials will be strictly
> harmonic. It doesn't matter whether the natural resonances of the
> system are harmonic or not.

Hah, it seems you're the one who has eventually found an explanation I understand. :-D Thanks.

Now it's upon me to look for the answer to the actual question of the amount of overtone stretching for struck strings (such as in acoustic pianos, for example) and that's it.

Petr

Kraig wrote:

> Funny it was Wolfe who told me two weeks ago that they were basically
> harmonic. and i questioned him about it further

Talking Wolfe, let's talk Wolfe's words rather than mine: www.phys.unsw.edu.au/jw/harmonics.html

Petr

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, Petr Paøízek <p.parizek@> wrote:
> >
> > Hi there,
> >
> > still wondering why acoustic pianos are tuned with stretched
> > octaves. Should someone get beating octaves, why aren't they
> > tuned with, for example, shrunk octaves ... Any idea?
> >
> > Petr
>
> One theory is to achieve concord with the partials of the
> instrument, which are stretched. Another (not mutually
> exclusive) theory is that the human ear prefers some
> stretch -- even for perfectly harmonic timbres -- over
> large distances (e.g. if you double a melody 6 octaves
> away it will sound flat if you use exact 2/1s). A final
> explanation might be that if you stretch the octaves, you
> can get pure 3:1 or even pure 3:2 intervals in all keys.
>
> As far as tuning them shrunk, I'm the only person in the
> world I know of doing that. The instrument loses its
> 'crisp' sound and some people balk at first... but the
> 3rds and 6ths definitely do benefit.
>
> -Carl
>

Check out shimmering octaves tuned sharp, and shimmering octaves
tuned flat, on Indonesian gamelan instruments.

-Cris

Cris Forster wrote:

> Check out shimmering octaves tuned sharp, and shimmering octaves
> tuned flat, on Indonesian gamelan instruments.

Audio examples would be more than welcome.

Petr

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
>
> Cris Forster wrote:
>
> > Check out shimmering octaves tuned sharp, and shimmering octaves
> > tuned flat, on Indonesian gamelan instruments.
>
> Audio examples would be more than welcome.
>
> Petr
>

Gender barungs, Gamelan Kyahi Kanjutmesem, Java.

-Cris

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
>
> Kraig wrote:
>
> > Funny it was Wolfe who told me two weeks ago that they were
basically
> > harmonic. and i questioned him about it further
>
> Talking Wolfe, let's talk Wolfe's words rather than mine:
> www.phys.unsw.edu.au/jw/harmonics.html
>
> Petr
>

I completely agree with all the descriptions at:

www.phys.unsw.edu.au/jw/harmonics.html

From _Musical Mathematics_, Chapter 3:

"Perfectly flexible strings do not exist. All strings must exhibit
a minimum amount of stiffness; otherwise, they could not resist the
force of tension."

With respect to flutes, organ pipes, etc., all columns of air are
subject to end corrections.

-Cris

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@...> wrote:
>
>
> Hi Petr,
>
> One simple rule is: If the system is being continuously driven, i.e.
> energy is being put into the system continuously, as opposed to being
> put in in a single impulse, then the partials will be strictly
> harmonic. It doesn't matter whether the natural resonances of the
> system are harmonic or not.

Not ALWAYS true. For instance, you can drive a piano string
continuously with a jet of compressed air, and it is still inharmonic.
It depends whether or not there is a feedback loop between excitation
mechanism and resonator.
>
> If the natural resonances are inharmonic, as in the flute you
> described, these inharmonic resonances will only be heard when the
> input stops and the system decays.

A nigh to impossible situation. A better way to "hear" it is to use it
as a passive resonator, i.e. excite it with white noise produced by a
loudspeaker. The filtered noise you hear at the other end will contain
peaks at all of the resonances, be they harmonic or not.
>
> Examples of continuously driven systems are: voice, wind instruments,
> brass instruments, bowed strings.

The voice is a complicated system and shouldn't really be included
among these as there is no active resonator which cooperates in a
feedback loop stabilizing the vibration of the vocal chords, as there
is with valve-driven wind instruments, be they lip valve or reed
valve. A closer analogy to the voice would be a system using a free
reed as an oscillator.
>
> Struck or plucked instruments may or may not be inharmonic depending
> on stiffness and mass distribution and nonlinearities in same. There
> will be some formulae for simple cases. I'm sorry I can't direct you
> to them.

Yes, you can play clever tricks with the modal frequencies by
introducing nonlinear mass or rigidity. But normally this is not the
case in real musical instrument (excepting gut strings, which can
develop some pretty serious nonlinear aspects with time, which
generally means they get so funky in response you have to change
them). So basically, struck/plucked strings ALWAYS are inharmonic. The
only question is, is it severe enough to cause any problems. With
harpsichords, clavichords, and early fortepianos, and modern steel
strung guitars, the strings are thin enough/tensioned enough that the
inharmonicity is for all practical purposes nil, even though it IS there.

If you look around for "coefficient of inharmonicty" you'll quickly
come across formulas that predict it. They even work more or less
correctly some of the time.

;-)

Ciao,
P

Cris wrote:

> Gender barungs, Gamelan Kyahi Kanjutmesem, Java.

The only thing Google has suggested are articles on JStor which says "You are not currently authorized to view this article".

Petr

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
>
> Cris wrote:
>
>
>
> > Gender barungs, Gamelan Kyahi Kanjutmesem, Java.
>
>
>
> The only thing Google has suggested are articles on JStor which
says "You are not currently authorized to view this article".
>
>
>
> Petr
>

Surjodiningrat, W., Sudarjana, P.J., and Susanto, A. (1972). Tone
Measurements of Outstanding Javanese Gamelans in Jogjakarta and
Surakarta, 2nd ed. Gadjah Mada University Press, Jogjakarta,
Indonesia.

Cris

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
>
>
> I know. And I don't say that I disagree. I was just trying to find
out if there is some kind of rule according to which it's possible to
tell something like an "average amount" of overtone detuning, for
example, for a particular string thickness of X (something) and
tension of Y (something else).

As I said in the other post, there are formulas for the coefficient of
inharmoncity. You need to know Young's modulus for the string
material. I've got one up and running in a spreadsheet, but I don't
remember where I got it originally.
>
>
> I don't think I need to explain how an overtone flute works ---

Yes, I know about the seljefløyte and it's various relatives. I just
couldn't figure out exactly what you were saying with the cork business.

> briefly, it's the one with no finger holes (you change the tones
just by changing the speed and thickness of the air stream or by
opening/closing the opposite end of the flute with your hand). A
hollow pipe with a hole near one end can serve as such a flute if you
put a piece of cork (or simply something which prevents the air from
getting to places you don't want) near the blowing hole. But if the
cork is not close enough to the blowing hole, than the overtone series
is rather mistuned as compared to regular harmonics -- to be more
precise, the intervals are "audibly" smaller.

Right, now I know what you were getting at. Much as I suspected, you
are confusing impedance minima/maxima with spectral inharmonicity, a
common misconception. While the two are closely related, they are in
fact two completely different things.

Impedance determines the frequencies to which the oscillation will
jump when the tube is overblown, but it does NOT affect the component
frequencies of individual complex tones produce by the instrument;
these tones are always perfectly harmonic. This is due to "mode
locking"; when a tube has impedance peaks or dips which are
inharmonic, when it is driven by an excitation mechanism that is
couple to the tube and has an undefined frequency of oscillation of
its own, the totality of the complex tone produced by the
excitor/resonator duality (both forming the oscillator) is a harmonic
"best fit" of them all. The way to experience this on your overtone
flute is to use a setup which produces a very inharmonic register
shift pattern, and then make recordings of the individual notes. Exam
them with any spectral analysis software and I am absolutely certain
you will find that the spectral content of each tone is exactly harmonic.

Clarinet is a perfect example. When it is overblown, it follows a
register shift pattern which approximates only the odd harmonics, but
the notes are rather out of tune, and the higher you go, the more out
of tune they are, and quite so. Yet a spectral analysis will
demonstrate that any one of the tones in any register is completely
harmonic. Furthermore, contrary to what most people think, the
spectral content is NOT limited to only the odd harmonics. The common
misconception that even harmonics are completely absent in clarinet
tone is only vaguely true for the notes in the first register. Even
these are only missing the 2 and 4th harmonic; above 5, they are all
there, even and odd, though odd tend to be stronger. In the second
register already the tone has a very strong presence of all harmonics,
even and odd. This is due once again to the different acoustic
functions of the tube: the primary function of determining the
fundamental frequency and the secondary function of filtering the
spectral content produced by the excitation mechanism. Because the
feedback loop of mode locking, the reed can only fundamentally excite
the tube at one of its resonant frequencies, but the reed itself is
producing a tone which is fairly rich in harmonics because it doesn't
open and close in a perfect sin wave. This tone is in turn filtered by
the resonances of the tube to a greater or lessor extent, but the even
harmonics produced by the reed are by no means completely eliminated
by this filtering process. Confusing these to separate functions leads
to much common misconception about the function of wind instruments.

One of the things I always stress in my classes is that
"inharmonicity" in strings and in wind instruments manifests in
fundamentally different ways, so any time ones hears the word one
needs to be attentive to context.

Ciao,

P

> If you look around for "coefficient of inharmonicty" you'll quickly
> come across formulas that predict it. They even work more or less
> correctly some of the time.
>
> ;-)
>
> Ciao,
> P

The coefficient of inharmonicity is a function of the stiffness
parameter J, and so not easy to calculate.

Having restrung many pianos, I find the coefficient of inharmonicity
for plain strings complete reliable However, for wound strings, all
such calculations are problematic.

-Cris

--- In tuning@yahoogroups.com, "Cris Forster" <cris.forster@...> wrote:

>
> The coefficient of inharmonicity is a function of the stiffness
> parameter J, and so not easy to calculate.

Don't you mean E? Young's modulus? Yeah, I know there are slight
variations depending on which direction, but it's more or less the same.
>
> Having restrung many pianos, I find the coefficient of inharmonicity
> for plain strings complete reliable However, for wound strings, all
> such calculations are problematic.

Which formula are you using? How do you get your values of E (or J)
for the wire you are using? How do you verify partial detuning? How
high have you checked in the partial series?

Ciao,

P

Paul Poletti wrote:

> Much as I suspected, you
> are confusing impedance minima/maxima with spectral inharmonicity, a
> common misconception. While the two are closely related, they are in
> fact two completely different things.

I don't know why I didn't understand for so long; I just had to read Dave Keenan's post a few hours ago and, suddenly, the light came. Now I'm realizing I actually wasn't reading Wolfe's webpage with much attention about 7 years ago and that he, in fact, explains the matter similarly to the way you do -- but in such a way I was unable to understand in 2001 when I first discovered the website; I'm convinced that if I had got the answer to my question from you or dave at that time, I would probably have understood the explanation better.

Petr

Hello, there, Petr and Cris and all.

Please let me provide a brief example of a stretched-octave sound in a
gamelan style. In this version of slendro, there are also some complex
unisons:

<http://www.bestII.com/~mschulter/ForErin.mp3>

Of course, I'd be delighted to get into intonational specifics if
people are interested, but the actual sound is the main thing.

Most appreciatively,

Margo Schulter
mschulter@...

Hello, Dave Keenan, and all.

First, Dave, thank you so much for posting that article with a link to
my sampling of metastable intervals and progressions in Zest-24, which
had been temporarily unavailable during some changes at the ISP where
it is located, but is now again accessible, thanks to your catalytic
interest and also the outstanding and friendly technical service at
www.bestII.com.

While there are many responses one could make to your "metallic"
scheme suggesting one possible classification of metastable, or
merciful, or cloudy, or grey intervals -- henceforth "MI" -- here I'd
like to offer some comments with an eye to theory and one hand planted
firmly on the keyboard, so to speak.

Above all, I'd like to emphasize my pleasure and joy to see that not
quite eight years after our paper on the noble mediant, the concept is
attracting musical interest in theory and practice. Your intuitive
insight, in the midst of our conversations about neo-Gothic music and
complex ratios, that Keenan Pepper's focus on Phi might afford us a
mathematical tool for identifying approximate regions of maximum
complexity, was a highlight at once to inspire intellectual delight
and to motivate new musical explorations.

My purpose here is to express my appreciation while suggesting one
possible perspective to put this and related questions of harmonic
entropy or tension in a larger musical context. This isn't to tie the
general kind of theory we're discussing to any specific tuning system
or style, only suggest that the theory may take on greater
concreteness as people do explore the JI-to-MI continuum in a variety
of systems and styles. The challenge seems always to be taking into
account a range of specifics while striving not to preclude new
systems or styles from emerging which might, in turn, both enrich and
modify the established general theory.

One drawback of theorizing in a "rich context" of style and
counterpoint is that not only might it lead to overgeneralizations,
but it tends to lead to longer posts, not always ideal for easy and
relaxed dialogue. I apologize for this possible drawback, and seek as
a partial remedy to invite you or anyone to "break in" at any point
and offer a comment or ask a question.

------------------------------------
1. The importance of musical context
------------------------------------

What I would emphasize at the outset is that how we view tunings or
specific intervals may depend on how we make music, and what styles we
are accustomed to, as has been suggested in certain recent discussions
here. Thus I find 17-EDO or George Secor's masterful 17-WT a "natural"
system both because the structure nicely fits a 13th-14th century
European style to which I'm accustomed, and because of the most
fruitful musical dialogue and friendship which George has shared with
me in an incredibly rich "cross-pollination," to use his felicitous
term.

Also, I might guess that intervals in the general region of 950 cents
are very attractive to me because they fit with a style where a major
sixth expands by stepwise contrary motion to an octave, while a minor
seventh likewise contracts to a fifth, with either type of resolution
often being possible in this "interseptimal" region between 12:7 and
7:4. Also, anything in the range of around 940-960 cents is likely to
delight me, because of these contrapuntal and melodic possibilities,
quite apart from the niceties of harmonic entropy or "just how
metastable is really metastable?" Further, I would happily use a 12:7
major sixth or 7:4 minor seventh in similar contexts: the different
shadings are largely "musically interchangeable," like nuances of a
well-temperament, with variety itself the spice of style.

Someone trying out 17-EDO or George's 17-WT from a perspective
grounded in 18th-19th century European styles, or using 950 cents as a
representation of 4:7 in styles of this kind or others where 4:5:6:7
is intended rather than a stimulating variation on something like
4:6:7 or 14:21:24 or 12:14:18:21 or 14:18:21:24 resolving by stepwise
contrary motion, may have a quite different response. The tuning
environment is likely to seem not su much a stimulating variation on
the familiar as a more or less radical departure or dislocation,
whether this is taken as a clue to consider a different tuning, or a
fascinatingly "warped" intonational space with new possibilities.

----------------------------------------------------------------
2. A continuum: MI, JI, RI, TI -- and VI (Variegated Intonation)
----------------------------------------------------------------

This brings me to a main point about MI: however it is defined, I
would regard it simply as one part of a useful continuum where all
points have musical value, and fine shadings of harmonic entropy serve
mostly to adorn a contrapuntal and melodic texture rather than to
dictate its structure. Thus I'd consider the MI region of 422 cents a
great neomedieval major third (often expanding to a fifth, or
contracting to a unison) -- and likewise anything within 15 cents on
either side. In Zest-24, for example, the usual sizes would be 408,
409, 422, or 434 cents. The nuances of entropy or color ornament the
happy fact of musical interchangeability. Further, in this
temperament, I often use sizes of 441, 442, 445, 454, or 455 cents in
a similar fashion as very large major thirds, relishing that the last
three values especially are in the interseptimal region between 9:7
and 21:16.

From my perspective of Variegated Intonation or VI, where lots of
subtle shadings are a big plus and indeed a system like Zest-24 seems
to specialize in being "all over the map" with a relatively modest
number of notes, one might envision MI as part of a continuum about as
follows, which may tie in with your metallic concepts.

If we start with JI involving small integer ratios, or "valleys" of
clarity and strong consonance like 1:1, 2:1, 3:2, and 4:3, and let the
ratios grow more complex, then we eventually reach the realm of
Rational Intonation or RI where large integer ratios may more or less
blend in with surrounding irrational ratios along the continuum --
adding, as you often have, that in special sonorities such as
isoharmonic ones very complex RI ratios may become "aurally JI." All
points are valued options on the spectrum.

If we start with noble mediants between small integer ratios or JI
valleys, then we arrive at approximate peak regions for harmonic
entropy, tension, ambiguity, cloudiness, or diffuse richness -- MI,
however it might be experienced. As the integer ratios being mediated
in a Phi-weighted fashion get more complex and RI-ish, however, the
connection between a mathematical noble mediant and aural MI becomes
more tenuous, and ultimately may point us to regions interesting
because of other musical considerations than "maximum entropy."

One term that occurs to me which might embrace the use both of complex
RI ratios and of noble mediants derived from such ratios would be
"Tele-Intonation" or TI -- using mathematical tools associated with
the simplest (JI) and most complex or ambiguous (MI) intervals to
arrive at some more "distant" and at least partially unpredictable
destination.

--------------------------------------------------
3. Mapping the contours -- tuning and counterpoint
--------------------------------------------------

An approach that much appeals to me might be a kind of "map" of the
continuum, with some main JI and MI regions -- and also, a la Paul
Erlich, some indication of the intervening contours.

For example, in Zest-24, here are the sizes available in the general
region around and between 12:7 (933 cents) and 7:4 (969 cents):

926 932 933 938 946 950 957 959 963 976 982 983

It's interesting to ask if 946 cents represents an approximate "peak"
of MI or complexity within this set of sizes, being near as it is to
the noble mediant at about 943 cents. Often I tend to think of 926-938
cents as mainly related to 12:7; 946 or 950 cents as ideally
interseptimal and flexible; 957 or 959 cents as leaning toward 7:4,
but still somewhat interseptimal; and 963-983 cents as 7:4 country.
What might a contour map look like?

In reflecting on these sizes, one thing occurs to me: often I might
regard the degree of "tension" or "excitement" as depending not only
on the intrinsic properties of an interval, but on how it is used.
Thus if asked to think of something "radical," I might focus on using
938 cents, a routine representation of 12:7, as a very small minor
seventh in something like F-G*-C-D* (0-243-696-938 cents) resolving to
the fifth C*-G* with melodic steps of 192 cents down (D*-C*, A*-G*)
and 50 cents up (C-C*, G-G*). Here the special excitement seems to
center on treating 243 cents as a small minor third and 938 cents as a
small minor seventh although they are closer to 8:7 and 12:7 than to
7:6 and 7:4. Since 243 cents is right around the noble mediant between
8:7 and 7:6, MI might be playing an important role; but I'd suspect
that the contrapuntal context is also central. For example, resolving
from F-A*-C-D* (0-434-696-938 cents, with C-D* at 241.4 cents), to
Eb-Bb-Eb is a standard progression featuring a tempered version of
14:18:21:24 -- although here, the arrangement of the intervals (might
one speak of "otonality" in a context of factors 2-3-7-9?) may play a
role as well as the melodic and contrapuntal action.

-------------------------
4. How wide an MI region?
-------------------------

A final point for now: as I recall, in our original concept, a noble
mediant between two significant JI valleys would likely mark the
general region of a maximum, or the central portion of a plateau,
rather than a specific point where "something happens" comparable to
locking in of partials at the focus of a JI valley. This suggests to
me that MI should involve regions of some considerable size, rather
than specific values, with the noble mediant as a general guide.

Thus, intuitively, I'd guess that if 943 cents is a significant noble
mediant -- "titanium" in your listing -- then 946 cents should be more
or less similar in its MI qualities, albeit with subtly different
shadings. More generally, I wonder if "MI" might embrace to a degree
almost anything between about 940 and 960 cents, say, a kind of
plateau between 12:7 and the deeper 7:4 valley, albeit with the
contours peaking at somewhere around 943 cents, or 10 cents from 12:7.

In making my sampling of "MI" or the like in Zest-24, I wanted to
explore some of these questions. How large is an MI region, and how
might we describe the shades somewhere between "JI" and "MI" in effect
(whether derived through RI, Phi-based TI, or pragmatic tempering of
other sorts)? How do MI properties, or shadings of entropy more
generally, interact with contrapuntal and melodic context?

Most appreciatively,

Margo Schulter
mschulter@...

Hi Margo,

Thanks for joining the discussion. I agree with all that you wrote.
And I understand you to be cautioning us not to see everything as an
approximation of either justness or metastability, as a man with new
hammer might see everything as a nail.

In regard to your proposed term TI (tele-intonation): I believe the
adoption of the term MI (metastable/merciful/murky intonation) frees
up the term NI (noble intonation) for exactly the purpose for which
you intended TI.

The analogy is: Rational is to Just as Noble is to Metastable. To
explain: Rational and Noble are mathematical properties while Just and
Metastable are aural. Only the simpler Rationals "explain" Just
intervals and only the simpler Nobles "explain" (most) Metastable
intervals.

The qualification "(most)" above, is to recognise that simple noble
numbers do not explain the metastable intervals closest to the unison,
octave, fifth and on the low side of the fourth (this last one
recently pointed out by Carl Lumma as affecting the accuracy of our
2000 article).

Harmonic entropy maxima appear to "explain" all the metastables, but
are not so mathematically convenient as noble numbers.

I agree that your question "just how metastable is really metastable?"
is something people must answer for themselves. However, I wonder if
the classic mediant of the last two attractor intervals might
conventionally form one boundary of the metastable region, and the
interval having the same harmonic entropy might form the boundary on
the other side. In your 12:7 to 7:4 case (with noble mediant at 942.5
cents) this would be 19:11 (946 cents) and about 939 cents.

You asked, "What might a contour map look like?"

Might it look like a Harmonic Entropy curve, such as one of these
three, made for you by Paul Erlich?
/tuning/files/dyadic/margo.gif
/tuning/files/dyadic/margo2.gif
/tuning/files/dyadic/margo3.gif

-- Dave Keenan

--- In tuning@yahoogroups.com, "Paul Poletti" <paul@...> wrote:
>
> --- In tuning@yahoogroups.com, "Cris Forster" <cris.forster@>
wrote:
>
> >
> > The coefficient of inharmonicity is a function of the stiffness
> > parameter J, and so not easy to calculate.
>
> Don't you mean E? Young's modulus? Yeah, I know there are slight
> variations depending on which direction, but it's more or less the
same.
> >
> > Having restrung many pianos, I find the coefficient of
inharmonicity
> > for plain strings complete reliable However, for wound strings,
all
> > such calculations are problematic.

>
> Which formula are you using? How do you get your values of E (or J)
> for the wire you are using? How do you verify partial detuning? How
> high have you checked in the partial series?
>
> Ciao,
>
> P
>

There are two different kinds of equations for the dimensionless
stiffness parameter (J): one based on a known tension (T), the
other, on a known frequency (F). I will give the latter:

J = (Pi^2 * E * S * K^2) / (2 * 4 * F^2 * M/u.l. * L^4)

where E is Young's modulus of elasticity, S is the cross-sectional
area of the string, K is the radius of gyration, F is the frequency
of the string, M/u.l. is the mass per unit length of the string, and
L is the vibrating length of the string.

This equation simplifies to the following formula:

J = [Pi^2 * (E/rho) * D^2] / (128 * F^2 * L^4)

where rho is the mass density of the stringing material, and D is
the diameter of the string.

The coefficient of inharmonicity is given in cents (c):

c = 1731 x J

-Cris

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

>
> >
> > Which formula are you using? How do you get your values of E (or J)
> > for the wire you are using? How do you verify partial detuning? How
> > high have you checked in the partial series?
> >

>
> There are two different kinds of equations for the dimensionless
> stiffness parameter (J): one based on a known tension (T), the
> other, on a known frequency (F). I will give the latter:
>
> J = (Pi^2 * E * S * K^2) / (2 * 4 * F^2 * M/u.l. * L^4)
>
> where E is Young's modulus of elasticity, S is the cross-sectional
> area of the string, K is the radius of gyration, F is the frequency
> of the string, M/u.l. is the mass per unit length of the string, and
> L is the vibrating length of the string.
>
> This equation simplifies to the following formula:
>
> J = [Pi^2 * (E/rho) * D^2] / (128 * F^2 * L^4)
>
> where rho is the mass density of the stringing material, and D is
> the diameter of the string.

Right. Thanks. But I still wonder where you get your value for E for
the piano wire you are using?
>
> The coefficient of inharmonicity is given in cents (c):
>
> c = 1731 x J
>
I don't understand. How can the coefficient be a constant in cents?
The formula I have, and my personal experience, is that the sharpening
increases with successively higher modes. The formula I have returns a
coefficient, B, which is then used thusly:

cents detuning = B*n^2

where n is the number of the harmonic. Thus, the detuning increase
with the value of n.

I'm still very much intersted in the rest of the questions:

How did you verify that the predicted detuning is equal to the real
detuning? The precise (to the degree required in this instance)
extraction of component frequencies in a sound with an envelope like
that of a struck piano string is not easy, so I'm wondering as to your
methodology.

How high up into the overtone series to you check? Or did you only
check the octave, based on the idea that detuning is a constant?

Where did you get this formula? I remember where mine came from, the
Martha Goodway book on the metallurgy of harpsichord strings.

Ciao,

P

--- In tuning@yahoogroups.com, "Paul Poletti" <paul@...> wrote:
>
> --- In tuning@yahoogroups.com, "Cris Forster" <cris.forster@>
wrote:
> >
> > --- In tuning@yahoogroups.com, "Paul Poletti" <paul@> wrote:
>
> >
> > >
> > > Which formula are you using? How do you get your values of E
(or J)
> > > for the wire you are using? How do you verify partial
detuning? How
> > > high have you checked in the partial series?
> > >
>
> >
> > There are two different kinds of equations for the dimensionless
> > stiffness parameter (J): one based on a known tension (T), the
> > other, on a known frequency (F). I will give the latter:
> >
> > J = (Pi^2 * E * S * K^2) / (2 * 4 * F^2 * M/u.l. * L^4)
> >
> > where E is Young's modulus of elasticity, S is the cross-
sectional
> > area of the string, K is the radius of gyration, F is the
frequency
> > of the string, M/u.l. is the mass per unit length of the string,
and
> > L is the vibrating length of the string.
> >
> > This equation simplifies to the following formula:
> >
> > J = [Pi^2 * (E/rho) * D^2] / (128 * F^2 * L^4)
> >
> > where rho is the mass density of the stringing material, and D
is
> > the diameter of the string.
>
> Right. Thanks. But I still wonder where you get your value for E
for
> the piano wire you are using?

Timoshenko, S.P. (1953). History of Strength of Materials, p. 92.
Dover Publications, Inc., New York, 1983.

"[Thomas] Young (1773–1829) had determined the weight of the
modulus of steel from the frequency of vibration of a tuning fork
and found it equal to 29 * 10^6 lb per in^2."

For steel, I use the following value: 29,000,000 psi.

-Cris

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:
>
>
> Hello, there, Petr and Cris and all.
>
> Please let me provide a brief example of a stretched-octave sound
in a
> gamelan style. In this version of slendro, there are also some
complex
> unisons:
>
> <http://www.bestII.com/~mschulter/ForErin.mp3>
>
> Of course, I'd be delighted to get into intonational specifics if
> people are interested, but the actual sound is the main thing.
>
> Most appreciatively,
>
> Margo Schulter
> mschulter@...
>

In general, Javanese gamelan music tends to be slow, soft, and
stately. In contrast, and in general, Balinese gamelan music tends
to be fast, loud, and brilliant.

If you ever have a chance to visit a library to listen to Javanese
music, you will find that hearing the beating of low bronze gongs
and high bronze bars is not only easy, but a thrill as well.

-Cris

--- In tuning@yahoogroups.com, "Cris Forster" <cris.forster@...> wrote:

> >
> > Right. Thanks. But I still wonder where you get your value for E
> for
> > the piano wire you are using?
>
>
>
> Timoshenko, S.P. (1953). History of Strength of Materials, p. 92.
> Dover Publications, Inc., New York, 1983.
>
> "[Thomas] Young (1773–1829) had determined the weight of the
> modulus of steel from the frequency of vibration of a tuning fork
> and found it equal to 29 * 10^6 lb per in^2."
>
> For steel, I use the following value: 29,000,000 psi.

Uh, don't really know where to start. I guess the most important thing
is that there are as many "steels" as there are mushrooms, and the
difference between their mechanical characteristics is about as vast
as the physical characteristics of fungi. Simply grabbing one value,
ESPECIALLY this value, is about as valid as saying, "Ah, just go out
in the forest and pluck the first mushroom like thing you come acroos
to make your soup." Your just as likely to end up dead as not.

Young's value is especially inappropriate because he dies before the
great change in steel making, from relatively small charcoal fired
fining to larger coal-fired processes culminating in the Bessemer
process of 1855. The steels produced by modern techniques are
completely different from c.1800 "steel". The primary difference was
the use of phosphorus and NOT carbon to increase strength.

Your value in scientific terminology is about 200 GPa. That's slightly
high for historical ferrous "steels", whcih tend to run between 180
and about 195 GPa. On the other hand, it is quite low for modern
high-strength steels, including piano wire, which tends to run about
220 GPa and above.

All that aside, you still haven't yet dealt with the other two very
big issues in your assertion:

(1) Why do you believe inharmonicity can be a constant in cents, when
everything else I have read, and my own test, indicate that it
increases with harmonic number.

(2) How are you extracting the frequencies of the inharmonic overtones?

Sorry to keep picking on you, but this is an issue I've been busy with
for about 15 years now. Plus I'm collaborating with Stephen Birkett in
his project to recreate pre-Bessemer ferrous wire, so we're pretty
much in this up to our necks and beyond.

Ciao,

P

> Hi Margo,

> Thanks for joining the discussion. I agree with all that you wrote.
> And I understand you to be cautioning us not to see everything as an
> approximation of either justness or metastability, as a man with new
> hammer might see everything as a nail.

Thank you, Dave, for a most gracious reply. To reciprocate your
recognition of this caution about JI/MI concepts, I want to
acknowledge that in fact the tools we're discussing are very
versatile, and can yield some striking insights often quite relevant
to the kinds of practical considerations I was raising.

For example, consider the noble mediant of 12:7-7:4 at 943 cents. If
asked to intuit the musical "midpoint" between a 12:7 major sixth and
a 7:4 minor seventh at 933 and 969 cents, I might guess around 950
cents. However, since 4*7 is considerably simpler than 7*12, it's not
so surprising that the harmonically entropic point of "equipoise" is
rather closer to 12:7, in fact only about 10 cents wider.

This could have real explanatory power for certain aspects of 14-EDO
and 24-EDO, for example. With 14-EDO, I approached it confident that
11/14 octave or 943 cents would make a nice wide "major sixth"
variation on 12:7 -- but would it be large enough persuasively to
suggest a very small "minor seventh" like 7:4? I tried it, and found
that it happily was. Knowing that I was right at the point of entropic
equipose might make this pleasant musical result more comprehensible.

Then, again, let's consider 24-EDO with its interval of 950 cents.
Here my point will be one of style and entropy. Too often, in my view,
24-EDO gets written about in rather general or unqualified terms when
the focus is on a rather specific application: some kind of classic
18th-19th century European style or derivative where 4:5:6:7 is a
primary ideal (if not necessarily of standard period practice, then of
certain modern offshoots). Please let me emphasize that I'm in no way
opposing an interest in the tuning of 4:5:6:7 or the choice of tunings
to realize this sonority, only to caution that there are various other
uses for ratios such as 4:7 and 6:7.

If we start with 4:5:6:7 as a premise, then I'd say it should quickly
become clear that 31-EDO would be dramatically more accurate for
everything except, of course, 3:2. However, let's suppose that instead
we're following a neomedieval kind of bent, say, and would like to
represent 4:6:7 in a kind of "manneristic" or "avant-garde" way.

If so, then 0-700-950 cents might nicely fit the purpose -- and the
noble mediant at 943 cents tells us that 950 cents is actually leaning
a bit in the direction of 7:4. Contrapuntally, in resolutions of this
0-700-950 sonority by stepwise contrary motion, voices are typically
going to move either by 200-cent tones or 50-cent steps serving as
very small semitones -- a beautiful sound. Thus D-A-B* (the * shows a
note raised by 1/24 octave, the 50-cent quartertone step) might
resolve to E-B or D*-A*, the 950-cent "seventh" contracting to a
fifth; or to C-G-C or C*-G*-C*, the same interval used as a "sixth"
expanding to an octave. Mathematically, 950 cents is the same distance
of 250 cents from either a 700-cent fifth or a 1200-cent octave, so
either progressive is equally "efficient."

Entropically, we learn from the noble mediant that 0-700-950 cents,
while thus flexible, has a bit of a tilt toward 7:4 -- an interesting
nuance of a system which, of course, simply and elegantly provides a
set of _interseptimal_ intervals (250, 450, 750, 950 cents) rather
than accurate representations of septimal ratios. Thus if the goal is
a kind of offbeat variation on 4:6:7 and the like in a setting where
ratios of 2-3-7-9 without 5 are the main focus, and quite possibly
neutral intervals also (150, 350, 850. 1050 cents), 24-EDO is an
engaging system.

Here I've taken a bit of space to show how entropy and counterpoint
can interact, the former not necessarily determining the latter, but
providing subtle gradations of nuance and color.

What I would urge is that we recognize a possible aesthetic, as in
some consonance/dissonance theories of composition, of a subtle
continuum, where we might now seek JI, now maximal MI, and now some
intermediate grade of blend/tension or clarity/ambiguity. This might
happen by deliberate calculation in the choice of interval sizes
during a composition or improvisation, or by a "prepared aleatory"
technique -- for example, an unequal temperament like Zest-24 where
the variety of colors "happens" as one moves around without
necessarily reasoning, "I'll use 946 cents here, 950 cents there, and
957 cents for this cadence."

Of course, a 17th-19th century well-temperament of the kind germinally
presented by Werckmeister is a fine example. I wouldn't necessarily
assume that Bach calculated every improvisatory move from a view of
precise harmonic tension, which could vary from instrument to
instrument, or that he used a single tuning scheme. However,
"variegation" was clearly in the air, and the noble mediant gives one
measure of the colors we may be painting with in composing or
improvising.

> In regard to your proposed term TI (tele-intonation): I believe the
> adoption of the term MI (metastable/merciful/murky intonation) frees
> up the term NI (noble intonation) for exactly the purpose for which
> you intended TI.

We agree -- that's a neat solution.

> The analogy is: Rational is to Just as Noble is to Metastable. To
> explain: Rational and Noble are mathematical properties while Just
> and Metastable are aural. Only the simpler Rationals "explain" Just
> intervals and only the simpler Nobles "explain" (most) Metastable
> intervals.

Nicely put! A fine point is that while RI encompasses JI as a subset,
in many contexts if I refer to a complex ratio like 176:117 as "RI,"
there's what Paul Grice might call a "conversational implicature" that
I am _not_ viewing it as aurally JI, but rather as a fifth "tempered
by ratio" at not quite 5 cents wide. Similarly, in a setting where we
are describing aural maxima as MI, to call some remote noble mediant
"NI" would carry a suggestion that aural significance as a maximum
(MI) is not alleged. Incidentally, MI could mean "Maximal Intonation"
in "harmonic entropy maximum" terms.

> The qualification "(most)" above, is to recognise that simple noble
> numbers do not explain the metastable intervals closest to the
> unison, octave, fifth and on the low side of the fourth (this last
> one recently pointed out by Carl Lumma as affecting the accuracy of
> our 2000 article).

Yes, I've always thought in terms of 4:5-7:9 (where 422 cents is a
logical and musically likely result) rather than 4:5-3:4. An
interesting and very likely culturally-linked or stylistically-linked
point is that for the 4:5-7:9 region -- at least for those of us used
to larger major thirds in the Pythagorean-septimal range of 408-435
cents, 422 cents is not a point of _categorical_ ambiguity. We clearly
have some kind of "major third," indeed _the_ representation of this
interval in a system like 17-EDO at 423.53 cents. What we have is a
point of maximum complexity or tension within a category.

In the interseptimal space of 7:9-3:4 from a viewpoint of valleys, or
something like 7:9-16:21 from my own perception that 21:16 is a
clearly recognizable "small fourth" at a 64:63 comma narrow of 3:4,
we're dealing not only with maximal entropy but, from a certain
stylistic perspective, with _categorical_ ambiguity: a large major
third, or a small fourth?

In a gamelan setting where the musical "phonemes" or basic categories
of sound are quite different -- say an interval of two slendro steps
-- categorical ambiguity very well might not apply: to speak
linguistically, the "opposition" of "European third/fourth" might be
quite extraneous. Of course, given the typical timbres, harmonic
entropy maxima based on harmonic timbres might also be inapplicable!

Then, again, native perceptions within different traditions can be
very interesting. For example, there's a view among some theorists in
the Arab world that the maqamat (plural of _maqam_ or mode) which have
a notably "ecstatic" effect are generally those with neutral steps and
intervals (e.g. Bayyati, Husseini, Saba); while those with only major
and minor steps are likely to be ecstatic more because of the skill of
the player or receptiveness of the listener than "by nature." Whatever
else this may say, it tells us that a special appreciation of neutral
intervals is not restricted to the outside listener unaccustomed to
them.

> Harmonic entropy maxima appear to "explain" all the metastables, but
> are not so mathematically convenient as noble numbers.

Yes, which is to say that shortcuts such as NI are at once valuable
and less than all-encompassing. From an intuitive perspective, Carl
and Dave, I might reason like this. Taking the noble mediant of
4:5-3:4 must inevitably give a result weighted toward the more
_complex_ ratio of 4:5, and thus nearer to it than the classic mediant
of

5+4
---
4+3

or 9:7. However, we readily guess that maximum harmonic entropy will
be closer to 3:4 than 7:9 is -- and thus we need something other than
NI to find it, or at least other than the 4:5-3:4 pairing.

Even if we try 7:9-4:3, this likewise will give something closer to the
more complex ratio, 7:9, than the classic mediant at 13:10. Like others
who have discussed this question here such as Cameron Bobro, I'd
intuitively place the maximum of complexity at somewhere between 13:10 and
21:16. Thus something other than a noble mediant would be needed to
address this question. This isn't to say that 448 cents isn't a charming
region.

> I agree that your question "just how metastable is really
> metastable?" is something people must answer for
> themselves. However, I wonder if the classic mediant of the last two
> attractor intervals might conventionally form one boundary of the
> metastable region, and the interval having the same harmonic entropy
> might form the boundary on the other side. In your 12:7 to 7:4 case
> (with noble mediant at 942.5 cents) this would be 19:11 (946 cents)
> and about 939 cents.

That's interesting -- here, about a 7-cent range for "clear MI." Of
course, from an outlook like that of 13th-century and some
20th-century European counterpoint theory, we could recognize lots of
degrees and shadings of "perfect blend" (JI), "maximal ambiguity"
(MI), and intermediate conditions such as "partial clarity" (e.g. a
major third at 400 cents in a meantone-like style, where the "partial"
might be underscored) or "partial ambiguity" (e.g. an interseptimal
950 cents as a very large major sixth or small minor seventh, not too
far from MI at around 939-946 cents but a different shading).

Here there might be at least two dimensions, both of which harmonic
entropy would take into account: the distance from a given JI or MI
focus or region; and the "depth" or "height" of that valley or
maximum.

As in the rules of intervallic counterpoint, we might sometimes want
JI or MI, or the strongest contrast emphasizing an optimization of
both: an analogous example in intervallic style would be the 12th-13th
century move by oblique motion from a major seventh, one of the most
"perfect" or acute dissonances, to an octave, a pure or "perfect"
concord.

However, we might also treasure a variety of intermediate shadings
between JI and MI, with various "rules" or preferences as to
transitions or juxtapositions. For example, 16th-century European
counterpoint rules actually treat thirds and sixths, classed as
"imperfect concords," as the most favored sounds, with rules which
avoid or "cushion" extreme contrasts. A general guideline for
significant sonorities is to move from discord to imperfect concord,
and from there in two-voice progressions to perfect concord, so that
the favored 13th-14th century move by stepwise contrary motion from a
minor seventh to a fifth is excluded as a prominent progression.
Rather the seventh must be prepared as a suspension, moderating its
impact, and normally in a cadential progression it resolves down
stepwise to the major sixth, an imperfect concord, which might then
expand to the perfect concord of the octave.

These patterns apply for other styles, for example in the 20th
century, where the "rules" may be less clearly defined, but a variety
of intervals and levels of clarity or tension is clearly sought and
achieved. Ludmila Ulehla has drawn graphs of consonance/dissonance in
compositions of different periods, seeking to map out the relevant
categories of sonorities according to the rules and practices of each
period. While we're speaking here of conventional interval categories
and the sonorities they form in a given style, the same approach could
be applied to harmonic entropy.

> You asked, "What might a contour map look like?"

> Might it look like a Harmonic Entropy curve, such as one of these
> three, made for you by Paul Erlich?
> [150]/tuning/files/dyadic/margo.gif
> [151]/tuning/files/dyadic/margo2.gif
> [152]/tuning/files/dyadic/margo3.gif

Ah, yes, I remember that we discussed these around 2002, with George
getting involved in the dialogue.

Anyway, I feel like apologizing for thinking out loud at this length,
but a lesson might be that harmonic entropy as conveniently reflected
in many regions by the noble mediant could indeed be approached from a
dynamic, graduated, and stylistically flexible standpoint. It's one
measuring tool, and doesn't have to be prescriptive, but can give us
interesting descriptions of practice, sometimes with results not so
obvious.

What I'd say is: let's think in terms of a continuum where JI, MI, and
all the intermediate nuances are open, and serve as possible goals
both of tuning design and actual compositions or improvisations.

The question is not just optimizing intervals at one or two points or
regions on the JI-MI scale, necessarily, but sometimes getting a
motley assortment of shadings and degrees. In actual music, such
tunings might support various preferences for the successions,
transitions, or juxtapositions of different entropy levels -- by
calculation, by habitual and at least partly unformulated intuition,
or even by chance in an unequal temperament or the like (with the
element of calculation possibly coming in the choice of a temperament
which routinely produces a variety of colors even if the performer is
concentrating on other things).

In short, while I'm sure we'd agree that harmonic entropy is not a
theory of tuning or composition, I suspect we'd also agree that it can
greatly enrich and enhance such theories.

> -- Dave Keenan

Most appreciatively,

Margo Schulter
mschulter@...

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:

> For example, consider the noble mediant of 12:7-7:4 at 943 cents. If
> asked to intuit the musical "midpoint" between a 12:7 major sixth
and
> a 7:4 minor seventh at 933 and 969 cents, I might guess around 950
> cents. However, since 4*7 is considerably simpler than 7*12, it's
not
> so surprising that the harmonically entropic point of "equipoise" is
> rather closer to 12:7, in fact only about 10 cents wider.

Hi Margo! I must say that I seek simpler explanations and believe
that seeking a midpoint between 12/7 and 7/4 most likely would occur
when these intervals are already in a tuning. Far more likely in an
out-of-the-blue or more directly harmonic-series oriented situation
would be to seek out or find pleasing an interval which strikes a
poise between 8:5 and 9:5.

According to my "wrong" method, that's at 942.517 cents, ie, might as
well be the same thing as your example above. But I find my
explanation simpler.

Either way, we get the inversion of the "consant interval" of Ibn
Sin, and I don't think it's sheer coincidence. Note that the interval
you give above is even closer to Ibn Sin's than mine is, a tiny
fraction of a cent rather than .9 cents. I believe that we are
dealing with a fuzzy region a cent or two wide.

I also believe that a great big catalog of these kinds of intervals
has been around for centuries and more, for no less than three
intervals I've identified as "shadows" have showed up in the Scala
archives or on Gann's nifty anatomy of an octave page labelled
"Zalzal" or "Avicenna".

For Carl: you just multiply the difference between two frequencies by
the phi conjugate 1/phi, then add or subtract that figure to or from
one of the original figures in order to weight it toward the interval
which is weaker in terms of the harmonic series, which is almost
always the more complex number.

No different than you'd reckon the golden section in the visual arts.

I'll go step by step in a bit, trying to multitask making backups,
arg.

-Cameron Bobro

> Margo's "Variegated Intonation", of her Zest-24 scale, made me think
> about taking that idea to a ridiculous extreme, namely perfect
> difference scales. These are based on perfect difference sets, which
> I'm sure have been discussed before on this list or on tuning-math.

Thank you, Dave, for a neat understanding of the Variegated Intonation or
VI concept. I agree that perfect difference scales are an ideal way to
distill the idea to its essence, and give a really dramatic demonstration!

> The idea is simple enough. The smallest set of pitches in some EDO
> such that every possible interval-size in that EDO can be generated
> exactly once per octave (except of course the unison which occurs
> once per note).

Your examples are excellent, and fun to run in Scala with a command
like 'show /line intervals' to get a kind of spreadsheet-like layout
displaying an overview of all the varied intervals and positions. As
you point out, your 14-note scale for 183-EDO is especially delightful:

> ! PerfDif14.scl
> !
> Perfect difference scale for 183-EDO
> 14
> !
> 32.78688525
> 183.6065574
> 249.1803279
> 268.852459
> 321.3114754
> 327.8688525
> 445.9016393
> 491.8032787
> 603.2786885
> 701.6393443
> 793.442623
> 806.557377
> 832.7868852
> 2/1

Here it's very appropriate, of course, that above your 1/1 we have
both a virtually just 3/2 and an almost exact Phi! Surveying the aural
JI/MI spectrum indeed!

These perfect difference scales are, I would agree, an ideal way to
get across the concept of VI.

With many thanks,

Margo

> Hi Margo! I must say that I seek simpler explanations and believe
> that seeking a midpoint between 12/7 and 7/4 most likely would
> occur when these intervals are already in a tuning. Far more likely
> in an out-of-the-blue or more directly harmonic-series oriented
> situation would be to seek out or find pleasing an interval which
> strikes a poise between 8:5 and 9:5.

Hi, Cameron. Here I would guess that our comments are coming from
rather different musical frames of reference. My first reaction to
your reference to 8:5 and 9:5 was: "But that's from a large minor
sixth to a large minor seventh, with lots of categories between,
presumably including some kind of major sixth, and maybe a neutral
sixth also."

However, if you are thinking literally of a "harmonic series" --
i.e. 5:6:7:8:9..., then your comment would, of course, hold.

For me, the idea of a middle or "interseptimal" ground between 12:7
and 7:4 can apply in two situations. The first is in tuning systems
like 14-EDO, 24-EDO or 29-EDO (or 19-EDO) where there isn't any close
representation of 12:7 or 7:4, but an intriguingly intermediate size
that might play a role comparable to either. The purpose of using such
a system wouldn't be to get accurate septimal ratios, but precisely to
get interseptimal ones, whether we speak of integer ratios as
landmarks like 19:11 or 26:15, or simply sizes in cents in the 940-960
zone, more or less.

The other situation is where we have some fairly close septimal
representations plus intermediate intervals as well: this happens in
Zest-24. Here's an article on a 19-note subset of this temperament
emulating a septimal JI (2-3-7-9) matrix, with fairly close
representations of 12:7 or 7:4 in some positions, and intermediate
ones like 946 or 950 cents in others:

<http://www.bestII.com/~mschulter/zest24-aaron_akj.txt>

> According to my "wrong" method, that's at 942.517 cents, ie, might
> as well be the same thing as your example above. But I find my
> explanation simpler.

Please let me emphasize that I don't think there is any one "right" or
"wrong" method. The noble mediant is merely one way to approach the
problem. I would consider seeing what interval sounds best for your
purposes as the right way.

> Either way, we get the inversion of the "consant interval" of Ibn
> Sin, and I don't think it's sheer coincidence. Note that the
> interval you give above is even closer to Ibn Sin's than mine is, a
> tiny fraction of a cent rather than .9 cents. I believe that we are
> dealing with a fuzzy region a cent or two wide.

Actually I'd suspect we're seeking a convenient number as a guide to
something notably fuzzy. Things like variations of timbre are going to
have an effect also, whether subtle or dramatic.

> I also believe that a great big catalog of these kinds of intervals
> has been around for centuries and more, for no less than three
> intervals I've identified as "shadows" have showed up in the Scala
> archives or on Gann's nifty anatomy of an octave page labelled
> "Zalzal" or "Avicenna".

The file octave.txt -- nifty, I'd say also! -- shows a ratio of
196/169 or 256.596 cents which is listed as an interval deemed
"consonant" by Ibn Sina (or in Latin Europe, Avicenna). This is
interesting, and I'd agree with this great philosopher and music
theorist in his appraisal made something like a millennium ago. I'd be
interested to know his context for this interval. It is very close
also to 297/256, or 257.182 cents. Thanks to Kyle Gann for making that
page "Anatomy of an Octave" available, and to you for reminding me of
how much fun I've had with that list of intervals.

Ibn Sina also described a scale with simple septimal ratios, which
George Secor and I discussed in our exploration of his 17-WT tuning,
which has some nice approximations:

1/1 14/13 7/6 4/3 3/2 21/13 7/4 2/1

This is in the Scala archive as avicenna_diat.scl.

Most appreciatively,

Margo
mschulter@...

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:
>
>
> > Margo's "Variegated Intonation", of her Zest-24 scale, made me think
> > about taking that idea to a ridiculous extreme, namely perfect
> > difference scales. These are based on perfect difference sets, which
> > I'm sure have been discussed before on this list or on tuning-math.
>
> Thank you, Dave, for a neat understanding of the Variegated
Intonation or
> VI concept. I agree that perfect difference scales are an ideal way to
> distill the idea to its essence, and give a really dramatic
demonstration!
>

Thanks Margo,

Below I've given the 24-note perfect all-interval scale. It's in
553-EDO which doesn't have good JI approximations in relative terms,
but since in absolute terms it's within 1.1 cents of anything, perhaps
that doesn't matter.

However I'm going to beat Kraig to the draw here and point out that
while it's neat to be able to have all possible dyads in a scale
having so few notes, that doesn't get us anywhere near having all
possible triads and larger. As far as I know, the only way to have all
possible triads is to have all the notes of the EDO. And therefore VI
scales like your Zest-24 are likely to be of far more value in
creating real music.

! PerfDif24.scl
!
Perfect difference scale for 553-EDO
24
!
19.52983725
71.60940325
80.28933092
82.45931284
210.4882459
264.7377939
279.9276673
303.7974684
308.1374322
329.8372514
414.4665461
444.8462929
451.3562387
546.835443
603.2549729
620.6148282
707.4141049
720.4339964
766.0036166
798.5533454
833.2730561
874.5027125
922.2423146
2/1

In another message in this thread, I mentioned the possible use of
perfect difference sets in the linear frequency domain. In that case
they represent the complete opposite of Kraig's favourite isoharmonic
chords. All the tones could be multiples of a single virtual
fundamental and yet no pair of tones would have the same difference
tone as any other pair. e.g. 8:10:11:15 16:20:21:23.

Now if you took one of these chords and stretched or compressed it
until all the dyads were near metastable, maybe you'd have some kind
of "most discordant" chord. Not even the benefit of coinciding
difference tones.

It's interesting that the first practical application of perfect
difference sets was for exactly this purpose, only with radio waves,
not sound waves. In the early days of radio, non-linearity and the
resulting intermodulation distortion were a real problem. Wallace
Babcock (1952) showed that by arranging channel frequencies as a
perfect difference set much interference could be avoided.

-- Dave Keenan

I never found it fruitful say in 12 or 31 to pursue past the dyads. these seem to be more important melodically than harmonically.

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Dave Keenan wrote:
>
>
>
> However I'm going to beat Kraig to the draw here and point out that
> while it's neat to be able to have all possible dyads in a scale
> having so few notes, that doesn't get us anywhere near having all
> possible triads and larger.
>

Hello, everyone.

Please let me propose a possible solution to the question of
terminology for harmonic entropy discussions.

As Carl has remarked, attempts to use terms related to words such as
consonance/dissonance or concord/discord for style-neutral phenomena
have not evidently won universal acceptance. What I suggest are two
pairs of terms which may avoid or minimize connotations of "goodness"
or "badness," or of style-related associations.

For levels of harmonic entropy, "transparency/translucency" might be
synonyms for "simplicity/complexity": a more "transparent" interval
more simply maps to a "valley" ratio than a more complex or
"translucent" one.

"Does 9:11, or 347 cents, itself represent a
valley of simplicity with its own transparent
identity, or an approximate peak of translucency
in the region between 6:5 and 5:4? Either way,
neutral thirds of around this size are very popular
in a range of world musics, and initiated listeners
have no problem in recognizing and relishing this
shade of tuning."

For questions of beatfulness within the critical band or the like, how
about "smoothness/friction"? People may have varying degrees of taste
for smoothness, friction, or mixtures of the two.

By choosing terms like these, harmonic entropy theory might better
maintain the kind of modest posture which I would see as incumbent,
given the variety of world styles and tuning practices as emphasized
by Kraig, and also the learning and acculturation process which Mike
has nicely conveyed.

Most appreciatively,

Margo Schulter
mschulter@...

Hello Margo, and thank you for weighing in on this! I'll address your
points one at a time:

> Hello, everyone.
>
> Please let me propose a possible solution to the question of
> terminology for harmonic entropy discussions.
>
> As Carl has remarked, attempts to use terms related to words such as
> consonance/dissonance or concord/discord for style-neutral phenomena
> have not evidently won universal acceptance. What I suggest are two
> pairs of terms which may avoid or minimize connotations of "goodness"
> or "badness," or of style-related associations.
>
> For levels of harmonic entropy, "transparency/translucency" might be
> synonyms for "simplicity/complexity": a more "transparent" interval
> more simply maps to a "valley" ratio than a more complex or
> "translucent" one.

Another idea related to this one that that of describing intervals in
terms of "depth;" that is, 3/2 is of shallow depth, that 5/4 is
"deeper," etc. It's sort of synesthetically how I perceive intervals
anyway. I intuitively feel the process as being analogous to the
visual processing in the brain whereas two separate left/right images
are combined to make a 3d picture. Intervals high in harmonic entropy
are simply harder to place, depth-wise - it's harder for the brain to
orient itself and get the depth of 9/11 than it is to get the depth of
3/2, for example.

> "Does 9:11, or 347 cents, itself represent a
> valley of simplicity with its own transparent
> identity, or an approximate peak of translucency
> in the region between 6:5 and 5:4? Either way,
> neutral thirds of around this size are very popular
> in a range of world musics, and initiated listeners
> have no problem in recognizing and relishing this
> shade of tuning."

Dave pointed out that the size of that S parameter determines whether
or not there is a valley at 9/11... So that's another concern when it
comes to labeling these things. For one listener, they might well have
a valley there (meaning they have succeeded in perceiving the depth of
that interval). Another listener who is much more used to 6/5 and 5/4
might only hear it as an out of tune, multistable version of both of
them (aka they'd have a valley).

So the question of whether there exists an absolute peak or trough for
any interval is already fairly ambiguous... It depends on that S
parameter.

> For questions of beatfulness within the critical band or the like, how
> about "smoothness/friction"? People may have varying degrees of taste
> for smoothness, friction, or mixtures of the two.
>
> By choosing terms like these, harmonic entropy theory might better
> maintain the kind of modest posture which I would see as incumbent,
> given the variety of world styles and tuning practices as emphasized
> by Kraig, and also the learning and acculturation process which Mike
> has nicely conveyed.

I agree with the concept behind it. Also, I would say we can use any
terms at all as long as they're not used to make statements such as
that 11/9 absolutely IS a peak of harmonic entropy. After all, 5/4
wasn't a harmonic entropy valley back in early western music. Or maybe
it was, and it was a purely cognitive and intellectual limit that
musicians of that age placed on themselves - that they "pretended" or
"feared" that it was a dissonance. We may never know.

Thank you for your input though - I appreciate that you have pieced
the whole picture together clearly... I like the sound of the
transparency and translucency concept, and it do see it tieing in with
the depth concept as well.

-Mike

> Hello Margo, and thank you for weighing in on this! I'll address
> your points one at a time:

Hello, Mike, and thank you for joining this thread. You raise some
questions about the status of 5:4 in medieval European music where, as
a medievalist, I'd like to flesh things out a bit. Also, while no one
today has audio tapes with 13th-14th century performances on them, I
can also share some very concrete experience about some of my own
20th-21st century tuning systems and practices have been interpreted
on this list. My challenge is to keep this reasonably concise, and
reasonably "transparent" (as much as I love complex intervals)!

> Another idea related to this one that that of describing intervals
> in terms of "depth;" that is, 3/2 is of shallow depth, that 5/4 is
> "deeper," etc. It's sort of synesthetically how I perceive intervals
> anyway. I intuitively feel the process as being analogous to the
> visual processing in the brain whereas two separate left/right
> images are combined to make a 3d picture. Intervals high in harmonic
> entropy are simply harder to place, depth-wise - it's harder for the
> brain to orient itself and get the depth of 9/11 than it is to get
> the depth of 3/2, for example.

This is an evocative metaphor, "depth/shallowness," and the main
possible complication is that we're already speaking of "valleys" for
simple intervals and "plateaus" or "peaks" for complex ones, not a
usage that I necessarily would have originated, but one which is
there. Some terms can have different and even "opposite" meanings --
"to sanction" as either "to authorize" or "to penalize" -- so that
might not be a bar to a description I like. However, I'm not sure that
"shallowness" is the term for what I feel in a fifth, the most complex
kind of stable concord in lots of world musics (including medieval
European) and in much of my own. Maybe I'd speak of a "depth of
clarity" for the fifth and a "depth of intricacy" for more complex
intervals. This could be "clarity/intricacy" for short.

> Dave pointed out that the size of that S parameter determines
> whether or not there is a valley at 9/11... So that's another
> concern when it comes to labeling these things. For one listener,
> they might well have a valley there (meaning they have succeeded in
> perceiving the depth of that interval). Another listener who is much
> more used to 6/5 and 5/4 might only hear it as an out of tune,
> multistable version of both of them (aka they'd have a valley).

May I suggest that we make a vital distinction by using "valley" for
an interval deemed to be acoustically an entropic minimum, or ratio of
"simplicity": a fact of physics rather than style. The term "oasis"
might nicely express an interval considered pleasant and congenial in
a given cultural tradition or style, whether stable or mildly
unstable, and whether it happens or not to be an entropic valley.

Thus in a 21st-century neomedieval style, 4:3 and 3:2 are both
entropic valleys and stylistically stable oases, as is the complete
three-voice concord 2:3:4; while 4:6:9 or 6:8:9 are valleys that serve
as stylistically unstable but relatively blending and very pleasant
oases. Septimal intervals such as 7:6, 9:7, and 7:4 are likewise
valleys serving as mildly unstable oases -- as are 13:11 and 14:11,
not necessarily entropic valleys. Neutral thirds in the range of
around 330-372 cents are also important, whether as special elements
of color or as basic elements of styles with a Near Eastern flavor:
while sizes around the possible 11:9 valley are common, especially
favored mildly unstable oases are around 17:14 and 21:17, the former
at least quite possibly closer to a peak than a valley.

Note that 5:4 simply isn't important to me in most neomedieval
styles. It might be essentially absent from a system (e.g. 17-EDO or
George Secor's brilliant well-tempered variation, 17-WT), or be
approximated in a few exotic positions of a 24-note system such as
Peppermint, or be very closely represented in the diminished fourths
of a medieval or later Pythagorean tuning, or appear at many locations
as a primary feature of the tuning, as in a modified meantone used for
neomedieval purposes.

George Secor and I have both written extensively about modern styles
(neomedieval and otherwise) in systems such as 17-EDO or 17-WT where
there are no close counterparts of 5:4 or 6:5, Here are links to his
article appearing in _Xenharmonikon_ 18, followed by a version of mine:

<http://xenharmony.wikispaces.com/space/showimage/17puzzle.pdf>
<http://www.bestII.com/~mschulter/Secor_17-WT_draft.zip>

George's perspectives and insights are absolutely vital to this
discussion. While I've been involved with medieval and neomedieval
music for 40 years, he approaches it from another musical viewpoint,
based on a vastly greater knowledge of tuning systems than mine, with
great artistic discernment and empathy.

[On my proposed concepts of "transparency/translucency" and
"smoothness/friction"]

> I agree with the concept behind it. Also, I would say we can use any
> terms at all as long as they're not used to make statements such as
> that 11/9 absolutely IS a peak of harmonic entropy. After all, 5/4
> wasn't a harmonic entropy valley back in early western music. Or
> maybe it was, and it was a purely cognitive and intellectual limit
> that musicians of that age placed on themselves - that they
> "pretended" or "feared" that it was a dissonance. We may never know.

Here I'd say, strictly speaking that 5:4 is a "harmonic entropy valley"
under the laws of physics in any time or place, but it isn't evidently
a stylistic oasis in many medieval or 21st-century neomedieval styles,
the latter of which I can attest at first hand, or in a variety of
other world musics which happen to focus on other intervals or regions.

This doesn't mean that I "fear" 5:4, or "pretend" that it's not an
acoustical valley, and indeed an oasis in other styles including those
of the late 15th to early 17th centuries in Europe that I also
practice. Rather it may mean that the interval doesn't seem so central
to a given style, whether or not it happens to be present in the
tuning system at hand -- with modified meantone used for neomedieval
idioms where the regular meantone thirds are absent or rare as a
dramatic illustration.

As to "pretending" and "fearing," I would strongly urge that we
approach the fact that a given style happens not to regard or use a
given entropic valley as an oasis in other terms.

In most 13th-14th century styles of Continental Europe, or similar
neomedieval styles today, 5:4 is not an oasis -- and all thirds, of
course, are definitely unstable, although relatively blending. For
more on historical practice and theory. Note that in these documents
synbols like 8/5 or 5/4 mean figured bass notation for sonorities like an
octave and fifth, or fifth and fourth, above the lowest note, rather than
tuning ratios, something I quickly learned could cause misunderstandings
among people for whom the latter reading is more common!

<http://www.medieval.org/emfaq/harmony/13c.html>
<http://www.medieval.org/emfaq/harmony/pyth.html>

Similarly, the dauntless Nicola Vicentino in 1555 explores every
interval in a 31-note meantone circle around 1/4-comma or 31-EDO, and
finds neutral thirds for which he gives an approximate ratio of 11:9 as
very pleasant, but intervals evidently near 7:6 and 7:6 as rather on
the dissonant side, and mostly to be "put aside." During Bach's early
childhood, Christiaan Huygens celebrated 7:6 and 7:4 as concords and
special adornments of such a 31-note cycle, but I doubt that any close
equivalents occurred in Bach's typical circulating tunings -- let
alone the neutral thirds favored by Vicentino and already embraced as
a basic feature of Near Eastern styles for at least about a millennium
(starting with Zalzal in the 8th century).

To make some of these points more concrete, here is a piece in a
21st-century tuning system:

<http://www.bestII.com/~mschulter/MMMYear001.mp3>

We have the 3:2 and 4:3 valleys as prominent stable oases, and various
sizes for unstable intervals such as seconds, thirds, sixths, and
sevenths, including such relatively blending oases as 7:6 and 7:4.

Here's another piece in the same 12-note tuning system:

<http://www.bestII.com/~mschulter/IntradaFLydian.mp3>
<http://www.bestII.com/~mschulter/IntradaFLydian.pdf>

In the style based largely on a Manneristic meantone idiom around
1600, 5:4 is, of course, a stable oasis. As a modern libery which
might reflect _some_ adventurous period trends, there are touches of
septimal color, and also the use of a major sixth sonority with Eb-G-C
(0-408-913 cents in this modified meantone tuning) in a cadential
context where it leads to a conventional 4-3 suspension and
resolution.

Here is another piece in a 24-note expansion of this system:

<http://www.bestII.com/~mschulter/Baran-GiftOfRain.mp3>

In this piece inspired by Persian music, the 3:2 and 4:3 valleys are
vital oases, and neutral thirds are a basic element -- notably absent
from the previous two pieces! Also some septimal valleys occur as
oases, while 5:4 is unimportant, and if present would occur more or
less by accident.

Well, I've kept it under 200 lines, at least. Of course, we can
discuss medieval theory and practice more, as well as interesting
variant views (e.g. Walter Odington on 5:4 and 6:5 around 1300, who
may be reflecting a lot of English style and practice; Vicentino on
11:9; Huygens and later Euler on 7:6 and 7:4).

Most appreciatively,

Margo Schulter
mschulter#calweb.com

> Indeed, the neutral thirds of Maqam music are more diversified than
> the ratio 11:9 can encompass. As far as I can tell, middle second
> intervals roam the mujannab zone between 14:13 and 11:10. This is a
> region 37 cents wide. Then again, it is possible to broaden the zone
> and take 15:14 and 10:9 as the extremes. In that case, we acquire a
> zone that is 63 cents wide!

Dear Oz (and Dave, and all),

Thank you for these remarks so nicely summing up the flexibility of
intonation in Maqam music, which I have relished as someone seeking
an appreciation of this high art. Please let me try, with due humility
as someone in the early stages of learning to hear and most
imperfectly to approximate this music, both to comment on your
observations and to address the question of polyphony or harmony and
the neutral third.

Interestingly, your suggestion of a basic range from 14:13 to 11:10
very much agrees with my own perceptions. In a paper to which I am now
making some small corrections and revisions with many thanks to Dave,
I suggest a slightly wider range of about 125-170 cents, which I
recall that Farhat has suggested in his study of Persian Dastgah
music.

<http://www.bestII.com/~mschulter/IntervalSpectrumRegions.txt>

At times, I may use 121/122 or 171/172 cents in the place of a neutral
second, and the effect can approximate that of a neutral second -- for
example, in certain polyphonic progressions which are, of course,
outside the scope of traditional Maqam or Dastgah music. However, as I
have learned, to say that an interval around 170 cents has a generally
"neutral effect," and that it aptly and smoothly fits a given Maqam
for a listener who speaks this musical language as a native, may be
two quite different things!

> I conjecture that high prime limit and relatively simple integer
> ratios such as 14:13, 27:25, 13:12, 12:11, 35:32, 11:10, 54:49 are
> at play. Then again, instead of middle seconds, we might be looking
> at neutral thirds, in which case, the performer could be searching
> for 17:14, 39:32, 11:9, 27:22, 16:13, 21:17 and 31:25.

Please let me warmly agree that these are evocative ratios. Thinking
of a neutral third as a major second somewhere around 9:8 plus some
kind of neutral second is one approach. Another, which Dan Stearns
interestingly suggested in a recent post, is to take the 8-step
interval of 29-EDO at 331 cents as a very small neutral third at the
lower end of the range (maybe 23:19 or, more classically, 63:52). I
would say likewise that 9/29 octave or 372 cents, a virtually exact
realization of your 31:25, is at about the upper end of the range.

In fact, your ratios yield what I would consider a handy map of the
region, to which I'll add one ratio for symmetry, 75:62 at the lower
boundary (which could also be 98:81, or 4:3 less the classic 54:49
which Scala calls "Zalzal's mujannab," in which case the upper
boundary could be 243:196, or 9:8 plus 54:49, a tiny superparticular
difference of 6076:6075):

small or large
supraminor central or submajor
|---------------|------------------------|---------------|
75:62 17:14 39:32 11:9 27:22 16:13 21:17 31:25
330 336 342 347 355 360* 366 372

*16:13 is actually about 359.47 cents, but I have
rounded upward to 360 cents so as to make 39:32
and 16:13 add up to a rounded 702 cents.

It would seem that often central neutral thirds are typical of Maqam
music, while the small and large or supraminor/submajor sizes are
often preferred in Dastgah music. It would interesting to ask whether
this fits with your own impressions of the music, and how the various
regional variations of these styles may have evolved, given the rich
cross-pollination and common elements for these manifestations of the
Islamic tradition in the Near East.

If one seeks historical precedents, then Zalzal's scale according to
al-Farabius gives us a subtle 27:22-11:9 division of the fifth. I seem
to recall that Ibn Sina (or Avicenna in Latin Europe) placed Zalzal's
wosta fret at around 39:32; and, of course, 13:12 is a common ratio in
the medieval Islamic era, which would yield 39:32 when added to 9:8 or
16:13 when subtracted from 4:3. His wonderful scale based on a division
of a 7:6 third into 14:13:12 also has supraminor thirds at 63:52 or
332 cents, and submajor thirds at 26:21 or 369 cents, equal to 9:8
plus 14:13 or 4:3 less 14:13.

When I shared with George Secor my taste for 17:14 and 21:17, he
pointed out to me that 14:17:21 is actually the simplest neutral third
division of the fifth; either 32:39:48 or 18:22:27 requires a higher
odd-limit.

> So many ratios imply the necessity to temper pitches and represent
> diverse middle seconds as well as neutral thirds with few, yet,
> variagated microtones that are reasonably spaced. That is how I
> arrived at 79 MOS 159-tET. Check out page 95 of my doctorate
> dissertation. There, you will see how the 79-tone tuning
> approximates many RI intervals.

Oz, this is truly a brilliant system, both the way in which pure 3:2
fifths and 4:3 fourths are combined with temperament, and the array of
delicious flavors generated.

Curiously, in the kinds of polyphonic styles where I use neutral
thirds toward with smaller minor or larger major thirds, the
contrapuntal rules for their use are rather alike. Here _all_ thirds
are relatively blending but unstable, or "quasi-consonant" as you well
put it -- the neutral ones, but major and minor also! And all types of
thirds may vary in flavor over a wide range. In fact, I might gives
ranges of about 254-300 for minor; 330-372 for neutral; and 408-442
for major. Note that in this kind of style, the areas immediately
around 6:5 and 5:4 are less typical -- although, as we have noted,
they are often important in varieties of Maqam music. However, the
simple ratio or "valley" of 7:6 often figures very prominently.

In directed progressions, a general guideline is that a minor or
supraminor (small neutral) third will often resolve to a unison, and
submajor (large neutral) or major third to a fifth, one voice moving
by a major second and the other by a semitone or small neutral
second. The central neutral range of thirds has a special quality
because in resolving to either a unison or a fifth, one voice
typically moves by a major second and the other by a central neutral
second rather different from either a tone or a semitone.

If you are curious, here is an article about a 19-note tempered
system based on two JI tunings by Gene Ward Smith for Aaron
Johnson. While the main idea is to get lots of minor thirds within
about 20 cents of 7:6, maybe the most important simple ratio after 3:2
and 4:3, there are lots of neutral flavors also, although not ones
necessarily fitting any standard scheme of Maqamat or Dastgah-ha. It
mibht be interesting, however, as example of flexible tuning for
thirds (minor, neutral, and major).

<http://www.bestII.com/~mschulter/zest24-aaron_akj.txt>

Most appreciatively,

Margo Schulter
mschulter@...

Hello, all.

Mike's comment in a related dialogue the two of us are
having moved me to offer some comments here about some very
basic questions on the direction for harmonic entropy theory
to go. The question of the "depth" of an interval, as he terms
it, is to me an outstanding example of how what we may need
here is a theory of relativity, even if that outside of
"harmonic entropy proper."

Specifically, I was reflecting on Mike's comparison of the
"depth" -- which might here mean "richness" or possibly
"intricacy" -- of an interval such as 3:2 or 5:4.

This brings me to something I was vividly aware of for about
three decades before I got involved in our wonderful world of
alternative tunings.

To me, a 3:2 fifth or complete 2:3:4 was at once a simple concord,
or "valley," and a very rich sonority. In medieval European music,
of course, the latter is a complete stable sonority, the goal of
motion and the standard of a "perfect" sound. And so it still is.

However, when immersed in the different universe of 16th-century
European sound where 4:5:6 is a complete stable concord -- or,
more aptly spaced, something like 2:3:4:5 or 2:4:5:6, to put
Zarlino's or Morley's suggestions into frequency ratio form --
2:3:4 or the common four-voice variation of 1:2:3:4 can sound
quite "sparse" and a bit austere -- radically different from
2:3:4 or 1:2:3 in a 13th-14th century setting!

Long before getting wrapped up in the fine points of ratios,
I had a vivid experience in college. I have composed a simple
three-voice piece in a style around 1200, and was fortunate
enough to find three other students ready to try singing it.
They reached the end of the first phrase, where I had written
the sonority of a fifth with two voices converging on a unison.
However, two of them arrived at the fifth and the other at
a major third above the lowest voice -- especially likely in
reading at sight. I'm not sure how that major third was tuned,
but I vividly felt how an unstable third simply was "out of
place" there to my ears -- but obviously quite natural to
theirs. Since a major third would have been equally natural
to me in a Renaissance style, this was a matter of contextual
grammar rather than "entropy" in the abstract.

Thus, to borrow Mike's term, I would say that a 3:2 fifth or
2:3:4 sonority has greater "depth" -- or maybe better yet,
given his binocular vision analogy, "dimension" -- in a medieval
than in a Renaissance context.

Might the different modes of seeing with cones (day vision) and rods
(night vision) provide an analogy for shifting between styles. When one's
eyes are dark-adapted physically and biochemically, a level of light
normally routine can be painfully bright, as well as undoing the
adaptation -- thus the careful rules of etiquette for amateur
astronomical gatherings governing automobile lights and even small
torches or flashlights.

I'm not sure where these metaphors might lead us, but the situational
perception of entropy in context, if that's not an oxymoron, or if it
_is_ the kind of oxymoron to provoke further thought, might be one
direction to consider.

Most appreciatively,

Margo
mschulter@...

Dear Tom, Brad, Johnny, and all,

Thank you for some most stimulating discussions which give much to
ponder, combined with some poetic verbal music. Tom, your vivid
metaphor for meantone has led me to the title "Alternative Boulevards"
for some musical commentaries I plan to write, with full credit to you
for that title.

</tuning/topicId_77237.html#77655>

One thing I've found over the years is that we tend to be at our best
when campaigning positively, whether it's describing the special
charms of 17th-century meantone; or the genius of Werckmeister and his
scheme of well-temperaments; or the possibilities of a different kind
of tuning circle like Sorge's for Bach's music. The secret is that
traffic on one artistic avenue need not collide with that on another:
there's enough space for different tastes. And yes, Zarlino compares
an appreciation of good counterpoint with an informed taste for the
right kind of healthy food, so I guess "taste" is a period concept.

As someone who typically approaches meantone (regular or modified)
from a 16th-century to early 17th-century perspective, maybe my most
helpful substantive contribution to this discussion would be to
emphasize the vast range of epochs, places, and styles you are
discussing and sometimes rather pointedly debating. Since Frescobaldi
and the period around 1600 has been raised, I would emphasize one
specific point that becomes less applicable as we move into the realm,
still very broad, that you seem mainly to be focusing on.

In the era around 1600, and toward mid-century too, modes or "tones"
are still a vital focus of practice, as naturally I would emphasize
since it happens to be my main focus in lots of musicmaking <grin>.
For example, I once heard what sounded like a cadence on E with an
ascending rather than descending semitone in an opera of Monteverdi as
a kind of rhetorical figure going outside the usual nature of
Phrygian. In another words, what I heard as D# seemed to me "exotic"
-- a striking modern gesture, and just what we would expect from this
composer, of course!

Does thinking in terms of 12 modes give me a different perspective on
meantone than people oriented mainly to major/minor -- and would
people in the early 17th century, or some of them, have heard
Monteverdi's cadence as I did? Whatever, I'd say that not only the
range of consonance/dissonance which you have pointed out, Tom, but
the variety of modes, makes either a regular or modified meantone in
12 or more notes seem quite spacious.

As to there being no recordings from the period, that is true -- and
would also be true for an organist or harpsichordist of that time in
Sicily considering how to read and realize a piece of Monteverdi or
Frescobaldi without the benefit of a tape or CD hot from Mantua or
Rome. A notated piece of music in score, part-books, or whatever, of
course, is itself a kind of document -- so then, as now, inference and
interpretation can't be avoided. And it's fun.

Debates might be more affable if we focus on the debates in that era,
sometimes with both sides represented. Thus Jean Denis in 1643 and
1650 is out to stop the "black marketeering" of using augmented or
diminished meantone intervals in place of the "proper" ones: an
organist must on no account make modal transpositions like that of the
First Tone (more or less D Dorian) to F -- with F-G# as the third!
Don't let choristers who wish to "sing at their convenience" steer you
to such a result! Otherwise, people will not only consider your
otherwise most skillful playing as incompetent, but will conclude that
you cannot keep the instrument in tune.

However, Huygens in 1691 finds 7/6, along with 7/4, a prize jewel of
the 31-note cycle (1/4-comma or 31-EDO), and the comments in the
French literature about how people relish "transposed modes" (i.e. in
meantone, regular or modified) indicate that Denis may not have been
voicing a universal view for either the earlier or later part of the
century. Of course, as Denis would likely acknowledge, those augmented
and diminished intervals themselves are part of the standard idiom: he
may have decrying the tendency of people to stumble into them
unawares, possibly giving us a hint that people were nevertheless
doing it and sometimes liking it (with or without help from choristers
asking a keyboardist with 12 notes per octave to "play at their
convenience").

In short, as one following and favoring a modal kind of meantone
music, I want that diesis, the delightful disparity of major and minor
semitones, and those colorful augmented and diminished intervals.
However, far be it from me to assume that people moving into a tonal
universe in the decades around 1700 would have these same priorities,
although the discussions of "transposed modes" in French theory
suggest that the older perspectives on modality and meantone still
exerted a certain traction.

At the same time, I must acknowledge a radical historical
discontinuity or indeed "space warp" between any likely 17th-century
perspective on modified meantone (mentioned by Praetorius) and mine.
For "12-note circulation," all I need to do is tune F-C# in a regular
1/4-comma or 2/7-comma or the like and place the other three notes so
as share out the harmonic excess equally among the remaining four
fifths -- to paraphrase Brad, "Average it out, sister!"

The reason for this is that Tom's "black market" in those once not so
prestigious neighbhorhoods has become Saks Fifth Avenue, an
establishment of the most respectable kind with beautiful neo-Gothic
architecture inspired, however inexactly, by Marchettus and Machaut
and the usual 14th-century suspects, where major thirds are indeed
broad and expansive avenues leading to a fifth.

"Two just 5:4 thirds plus a just 9:7 third happily come quite
close to a 2:1 octave, with 2/7-comma nicely splitting the difference"
may be a precept of 21st-century "urban planning," but 16th-17th
century theory it's not.

Getting back to the main dialogue, however, one thing should grace and
temper our discussion: an awareness that musicians then and now can
differ in taste, and, in short, "hear things differently." Could the
debate about Werckmeister-Neidhardt-Sorge be analogous to arguing
whether a keyboard piece around 1600 was conceived for 1/4-comma,
2/7-comma, or 1/5-comma meantone? In other words, rather than seeking
the "correct" practice for a given period, we're canvassing the
diverse ways a piece could been and quite likely was played, and
exploring the question of which we like best, and why.

With many thanks,

Margo
mschulter@...

Dear Andreas,

Thank you for posting your temperament with one wide fifth based on
complex integer ratios, a technique that fascinates me, and which I
have used in rather different stylistic settings.

There is one small revision that might appeal to me, and I would be
very curious as to your opinion on what I might propose.

Looking at the sizes of fifths, I noticed that at D-A there is a fifth
considerably narrower than any other, at only about 694.1 cents, or
almost 8 cents smaller than pure, with a ratio of 884/592. This is a
bit more than 1/3 Pythagorean comma of tempering, and I wondered if
the step A (Scala step 9) might be raised slightly to make this fifth
a bit closer to pure without seriously compromising any other
interval.

In this proposed variation on your tuning, A is raised to 885/529 with
respect to C, or 885/592 with respect to D, forming a fifth D-A which
is narrow by almost exactly 1/4 Pythagorean comma -- indeed, the
accuracy of 885/592 in approximating this common degree of temperament
is amazing! The main complication I would see, if you like the result,
is that your pitch standard would then become a4 = 442.5.

! ProposedVariationOnSparschuh442wideFrench5th.scl
!
Proposed revision: step 9 (A) at 885/529, 890.9 cents -- Margo Schulter
12
!
558/529
592/529
628/529
662/529
706/529
744/529
792/529
837/529
885/529
942/529
992/529
2/1

0: 1/1 0.000 unison, perfect prime
1: 558/529 92.397
2: 592/529 194.795
3: 628/529 296.996
4: 662/529 388.276
5: 706/529 499.681
6: 744/529 590.442
7: 792/529 698.679
8: 837/529 794.352
9: 885/529 890.892
10: 942/529 998.951
11: 992/529 1088.487
12: 2/1 1200.000 octave

Most appreciatively,

Margo Schulter
mschulter@...

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:
>
> > Hi Margo! I must say that I seek simpler explanations and
believe
> > that seeking a midpoint between 12/7 and 7/4 most likely would
> > occur when these intervals are already in a tuning. Far more
likely
> > in an out-of-the-blue or more directly harmonic-series oriented
> > situation would be to seek out or find pleasing an interval
which
> > strikes a poise between 8:5 and 9:5.
>
> Hi, Cameron. Here I would guess that our comments are coming from
> rather different musical frames of reference. My first reaction to
> your reference to 8:5 and 9:5 was: "But that's from a large minor
> sixth to a large minor seventh, with lots of categories between,
> presumably including some kind of major sixth, and maybe a neutral
> sixth also."
>
> However, if you are thinking literally of a "harmonic series" --
> i.e. 5:6:7:8:9..., then your comment would, of course, hold.

Yes, exactly. I think the "first families" are always going to be
direct descendants and relatives of the audible harmonic series. Even
in heavily beating tunings, the harmonic series is the reference
structure. There is no aesthetic judgement here of course. Intervals
directly derived from the harmonic series may be exactly what someone
does NOT desire, and so on.
>
> For me, the idea of a middle or "interseptimal" ground between 12:7
> and 7:4 can apply in two situations. The first is in tuning systems
> like 14-EDO, 24-EDO or 29-EDO (or 19-EDO) where there isn't any
close
> representation of 12:7 or 7:4, but an intriguingly intermediate
size
> that might play a role comparable to either. The purpose of using
such
> a system wouldn't be to get accurate septimal ratios, but
precisely to
> get interseptimal ones, whether we speak of integer ratios as
> landmarks like 19:11 or 26:15, or simply sizes in cents in the 940-
960
> zone, more or less.

Yes- for example, I remember mentioning to you a 23-o and u- tonality
based 17 tone tuning or tetrachord. If we were for example to take
one of your tunings in the Scala archive which is "elevensy" with a
14/11 for example rather than the 23/18 I had chosen, we could strike
a maximally both-and-neither "floating" tuning between the two by
using a number of different classical means, including the golden
section.

Each would have its own subtly different character and cohesiveness-
the arithmetic mean would strike a mean which subsumes both of the
original intervals into a shared harmonic series so to speak, etc.

Oops gotta run, child calling...

>
> The other situation is where we have some fairly close septimal
> representations plus intermediate intervals as well: this happens
in
> Zest-24. Here's an article on a 19-note subset of this temperament
> emulating a septimal JI (2-3-7-9) matrix, with fairly close
> representations of 12:7 or 7:4 in some positions, and intermediate
> ones like 946 or 950 cents in others:
>
> <http://www.bestII.com/~mschulter/zest24-aaron_akj.txt>
>
> > According to my "wrong" method, that's at 942.517 cents, ie,
might
> > as well be the same thing as your example above. But I find my
> > explanation simpler.
>
> Please let me emphasize that I don't think there is any one
"right" or
> "wrong" method. The noble mediant is merely one way to approach the
> problem. I would consider seeing what interval sounds best for your
> purposes as the right way.
>
> > Either way, we get the inversion of the "consant interval" of
Ibn
> > Sin, and I don't think it's sheer coincidence. Note that the
> > interval you give above is even closer to Ibn Sin's than mine
is, a
> > tiny fraction of a cent rather than .9 cents. I believe that we
are
> > dealing with a fuzzy region a cent or two wide.
>
> Actually I'd suspect we're seeking a convenient number as a guide
to
> something notably fuzzy. Things like variations of timbre are
going to
> have an effect also, whether subtle or dramatic.
>
> > I also believe that a great big catalog of these kinds of
intervals
> > has been around for centuries and more, for no less than three
> > intervals I've identified as "shadows" have showed up in the
Scala
> > archives or on Gann's nifty anatomy of an octave page labelled
> > "Zalzal" or "Avicenna".
>
> The file octave.txt -- nifty, I'd say also! -- shows a ratio of
> 196/169 or 256.596 cents which is listed as an interval deemed
> "consonant" by Ibn Sina (or in Latin Europe, Avicenna). This is
> interesting, and I'd agree with this great philosopher and music
> theorist in his appraisal made something like a millennium ago.
I'd be
> interested to know his context for this interval. It is very close
> also to 297/256, or 257.182 cents. Thanks to Kyle Gann for making
that
> page "Anatomy of an Octave" available, and to you for reminding me
of
> how much fun I've had with that list of intervals.
>
> Ibn Sina also described a scale with simple septimal ratios, which
> George Secor and I discussed in our exploration of his 17-WT
tuning,
> which has some nice approximations:
>
> 1/1 14/13 7/6 4/3 3/2 21/13 7/4 2/1
>
> This is in the Scala archive as avicenna_diat.scl.
>
> Most appreciatively,
>
> Margo
> mschulter@...
>

Dear Margo,

First, let me thank you for bringing to my attention your well-prepared article titled "Regions of the Interval Spectrum". I am glad to observe that the chapters on neutral intervals agree with my views on Maqam music theory. Please note, that 14:13 is also given by Urmavi as the approximation to all middle melodic (lahni) intervals (supraminor seconds) in his treatise (Risale al-Sharafiyyah).

I think the theory of Maqam music and other "ethnic" genres around the world are much neglected by the alternative tuning list community. Most of the discussions are centered around either historical or contemporary microtonalisms for furthering Western music culture alone. While I appreciate the contributions by the West to musical art, I believe the Western quarter (pun intended) can account for only a fraction of the actual music-making in the globe. One of the greatest traditions is right next door: A venerable monophonal Middle Eastern culture based on maqamat, destgaha and raga. This "exotic" culture has been influenced by a thousand years of Islamic atmosphere to inspire such styles and practices as Mevlevi rites, Qawwali performances, peshrevs, taqsims, gazels, etc... Your penchant to discover more of the theories and styles of exotic traditions is admirable.

Though my experience is most inadequate to describe the musical wonders of the Islamic Civilization, my presence in the tuning list as a fresh academician should be construed as an oppurtunity to discover a glimpse of at least the Turkish branch of this grand culture.

You are correct in establishing 121 and 171 cents as producing "neutral effects". However, most of the time 168+ cents won't do at all. I myself shun 170 cents up to 180 cents as a useless region, a sort of "dead zone" if you prefer, and avoid it at all costs. You call the gap between these values the "equitable heptatonic". It may be useful for some African, Burmese or Gamelan ensembles, but I do not believe much can be done with such intervals in Maqam music.

You might remember, that in the first year of my presence in the tuning list, I advocated 29-EDO as a temperament suitable for Maqam music for the very reasons explained by Dan and yourself.

I acknowledge 98:81 or 75:62 as a supraminor second left out in my previous post. Also, 26/21 is also important as a submajor third. Thanks for mentioning these ratios.

Ah, but supraminor thirds are not restricted to Persian music. For you see, they are important flavours in such maqams as Hijaz, Nikriz and Neveser. I admit, though, that Persians are more fond of supraminor seconds compared to middler seconds.

14:17:21 makes a wonderful neutral chord.

I am glad to hear your appreciation of 79 MOS 159-tET. What do you think of my new 24-tone tuning for Maqam music based on a modified meantone temperament? I have explained it in the 76333rd message to this list.

I'm sorry, I am too tired to read the last article on 19-note tempered system. Later inshallah.

Cordially,
Oz.

On Jun 26, 2008, at 1:17 AM, Margo Schulter wrote:

>
>> Indeed, the neutral thirds of Maqam music are more diversified than
>> the ratio 11:9 can encompass. As far as I can tell, middle second
>> intervals roam the mujannab zone between 14:13 and 11:10. This is a
>> region 37 cents wide. Then again, it is possible to broaden the zone
>> and take 15:14 and 10:9 as the extremes. In that case, we acquire a
>> zone that is 63 cents wide!
>
> Dear Oz (and Dave, and all),
>
> Thank you for these remarks so nicely summing up the flexibility of
> intonation in Maqam music, which I have relished as someone seeking
> an appreciation of this high art. Please let me try, with due humility
> as someone in the early stages of learning to hear and most
> imperfectly to approximate this music, both to comment on your
> observations and to address the question of polyphony or harmony and
> the neutral third.
>
> Interestingly, your suggestion of a basic range from 14:13 to 11:10
> very much agrees with my own perceptions. In a paper to which I am now
> making some small corrections and revisions with many thanks to Dave,
> I suggest a slightly wider range of about 125-170 cents, which I
> recall that Farhat has suggested in his study of Persian Dastgah
> music.
>
> <http://www.bestII.com/~mschulter/IntervalSpectrumRegions.txt>
>
> At times, I may use 121/122 or 171/172 cents in the place of a neutral
> second, and the effect can approximate that of a neutral second -- for
> example, in certain polyphonic progressions which are, of course,
> outside the scope of traditional Maqam or Dastgah music. However, as I
> have learned, to say that an interval around 170 cents has a generally
> "neutral effect," and that it aptly and smoothly fits a given Maqam
> for a listener who speaks this musical language as a native, may be
> two quite different things!
>
>> I conjecture that high prime limit and relatively simple integer
>> ratios such as 14:13, 27:25, 13:12, 12:11, 35:32, 11:10, 54:49 are
>> at play. Then again, instead of middle seconds, we might be looking
>> at neutral thirds, in which case, the performer could be searching
>> for 17:14, 39:32, 11:9, 27:22, 16:13, 21:17 and 31:25.
>
> Please let me warmly agree that these are evocative ratios. Thinking
> of a neutral third as a major second somewhere around 9:8 plus some
> kind of neutral second is one approach. Another, which Dan Stearns
> interestingly suggested in a recent post, is to take the 8-step
> interval of 29-EDO at 331 cents as a very small neutral third at the
> lower end of the range (maybe 23:19 or, more classically, 63:52). I
> would say likewise that 9/29 octave or 372 cents, a virtually exact
> realization of your 31:25, is at about the upper end of the range.
>
> In fact, your ratios yield what I would consider a handy map of the
> region, to which I'll add one ratio for symmetry, 75:62 at the lower
> boundary (which could also be 98:81, or 4:3 less the classic 54:49
> which Scala calls "Zalzal's mujannab," in which case the upper
> boundary could be 243:196, or 9:8 plus 54:49, a tiny superparticular
> difference of 6076:6075):
>
> small or large
> supraminor central or submajor
> |---------------|------------------------|---------------|
> 75:62 17:14 39:32 11:9 27:22 16:13 21:17 31:25
> 330 336 342 347 355 360* 366 372
>
> *16:13 is actually about 359.47 cents, but I have
> rounded upward to 360 cents so as to make 39:32
> and 16:13 add up to a rounded 702 cents.
>
> It would seem that often central neutral thirds are typical of Maqam
> music, while the small and large or supraminor/submajor sizes are
> often preferred in Dastgah music. It would interesting to ask whether
> this fits with your own impressions of the music, and how the various
> regional variations of these styles may have evolved, given the rich
> cross-pollination and common elements for these manifestations of the
> Islamic tradition in the Near East.
>
> If one seeks historical precedents, then Zalzal's scale according to
> al-Farabius gives us a subtle 27:22-11:9 division of the fifth. I seem
> to recall that Ibn Sina (or Avicenna in Latin Europe) placed Zalzal's
> wosta fret at around 39:32; and, of course, 13:12 is a common ratio in
> the medieval Islamic era, which would yield 39:32 when added to 9:8 or
> 16:13 when subtracted from 4:3. His wonderful scale based on a > division
> of a 7:6 third into 14:13:12 also has supraminor thirds at 63:52 or
> 332 cents, and submajor thirds at 26:21 or 369 cents, equal to 9:8
> plus 14:13 or 4:3 less 14:13.
>
> When I shared with George Secor my taste for 17:14 and 21:17, he
> pointed out to me that 14:17:21 is actually the simplest neutral third
> division of the fifth; either 32:39:48 or 18:22:27 requires a higher
> odd-limit.
>
>> So many ratios imply the necessity to temper pitches and represent
>> diverse middle seconds as well as neutral thirds with few, yet,
>> variagated microtones that are reasonably spaced. That is how I
>> arrived at 79 MOS 159-tET. Check out page 95 of my doctorate
>> dissertation. There, you will see how the 79-tone tuning
>> approximates many RI intervals.
>
> Oz, this is truly a brilliant system, both the way in which pure 3:2
> fifths and 4:3 fourths are combined with temperament, and the array of
> delicious flavors generated.
>
> Curiously, in the kinds of polyphonic styles where I use neutral
> thirds toward with smaller minor or larger major thirds, the
> contrapuntal rules for their use are rather alike. Here _all_ thirds
> are relatively blending but unstable, or "quasi-consonant" as you well
> put it -- the neutral ones, but major and minor also! And all types of
> thirds may vary in flavor over a wide range. In fact, I might gives
> ranges of about 254-300 for minor; 330-372 for neutral; and 408-442
> for major. Note that in this kind of style, the areas immediately
> around 6:5 and 5:4 are less typical -- although, as we have noted,
> they are often important in varieties of Maqam music. However, the
> simple ratio or "valley" of 7:6 often figures very prominently.
>
> In directed progressions, a general guideline is that a minor or
> supraminor (small neutral) third will often resolve to a unison, and
> submajor (large neutral) or major third to a fifth, one voice moving
> by a major second and the other by a semitone or small neutral
> second. The central neutral range of thirds has a special quality
> because in resolving to either a unison or a fifth, one voice
> typically moves by a major second and the other by a central neutral
> second rather different from either a tone or a semitone.
>
> If you are curious, here is an article about a 19-note tempered
> system based on two JI tunings by Gene Ward Smith for Aaron
> Johnson. While the main idea is to get lots of minor thirds within
> about 20 cents of 7:6, maybe the most important simple ratio after 3:2
> and 4:3, there are lots of neutral flavors also, although not ones
> necessarily fitting any standard scheme of Maqamat or Dastgah-ha. It
> mibht be interesting, however, as example of flexible tuning for
> thirds (minor, neutral, and major).
>
> <http://www.bestII.com/~mschulter/zest24-aaron_akj.txt>
>
> Most appreciatively,
>
> Margo Schulter
> mschulter@...
>
>
> ------------------------------------
>
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>

Ozan Yarman wrote:
> Dear Margo,
> > First, let me thank you for bringing to my attention your well- > prepared article titled "Regions of the Interval Spectrum". I am glad > to observe that the chapters on neutral intervals agree with my views > on Maqam music theory. Please note, that 14:13 is also given by Urmavi > as the approximation to all middle melodic (lahni) intervals > (supraminor seconds) in his treatise (Risale al-Sharafiyyah).
> > I think the theory of Maqam music and other "ethnic" genres around the > world are much neglected by the alternative tuning list community. I think you may be right about that, unfortunately. I think it may be the case that few of us in the English-speaking parts of the world have much detailed exposure to the theory of Maqam music, or the ragas of India, or Georgian vocal music, or any number of other examples. I've read a little about the pelog and slendro scales of Indonesia (mainly the Balinese versions), but the variations of those tunings are extensive. Pretty much everything I know about Maqam tuning I've learned from this list and sites like maqamworld.com, but that doesn't stop me from enjoying music performed in these tuning systems.

So if I don't always comment on these posts, it's not necessarily from any lack of interest, but mainly that I don't have the background to contribute anything new (beyond "I like the sound of neutral seconds" or the like).

On Fri, Jun 27, 2008 at 8:15 PM, Ozan Yarman <ozanyarman@...> wrote:

> I think the theory of Maqam music and other "ethnic" genres around the
> world are much neglected by the alternative tuning list community.
> Most of the discussions are centered around either historical or
> contemporary microtonalisms for furthering Western music culture
> alone. While I appreciate the contributions by the West to musical
> art, I believe the Western quarter (pun intended) can account for only
> a fraction of the actual music-making in the globe. One of the
> greatest traditions is right next door: A venerable monophonal Middle
> Eastern culture based on maqamat, destgaha and raga. This "exotic"
> culture has been influenced by a thousand years of Islamic atmosphere
> to inspire such styles and practices as Mevlevi rites, Qawwali
> performances, peshrevs, taqsims, gazels, etc... Your penchant to
> discover more of the theories and styles of exotic traditions is
> admirable.

> Though my experience is most inadequate to describe the musical
> wonders of the Islamic Civilization, my presence in the tuning list as
> a fresh academician should be construed as an oppurtunity to discover
> a glimpse of at least the Turkish branch of this grand culture.

Well hey man, if you have a listening list of stuff you can recommend,
I think we'd all love to check it out. World music is one of the most
fascinating things in the, well, the world. Mainly because you have
thousands of years of musical development behind most of these
cultures and styles, and so they are usually very much advanced.

Jeff Buckley did a Qawwali-inspired song, "Dream Brother," in which he
mixed pop/rock with traditional Qawwali elements, and it's one of my
favorite songs. I started looking for some traditional Qawwali
recordings when I heard that song, and I didn't really find much.

Any time there is an old, ancient branch of music that has reached as
high of a level of artistic development as the one we're talking about
here, people will be interested. I just think many don't know about it
yet.

One interesting thing to note is that the religious music of all of
the world sounds very, very, very similar. Perhaps not the music that
is "associated" with various churches and such - but the music that
monks sing, the music that is sung to draw people closer to the
experience of God.

-Mike

--- In tuning@yahoogroups.com, "Mike Battaglia" <battaglia01@...>
wrote:
>
> On Fri, Jun 27, 2008 at 8:15 PM, Ozan Yarman <ozanyarman@...>
wrote:
>
> > I think the theory of Maqam music and other "ethnic" genres
around the
> > world are much neglected by the alternative tuning list
community.
> > Most of the discussions are centered around either historical or
> > contemporary microtonalisms for furthering Western music culture
> > alone. While I appreciate the contributions by the West to
musical
> > art, I believe the Western quarter (pun intended) can account
for only
> > a fraction of the actual music-making in the globe. One of the
> > greatest traditions is right next door: A venerable monophonal
Middle
> > Eastern culture based on maqamat, destgaha and raga.
This "exotic"
> > culture has been influenced by a thousand years of Islamic
atmosphere
> > to inspire such styles and practices as Mevlevi rites, Qawwali
> > performances, peshrevs, taqsims, gazels, etc... Your penchant to
> > discover more of the theories and styles of exotic traditions is
> > admirable.
>
> > Though my experience is most inadequate to describe the musical
> > wonders of the Islamic Civilization, my presence in the tuning
list as
> > a fresh academician should be construed as an oppurtunity to
discover
> > a glimpse of at least the Turkish branch of this grand culture.
>
> Well hey man, if you have a listening list of stuff you can
recommend,
> I think we'd all love to check it out. World music is one of the
most
> fascinating things in the, well, the world. Mainly because you have
> thousands of years of musical development behind most of these
> cultures and styles, and so they are usually very much advanced.
>
> Jeff Buckley did a Qawwali-inspired song, "Dream Brother," in
which he
> mixed pop/rock with traditional Qawwali elements, and it's one of
my
> favorite songs. I started looking for some traditional Qawwali
> recordings when I heard that song, and I didn't really find much.
>
> Any time there is an old, ancient branch of music that has reached
as
> high of a level of artistic development as the one we're talking
about
> here, people will be interested. I just think many don't know
about it
> yet.
>
> One interesting thing to note is that the religious music of all of
> the world sounds very, very, very similar. Perhaps not the music
that
> is "associated" with various churches and such - but the music that
> monks sing, the music that is sung to draw people closer to the
> experience of God.
>
> -Mike
>

One of the best collections of music from the Mideast is a 15-CD set
called _The Music of Islam_, by Celestial Harmonies, 1998.

-Cris

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:
>
Dear Margo:
>
> Thank you for posting your temperament with one wide fifth based on
> complex integer ratios, a technique that fascinates me, and which I
> have used in rather different stylistic settings.
>
My intrest comes form Werckmeister's:
http://www.rzuser.uni-heidelberg.de/~tdent/septenarius.html
http://en.wikipedia.org/wiki/Werckmeister_temperament#Werckmeister_IV_.28VI.29:_the_Septenarius_tunings
/tuning/topicId_69724.html#69750
in his
"Musicalische Temperatur"
http://diapason.xentonic.org/ttl/ttl01.html
there on p72.

! septenarius.scl
! Werckmeister's #6 in string-lenghts on the monochord
!
C196 C#186 D176 D#165 E156 F147 F#139 G131 G#124 A117 B110 H104...
!
12
!
98/93 ! := 196/186 = C/C#
196/176 ! = C/D
196/165 ! = C/D#
49/39 ! := 196/156 = C/E = (5/4)*(196/195)
4/3 ! := 196/147 = C/F
196/139 ! = C/F#
196/131 ! = C/G = (3/2)*(392/393)
49/31 ! :=196/124 = C/G#
196/117 ! = C/A
98/55 ! =196/110 = C/B
49/26 ! =196/104 = C/H
2/1
!

later
http://www.tuningforktherapy.com/about.html
"In 1834, J.H. Scheibler presented a set of 54 tuning forks covering
ranges from 220 Hz to 440Hz."
http://www.apogeelearning.com/acutone/historytuningfork_1_2.html
"J. H. Scheibler in Germany in 1834 presented a set of 54 tuning forks
covering the range from 220 Hz to 440 Hz, at intervals of 4 Hz."
or
http://www.cosmeo.com/viewArticle.cfm?guidAssetId=A44930C3-53BA-4646-AD2C-B6AA4F1A62ED&&nodeid=
"The German physicist Johann Heinrich Scheibler (1777-1838) made the
first accurate determination of pitch corresponding to frequency and
proposed the standard A equals 440 in 1834."

for that purpose of defineing his 440Hz standard
S. appearently referred to above Werckmeister's
original Monochord-lenghts C196...C'98 (Upper-case letttes) on p.73
but now instead in reverse pitch-order
interpreted as absolute frequncies
(here denotated in lower-case letters):

g1 = 49 C'98 C196
d3 = F147 := 49*3
a2 = 55 B110 220 440 (<441 = 147*3) hence Scheibler's choice of 440cps
e3 = D#165 := 55*3
b0 = 31 62 G#124 248 496 (!>! 497 = 165*3) attend the wide 5th !
f#2 = 93 186C#
c#3 = F#139 278 (<279 = 93*3)
g#1 = 52 H104 208 416 (<417 = 139*3)
eb1 = 39 78 E156 := 52*3
bb2 = A117 := 39*3
f1 = 44 88 D176 352 (!>! 351 = 117*3) another wirde 5th !
c3 = G131 (<132 = 44*3)
g1 = 49 C'98 C196 392 (<393 = 131*3)

with the invariant 5ths against change from lenghts to frequency:
1. 392:393 for G-C & c-g
2. 278:279 for F#-C# & f#-c#

completely:
C392:393~G131:132~D352:351~AEH416:417~F#278:279~C#G#496:495~D#B440:441~F-C

c392:393~gd440:441~ae496:495~bf#278:279~c#416:417~g#eb-bb352:351~f131:132c

!reverseSeptenarius.scl
!
W's monochord-lengths backwards in reverse direction as frequencies
!
139/131 ! c#3/c3
147/131 ! d3 /c3
156/131 ! eb3/c3
165/131 ! e3 /c3 =(5/4)*(132/131)
176/131 ! f3/ c3 =(4/3)*(132/131) some scholars prefer 175 instead 176
186/131 ! f#3/c3
196/131 ! g3 /c3 = (3/2)*(392/393)
208/131 ! g#3/c3
220/131 ! a3 /c3 in order to meet Scheibler's absolute preference
234/131 ! bb3/c3
246/131 ! b3 /c3
2/1
!

get rid of the 2 wide-5ths ib that
in order to obtain a real 'well-temerature' "Wohl-Temperatur"
by the modified chain of 5ths:

C 392:393 G D 440:441 A E 494:495 B 740:741 F# C# 1664:1665 G#
Ab Eb Bb 350:351 F 524:525 C

!rev_Sept_Well_mod.scl
!
without the wide 5ths in Werckmeister's original ratios
12
!
555/524 ! c#5/c5
147/131 ! d3 /c3
156/131 ! eb3/c3
165/131 ! e3 /c3 =(5/4)*(132/131)
175/131 ! f3/ c3 =(4/3)*(525/524) some scholars prefer 175 instead 176
185/131 ! f#3/c3
196/131 ! g3 /c3 = (3/2)*(392/393)
208/131 ! g#3/c3
220/131 ! a3 /c3 Scheibler's fork a4=440cps
234/131 ! bb3/c3
247/131 ! b3 /c3
2/1
!

but that sounds
-at least in my ears-
all to much alike 12-EDO.

> There is one small revision that might appeal to me, and I would be
> very curious as to your opinion on what I might propose.
>
> Looking at the sizes of fifths, I noticed that at D-A there is a fifth
> considerably narrower than any other, at only about 694.1 cents, or
> almost 8 cents smaller than pure, with a ratio of 884/592. This is a
> bit more than 1/3 Pythagorean comma of tempering, and I wondered if
> the step A (Scala step 9) might be raised slightly to make this fifth
> a bit closer to pure without seriously compromising any other
> interval.
>
> In this proposed variation on your tuning, A is raised to 885/529 with
> respect to C, or 885/592 with respect to D, forming a fifth D-A which
> is narrow by almost exactly 1/4 Pythagorean comma -- indeed, the
> accuracy of 885/592 in approximating this common degree of temperament
> is amazing! The main complication I would see, if you like the result,
> is that your pitch standard would then become a4 = 442.5.

fully agreed!
that's an good idea,
as already mentioned in:
/tuning/topicId_75816.html#76113
"!well_Violin2Piano.scl
...
885/523 ! A = 442.5Hz*2 absolute a4
"
>
hence i do accept yours improvement as welcome:
> ! ProposedVariationOnSparschuh442wideFrench5th.scl
> !
> Proposed revision: step 9 (A) at 885/529, 890.9 cents -- Margo Schulter
...
> 885/529
...
meanwhile i do consider
264.5C4 as to harsh for the middle-C4,
a better choice -in my ears, at least on my piano- would be:
264.3

Perhaps you also like an further "septenarian" refinement too:
Start at an minor-tone (10:9) below Scheibler:

G; (7*7=E6:27=49 E8:27=98 <) 99G2 198G3 396G4 := 440Hz*(9:10)
D; (E6:9=147 E8:9=294 < 295=A5:3 <) 296D4 (<297 = G2*3)
A; (E6:3=441 E8:3=882 <) 885A5 just as Margo proposes
E; 1323E6
B; 31B0.....3968B7 (<3969=E6*3) through all 8 Bs on the piano-keys
F#; 93F#2 := B0*3
C#; 279C#4 := F#2*3
G#; 209G#3 418G#4 836G#5 (< 887 = C#4*3)
Eb; 627Eb5 := G#3*3
Bb; 235Bb3 470Bb4 940Bb5 1880Bb6 (< 1881 = Eb5*3)
F; (C4:3=88.1 C5:3=176.2 C6:3=352.4 <) 352.5F4 705F5 := Bb3*3
C; (G1:3=33 ... G4:3=264 <) 264.3C4
G; 99G2 = 33*3

Chromatically on the keys in frequencies as absolute-pitchs:

264.3_C4
279___C#4
296___D4
313.5_Eb4
330.75E4
352.5_F4
372___F#4
396___G4
418___G#4
442.5_A4
470___Bb4
528.6_C5

!sparschuh885A5.scl
!
c880:881g296:297d295:296a3968:3969bf#c#836:837g#eb1880:1881bbf3524:3525c
!
12
!
2790/2463
2960/2463
3135/2463
6615/4926 ! (5:4)(882:881)
3525/2463 ! (4:3)(3525:3524)
3720/2463
3960/2463 ! (3:2)(880:881)
4180/2463
4425/2463
4700/2463
4960/2463
2/1
!

What do you think about that now?

Yours Sincerely
A.S.

> Dear Margo,

> First, let me thank you for bringing to my attention your well-
> prepared article titled "Regions of the Interval Spectrum". I am
> glad to observe that the chapters on neutral intervals agree with my
> views on Maqam music theory. Please note, that 14:13 is also given
> by Urmavi as the approximation to all middle melodic (lahni)
> intervals (supraminor seconds) in his treatise (Risale
> al-Sharafiyyah).

Dear Ozan,

Thanks you for this very interesting observation about Urmavi and
14:13. There is a scale in the Scala archives (iran_diat.scl) edited
by Dariush Anooshfar which gives what is identified as a 125-EDO
version of a scale by Urmavi, one which may use some intervals which
are still featured in certain Iranian musics. This is discussed a bit
more below.

Also, while I am using less than perfect software for processing pdf
files that runs out of virtual memory on your dissertation file, there
is no problem with the doctoral reports and other related documents on
your Web site which I'm guessing may contain similar material. Thank
you for so generously making all of these material available!

> I think the theory of Maqam music and other "ethnic" genres around
> the world are much neglected by the alternative tuning list
> community. Most of the discussions are centered around either
> historical or contemporary microtonalisms for furthering Western
> music culture alone. While I appreciate the contributions by the
> West to musical art, I believe the Western quarter (pun intended)
> can account for only a fraction of the actual music-making in the
> globe. One of the greatest traditions is right next door: A
> venerable monophonal Middle Eastern culture based on maqamat,
> destgaha and raga. This "exotic" culture has been influenced by a
> thousand years of Islamic atmosphere to inspire such styles and
> practices as Mevlevi rites, Qawwali performances, peshrevs, taqsims,
> gazels, etc... Your penchant to discover more of the theories and
> styles of exotic traditions is admirable.

Please let me add, in response to some questions people have raised,
that there are now some excellent books in English by exponents of
some of these traditions: thus Touma and Racy on Maqam music as
practiced in the Arab world; and Farhat on the Persian Dastgah system,
as well as book on this Dastgah tradition with some companion CD's by
Dr. Tala`i. Racy is especially interesting on the "ecstatic" qualities
of traditional music which is lost when tuning is dictated by
arbitrary standard such as 24-EDO, rather than adapted to each Maqam.

> You are correct in establishing 121 and 171 cents as producing
> "neutral effects". However, most of the time 168+ cents won't do at
> all. I myself shun 170 cents up to 180 cents as a useless region, a
> sort of "dead zone" if you prefer, and avoid it at all costs. You
> call the gap between these values the "equitable heptatonic". It may
> be useful for some African, Burmese or Gamelan ensembles, but I do
> not believe much can be done with such intervals in Maqam music.

This may be rather analogous to larger major thirds around 408-440
cents, which may excel in a European style based on medieval or
neomedieval polyphony, but can hardly be aptly substituted as a
general rule in a Renaissance meantone style where 5:4 is expected!

I seem to recall having read a passage somewhere in your writings
about choosing steps so as to avoid an interval of 170 cents between
two significant degrees for Maqam music.

> You might remember, that in the first year of my presence in the
> tuning list, I advocated 29-EDO as a temperament suitable for Maqam
> music for the very reasons explained by Dan and yourself.

I should go back and read your posts: that the three of us all see
29-EDO in this way, coming from quite different perspectives, is
fascinating.

> I acknowledge 98:81 or 75:62 as a supraminor second left out in my
> previous post. Also, 26/21 is also important as a submajor third.
> Thanks for mentioning these ratios.

One place 26/21 comes up is in Ibn Sina's 28:26:24:21 tetrachord.

> Ah, but supraminor thirds are not restricted to Persian music. For
> you see, they are important flavours in such maqams as Hijaz, Nikriz
> and Neveser. I admit, though, that Persians are more fond of
> supraminor seconds compared to middler seconds.

> 14:17:21 makes a wonderful neutral chord.

Agreed!

> I am glad to hear your appreciation of 79 MOS 159-tET. What do you
> think of my new 24-tone tuning for Maqam music based on a modified
> meantone temperament? I have explained it in the 76333rd message to
> this list.

This is a fascinating design: 12 notes from a circulating temperament
of Rameau, and the other 12 added to provide some delicious Maqam
flavors!

</tuning/topicId_76333.html#76333>

> I'm sorry, I am too tired to read the last article on 19-note
> tempered system. Later inshallah.

Please let me add that after looking at your ingenious Rameau-based
tuning, I thought I might post a less precise set of 24 steps from
Zest-24 which has some of its steps or _perdeler_ (plural of Turkish
_perde_ or "step" or "tone," if I'm correct) fitting a tuning such as
Anooshfar's iran_diat.scl which he attributes to Urmavi, rather than
the usual steps appearing your system.

! iran_diat.scl
!
Iranian Diatonic from Dariush Anooshfar, Safi-a-ddin Armavi's scale from 125 ET
7
!
220.800 cents
441.600 cents
489.600 cents
710.400 cents
931.200 cents
979.200 cents
2/1

The Anooshfar-like intervals that arise (I wonder what kind of tuning
by Urmavi was the basis for iran_diat.scl) are the large major third
and sixth at 441 and 932 cents, and also the 50-cent steps (for
example 440-491, 931-982). These seem to me typical of iran_diat.scl
rather than standard Maqam practice, but I wonder how you would see
this.

An interesting question is how one might name the perdeler in this
Zest-24 tuning.

0 224 441 491 708 932 982 1200

What I would emphasize is that Zest-24 is a radically different way of
going from a 12-note circulating tuning to a 24-note system. You took
Rameau's 12-note circle and added customized steps designed for ideal
placement in a Maqam style. My method was much more crude: simply add
a second identical circle at a distance suggested by the structure of
the tuning rather than the needs of Maqam intonation.

As one might say, this is the difference between system where Maqam
music shapes the finely calculated placement of the notes or perdeler;
and one where the notes bend Maqam music to their measure, not
necessarily with any predictability of results!

When I try 'show /line intervals' in Scala, I see some nice versions
of Rast in various forms and Ushshaq, for example -- but not
necessarily in the _relations_ than traditional schemes of modulation
would require.

Also, the structure of the Zest-24 tuning itself imposes some
arbitrary limitations. For example, your 24-note system includes
neutral thirds at both 17/14 and 16/13; or, in your 79-MOS, sizes of
both 332 and 362 cents above Rast. In Zest-24, however, I realized
that it is impossible to have two sizes of neutral thirds above the
same perde or step. The situation might be analogous to that of some
equal temperaments where only a single size of middle or neutral third
is available -- although here, the size may change as one moves around
the system.

In a table following the Scala file, I attempt to guess at some
possible names for the steps or perdeler; but I would invite your
comments on this. The question of the perdeler and names for various
shades of modification is one most interesting.

! zest24-Bbup.scl
!
Tuning set starting from Bb* (24)
24
!
32.81250
83.20312
153.51562
203.90624
223.82812
274.21874
344.53124
394.92187
440.62500
491.01562
536.71875
587.10937
657.42187
707.81249
727.73437
778.12500
849.60937
899.99999
931.64062
982.03124
1040.62500
1091.01562
1149.60937
2/1

--------------------------------------------------------------------
approx tentative approx approx
degree cents 159-EDO perde name 288-EDO RI
--------------------------------------------------------------------
0: 0.00000 0 Rast 0 1/1
1: 32.81250 4 (Sarp Rast) 8 64/63
2: 83.20312 11 Shuri 20 22/21
3: 153.51562 20 (Zengule cluster) 37 12/11
4: 203.90624 27 Dugah 48 9/8
5: 223.82812 30 (Sarp Dugah) 54 8/7
6: 274.21874 37 Nerm (soft) Kurdi 66 75/64
7: 344.53124 46 Segah of Ushshaq 83 11/9
8: 394.92187 52 Segah 95 5/4
9: 440.62500 58 Dik Nishabur 106 9/7
10: 491.01562 65 Chargah 118 4/3
11: 536.71875 71 Nim Hijaz 129 15/11
12: 587.10937 78 Hijaz 141 7/5
13: 657.42187 87 (Saba cluster) 158 19/13
14: 707.81249 94 Neva 170 3/2
15: 727.73437 96 (Sarp Neva) 175 32/21
16: 778.12500 103 Bayyati 187 11/7
17: 849.60937 113 (Hisar/Huzzam cluster) 204 18/11
18: 899.99999 119 Huseyni 216 27/16
19: 931.64062 123 (Sarp Huseyni) 224 12/7
20: 982.03124 130 Koutchouk (little) Ajem 236 30/17
21: 1040.62500 138 (Evdj cluster) 250 51/28
22: 1091.01562 145 Evdj 262 15/8
23: 1149.60937 152 Mahurek 276 64/33
24: 1200.00000 159 Gerdaniye 288 2/1

With many thanks,

Margo
mschulter@...

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@> wrote:

Dear Margo,

Werckmeister's so called
11-limit "septenarian" 99:98 Comma amounts (81:80):(441:440)
11-limit "septenarian" 441:440 Schisma amounts (81:80):(99:98)
if understood that ratios as epimoric bisection of the SC=81:80.

the following well-tuning uses that as epimoric bi-scetion:

F 440:441 G = F 881:882 C 880:881 G inbetween F-C-G

and the trisection
inbetween the violin empty strings G-D-A-E consisting in

G 98:99 E = G 276:279 E subdivided into 3 epimoric factors:

G 296:297 D 295:296 A 296:297 E

and respectively for the schisma B 32768:32805 F within

E 3968:3969 B F# C# G# 2510:2511 Eb 3764:3765 Bb 4704:4705 F

into 4 another superparticular ratios
in order to conclude the cycle of 5ths by
the totally amount of an PC=3^12/2^19.

meanwhile my good-old piano got tuned in that refinement :

F 881:882 C 880:881 G 296:297 D 295:296 A 296:297 E

with the interim product inbetween FCGDAE = 80:81 = SC and

E 3968:3969 B F# C# G# 2510:2511 Eb 3764:3765 Bb 4704:4705 F

for yielding the Schisma=2^15/5/3^8=32768:32805.

that's expanded:
> Start at an minor-tone (10:9) below Scheibler:
>
> G; (7*7=E6:27=49 E8:27=98 <) 99G2 198G3 396G4 := 440Hz*(9:10)
> D; (E6:9=147 E8:9=294 < 295=A5:3 <) 296D4 (<297 = G2*3)
> A; (E6:3=441 E8:3=882 <) 885A5 just as Margo proposes
> E; 1323E6
> B; 31B0...3968B7(<3969=E6*3=49) lowest pitch on 5string doublebass
> F#; 93F#2 := B0*3
> C#; 279C#4 := F#2*3
attend the modifications from here on:
G#; 887G#5 := C#4*3
Eb; 1255Eb6 2510Eb7 (< 2511 := G#5*3)
Bb; (F2:3=29.4 .. F6:3=470.4<)470.5Bb4 941Bb5 1882Bb6 (3765:=Eb6*3)
F; (C4:3 = 88.1 <) 88.2F2
> C; (G1:3=33 ... G4:3=264 <) 264.3C4
> G; 99G2 = 33*3
>
Hence the
> Chromatically
improved values
>on the keys in frequencies as absolute-pitchs:
do arise a tiny little bit on the corresponding notes:
>
Correction:
> 264.3_C4
> 279___C#4
> 296___D4
313.75 > instead formerly 313.5_Eb4
> 330.75E4
> 352.5_F4
> 372___F#4
> 396___G4
418.5 > instead formerly 418___G#4
> 442.5_A4
470.5 > instead fromerly 470___Bb4
> 528.6_C5
>
all others pitches remain unchanged.

please forget about my obsolte
> !sparschuh885A5.scl
...
and replace it by the more elaborate:

!ef_JI_MiFa.scl
!
Sparschuh's 80:81 inbetween F-C-G-D-A-E and 32768:32805 in E B...Bb F
!
12
!
2790/2463
2960/2463
6275/2463 ! was > 3135/2463 formerly
6615/4926 ! (5:4)(882:881)
3524/2463 ! (4:3)(882:881) was > 3525/2463 ! (4:3)(3525:3524)
3720/2463
3960/2463 ! (3:2)(880:881)
4185/2463 ! was > 4180/2463 formerly
4425/2463
4705/4700 ! was > 4700/2463 fromerly
4960/2463
2/1
!

attend the exact JI haltone E 16:15 F that is
shifteted 882:881 upwards versus the unison 1:1,
because the 4th F-C and the 3rd C-E
turn out to be the same 882:881 sharper than JI.

Quests:
Who in that group here dares to try that on his/hers own instrument?
Are there any proposals in order to improve even that once more again?

Yours Sincerely
A.S.

--- In tuning@yahoogroups.com, "Andreas Sparschuh" <a_sparschuh@...>
wrote:

> Sparschuh's 80:81 inbetween F-C-G-D-A-E and 32768:32805 in E B...Bb F
> !
> 12
> !
> 2790/2463
> 2960/2463
> 6275/2463 ! was > 3135/2463 formerly
> 6615/4926 ! (5:4)(882:881)
> 3524/2463 ! (4:3)(882:881) was > 3525/2463 ! (4:3)(3525:3524)
> 3720/2463
> 3960/2463 ! (3:2)(880:881)
> 4185/2463 ! was > 4180/2463 formerly
> 4425/2463
> 4705/4700 ! was > 4700/2463 fromerly
> 4960/2463
> 2/1
> !
>
> Quests:
> Who in that group here dares to try that on his/hers own instrument?

I have a straightforward practical question.

With only an A=440 tuning fork in one hand, a harpsichord tuning lever
in the other hand, and absolutely NO electronic devices of any kind:
how exactly should one proceed to get all twelve of your notes
correctly tuned onto a harpsichord, using this scheme? And with no
way of measuring integer frequencies, either, or knowing when they've
been achieved precisely?

To what precision are errors acceptable? And why?

> Are there any proposals in order to improve even that once more
> again?

It depends what the word "improve" means to you. Does one first have
to agree with your goal of proportional beating, and your constraint
of integer frequencies? All this stuff just looks like
nearly-meaningless tables of numerals to me, sorry; the only way I
know to assess its quality is to see if it agrees with *your own*
goals...which doesn't tell us one way or another about the usefulness
for anything else *but* your own goal of proportional beating (or
whatever it is).

If I'd somehow take the time and get this temperament set up on my
harpsichord, within some acceptable error tolerance but without using
any electronic devices: how would the resulting temperament sound in
playing (say) some late Couperin? What does it do for the music,
harmonically and melodically? That's the kind of thing I personally
care about: a temperament that sounds great in the music, and that can
be done entirely by ear in less than 10 minutes without having to
calculate (or even refer to) a page of numbers. How, please?

And, what happens if I'd want to start on A=430 or something else
(maybe not having anything to do with integers!), or on some C? Does
it all need to be recalculated? Help out the practical musicians who
just want to listen to the sounds of intervallic relationships,
calculating nothing....

Brad Lehman

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:
>
>
> > Indeed, the neutral thirds of Maqam music are more diversified
than
> > the ratio 11:9 can encompass. As far as I can tell, middle second
> > intervals roam the mujannab zone between 14:13 and 11:10. This is
a
> > region 37 cents wide. Then again, it is possible to broaden the
zone
> > and take 15:14 and 10:9 as the extremes. In that case, we acquire
a
> > zone that is 63 cents wide!
>
> Dear Oz (and Dave, and all),
>
> Thank you for these remarks so nicely summing up the flexibility of
> intonation in Maqam music, which I have relished as someone seeking
> an appreciation of this high art. Please let me try, with due
humility
> as someone in the early stages of learning to hear and most
> imperfectly to approximate this music, both to comment on your
> observations and to address the question of polyphony or harmony and
> the neutral third.
>
> Interestingly, your suggestion of a basic range from 14:13 to 11:10
> very much agrees with my own perceptions. In a paper to which I am
now
> making some small corrections and revisions with many thanks to
Dave,
> I suggest a slightly wider range of about 125-170 cents, which I
> recall that Farhat has suggested in his study of Persian Dastgah
> music.
>
> <http://www.bestII.com/~mschulter/IntervalSpectrumRegions.txt>
>
> At times, I may use 121/122 or 171/172 cents in the place of a
neutral
> second, and the effect can approximate that of a neutral second --
for
> example, in certain polyphonic progressions which are, of course,
> outside the scope of traditional Maqam or Dastgah music. However,
as I
> have learned, to say that an interval around 170 cents has a
generally
> "neutral effect," and that it aptly and smoothly fits a given Maqam
> for a listener who speaks this musical language as a native, may be
> two quite different things!
>
> > I conjecture that high prime limit and relatively simple integer
> > ratios such as 14:13, 27:25, 13:12, 12:11, 35:32, 11:10, 54:49 are
> > at play. Then again, instead of middle seconds, we might be
looking
> > at neutral thirds, in which case, the performer could be searching
> > for 17:14, 39:32, 11:9, 27:22, 16:13, 21:17 and 31:25.
>
> Please let me warmly agree that these are evocative ratios. Thinking
> of a neutral third as a major second somewhere around 9:8 plus some
> kind of neutral second is one approach. Another, which Dan Stearns
> interestingly suggested in a recent post, is to take the 8-step
> interval of 29-EDO at 331 cents as a very small neutral third at the
> lower end of the range (maybe 23:19 or, more classically, 63:52). I
> would say likewise that 9/29 octave or 372 cents, a virtually exact
> realization of your 31:25, is at about the upper end of the range.
>
> In fact, your ratios yield what I would consider a handy map of the
> region, to which I'll add one ratio for symmetry, 75:62 at the lower
> boundary (which could also be 98:81, or 4:3 less the classic 54:49
> which Scala calls "Zalzal's mujannab," in which case the upper
> boundary could be 243:196, or 9:8 plus 54:49, a tiny superparticular
> difference of 6076:6075):
>
> small or large
> supraminor central or submajor
> |---------------|------------------------|---------------|
> 75:62 17:14 39:32 11:9 27:22 16:13 21:17 31:25
> 330 336 342 347 355 360* 366 372
>
> *16:13 is actually about 359.47 cents, but I have
> rounded upward to 360 cents so as to make 39:32
> and 16:13 add up to a rounded 702 cents.
>
> It would seem that often central neutral thirds are typical of Maqam
> music, while the small and large or supraminor/submajor sizes are
> often preferred in Dastgah music. It would interesting to ask
whether
> this fits with your own impressions of the music, and how the
various
> regional variations of these styles may have evolved, given the rich
> cross-pollination and common elements for these manifestations of
the
> Islamic tradition in the Near East.
>
> If one seeks historical precedents, then Zalzal's scale according to
> al-Farabius gives us a subtle 27:22-11:9 division of the fifth. I
seem
> to recall that Ibn Sina (or Avicenna in Latin Europe) placed
Zalzal's
> wosta fret at around 39:32; and, of course, 13:12 is a common ratio
in
> the medieval Islamic era, which would yield 39:32 when added to 9:8
or
> 16:13 when subtracted from 4:3. His wonderful scale based on a
division
> of a 7:6 third into 14:13:12 also has supraminor thirds at 63:52 or
> 332 cents, and submajor thirds at 26:21 or 369 cents, equal to 9:8
> plus 14:13 or 4:3 less 14:13.
>
> When I shared with George Secor my taste for 17:14 and 21:17, he
> pointed out to me that 14:17:21 is actually the simplest neutral
third
> division of the fifth; either 32:39:48 or 18:22:27 requires a higher
> odd-limit.

Hi Margo,

I wish I had more time these to follow some of these discussions on
the tuning lists. Lately it's been restricted mostly to searching
for recent occurrences of my name to see if anything requires my
reply.

In light of the current discussion about harmonic entropy and
metastable (noble mediant) intervals, in case you haven't already
mentioned it, there's something else I happened to notice about the
14:17:21 triad and its 2nd inversion, 21:28:34. The noble mediant
between 4:5 and 5:6 is ~339.344c, which is within a few cents of
14:17 (~336.130c), found in both 14:17:21 and 21:28:34. The noble
mediant between 3:5 and 5:8 is ~833.090c, which is close to 21:34
(~834.175), found in 21:28:34.

--George

--- In tuning@yahoogroups.com, "Brad Lehman" <bpl@...> wrote:
>

Hi Brad,

simply short all nominators over denominators by the factor 10
in order to meet scala the convention of integral fractions:

!Sparschuh885A5.scl
SC=80:81 inbetween F~C~G~D~A~E & schisma in E~B_F#_C#_G#~Eb~Bb~F
12
2790/2643 ! 279.0C#4 / 264.3C4
2960/2643 ! 296.0D_4 / 264.3C4
6275/4926 ! 627.5Eb4 / 528.6C5
6615/4926 ! 661.5E_4 / 264.3C4 = (5:4)(882:881) ~+1.964 Cents sharp
3524/2643 ! 352.4F_4 / 264.3C4 = (4:3)(882:881) ~+1.964 Cents sharp
3720/2643 ! 372.0F#4 / 264.3C4
3960/2643 ! 396.0G_4 / 264.3C4 = (3:2)(880:881) ~-1.966 Cents flat
4185/2643 ! 418.5G#4 / 264.3C4
4425/2643 ! 442.5A_4 / 264.3C4 440Hz(+2.5Hz = 150 MetronomeBeats/min)
4705/2643 ! 470.5Bb4 / 264.3C4
4960/2643 ! 496.0B_4 / 264.3C4
2/1
!

That results in epimoric beating lowered 5ths, all amounts given
in rational, ~Cents~ & ~TUs~ for the Syntonic-comma in F~C~G~D~A~E

F 881:882 C 880:881 G 296:297 D 295:296 A 294:295 E = product 80:81
F ~-1.963 C ~-1.966 G ~-5.839 D ~-5.859 A ~-5.879 E = sum ~-21.506C
F ~-60.28 C ~-60.34 G ~-179.2 D ~-179.8 A ~-180.4 E = sum ~660.04TUs

and the schisma = 2^15/5/3^8 = 32768:32805 ~-1.954Cents ~-59.96TUs
inbetween E~B_F#_C#_G#~Eb~Bb~F

E 3968:3969 B_F#_C#_G# 2510:2511 Eb 3764:3765 Bb 4704:4705 F
E ~-0.43624 B_F#_C#_G# ~-0.68959 Eb ~-0.45988 Bb ~-0.36799 F
E ~-13.3886 B_F#_C#_G# ~-21.1641 Eb ~-14.1141 Bb ~-11.2940 F

>
> With only an A=440 tuning fork in one hand, a harpsichord tuning lever
> in the other hand, and absolutely NO electronic devices of any kind:
> how exactly should one proceed to get all twelve of your notes
> correctly tuned onto a harpsichord, using this scheme?

The wanted precision of accessible accuracy
depends on several factors:

1. Quality of tuneability for the instrument alike
deviations due to inhamonicty of the strings?

2. Counting beats barely by own heart-pulse of
under aid of an clock or even better an adjustable Metronome?

3. Tuner is rested/relaxed or fatigued/exhausted or
may be even incompetent?

> And with no
> way of measuring integer frequencies, either, or knowing when they've
> been achieved precisely?

In yours personal "squiggle" impreciseness
of barely PC^(1/12)= 60TUs
probable the following rouding would be sufficient
for yours personal taste?:

F -1 C -1 G -3 D -3 A -3 E for approximation of about an ~SC
F -60 C -60 G -180 D -180 A -180 E for exaclty 660TUs~=~SC
and
E -0.25 B_F#_C#_G# -0.25 Eb -0.25 Bb -0.25 F in PC^(1/12) units
E -~15~ B_F#_C#_G# -~15~ Eb -~15~ Bb -~15~ F with sum=60TUs=~schisma

>
> To what precision are errors acceptable? And why?
That approximation in yours personal style
-if you would achive 15TUs = PC^(1/48) = ~0.5 Cents
would deviate maximal even less than 6 TUs = PC^(1/120) ~0.2 Cents.
>
> Does one first have
> to agree with your goal of proportional beating, and your constraint
> of integer frequencies?
Never at all, due to the possibilty to translate into
your's terms within an error of less than ~1/5 Cents.

> All this stuff just looks like
> nearly-meaningless tables of numerals to me, sorry;
No problem.
Maybe
How about that "squiggle"-type notation with 4 different grades of
5ths with the cycle:

F'C'G'''D'''A'''E.B_F#_C#_G#.Eb.Bb.F

with extended legend that has additional an dot "." for PC(-1/48)

' = PC(-1/12)
''' = PC^(-1/4) as in W#3 on the average C'''G'''D'''A_E_B'''F#_..._C
. = PC^(-1/48) that 1/4 of yours usual unit '
_ = an JI 5th of exactly 3/2=1.5

If you don't accept PC^(1/48) = ~schisma^(1/4) as smallest unit
then try intead that modifiaction
without any subschismatic refinements:

F'C'G'''D'''A'''E'B_F#_C#_G#_Eb_Bb_F

> the only way I
> know to assess its quality is to see if it agrees with *your own*
> goals...

I.m.h.o. ~0.2 Cents on the average in precision will suffice enough.

> which doesn't tell us one way or another about the usefulness
> for anything else *but* your own goal of proportional beating (or
> whatever it is).

Meanwhile I'm more tolerant in that aspect:
Never mind if you persist in inprecise tuning-methods
without counting beats exactly.
>
> If I'd somehow take the time and get this temperament set up on my
> harpsichord,
Simply try it out!

> within some acceptable error tolerance but without using
> any electronic devices:

Surely with yours ability and daily practice
in hearing you should achieve at least for:

F'C'G'''D'''A'''E'B_F#_C#_G#_Eb_Bb_F

an exactness of
even less than about PC(-1/24)accuracy
or equivalent ~1 Cents precision an the average.

> how would the resulting temperament sound in
> playing (say) some late Couperin?
Works fine, due to the pronouced Baroque key-characteristics.

> What does it do for the music,
> harmonically and melodically?

C-major deviates the least from JI.

> That's the kind of thing I personally
> care about: a temperament that sounds great in the music, and that
> can
> be done entirely by ear in less than 10 minutes without having to
> calculate (or even refer to) a page of numbers.

even my modern 3-fold stringed piano with 88keys over 7 octaves
needs only a few minutes retuning when the weather changes.
>
> And, what happens if I'd want to start on A=430 or something else
Transposing is no problem for scala.

442.5 / 430 = 177/172 = 35.4/34.4 = ~1.02906977... or ~2.9 %

(1 200 * ln(430 / 442.5)) / ln(2) = -49.6089545...Cents

that's about an half of an 12-EDO semitone = 50Cents lower than 442.5.

Where's the problem there,
except that for A4=430Hz the string tension becomes to loose in an
modern standard piano?

> (maybe not having anything to do with integers!), or on some C?
> Does
> it all need to be recalculated?

Scala does that job quite well.

> Help out the practical musicians who
> just want to listen to the sounds of intervallic relationships,

Please let me know your's opinion
after you have tryed out some of the above versions.

I do agree with you, that without:
> calculating
the corresponding acustical ratios, there is almost
>nothing....
understood in tempering key-instruments in an properly way.

Yours Sincerely
A.S.

--- In tuning@yahoogroups.com, "Andreas Sparschuh" <a_sparschuh@...>
wrote:

/tuning/topicId_77364.html#77900
(That message has five pages of explaining and figuring.)

Hi Andreas,

I printed out your five pages and I boiled it all down to one simple
set of by-ear instructions that I had to derive for myself...and it
has to be from a C fork, unfortunately, although I asked you
specifically for instructions from an A fork. I then set it up on a
harpsichord to give it a fair spin, and I've been playing some late
Couperin on it (which, again, is what I asked you to explain, and not
in generalities).

Here's what I've got. These were the most helpful several lines from
your five pages, where you allowed us to round off your 5ths to "only"
four different sizes, instead of micro-managing decimal fragments of
cents:

> F -1 C -1 G -3 D -3 A -3 E for approximation of about an ~SC
> and
> E -0.25 B_F#_C#_G# -0.25 Eb -0.25 Bb -0.25 F in PC^(1/12) units

In other words, we're supposed to set it up almost the same as in
"Vallotti", but make the core set of tempered 5ths on the naturals
*uneven* instead of even. F-C-G get only 1/11 SC apiece, and all the
rest of the syntonic comma goes into G-D-A-E, 3/11 per. And then the
one schisma that's left over gets spread carefully across *four* of
the six remaining 5ths. Golly.

(And we're supposed to ignore the fact that for Neidhardt and Sorge,
the 18th century experts, the schisma was the smallest practical unit?
They were right: it *is* hard to control any 5ths that are less than
a schisma off pure, let alone anything as fine as 1/4 schisma...or
decimal bits and pieces grinding it even more finely than that. But,
let's continue.)

Anyway...I did it like this:

C from fork.

E in the C-E major 3rd nudged one schisma sharp of pure 5:4.

C-F 5th one schisma narrow by knocking the F upward the slightest
audible nudge from the pure spot.

C-G 5th one schisma narrow by knocking the G downward the slightest
audible nudge from pure.

Set D and A so all three of G-D, D-A, and A-E are averaged out in
quality. They're rough. They're as bad as almost 1/4 SC each!

There's one schisma left to burn off, doing all the notes from
E-B-F#-C#-G#-D#-A#-F, where E and F are the fixed endpoints. From E,
I gave the B the very slightest smudge flatward (1/4 schisma!), which
is just about impossible to control with non-threaded pins, but my
wrist knows I did it. Then I made B-F#-C#-G# pure as prescribed.
This left D# and A# to be finessed in between G# and F, so each of the
three intervals gets another one of these 1/4 schisma smudges. It
works out, but again it's all in the wrist: amounts this tiny are just
about impossible to HEAR with any accuracy. A quarter of a schisma
off pure?!

And that's all 12 notes.

I played through some music from the last four Ordres by Couperin, and
I'll do some more. The results sound reasonable, although I can't say
they're any noticeable improvement ahead of normal "Vallotti". The
shape is almost identical, other than being done unevenly downtown:
too much tempering in G-D-A-E and not enough in F-C-G. The several
worst triads (B major, F# major, C# major) are still as bad as they
are in "Vallotti", give or take less than a schisma. And E major, A
major, and D major aren't as good as Vallotti's; the major 3rds are
higher, and the D-A-E 5ths are rougher. Oh well!

It's a passable sound, but I'd need to be given some compelling
reasons why I (or anyone else) should fuss with this uneven
F-C-G-D-A-E business downtown, instead of simply making all of them
the same quality as one another. Why don't we just square off all the
schismatic and sub-schismatic stuff, and give a straightforward
F-C-G-D-A-E of 1/5 PC each, instead of ratios or syntonic comma stuff?
The same major 3rds will be good or bad by virtually "the same"
amounts, when listening to and playing normal 18th century music.
And, after the harpsichord has sat there for an hour or so, it will
have drifted off spots enough already that your 1/4 schisma business
will all be negligible at best. How is anyone going to know, or care,
that the little 1/4 schisma fragments are in the right spots around
the back, and not between F#-C# or wherever?

=====

Meanwhile, your five-page (five!) batch of reckoning still doesn't
help much with a start from an A fork, which was the question. (The
note A is in the middle of the heaviest tempering....) Assume you're
wanting some other harpsichordist to try this on a real harpsichord,
but they're less mathematically savvy than I am. They don't want to
know anything about beats, but they want to get it "right enough" so
they can go ahead and play their music WITHOUT CALCULATING ANYTHING.
They don't have Scala and they couldn't care less about any numbers.
Please give some step-by-step practical instructions that would
satisfy the accuracy you desire.

Thanks,
Brad Lehman

--- In tuning@yahoogroups.com, "Brad Lehman" <bpl@...> wrote:
>

> Meanwhile, your five-page (five!) batch of reckoning still doesn't
> help much with a start from an A fork, which was the question.

Why on earth are you insisting on this modern way of doing things? I
thought you were into reproducing historical methods! Start with a C
fork (or maybe F), like the old guys did! You can't have it both ways,
continuously pointing to the old music and then chiding Andreas for
not constructing a system starting from the mid to late 20th century
standard. After all, J. Cree Fisdcher in 1908 is still telling the
piano tuner to use a C fork!

That said, I agree with you completely, this seems like nothing more
than "having fun with numbers". What's the point?

Ciao,

P

As I said, perfectly clearly yesterday, when I tried out Andreas's
temperament yesterday I DID use a C fork.

My question about A was to hear what he had to say about making things
*exactly* on 440 or whatever, as integer frequencies. Wasn't that his
point, the hitting of exact integers along with proportional beats,
somehow?

And I agree with you, Paul: it all does look like just having a bunch
of fun with numbers, as far as that goes.

Brad Lehman

--- In tuning@yahoogroups.com, "Paul Poletti" <paul@...> wrote:
>
> --- In tuning@yahoogroups.com, "Brad Lehman" <bpl@> wrote:
> >
>
> > Meanwhile, your five-page (five!) batch of reckoning still doesn't
> > help much with a start from an A fork, which was the question.
>
> Why on earth are you insisting on this modern way of doing things? I
> thought you were into reproducing historical methods! Start with a C
> fork (or maybe F), like the old guys did! You can't have it both ways,
> continuously pointing to the old music and then chiding Andreas for
> not constructing a system starting from the mid to late 20th century
> standard. After all, J. Cree Fisdcher in 1908 is still telling the
> piano tuner to use a C fork!
>
> That said, I agree with you completely, this seems like nothing more
> than "having fun with numbers". What's the point?
>
> Ciao,
>
> P
>

Dear Margo,

My apologies for the very late reply. I have been enjoying a well-earned summer's rest. I hope the importance of the topic has not faded while I was away. My comments are below:

On Jul 1, 2008, at 1:57 AM, Margo Schulter wrote:

>> Dear Margo,
>
>> First, let me thank you for bringing to my attention your well-
>> prepared article titled "Regions of the Interval Spectrum". I am
>> glad to observe that the chapters on neutral intervals agree with my
>> views on Maqam music theory. Please note, that 14:13 is also given
>> by Urmavi as the approximation to all middle melodic (lahni)
>> intervals (supraminor seconds) in his treatise (Risale
>> al-Sharafiyyah).
>
> Dear Ozan,
>
> Thanks you for this very interesting observation about Urmavi and
> 14:13. There is a scale in the Scala archives (iran_diat.scl) edited
> by Dariush Anooshfar which gives what is identified as a 125-EDO
> version of a scale by Urmavi, one which may use some intervals which
> are still featured in certain Iranian musics. This is discussed a bit
> more below.
>

Ah, I am sad to say that the Scala 125-EDO subset does great injustice to the original Ushshaq scale of Urmavi, which I believe goes:

9:8 x 9:8 x 256:243 x 9:8 x 9:8 x 256:243 x 9:8

Notice how the limma has been reduced in size by half in the Scala file so as to destroy the qualities of the half-tone step, making it a quarter-tone step. Bear in mind that this Ushshaq has nothing to do with the Ushshaq we perform today. I am at a lost to explain how a diatonic major scale got transformed to a neutral minor during the centuries that followed Urmavi.

> Also, while I am using less than perfect software for processing pdf
> files that runs out of virtual memory on your dissertation file, there
> is no problem with the doctoral reports and other related documents on
> your Web site which I'm guessing may contain similar material. Thank
> you for so generously making all of these material available!
>

You are most welcome. I do need to re-organize the clutter in my website however.

>> I think the theory of Maqam music and other "ethnic" genres around
>> the world are much neglected by the alternative tuning list
>> community. Most of the discussions are centered around either
>> historical or contemporary microtonalisms for furthering Western
>> music culture alone. While I appreciate the contributions by the
>> West to musical art, I believe the Western quarter (pun intended)
>> can account for only a fraction of the actual music-making in the
>> globe. One of the greatest traditions is right next door: A
>> venerable monophonal Middle Eastern culture based on maqamat,
>> destgaha and raga. This "exotic" culture has been influenced by a
>> thousand years of Islamic atmosphere to inspire such styles and
>> practices as Mevlevi rites, Qawwali performances, peshrevs, taqsims,
>> gazels, etc... Your penchant to discover more of the theories and
>> styles of exotic traditions is admirable.
>
> Please let me add, in response to some questions people have raised,
> that there are now some excellent books in English by exponents of
> some of these traditions: thus Touma and Racy on Maqam music as
> practiced in the Arab world; and Farhat on the Persian Dastgah system,
> as well as book on this Dastgah tradition with some companion CD's by
> Dr. Tala`i. Racy is especially interesting on the "ecstatic" qualities
> of traditional music which is lost when tuning is dictated by
> arbitrary standard such as 24-EDO, rather than adapted to each Maqam.
>

I should state further, that it is most unfortunate that Turks have not written a single academic work on their conception of Maqam music for the auspices of the international music community since Rauf Yekta. My thesis should serve as a brick that closes the centennial gap.

>> You are correct in establishing 121 and 171 cents as producing
>> "neutral effects". However, most of the time 168+ cents won't do at
>> all. I myself shun 170 cents up to 180 cents as a useless region, a
>> sort of "dead zone" if you prefer, and avoid it at all costs. You
>> call the gap between these values the "equitable heptatonic". It may
>> be useful for some African, Burmese or Gamelan ensembles, but I do
>> not believe much can be done with such intervals in Maqam music.
>
> This may be rather analogous to larger major thirds around 408-440
> cents, which may excel in a European style based on medieval or
> neomedieval polyphony, but can hardly be aptly substituted as a
> general rule in a Renaissance meantone style where 5:4 is expected!
>

Nicely put.

> I seem to recall having read a passage somewhere in your writings
> about choosing steps so as to avoid an interval of 170 cents between
> two significant degrees for Maqam music.
>

Indeed so. I remember having stated the necessity to avoid the equitable heptotonic in Yarman24.

>> You might remember, that in the first year of my presence in the
>> tuning list, I advocated 29-EDO as a temperament suitable for Maqam
>> music for the very reasons explained by Dan and yourself.
>
> I should go back and read your posts: that the three of us all see
> 29-EDO in this way, coming from quite different perspectives, is
> fascinating.
>

Righteously.

>> I acknowledge 98:81 or 75:62 as a supraminor second left out in my
>> previous post. Also, 26/21 is also important as a submajor third.
>> Thanks for mentioning these ratios.
>
> One place 26/21 comes up is in Ibn Sina's 28:26:24:21 tetrachord.
>

I checked Ibn Sina's passages. The actual tetrachord is given as 8:7 x 14:13 x 13:12. The tetrachord you have mentioned permutes the last two intervals. In the original form, we have 16:13 instead of 26:21. But I suspect the tetrachord should be read the other way around, which yields 12:13:14:16. Ibn Sina calls this a Kavi genus which is of quality and nobility, I am not sure what this corresponds to in music theory of the Ancient world. Ibn Sina says that this is the genus preferred by Ptolemy.

>> Ah, but supraminor thirds are not restricted to Persian music. For
>> you see, they are important flavours in such maqams as Hijaz, Nikriz
>> and Neveser. I admit, though, that Persians are more fond of
>> supraminor seconds compared to middler seconds.
>
>> 14:17:21 makes a wonderful neutral chord.
>
> Agreed!
>
>> I am glad to hear your appreciation of 79 MOS 159-tET. What do you
>> think of my new 24-tone tuning for Maqam music based on a modified
>> meantone temperament? I have explained it in the 76333rd message to
>> this list.
>
> This is a fascinating design: 12 notes from a circulating temperament
> of Rameau, and the other 12 added to provide some delicious Maqam
> flavors!
>
> </tuning/topicId_76333.html#76333>
>

Don't forget the 17-tone closed cycle achieved via superpythagorean fifths for 7-limit major and minor chords!

>> I'm sorry, I am too tired to read the last article on 19-note
>> tempered system. Later inshallah.
>
> Please let me add that after looking at your ingenious Rameau-based
> tuning, I thought I might post a less precise set of 24 steps from
> Zest-24 which has some of its steps or _perdeler_ (plural of Turkish
> _perde_ or "step" or "tone," if I'm correct) fitting a tuning such as
> Anooshfar's iran_diat.scl which he attributes to Urmavi, rather than
> the usual steps appearing your system.
>
> ! iran_diat.scl
> !
> Iranian Diatonic from Dariush Anooshfar, Safi-a-ddin Armavi's scale > from 125 ET
> 7
> !
> 220.800 cents
> 441.600 cents
> 489.600 cents
> 710.400 cents
> 931.200 cents
> 979.200 cents
> 2/1
>

I urge you not to consider this subset of 125-EDO as a basis for any scale construction. The correct Urmavi scale is the one I provided above.

> The Anooshfar-like intervals that arise (I wonder what kind of tuning
> by Urmavi was the basis for iran_diat.scl) are the large major third
> and sixth at 441 and 932 cents, and also the 50-cent steps (for
> example 440-491, 931-982). These seem to me typical of iran_diat.scl
> rather than standard Maqam practice, but I wonder how you would see
> this.
>

I view the 125-EDO subset as a corruption and misrepresentation of Urmavi's original scale.

> An interesting question is how one might name the perdeler in this
> Zest-24 tuning.
>
> 0 224 441 491 708 932 982 1200
>
> What I would emphasize is that Zest-24 is a radically different way of
> going from a 12-note circulating tuning to a 24-note system. You took
> Rameau's 12-note circle and added customized steps designed for ideal
> placement in a Maqam style. My method was much more crude: simply add
> a second identical circle at a distance suggested by the structure of
> the tuning rather than the needs of Maqam intonation.
>
> As one might say, this is the difference between system where Maqam
> music shapes the finely calculated placement of the notes or perdeler;
> and one where the notes bend Maqam music to their measure, not
> necessarily with any predictability of results!
>

Agreed!

> When I try 'show /line intervals' in Scala, I see some nice versions
> of Rast in various forms and Ushshaq, for example -- but not
> necessarily in the _relations_ than traditional schemes of modulation
> would require.
>
> Also, the structure of the Zest-24 tuning itself imposes some
> arbitrary limitations. For example, your 24-note system includes
> neutral thirds at both 17/14 and 16/13; or, in your 79-MOS, sizes of
> both 332 and 362 cents above Rast. In Zest-24, however, I realized
> that it is impossible to have two sizes of neutral thirds above the
> same perde or step. The situation might be analogous to that of some
> equal temperaments where only a single size of middle or neutral third
> is available -- although here, the size may change as one moves around
> the system.
>

17/14 is not only requisite of Ushshaq as a very low perde segah, but also as the second degree of Hijaz as a fairly steep perde kurdi.

> In a table following the Scala file, I attempt to guess at some
> possible names for the steps or perdeler; but I would invite your
> comments on this. The question of the perdeler and names for various
> shades of modification is one most interesting.
>
>
> ! zest24-Bbup.scl
> !
> Tuning set starting from Bb* (24)
> 24
> !
> 32.81250
> 83.20312
> 153.51562
> 203.90624
> 223.82812
> 274.21874
> 344.53124
> 394.92187
> 440.62500
> 491.01562
> 536.71875
> 587.10937
> 657.42187
> 707.81249
> 727.73437
> 778.12500
> 849.60937
> 899.99999
> 931.64062
> 982.03124
> 1040.62500
> 1091.01562
> 1149.60937
> 2/1
>
>
> --------------------------------------------------------------------
> approx tentative approx approx
> degree cents 159-EDO perde name 288-EDO RI
> --------------------------------------------------------------------
> 0: 0.00000 0 Rast 0 1/1
> 1: 32.81250 4 (Sarp Rast) 8 64/63
> 2: 83.20312 11 Shuri 20 22/21
> 3: 153.51562 20 (Zengule cluster) 37 12/11
> 4: 203.90624 27 Dugah 48 9/8
> 5: 223.82812 30 (Sarp Dugah) 54 8/7
> 6: 274.21874 37 Nerm (soft) Kurdi 66 75/64
> 7: 344.53124 46 Segah of Ushshaq 83 11/9
> 8: 394.92187 52 Segah 95 5/4
> 9: 440.62500 58 Dik Nishabur 106 9/7
> 10: 491.01562 65 Chargah 118 4/3
> 11: 536.71875 71 Nim Hijaz 129 15/11
> 12: 587.10937 78 Hijaz 141 7/5
> 13: 657.42187 87 (Saba cluster) 158 19/13
> 14: 707.81249 94 Neva 170 3/2
> 15: 727.73437 96 (Sarp Neva) 175 32/21
> 16: 778.12500 103 Bayyati 187 11/7
> 17: 849.60937 113 (Hisar/Huzzam cluster) 204 18/11
> 18: 899.99999 119 Huseyni 216 27/16
> 19: 931.64062 123 (Sarp Huseyni) 224 12/7
> 20: 982.03124 130 Koutchouk (little) Ajem 236 30/17
> 21: 1040.62500 138 (Evdj cluster) 250 51/28
> 22: 1091.01562 145 Evdj 262 15/8
> 23: 1149.60937 152 Mahurek 276 64/33
> 24: 1200.00000 159 Gerdaniye 288 2/1
>
>

Since there are no more steps between rast and shuri, you should name the second degree as dik rast. The same goes for sarp dugah, sarp neva, and sarp huseyni. They should all be dik in parantheses. Also, you should name 154 cents as Zengule without the paranthesis. 274 cents makes a very low kurdi, but kurdi nonetheless. Similarly, 345 cents is could be a dik kurdi if you prefer. 441 cents is buselik and nishabur at the same time since there are no other options. I'd rather you called it a buselik. Saba cluster should be plainly saba. You could drop one y from Bayyati to make it more Turkish. 845 cents is plainly hisar without the paranthesis. Little ajem is needless since we took kurdi plainly. Let's call it simply ajem. Evdj cluster is a misnomer here, it should be a dik ajem. Mahurek ought to be mahur.

But you see, you cannot play a decent Hijaz with this setup, because you are either consigned to a very low kurdi or a very high dik kurdi.

> With many thanks,
>
> Margo
> mschulter@...
>

Cordially,
Oz.

Dear Mike, my apologies for the very late reply. As I have stated in my recent message to Margo Schulter, I had been enjoying a well deserved summer's rest.

To answer your enquiry, here are some links to older messages to this forum on the subject of masters of Turkish maqam music:

/tuning/topicId_62844.html#62904

/tuning/topicId_61173.html#62039

/tuning/topicId_56632.html#56664

The names of some of the prominent masters have been listed in these messages. A search in amazon.com could yield links to the performances of masters themselves.

Fusion type endeavours in "world music" does occasionally result in original productions worthy of approval. However, for a crash course in maqam music, you need to listen to acclaimed executants and venerable exponents of the tradition, not syntheses.

Direct personal experience of Allah is very much ingrained in Sufi music. Most of the known neyzens in Turkiye are into tasavvuf. You are likely to enjoy the Erguner brothers, the elder of which, Kudsi, has done world fusion too if I heard correctly:

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

If you are into the Turkish ney for the love of its trancendental sound, here are acknowledged quotidian performers of the instrument:

http://www.neyzenim.com/neyzenler.htm

Oz.

/tuning/topicId_34979.html#36343

On Jun 28, 2008, at 7:42 AM, Mike Battaglia wrote:

> On Fri, Jun 27, 2008 at 8:15 PM, Ozan Yarman <ozanyarman@...> > wrote:
>
>> I think the theory of Maqam music and other "ethnic" genres around >> the
>> world are much neglected by the alternative tuning list community.
>> Most of the discussions are centered around either historical or
>> contemporary microtonalisms for furthering Western music culture
>> alone. While I appreciate the contributions by the West to musical
>> art, I believe the Western quarter (pun intended) can account for >> only
>> a fraction of the actual music-making in the globe. One of the
>> greatest traditions is right next door: A venerable monophonal Middle
>> Eastern culture based on maqamat, destgaha and raga. This "exotic"
>> culture has been influenced by a thousand years of Islamic atmosphere
>> to inspire such styles and practices as Mevlevi rites, Qawwali
>> performances, peshrevs, taqsims, gazels, etc... Your penchant to
>> discover more of the theories and styles of exotic traditions is
>> admirable.
>
>> Though my experience is most inadequate to describe the musical
>> wonders of the Islamic Civilization, my presence in the tuning list >> as
>> a fresh academician should be construed as an oppurtunity to discover
>> a glimpse of at least the Turkish branch of this grand culture.
>
> Well hey man, if you have a listening list of stuff you can recommend,
> I think we'd all love to check it out. World music is one of the most
> fascinating things in the, well, the world. Mainly because you have
> thousands of years of musical development behind most of these
> cultures and styles, and so they are usually very much advanced.
>
> Jeff Buckley did a Qawwali-inspired song, "Dream Brother," in which he
> mixed pop/rock with traditional Qawwali elements, and it's one of my
> favorite songs. I started looking for some traditional Qawwali
> recordings when I heard that song, and I didn't really find much.
>
> Any time there is an old, ancient branch of music that has reached as
> high of a level of artistic development as the one we're talking about
> here, people will be interested. I just think many don't know about it
> yet.
>
> One interesting thing to note is that the religious music of all of
> the world sounds very, very, very similar. Perhaps not the music that
> is "associated" with various churches and such - but the music that
> monks sing, the music that is sung to draw people closer to the
> experience of God.
>
> -Mike