Yahoo Tuning Groups Ultimate Backup tuning Microtonal meeting in France

Hi all !
I'm back for a while !
I wonder how many subscribers to the tuning list are around France those
days... ?
Anyway if any of them are interested, here is a small info on a microtonal
meeting I do organize next week-end in the south of France, near the Abbey
of Thoronet.
(my excuses for the translation) :

><<<>>><<<>>><<<>>><<<>>><<<>>><<<>>><<<>>><<<>>><<<>>><<<>>><<<>>><<<>>><<<>>

Microtonal Tuning ("L¹Accord Microtonal")

- conference, workshop, concert -

with Jacques Dudon, François Breton, Jean-Pierre Poulin, Jérôme Désigaud
at ³La Source de Vie², chemin des Riaux, Carcès (Var, France)

15, 16, 17 of july 2005

CONFERENCE ³New tools for just intonation², friday 15th of july, 20h (free)
Sound performance by Jerome Desigaud (double flute from Rajasthan, circular
breathing)
³From a harmonic simplicity² by Jacques Dudon, just intonation composer,
instrument maker, director of the AEH and the "Ensemble de Musique
Microtonale du Thoronet"
³Alternatives to the unique intonation² by Jean-Pierre Poulin, author of ³La
petite encyclopédie des échelles et des modes², creator of the "Poulin
flûte"
³The untempered keyboard" by François Breton, Atelier des Muses, musician in
the "Ensemble de Musique Microtonale du Thoronet"

WORKSHOP "L¹Accord microtonal", saturday 16th & sunday 17th of july
(9h-18h):
Practical initiation to keyboards microtuning, on various hardware or
virtual synthesizers, and microtonal guitar re-fretting, other instruments
on request.
Basic notions ; what types of tunings can we use, etc. (with practical
examples to the requests of participants)
World traditionnal music models developped by the AEH, scales
classification, timbre and intonation adequation, etc.

CONCERT in just intonation, sunday 17th of july, 20h (free) :
by François Breton, Jean-Pierre Poulin, Jacques Dudon, Jérôme Désigaud,
along with the workshop participants.

-----------------------------------------------------------------------
Jacques Dudon
Atelier d'Exploration Harmonique - Les Camails 83340 LE THORONET France
tel 33 4 94 73 87 78
http://aeh.free.fr

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

Jacques Dudon wrote:
><<<>>><<<>>><<<>>><<<>>><<<>>><<<>>><<<>>><<<>>><<<>>><<<>>><<<>>><<<>>><<<
>>
> Microtonal Tuning ("Lï¿½Accord Microtonal")
> - conference, workshop, concert -
>
> with Jacques Dudon, Franï¿½ois Breton, Jean-Pierre Poulin, Jï¿½rï¿½me Dï¿½sigaud
> at ï¿½La Source de Vieï¿½, chemin des Riaux, Carcï¿½s (Var, France)
>
> 15, 16, 17 of july 2005
>
> CONFERENCE ï¿½New tools for just intonationï¿½, friday 15th of july, 20h
(free)
> Sound performance by Jerome Desigaud (double flute from Rajasthan,
circular
> breathing)
> ï¿½From a harmonic simplicityï¿½ by Jacques Dudon, just intonation composer,
> instrument maker, director of the AEH and the "Ensemble de Musique
> Microtonale du Thoronet"
> ï¿½Alternatives to the unique intonationï¿½ by Jean-Pierre Poulin, author of
ï¿½La
> petite encyclopï¿½die des ï¿½chelles et des modesï¿½, creator of the "Poulin
> flï¿½te"
> ï¿½The untempered keyboard" by Franï¿½ois Breton, Atelier des Muses, musician
in
> the "Ensemble de Musique Microtonale du Thoronet"
>
> WORKSHOP "Lï¿½Accord microtonal", saturday 16th & sunday 17th of july
> (9h-18h):
> Practical initiation to keyboards microtuning, on various hardware or
> virtual synthesizers, and microtonal guitar re-fretting, other instruments
> on request.
> Basic notions ; what types of tunings can we use, etc. (with practical
> examples to the requests of participants)
> World traditionnal music models developped by the AEH, scales
> classification, timbre and intonation adequation, etc.
>
> CONCERT in just intonation, sunday 17th of july, 20h (free) :
> by Franï¿½ois Breton, Jean-Pierre Poulin, Jacques Dudon, Jï¿½rï¿½me Dï¿½sigaud,
> along with the workshop participants.
>

Jacques,
That workshop sounds like dynamite! "Practical initiation to keyboard
microtuning", "World traditional music models", and "scale classification",
all sound very interesting ... wish I could be there! Will you be
publishing
any proceedings? Or does the AEH have any written resources available
on these subjects?

Wishing you every success for your conference, workshop and concert!

Regards,
Yahya

--
No virus found in this outgoing message.
Checked by AVG Anti-Virus.
Version: 7.0.323 / Virus Database: 267.8.9/41 - Release Date: 5/7/05

🔗Jacques Dudon (AEH) <fotosonix@wanadoo.fr>

le 11/07/05, Yahya Abdal-Aziz wrote :

> Jacques Dudon wrote:
>> <<<>>><<<>>><<<>>><<<>>><<<>>><<<>>><<<>>><<<>>><<<>>><<<>>><<<>>><<<>>><<<

>> Microtonal Tuning ("L¹Accord Microtonal")
>> - conference, workshop, concert -
>>
>> with Jacques Dudon, François Breton, Jean-Pierre Poulin, Jerome Désigaud
>> at Carces (Var, France)
>>
>> 15, 16, 17 of july 2005

> Jacques,
> That workshop sounds like dynamite! "Practical initiation to keyboard
> microtuning", "World traditional music models", and "scale classification",
> all sound very interesting ... wish I could be there! Will you be
> publishing
> any proceedings? Or does the AEH have any written resources available
> on these subjects?
> Wishing you every success for your conference, workshop and concert !
> Regards,
> Yahya

> Sounds absolutely delightful. I hope you give us a report on it.
(from Gene Ward Smith, 11 juillet 2005)

Hi all,
Thanks Gene & Yahya for your friendly thougths,
This 3-days workshop were indeed filled with happiness.
We had 13 participants among which 2 ladies, which I think is a performance
(what is the percentage of ladies among the tuning list subscribers today ?
;)
Sorry, no proceedings, no report on internet...
AEH has many written ressources, but almost all in french.
Nothing secret in our work, but no time to do it, as I have to prepare our
next annual main festival, "Les Noces Harmoniques" (25-26-27-28 of august
2005 - by the way open to microtonal travelers from the whole world).

In brief, something like 28 tunings for keybords and guitars were studied,
about all on traditional models, along with of course a maximum of basics.
Among the many nice things we did not expected, a young man, Nicolas Escot,
arrived from Paris (900 kms) with a spanish guitar refretted in a 17-limit
system of his own, all well done with curved frets and good sound.
So as an introduction in hearing the 17th harmonic in a simple 17-limit
scale I proposed my prefered (Dudon) version of morning raga Bhairav :
1/1 17/16 5/4 4/3 3/2 51/32 15/8 2/1
(which can have a few double notes as well)
Then (after the guitar passed from hand to hand and was judged "very
circular" by all the present guitar players), we studied the vectorial
structure of Nicolas's guitar, which was based on a cycle of pure fifths
between F and B, with two 17/16 up from F & C and three 17/16 down from A,
E, B :

1/1 17/16 9/8 81/68 81/64 729/544 729/512 3/2 51/32 27/16 243/136
243/128
(with 1/1 = F of the guitar, tuned in the standard tuning E A D G B E)
If any one has any comment on it, I will be glad to transmit to Nicolas.
My comment was that it was certainly a very circular tuning, even may be too
much. The three less-than 3/2 fifths C# G#, E F# & A# F are very acceptable,
but I found the three best approaching natural major thirds (by 64/51) could
have an improved "guitaristic" situation.
After a quick look this can be done by using 27/17 instead of 51/32 :
1/1 17/16 9/8 81/68 81/64 729/544 729/512 3/2 27/17 27/16 243/136
243/128 and having either A or D on 243/128 instead of E - then A or D have
an optimized diatonic major scale, which might be some of the best choices
on a standard E A D G B E guitar tuning ?
I need some classical guitarist's lights on the subject.

Apart from this and many other questions, one question was left without
definitive answer inside the workshop timings :
How to tune african heptatonic instruments like cora and balafons ?
(without using 8 or 9 notes by octave ! ;)
Same question, by the way, with a celtic harp...

----------------------------------------------------------------
Jacques Dudon
Atelier d'Exploration Harmonique - Les Camails 83340 LE THORONET
tel & fax 04 94 73 87 78 - tel & répondeur 04 94 73 80 25
fotosonix@wanadoo.fr
http://aeh.free.fr

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, "Jacques Dudon (AEH)" <fotosonix@w...>
wrote:

> Apart from this and many other questions, one question was left
without
> definitive answer inside the workshop timings :
> How to tune african heptatonic instruments like cora and balafons ?
> (without using 8 or 9 notes by octave ! ;)
> Same question, by the way, with a celtic harp...

My answer for the string instruments, of course, is meantone tuning
(in general, not necessarily the 1/4-comma version). In the West,
what we call meantone today was simply called "correct intonation" by
musicians for two or three centuries, including most of the
Renaissance, Baroque, and Classical periods. So it would be
historically quite appropriate for much Celtic harp music. When I
play diatonic music on my 31-equal guitar (a fine representative of
the meantone 'family'), I'm enchanted by the peaceful purity of the
intervals and chords, and it's very difficult indeed for my ears to
accept a rather different 7-note diatonic tuning (whether 12-equal or
5-limit JI), with its attendant harsh intervals, immediately
thereafter.

The Kora/cora is an instrument I love dearly, there being some
virtuoso players who've moved from Africa to the Boston area, and
though certainly not Western, I've been surprised to find that they
are definitely tuned with meantone tendencies -- even when playing
along with 12-equal Western instruments. I don't claim this is
universal for kora tuning, but the particular style, perhaps
consciously "world-music" oriented, that finds itself represented by
African musicians on the Boston streets and clubs seems to gravitate
towards meantone kora tunings. Being familiar with the sound of
meantone mainly through "authentic" recordings of Renaissance music,
the sound of the Kora makes me imagine a cultural exchange of
troubadours between Europe and Africa during the 16th century, the
gut strings and fluid polyphony of the Kora being oddly reminiscent
of the sound of lute music.

I wouldn't expect to find a *regular* meantone tuning on a Kora, an
instrument which is tuned by ear frequently during performance (most
often only one note per octave is retuned, to effect a different
diatonic scale one step away on the chain of fifths, but overall
tuning corrections seem necessary every few songs or so since the
instrument is not built with locking tuners). But the players I've
heard tune so that all the thirds and sixths in the diatonic scale
are closer to pure than they could be in a JI (by Dave's definition)
scale, so the indication is definitely that some kind of (perhaps
irregular) meantone tuning is being used.

Of course, it's quite possible to specify such tunings using high-
limit ratios, as Kirnberger and others were fond of doing, and as
Jacques Dudon's rotating discs might require one to do. Erv Wilson's
writings on meta-meantone provide just one way in which a regular
meantone tuning can be seen as the limit of a series of rational
scales defined by a simple recurrence relation. Meta-meantone is a
flavor of meantone with quite narrow perfect fifths (and wide perfect
fourths), and a rational scale with numbers as small as

75 112 167 250 374 558 834,

after reducing to one octave, gives a fairly good realization of a
meta-meantone diatonic:

0 cents
125.64 cents
311.5 cents
507.33 cents
695.74 cents
819.88 cents
1010 cents

For other varieties of meantone with fifths and fourths closer to
pure, one can find other recurrence relations which produce other
series of integers that similarly converge to the meantone fairly
quickly. Gene Ward Smith has written about this topic before and will
probably do so again . . .

As for the tuning of the balafon, and the many other traditions of
kora tuning that one may find in Africa, I recommend reading this
webpage:

http://tcd.freehosting.net/djembemande/bala.html

I'm also fascinated by other African tunings, such as the Chopi
tunings Erv Wilson refers to as "Mavila". I've written a lot about
that already, so I think I'll stop this long post right here for
now . . .

-Paul

🔗Jacques Dudon (AEH) <fotosonix@wanadoo.fr>

le 20/07/05 22:40, wallyesterpaulrus à wallyesterpaulrus@yahoo.com a écrit :

> --- In tuning@yahoogroups.com, "Jacques Dudon (AEH)" <fotosonix@w...>
> wrote:
>
>> Apart from this and many other questions, one question was left
>> without definitive answer inside the workshop timings :
>> How to tune african heptatonic instruments like cora and balafons ?
>> (without using 8 or 9 notes by octave ! ;)
>> Same question, by the way, with a celtic harp...

> ...so the indication is definitely that some kind of (perhaps
> irregular) meantone tuning is being used...

> As for the tuning of the balafon, and the many other traditions of
> kora tuning that one may find in Africa, I recommend reading this
> webpage:
>
> http://tcd.freehosting.net/djembemande/bala.html

This resumes well what I thought were the two extreme tendencies for the
balafon heptatonic tunings I heard : the Diatonic "able to play with
Western bands" as some african might say, and the equiheptatonic, which
should able other african balafonists to play also in a Thaï band - on more
rare occasions.
Playing myself some kind of heptaphonic koto of my own, I must admit, as
much as I love to play major and minor modes alltogether, I never stop
retuning and some kind of meantone finds its way.
I kind of agree with Lou Harrison that recommends the Kirnberger 2 as the
less destructive solution, except that the division of the major third
doesn't have to be equal.
We tested rapidly three of them in our workshop last sunday :
143/128, 161/144 and 272/243, which won the prize, because harmonic 17 was
very present that day probably.
To be able to define which kind of meantone you want between 10/9 and 9/8
not only helps to define a balance between minor and major modes, but also
allows some extensions to very different harmonic worlds such as 11 & 13 for
143/128, 7 & 23 for 161/144, or 17 for 272/243, and so on.

I worked a lot on heptatonic scales families, and in my point of wiew there
are at least 6 or 7 levels between diatonic and equiheptatonic scales :
1) Diatonic, mainly of 5-limit models (of course there are also many hyper-
-diatonics)
2) Next to diatonic I place some of the main Burmese scales, with semitones
slightly larger than 16/15
3) Then some type of common final for Rast (in Arab music, not the turkish
Rast) which is a Rast whose neutral thirds over the tonic become major
4) Then 13-limit kind of diatonic scales (Persian music, also Gnawa)
5) Then the semidiatonic neutral Rast family
6) Then but with a different structure the semidiatonic Mohajira family,
sligthly more circular ;
7) Then a very diffuse family using the 10/9 type of intervals as largest
tones and still one or two semitones of the 12/11 type ;
8) Then the Thaï or (almost)-equiheptatonic family, where the smallest type
of tone is no smaller than 11/10. Besides, semitone sensation becomes rare
in a Thaï scale.
9) In the end, an exactly-equal tempered heptaphone is more theorical but
can exist as a tuning tendency.

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

Jacques,

You wrote:
> ...
> I worked a lot on heptatonic scales families, and in my point of wiew
there
> are at least 6 or 7 levels between diatonic and equiheptatonic scales :
> 1) Diatonic, mainly of 5-limit models (of course there are also many
hyper-
> -diatonics)
> 2) Next to diatonic I place some of the main Burmese scales, with
semitones
> slightly larger than 16/15
> 3) Then some type of common final for Rast (in Arab music, not the turkish
> Rast) which is a Rast whose neutral thirds over the tonic become major
> 4) Then 13-limit kind of diatonic scales (Persian music, also Gnawa)
> 5) Then the semidiatonic neutral Rast family
> 6) Then but with a different structure the semidiatonic Mohajira family,
> sligthly more circular ;
> 7) Then a very diffuse family using the 10/9 type of intervals as largest
> tones and still one or two semitones of the 12/11 type ;
> 8) Then the Thaï¿½ or (almost)-equiheptatonic family, where the smallest
type
> of tone is no smaller than 11/10. Besides, semitone sensation becomes rare
> in a Thaï¿½ scale.
> 9) In the end, an exactly-equal tempered heptaphone is more theorical but
> can exist as a tuning tendency.

Thanks for this list and classification. It seems that you have actual
scales
observed "in the wild" for each class. The following questions occur to
me -
1. What are some good examples of the use of scales from each class?
(Naming country, culture, and song.)
2. Are recordings available?
3. Suggested explanations for how the scale arose?

Regards,
Yahya

--
No virus found in this outgoing message.
Checked by AVG Anti-Virus.
Version: 7.0.323 / Virus Database: 267.9.2/53 - Release Date: 20/7/05

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

Hi Paul,

You wrote:
> > How to tune african heptatonic instruments like cora and balafons ?
> > (without using 8 or 9 notes by octave ! ;)
>
> My answer for the string instruments, of course, is meantone tuning
[YA]----------8><-----Snip!

> The Kora/cora is an instrument I love dearly, there being some
> virtuoso players who've moved from Africa to the Boston area, and
> though certainly not Western, I've been surprised to find that they
> are definitely tuned with meantone tendencies -- even when playing
> along with 12-equal Western instruments.

[YA]
I'm surprised, too! Have you spoken to the musicians about their
tunings? If so, how do they explain them? And do we have any pitch
measurements to support this notion of meantone tuning?

> ... I don't claim this is
> universal for kora tuning, but the particular style, perhaps
> consciously "world-music" oriented, that finds itself represented by
> African musicians on the Boston streets and clubs seems to gravitate
> towards meantone kora tunings. Being familiar with the sound of
> meantone mainly through "authentic" recordings of Renaissance music,
> the sound of the Kora makes me imagine a cultural exchange of
> troubadours between Europe and Africa during the 16th century, the
> gut strings and fluid polyphony of the Kora being oddly reminiscent
> of the sound of lute music.

[YA]
Is there any evidence of such an exchange, I wonder?

I imagine cultural exchanges between Arabic and African musicians,
along trade routes both across the continent from Egypt to the Niger
and Timbuktu, and down the eastern coast and Rift Valley to Zanzibar.
Moreover, such exchanges would have occurred over a very long period.

However, if meantones were part of such exchanges, I would expect
them to have travelled from Africa to Arabia rather than the reverse.
My reasoning is that stringed instruments - more prevalent in Arabia
than in Africa - are easier to tune accurately by measurement than are
either pipes or idiophones. And many simple end-blown flutes I've seen
(I still have a couple) have apparently been tuned by boring equidistant
finger holes in the compass of the lower fourth of the flute, thus making
the first two (approximate) whole tones nearly equal. But I hasten to
add that I'm only speculating here.

> I wouldn't expect to find a *regular* meantone tuning on a Kora, an
> instrument which is tuned by ear frequently during performance (most
> often only one note per octave is retuned, to effect a different
> diatonic scale one step away on the chain of fifths, ...

[YA]
Interesting observation! Seems a practical way to add some variety
without taking too long over it. Do you know if there is a common
pattern to the keys (tonics) used in a single performance?

> ...... but overall
> tuning corrections seem necessary every few songs or so since the
> instrument is not built with locking tuners). But the players I've
> heard tune so that all the thirds and sixths in the diatonic scale
> are closer to pure than they could be in a JI (by Dave's definition)
> scale, so the indication is definitely that some kind of (perhaps
> irregular) meantone tuning is being used.

[YA]
Hmmm ... "thirds ... closer to pure than they could be in a JI" ???
What could be purer than an exact ratio? Or did "Dave's definition"
- "sensible JI", if I recall correctly - entirely dispense with ratios?

...
> As for the tuning of the balafon, and the many other traditions of
> kora tuning that one may find in Africa, I recommend reading this
> webpage:
>
> http://tcd.freehosting.net/djembemande/bala.html

[YA]
Thanks for this reference. Tho it seems that the expert cited on
this page was a little vaguer than we're used to on the tuning list!
Though your extended experience with meantones does make your
impression more plausible than that of any strictly 12-EDO Western
musician, I'd hold out for some measurements, preferably made from
actual performances, and ideally where the instruments were miked
separately, before concluding just what the tuning actually is.

Regards,
Yahya

--
No virus found in this outgoing message.
Checked by AVG Anti-Virus.
Version: 7.0.323 / Virus Database: 267.9.2/53 - Release Date: 20/7/05

🔗Dave Keenan <d.keenan@bigpond.net.au>

Paul Erlich:
> > ...... But the players I've
> > heard tune so that all the thirds and sixths in the diatonic scale
> > are closer to pure than they could be in a JI (by Dave's definition)
> > scale, so the indication is definitely that some kind of (perhaps
> > irregular) meantone tuning is being used.

Yahya Abdal-Aziz:
> Hmmm ... "thirds ... closer to pure than they could be in a JI" ???
> What could be purer than an exact ratio? Or did "Dave's definition"
> - "sensible JI", if I recall correctly - entirely dispense with ratios?
>

Hi Ya,

I think you missed an important word in what Paul wrote. He wrote that
"all" the thirds and sixth were close to pure. In a 5-limit JI
diatonic this is not possible. At least two of them (e.g. D:F and F:D
in C major) must be out by a whole comma.

My definition _could_ dispense with ratios, but in fact you'd be crazy
to refuse to make use of the common observation that perceptibly just
intervals occur near ratios of small whole numbers. You just have to
refuse to be nailed down on exactly how small the numbers in the ratio
have to be and how close to the ratio you have to be. Since these are
highly context dependent.

I note that "5-limit JI diatonic" means the same thing by either
definition of JI. But when we get to 17-limit or beyond it is quite
possible for a tuning to be rational but not perceptibly JI. In fact
its possible for it to be indistinguishable from 12-equal or 12-well
temperaments which few people consider to be JI.

While "sensible" is accurate (in both of its meanings) I prefer the
term "perceptible JI" lest folks think that I'm claiming that the
other, purely mathematical definition of JI is either senseless or
nonsensical. :-)

However, I suppose if people prefer to refer to it as "sensible JI"
then that's OK since I could point out that I do not object to the
term "rational JI" for the other kind, even though it might be thought
that "sensible JI" was then, by implication, irrational. :-)

-- Dave Keenan

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

Dave Keenan wrote:

> Paul Erlich:
> > > ...... But the players I've
> > > heard tune so that all the thirds and sixths in the diatonic scale
> > > are closer to pure than they could be in a JI (by Dave's definition)
> > > scale, so the indication is definitely that some kind of (perhaps
> > > irregular) meantone tuning is being used.
>
> Yahya Abdal-Aziz:
> > Hmmm ... "thirds ... closer to pure than they could be in a JI" ???
> > What could be purer than an exact ratio? Or did "Dave's definition"
> > - "sensible JI", if I recall correctly - entirely dispense with ratios?
> >
>
> Hi Ya,
>
> I think you missed an important word in what Paul wrote. He wrote that
> "all" the thirds and sixth were close to pure. In a 5-limit JI
> diatonic this is not possible. At least two of them (e.g. D:F and F:D
> in C major) must be out by a whole comma.
>
> My definition _could_ dispense with ratios, but in fact you'd be crazy
> to refuse to make use of the common observation that perceptibly just
> intervals occur near ratios of small whole numbers. You just have to
> refuse to be nailed down on exactly how small the numbers in the ratio
> have to be and how close to the ratio you have to be. Since these are
> highly context dependent.
>
> I note that "5-limit JI diatonic" means the same thing by either
> definition of JI. But when we get to 17-limit or beyond it is quite
> possible for a tuning to be rational but not perceptibly JI. In fact
> its possible for it to be indistinguishable from 12-equal or 12-well
> temperaments which few people consider to be JI.
>
> While "sensible" is accurate (in both of its meanings) I prefer the
> term "perceptible JI" lest folks think that I'm claiming that the
> other, purely mathematical definition of JI is either senseless or
> nonsensical. :-)
>
> However, I suppose if people prefer to refer to it as "sensible JI"
> then that's OK since I could point out that I do not object to the
> term "rational JI" for the other kind, even though it might be thought
> that "sensible JI" was then, by implication, irrational. :-)
>
> -- Dave Keenan

Thanks, Dave! What you wrote makes very good sense :-).
Yes, I certainly missed the _significance_ of that little "all".
In other words, those thirds and sixths were all adjusted -
if not necessarily _mathematically_ optimally, then at least
optimally for all practical purposes, since the results Paul heard
were ALL "perceptibly just", or perhaps "perceptibly just enough".

Let's eschew "rational" and "sensible" as adjectives then, and use
perhaps "mathematical" and "perceptible" instead. Sorry to coin
yet another term! but this one doesn't appear to assume its own
superiority, as "rational JI" does.

So, the "mathematical JI" provides us with a fairly simple criterion
for justness of intonation, namely that the fundamental tones are
in _exact_ small integer ratios, where the limits to "small" might be
fixed by some conventional agreement.

Compared to this, "perceptible JI" is a bit more complex and subtle,
isn't it? Let me quote the meat of your definition again:

> ... perceptibly just
> intervals occur near ratios of small whole numbers. You ...
> refuse to be nailed down on exactly how small the numbers in the ratio
> have to be and how close to the ratio you have to be. Since these are
> highly context dependent.

So, perceptibly just intonation (PJI) might be achieved using any one
of an enormous number of tempering schemes; can avoid the wolf
fifths, such as D:f and F:d in Cmajor (typically 680 cents instead of
702); can supply thirds and sixths that are indistinguishable from
mathematically just (MJ) intervals; and can produce regular scales
with fewer different step sizes.

Whereas MJ intonation (MJI) uses no tempering whatsoever; cannot
avoid the wolves; can supply exact thirds and sixths; and produces the
least regular scales possible.

(Ah! Woodsman! Come temper your axe, to save poor little Red Riding
Hood, bewildered in a forest of mathematically just intervals, and
pursued by wolves!)

... I begin to see the attractions of PJI ... I wonder if Aline Honingh
is still tuned in to this discussion? But her original question seems to
me, now, to combine the virtues ofSocratic questioning with the force
of the sucker punch!!! Anyway, we live and learn!

Tell me, if you can and will ... can a meantone temperament be
perceptibly just?

Regards,
Yahya

--
No virus found in this outgoing message.
Checked by AVG Anti-Virus.
Version: 7.0.323 / Virus Database: 267.9.2/55 - Release Date: 21/7/05

🔗Dave Keenan <d.keenan@bigpond.net.au>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, "Jacques Dudon (AEH)" <fotosonix@w...>
wrote:

> I worked a lot on heptatonic scales families, and in my point of
wiew there
> are at least 6 or 7 levels between diatonic and equiheptatonic
scales :
> 1) Diatonic, mainly of 5-limit models (of course there are also
many hyper-
> -diatonics)

I don't know what you mean by "hyper-diatonics" -- of course many of
my papers have been about generalizing diatonicity -- but in this
context, perhaps level 0 would be the Pythagorean and super-
Pythagorean diatonics? Diatonic scales where the semitone is less
than half the size of the whole tone are melodically delicious;
surely they have enough historical and/or geographical significance
to belong on this continuum?

> 2) Next to diatonic I place some of the main Burmese scales, with
semitones
> slightly larger than 16/15
> 3) Then some type of common final for Rast (in Arab music, not the
turkish
> Rast) which is a Rast whose neutral thirds over the tonic become
major
> 4) Then 13-limit kind of diatonic scales (Persian music, also Gnawa)
> 5) Then the semidiatonic neutral Rast family
> 6) Then but with a different structure the semidiatonic Mohajira
family,
> sligthly more circular ;
> 7) Then a very diffuse family using the 10/9 type of intervals as
largest
> tones and still one or two semitones of the 12/11 type ;
> 8) Then the Thaï or (almost)-equiheptatonic family, where the
smallest type
> of tone is no smaller than 11/10. Besides, semitone sensation
becomes rare
> in a Thaï scale.
> 9) In the end, an exactly-equal tempered heptaphone is more
theorical but
> can exist as a tuning tendency.

Cool . . . I see this isn't a smooth progression of levels (Manuel
has Mohajira as 3+4+3+4+3+4+3 in 24-equal), but in the spirit of what
you've proposed, I would be tempted to "continue" on "beyond"
equiheptatonic scales to heptatonic scales with two large steps and
five small steps, in the reverse of the diatonic pattern.
The "Mavila" scale of the Chopi (SW Africa) shows this pattern; and
more blatantly, the Pelog tunings of Indonesia do. I really love
playing with these kinds of scales; the pentatonics are wonderfully
flavorful, with a much enhanced differentiation between small and
large pentatonic step sizes, and standard triadic harmonic
habits "work" but with minor-sounding triads where you'd expect major
and vice versa.

There are also other distributionally even heptatonic scales in
my "Middle Path" paper that, like Mohajira, don't have this 5+2 or
2+5 distribution of step sizes; but I'm not sure how far afield you'd
like to go with this topic . . .

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

Yahya, the kora players I've spoken to tune strictly by ear, and
don't know any tuning theory themselves. I haven't made any
measurements but yes, that would be enlightening . . .

--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:

> > I wouldn't expect to find a *regular* meantone tuning on a Kora,
an
> > instrument which is tuned by ear frequently during performance
(most
> > often only one note per octave is retuned, to effect a different
> > diatonic scale one step away on the chain of fifths, ...
>
> [YA]
> Interesting observation! Seems a practical way to add some variety
> without taking too long over it. Do you know if there is a common
> pattern to the keys (tonics) used in a single performance?

I wasn't referring to keys or tonics, I was referring to the diatonic
collections. My observation is that typically two different key
signatures are used in a performance, with a single string per octave
being adjusted to switch between the two. For example, if the kora is
tuned to the white-key diatonic scale, the B string might be retuned
to B-flat for certain pieces. As the website I linked to mentioned,
the kora is typically tuned according to the vocal range of the
singer, so there's no standard pitch or key. I'm sure all this varies
a lot from player to player, depending on repertoire, the nature of
the accompanying ensemble, and other factors.

> > ...... but overall
> > tuning corrections seem necessary every few songs or so since the
> > instrument is not built with locking tuners). But the players
I've
> > heard tune so that all the thirds and sixths in the diatonic
scale
> > are closer to pure than they could be in a JI (by Dave's
definition)
> > scale, so the indication is definitely that some kind of (perhaps
> > irregular) meantone tuning is being used.
>
> [YA]
> Hmmm ... "thirds ... closer to pure than they could be in a JI" ???
> What could be purer than an exact ratio? Or did "Dave's definition"
> - "sensible JI", if I recall correctly - entirely dispense with
>ratios?

Not at all -- it's just that by "pure" I meant "5-odd-limit" in this
context: Thirds of 5:4 or 6:5, and sixths of 8:5 or 5:3. Getting all
the thirds and sixths pure in a heptatonic scale, by this definition,
is an impossibility, but by using temperament you can come pretty
close. I see Dave replied to you too so hopefully that helped
too . . .

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:
> So, the "mathematical JI" provides us with a fairly simple criterion
> for justness of intonation, namely that the fundamental tones are
> in _exact_ small integer ratios, where the limits to "small" might be
> fixed by some conventional agreement.

Well no, Kraig Grady and Daniel Wolf and others made it quite clear
that their definition of JI will brook no limits on the size of the
numbers in the ratios (or if they do, they are at least into 3
digits). So "mathematical JI" is simply having notes tuned to exact
ratios. But then they have the problem that very few real-world
instruments can guarantee _exact_ ratios, so then it comes down to
something like the _intention_ to tune to exact ratios.

They made it quite clear that you could have something that was
audibly (or even measurably) indistinguishable from 12-equal, and they
would still call it JI, provided it was intended to be tuned to exact
ratios (no matter how large the numbers in the ratios).

> Compared to this, "perceptible JI" is a bit more complex and subtle,
> isn't it?

That depends on whether you are considering the actual psycho-physical
musical event, or the mathematical modelling of it.

Mathematical JI is of course extremely simple to model mathematically.
But as a psycho-physical event it is so complex and subtle as to be
completely non-existent. :-) i.e. As Kraig said, there's no way to
tell by listening whether or not something is (mathematical) JI.

Apparently one must simply take the composer's word for it.

Perceptible JI is complex and subtle to model mathematically (so is
colour). But as a psycho-physical event it is quite easy to
demonstrate "that special sound" to someone on any instrument capable
of playing two notes simultaneously and continuously varying the pitch
of one of them. This is little different from the way a child learns
how dark a colour has to be before it can be called black. (Although
to me, synaesthetically, (perceptible) JI harmonies are more akin to
_saturated_ colours, a term children are rarely taught).

> > Let me quote the meat of your definition again:
> > ... perceptibly just
> > intervals occur near ratios of small whole numbers. You ...
> > refuse to be nailed down on exactly how small the numbers in the ratio
> > have to be and how close to the ratio you have to be. Since these are
> > highly context dependent.

That's not a definition.

My definition is currently injuctive. It is of the form "Go and do
things A, B, C; L, M, N and X, Y, Z and listen, then know that
Justness is the most salient audible quality that is common to A, B
and C and is absent from X, Y and Z and is almost or just barely
present in L, M and N."

How would you define Redishness?

> So, perceptibly just intonation (PJI) might be achieved using any one
> of an enormous number of tempering schemes; can avoid the wolf
> fifths, such as D:f and F:d in Cmajor (typically 680 cents instead of
> 702);

Any tempering scheme that is gentle enough to leave a tuning
perceptibly just is unlikely to be able to eliminate all wolves. It
may reduce their number however. But microtemperaments can also be
used to eliminate the need for very small steps in a JI scale or to
allow it to modulate a little more widely, or as you saw in my
microtempered guitar article, to straighten out frets, while doing
minimal damage to the existing JI harmonies.

> can supply thirds and sixths that are indistinguishable from
> mathematically just (MJ) intervals; and can produce regular scales
> with fewer different step sizes.

Yes. Although you simply can't make a diatonic scale with all thirds
and sixths perceptibly just. A meantone somewhere near 2/7-comma is
the best you can do, with a little over 3 cents error in all thirds
and sixths.

> Whereas MJ intonation (MJI) uses no tempering whatsoever; cannot
> avoid the wolves; can supply exact thirds and sixths; and produces the
> least regular scales possible.

Oh no. That's quite wrong. You can do absolutely anything you like in
mathematical JI, since the only requirement is that the result is
expressed using (frequency or wavelength) ratios, never cents or other
logarithmic units. You can even do _tempering_ in mathematical JI, but
you'd better not call it that. :-)

> (Ah! Woodsman! Come temper your axe, to save poor little Red Riding
> Hood, bewildered in a forest of mathematically just intervals, and
> pursued by wolves!)

Hee hee.

-- Dave Keenan

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

Paul,

You wrote:
> > can avoid the wolf fifths, such as D:f and F:d in Cmajor
> > (typically 680 cents instead of 702);
>
> These are a wolf minor third and wolf major sixth, not wolf
> fifths, and in cents they'd be 294 & 906, not 680.

I plead idiocy! (due to fatigue, induced by overwork.
Well that's my excuse, and I'm sticking to it...)

Of course, I was talking about the intervals between
nominals D and A.

> > and can produce regular scales
> > with fewer different step sizes.
>
> This is certainly *not* a characteristic of being perceptibly
> just, though perhaps I've lost your meaning and/or train of
> thought.

Sorry! If you find my train of thought, please bring it home,
as it's been wandering lately ....

Not more regular? Not fewer step sizes? Surely both are
at least possible in some temperings ...

> > Tell me, if you can and will ... can a meantone temperament
> > be perceptibly just?
>
> I would say no, because the errors involved, typically 5 or 6
> cents, are quite a bit larger than the just noticeable
> difference for *harmonic* (though not for melodic)
> intervals; and it's only for harmonic, or vertical, intervals
> where exact, simple-integer JI has a sound perceptibly
> different from any slightly different tuning.

Thanks for the explanation. It makes sense.

I suppose that it is still possible, even without using
meantones, to have a PJI heptatonic scale with octave
equivalence with only one size of whole-tone step. For
example, a cycle of pure fifths, octave -reduced:
F 4/3
C 1/1
G 3/2
D 9/8
A 27/16
E 81/64
B 243/128
has all its 5 whole-tones equal to 9/8 and all its 2
semitones equal to 256/243. This would be perceptibly
just, wouldn't it?

What about this: a similar scale composed of a cycle of
impure fifths, of size say 299/200 rather than 3/2,
octave-reduced:
F 400/299
C 1/1
G 299/200
D 89 401/80 000
A 26 730 899/16 000 000
E 7 992 538 801/6 400 000 000
B 2 389 769 101 499/1 280 000 000 000
which has all its 5 whole-tones equal to 89 401/80 000
and all its 2 semitones equal to
2 560 000 000 000/2 389 769 101 499.

Would you classify this as perceptibly just? If not,
why not? Perhaps ONLY because the sound is noticeably
different - as you wrote -
> ... it's only for harmonic, or vertical, intervals where
> exact, simple-integer JI has a sound perceptibly
> different from any slightly different tuning.

To provide a mathematical test for perceptible JI, we
would have to quantify what is meant by "slightly
different", perhaps under stated conditions.

A pure fifth of ratio 3/2 has 701.955000865... cents,
while an impure fifth of ratio 299/200 has
696.174581308... cents. The difference between them
is about 5.78 cents. These imperfect fifths would be
perceptibly ... unjust when used as harmonic intervals.

Regards,
Yahya
--
No virus found in this outgoing message.
Checked by AVG Anti-Virus.
Version: 7.0.323 / Virus Database: 267.9.2/55 - Release Date: 21/7/05

🔗Jacques Dudon (AEH) <fotosonix@wanadoo.fr>

le 22/07/05 20:50, wallyesterpaulrus à wallyesterpaulrus@yahoo.com a écrit :

> --- In tuning@yahoogroups.com, "Jacques Dudon (AEH)" <fotosonix@w...>
> wrote:
>
>> I worked a lot on heptatonic scales families, and in my point of
>> wiew there
>> are at least 6 or 7 levels between diatonic and equiheptatonic
>> scales :
>> 1) Diatonic, mainly of 5-limit models (of course there are also
>> many hyper-diatonics)
>
> I don't know what you mean by "hyper-diatonics" -- of course many of
> my papers have been about generalizing diatonicity -- but in this
> context, perhaps level 0 would be the Pythagorean and super-
> Pythagorean diatonics?

Level 0 for the Pythagorean of course, yes - along with 135/128 and 19/18
semitones may be - ?
Then 17/16 would be level 0,5 ? gosh, we need one more faimly already in the
list !
what I meant by "hyperdiatonics" are diatonics with semitones ranging from
17/16 or so, down to 1 comma - but I don't know if it is a correct
appelation.

> Diatonic scales where the semitone is less
> than half the size of the whole tone are melodically delicious;
> surely they have enough historical and/or geographical significance
> to belong on this continuum ?

absolutely, we could find examples in japanese music, and also in ethiopian,
I would think.
I didn't mention those because we were talking of the heptatonic balafon or
cora tunings.
In one piece of my last ensemble creation that was intitled "La Confusion
des Genres" (The Confusion of Genera ?) I am playing on that : how, when you
double two of the notes of a slendro (one fourth apart) by a comma or so,
you start to have this diatonic feeling, with the sensible and tonic almost
united ... then at some point you don't know if you are in a pentatonic or
heptatonic system, and I love that.

>> 2) Next to diatonic I place some of the main Burmese scales, with
>> semitones slightly larger than 16/15
>> 3) Then some type of common final for Rast (in Arab music, not the
>> turkish Rast) which is a Rast whose neutral thirds over the tonic become
>> major
>> 4) Then 13-limit kind of diatonic scales (Persian music, also Gnawa)
>> 5) Then the semidiatonic neutral Rast family
>> 6) Then but with a different structure the semidiatonic Mohajira
>> family, sligthly more circular ;
>> 7) Then a very diffuse family using the 10/9 type of intervals as
>> largest tones and still one or two semitones of the 12/11 type ;
>> 8) Then the Thaï or (almost)-equiheptatonic family, where the
>> smallest type
>> of tone is no smaller than 11/10. Besides, semitone sensation
>> becomes rare in a Thaï scale.
>> 9) In the end, an exactly-equal tempered heptaphone is more
>> theorical but can exist as a tuning tendency.
>
> Cool . . . I see this isn't a smooth progression of levels (Manuel
> has Mohajira as 3+4+3+4+3+4+3 in 24-equal), but in the spirit of what
> you've proposed, I would be tempted to "continue" on "beyond"
> equiheptatonic scales to heptatonic scales with two large steps and
> five small steps, in the reverse of the diatonic pattern.
> The "Mavila" scale of the Chopi (SW Africa) shows this pattern; and
> more blatantly, the Pelog tunings of Indonesia do. I really love
> playing with these kinds of scales; the pentatonics are wonderfully
> flavorful, with a much enhanced differentiation between small and
> large pentatonic step sizes, and standard triadic harmonic
> habits "work" but with minor-sounding triads where you'd expect major
> and vice versa.

Yes ! I don't know the "Mavila" Scale, but the tendency is already potential
in what I mentionned rapidly in my 7th family, where from a Thaï scale you
trade two "11/10" for another "10/9" and one "12/11", or like in other
"Rast-attracted" Thaï scales.
Defective pentatonics from these scales are numerous, and I place the
delicate Gnawa scales in that family too.
Then on the "other side" of Mohajira, you find the medieval persian "Buzurg"
which could also
be near what you describe, when the two medium whole-tones of the Mohajira
tetrachords inflate to reach 8/7 and possibly 7/6. With the semitones
becoming smaller (13/12 and 14/13), I found many Pelog arise then at various
keys.

> There are also other distributionally even heptatonic scales in
> my "Middle Path" paper that, like Mohajira, don't have this 5+2 or
> 2+5 distribution of step sizes; but I'm not sure how far afield you'd
> like to go with this topic . . .

Well, all this is inspiring, even if I should be entering into more
"organisation work" now untill the end of august, for our music festival...

Where do we find your writtings ?

🔗Dave Keenan <d.keenan@bigpond.net.au>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:
>
> Paul,
>
> You wrote:
> > > can avoid the wolf fifths, such as D:f and F:d in Cmajor
> > > (typically 680 cents instead of 702);
> >
> > These are a wolf minor third and wolf major sixth, not wolf
> > fifths, and in cents they'd be 294 & 906, not 680.
>
> I plead idiocy! (due to fatigue, induced by overwork.
> Well that's my excuse, and I'm sticking to it...)
>
> Of course, I was talking about the intervals between
> nominals D and A.

Well, that's another wolf interval you'd have to avoid.

> > > and can produce regular scales
> > > with fewer different step sizes.
> >
> > This is certainly *not* a characteristic of being perceptibly
> > just, though perhaps I've lost your meaning and/or train of
> > thought.
>
> Sorry! If you find my train of thought, please bring it home,
> as it's been wandering lately ....
>
> Not more regular? Not fewer step sizes? Surely both are
> at least possible in some temperings ...

Yes, exactly. So why would these be characteristics of being
perceptibly just? Normally, being just and being tempered are
mutually exclusive, though we do enter a gray area when the
temperings gets really really tiny . . .

> I suppose that it is still possible, even without using
> meantones, to have a PJI heptatonic scale with octave
> equivalence with only one size of whole-tone step.

Yes, that's Pythagorean, or 3-limit JI. There are only two dimensions
here (you could think of the basis as consisting of the two primes 2
and 3), so scales with only two step sizes are what you'd expect.

> For
> example, a cycle of pure fifths, octave -reduced:
> F 4/3
> C 1/1
> G 3/2
> D 9/8
> A 27/16
> E 81/64
> B 243/128
> has all its 5 whole-tones equal to 9/8 and all its 2
> semitones equal to 256/243. This would be perceptibly
> just, wouldn't it?

In a 3-limit sense, yes. But you wrote "fewer different step sizes".
Fewer than what?

> What about this: a similar scale composed of a cycle of
> impure fifths, of size say 299/200 rather than 3/2,
> octave-reduced:
> F 400/299
> C 1/1
> G 299/200
> D 89 401/80 000
> A 26 730 899/16 000 000
> E 7 992 538 801/6 400 000 000
> B 2 389 769 101 499/1 280 000 000 000
> which has all its 5 whole-tones equal to 89 401/80 000
> and all its 2 semitones equal to
> 2 560 000 000 000/2 389 769 101 499.

This is incredibly close to the Woolhouse optimal meantone, 7/26-
comma meantone.

> Would you classify this as perceptibly just?

No.

> If not,
> why not?

None of the intervals sound just -- any of the thirds, sixths, or
perfect fourths or thirds can be retuned, by ear, by 3 or 6 cents, so
that the sound becomes just, and this happens at a small-integer
frequency ratio. Of course, it's impossible to do this for *all* the
intervals in the scale at once.

> Perhaps ONLY because the sound is noticeably
> different - as you wrote -
> > ... it's only for harmonic, or vertical, intervals where
> > exact, simple-integer JI has a sound perceptibly
> > different from any slightly different tuning.

Right.

> To provide a mathematical test for perceptible JI, we
> would have to quantify what is meant by "slightly
> different", perhaps under stated conditions.
>
> A pure fifth of ratio 3/2 has 701.955000865... cents,
> while an impure fifth of ratio 299/200 has
> 696.174581308... cents. The difference between them
> is about 5.78 cents. These imperfect fifths would be
> perceptibly ... unjust when used as harmonic intervals.

Right. They sound just fine in the context of Renaissance, Baroque,
or Classical music, where bare fifths are rarely dwelt upon; when the
triad is completed, the impurity of the fifth is hardly bothersome at
all.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

Hi Jacques,

> what I meant by "hyperdiatonics" are diatonics with semitones
ranging from
> 17/16 or so, down to 1 comma - but I don't know if it is a correct
> appelation.

Perhaps. We've been using the term "Superpythagorean" to mean that
the minor seconds are smaller, and the major seconds larger, than in
Pythagorean. And sometimes the Superpythagorean appelation
specifically indicates that 64:63 is tempered out, so that stacking
three (4:3) fourths approximates a 7:3, etc. . . .

It's worth noting that "diatonic" means "through the tones", the
prefix "dia-" meaning through. It does *not* mean "two tones" -- that
would be "ditonic", not "diatonic".

> then at some point you don't know if you are in a pentatonic or
> heptatonic system, and I love that.

Cool! How can I hear this?

> > Cool . . . I see this isn't a smooth progression of levels (Manuel
> > has Mohajira as 3+4+3+4+3+4+3 in 24-equal),

BTW, I know I've asked you before, but where did you get the
name "Mohajira" and its association to particular scales?

> > but in the spirit of what
> > you've proposed, I would be tempted to "continue" on "beyond"
> > equiheptatonic scales to heptatonic scales with two large steps
and
> > five small steps, in the reverse of the diatonic pattern.
> > The "Mavila" scale of the Chopi (SW Africa) shows this pattern;
and
> > more blatantly, the Pelog tunings of Indonesia do. I really love
> > playing with these kinds of scales; the pentatonics are
wonderfully
> > flavorful, with a much enhanced differentiation between small and
> > large pentatonic step sizes, and standard triadic harmonic
> > habits "work" but with minor-sounding triads where you'd expect
major
> > and vice versa.
>
> Yes ! I don't know the "Mavila" Scale,

I'll let Kraig respond to this before I try to.

>but the tendency is already potential
> in what I mentionned rapidly in my 7th family, where from a Thaï
scale you
> trade two "11/10" for another "10/9" and one "12/11", or like in
other
> "Rast-attracted" Thaï scales.
> Defective pentatonics from these scales are numerous, and I place
the
> delicate Gnawa scales in that family too.

Can you elaborate? Haven't heard of Gnawa, and I don't know what you
mean by "defective pentatonics". I was referring to pentatonics like
1+1+3+1+3 in 9-equal, for example, a very Pelog-like scale, where the
encapsulating "Pelog tuning" would be 1+1+2+1+1+1+2.

> > There are also other distributionally even heptatonic scales in
> > my "Middle Path" paper that, like Mohajira, don't have this 5+2 or
> > 2+5 distribution of step sizes; but I'm not sure how far afield
you'd
> > like to go with this topic . . .
>
> Well, all this is inspiring, even if I should be entering into more
> "organisation work" now untill the end of august, for our music
festival...
>
> Where do we find your writtings ?

Many of them are available through this page:

http://www.lumma.org/tuning/erlich/

but the more recent "Middle Path" paper I'm referring to is one I've
been snail-mailing out instead. If you'll e-mail me your home
address, I'd love to make a copy and drop it in the mail for you. I
believe this paper will also give you a much better idea of what is
meant when we say a comma is "tempered out".

Best,
Paul

🔗Afmmjr@aol.com

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Kurt Bigler <kkb@breathsense.com>

on 7/22/05 1:54 PM, wallyesterpaulrus <wallyesterpaulrus@yahoo.com> wrote:

> --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:
>
>> Tell me, if you can and will ... can a meantone temperament be
>> perceptibly just?
>
> I would say no, because the errors involved, typically 5 or 6 cents,
> are quite a bit larger than the just noticeable difference for
> *harmonic* (though not for melodic) intervals; and it's only for
> harmonic, or vertical, intervals where exact, simple-integer JI has a
> sound perceptibly different from any slightly different tuning.

I know I'm stepping late into a discussion of an existing definition
(originally "sensibly just", currently "perceptibly just"), but there is
something in this usage which is counter-intuitive to me. What you are
currently talking about I would call "not perceptibly unjust". But that
seems different to me from "perceptibly just" which to me should mean that
qualities associated specifically with JI (e.g. a quite distinctly audible
implied fundamental) should be clearly present. I think such qualities are
distinctly present with errors of 5 or 6 cents.

I don't really need to argue about which term means what, as long as there
are ways to cover the different meanings. The positive sense of perceptibly
just I just described needs some way of being referred to.

-Kurt

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Kurt Bigler <kkb@breathsense.com>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Afmmjr@aol.com

🔗Ozan Yarman <ozanyarman@superonline.com>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗Jacques Dudon (AEH) <fotosonix@wanadoo.fr>

Hi Paul,
Sorry for the delay,
Here is my postal address, where you / and anybody else can send me paper
documents :
-------------
Jacques Dudon
Atelier d'Exploration Harmonique - Les Camails 83340 LE THORONET - France

http://aeh.free.fr
tel : 33 - 4 94 73 87 78
-------------

le 25/07/05 21:46, wallyesterpaulrus à wallyesterpaulrus@yahoo.com a écrit :

> Hi Jacques,
>
>> what I meant by "hyperdiatonics" are diatonics with semitones
>> ranging from
>> 17/16 or so, down to 1 comma - but I don't know if it is a correct
>> appelation.
>
> Perhaps. We've been using the term "Superpythagorean" to mean that
> the minor seconds are smaller, and the major seconds larger, than in
> Pythagorean. And sometimes the Superpythagorean appelation
> specifically indicates that 64:63 is tempered out, so that stacking
> three (4:3) fourths approximates a 7:3, etc. . . .
>
> It's worth noting that "diatonic" means "through the tones", the
> prefix "dia-" meaning through. It does *not* mean "two tones" -- that
> would be "ditonic", not "diatonic".
Yes I know. I think "hyperdiatonics" was inspired to me by "Divisions of the
Tetrachord" by John Chalmers, who mentions "Hyperenharmonic" from Erv Wilson
(56/55 55/54 9/7) - I like "hyperdiatonics", being on the opposite side of
"semidiatonics", that we all know.
Besides, but that's my own feeling, I would rather leave Pythagoras, who is
beyond all that, and even beyond the 3-limit, out any stuff he was not for
sure specially into.

>> then at some point you don't know if you are in a pentatonic or
>> heptatonic system, and I love that.
>
> Cool! How can I hear this ?

In any Indonesian Gamelan in slendro tuning with bars a little distuned
between different instruments, wherever you happen to catch a "semitone
feeling", that's it ! then you have an hexatonic, and if you hear another
semitone you have good chances to be in front of some kind of a
"hyperdiatonic" !
ex : slendro 1 2 3 5 6 > add one comma under 1 & the other over 3 =
hyperdiatonic from 1
If you want a precise exemple, take a "S-J" (Dudon) slendro :
21 3 55 63 9 and add 165 & 7 = hyperdiatonic from 21
This is the contrary of "tempering out", I guess.

>>> Cool . . . I see this isn't a smooth progression of levels (Manuel
>>> has Mohajira as 3+4+3+4+3+4+3 in 24-equal),
this must be the Egyptian version then ! ;)

> BTW, I know I've asked you before, but where did you get the
> name "Mohajira" and its association to particular scales?
Mohajira is an arabian word that means "migratory". That was the first
feeling I got when I discovered it, trying the different modes of a 13-limit
Mohajira and finding they were reminding me, in my sense, of many different
cultures : arabian, persian, malagasy, even celtic sometimes ...
Mohajira is basically formed by the succession of six neutral thirds.
The two extremities are then, octave-reduced, one minor third apart.
That is why it is one step more circular than Rast, who has 1 major third,
2 minor thirds, and four neutral thirds.

>>> but in the spirit of what
>>> you've proposed, I would be tempted to "continue" on "beyond"
>>> equiheptatonic scales to heptatonic scales with two large steps
>>> and five small steps, in the reverse of the diatonic pattern.
>>> The "Mavila" scale of the Chopi (SW Africa) shows this pattern;
>>> and more blatantly, the Pelog tunings of Indonesia do. I really love
>>> playing with these kinds of scales; the pentatonics are
>>> wonderfully
>>> flavorful, with a much enhanced differentiation between small and
>>> large pentatonic step sizes, and standard triadic harmonic
>>> habits "work" but with minor-sounding triads where you'd expect
>>> major and vice versa.
>>
>> Yes ! I don't know the "Mavila" Scale,
>
> I'll let Kraig respond to this before I try to.
>
>> but the tendency is already potential
>> in what I mentionned rapidly in my 7th family, where from a Thaï
>> scale you
>> trade two "11/10" for another "10/9" and one "12/11", or like in
>> other "Rast-attracted" Thaï scales.
>> Defective pentatonics from these scales are numerous, and I place
>> the delicate Gnawa scales in that family too.
>
> Can you elaborate? Haven't heard of Gnawa, and I don't know what you
> mean by "defective pentatonics". I was referring to pentatonics like
> 1+1+3+1+3 in 9-equal, for example, a very Pelog-like scale, where the
> encapsulating "Pelog tuning" would be 1+1+2+1+1+1+2.

I was thinking of that too, but rather in non-equal 9.
The Gnawa are an ethnical brotherhood living nowadays mostly in Marocco, who
used to be slaves from Sudan. Their main instruments are the guimbri (or
Hajhouj), a 3-trings bass lute with skin, and the qarqabat (plur. qaraqeb),
kind of iron castanets, plus various drums. Their music is known to be part
of a healing ritual, the Lila, that lasts during the whole night.
The qaraqeb plays some kind of strange polyrythm between 3 and for 4 beats
per cycle, while they sing on some pentatonic scale that could be
interpreted as a 2+3+2+3+2 form, with the guimbri producing extatic bass
accompaniment. However, I found that the scale they played had rather often
a defective-Mohajira tendency (Mohajira with two notes ommited) : for
exemple a 4+7+3+7+3 form,
which could also be considered as a modal form of a 7+3+4+7+3 Pelog

>>> There are also other distributionally even heptatonic scales in
>>> my "Middle Path" paper that, like Mohajira, don't have this 5+2 or
>>> 2+5 distribution of step sizes; but I'm not sure how far afield
>>> you'd like to go with this topic . . .
>>
>> Well, all this is inspiring, even if I should be entering into more
>> "organisation work" now untill the end of august, for our music
>> festival...
>>
>> Where do we find your writtings ?

> Many of them are available through this page:
>
> http://www.lumma.org/tuning/erlich/
>
> but the more recent "Middle Path" paper I'm referring to is one I've
> been snail-mailing out instead. If you'll e-mail me your home
> address, I'd love to make a copy and drop it in the mail for you. I
> believe this paper will also give you a much better idea of what is
> meant when we say a comma is "tempered out".
>
> Best,
> Paul

🔗Dave Keenan <d.keenan@bigpond.net.au>

🔗Jacques Dudon (AEH) <fotosonix@wanadoo.fr>

le 31/07/05 2:33, Dave Keenan à d.keenan@bigpond.net.au a écrit :

> --- In tuning@yahoogroups.com, "Jacques Dudon (AEH)" <fotosonix@w...>
> wrote:
>> Yes I know. I think "hyperdiatonics" was inspired to me by
>> "Divisions of the
>> Tetrachord" by John Chalmers, who mentions "Hyperenharmonic" from
>> Erv Wilson
>> (56/55 55/54 9/7) - I like "hyperdiatonics", being on the opposite
>> side of "semidiatonics", that we all know.

> Hello Jacques,
>
> Hyperenharmonic is appropriate for Wilson's tetrachords since they are
> _beyond_ enharmonic in the diatonic-chromatic-enharmonic dimension.
> But the extreme in the other direction would be the division of the
> tetrachord into 3 equal parts, which is still a diatonic, and not the
> kind you are referring to.
- the extreme in the other direction, only from the point of wiew of this
precise representation (Chalmer's) ; I did draw some other representations
where enharmonic, chromatic, 3 equal parts, diatonic are also in continuity
and followed by these "diatonics with semitones ranging from 17/16 to a
comma" I call hyperdiatonics - may be an improper name, but then, on the
other side, "semi-diatonics" also...

Consider the following representation :
a square with a basis = Log (4/3)
(and appropriate scaling for cents if you wish)
The square's diagonal passing by zero,
Two other lines passing by zero and points Log (8/7) and Log (7/6) on the
upper side of the square.
parralal lines to the basis cut the square for many famous tetrachord
divisions,
and showing a total continuity between (from bottom to top) enharmonics to
different slendros, passing by "hyperchromatics", chromatics, Hijaz, Buzurg,
Bayati, Homalon, semidiatonics, diatonics, "hyperdiatonics", slendros...
Sometimes you have different modes, but all main greek and arab and more
tetrachords are there.
The Log (8/7) line shows the aliquote division of interval a + b
The Log (7/6) line shows the arithmetic division of interval a + b

> The kind you are referring to, differ from
> more common diatonics in a dimension "at right angles" to the
> diatonic-chromatic-enharmonic dimension.
Sorry, I don't understand this sentance. Can you develop ? Where are the
right angles you're talking about ?

> John Chalmers calls them "improper diatonics" here
> http://sonic-arts.org/chalmers/diagrams.htm
right ; improper diatonics, and neither enharmonic nor chromatic - that's
why they deserve to be better defined than only "improper diatonics"...

example : 9/8 34/27 4/3 = because of a semitone smaller than 100 cents you
would call this "improper diatonic" ?

> And somewhat oddly, Byzantine practitioners call them "enharmonic".
> -- Dave Keenan
Nice one, I will add it to my collection of "confused genera"...

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, "Jacques Dudon (AEH)" <fotosonix@w...>
wrote:

> Hi Paul,
> Sorry for the delay,
> Here is my postal address, where you / and anybody else can send me
paper
> documents :
> -------------
> Jacques Dudon
> Atelier d'Exploration Harmonique - Les Camails 83340 LE THORONET -
France

Thanks! I just made two copies for you and Gene . . . now I just need
to receive or find Gene's address . . .

> >> then at some point you don't know if you are in a pentatonic or
> >> heptatonic system, and I love that.
> >
> > Cool! How can I hear this ?
>
> In any Indonesian Gamelan in slendro tuning with bars a little
distuned
> between different instruments, wherever you happen to catch
a "semitone
> feeling", that's it ! then you have an hexatonic, and if you hear
another
> semitone you have good chances to be in front of some kind of a
> "hyperdiatonic" !
> ex : slendro 1 2 3 5 6 > add one comma under 1 & the other over 3 =
> hyperdiatonic from 1
> If you want a precise exemple, take a "S-J" (Dudon) slendro :
> 21 3 55 63 9 and add 165 & 7 = hyperdiatonic from 21
> This is the contrary of "tempering out", I guess.

Well, this kind of effect can result from tempering things out
too . . . for example the "Superpyth" scale in my paper (that I'm
about to send you) has a pretty even pentatonic, and then the
two 'semitones' in the diatonic are only 52 cents. Yet this system
arises *entirely* from JI by tempering out a certain group of commas
while doing the least possible damage to the concordance of the
underlying JI ratios. Hopefully my paper will help make this clear.

> >>> Cool . . . I see this isn't a smooth progression of levels
(Manuel
> >>> has Mohajira as 3+4+3+4+3+4+3 in 24-equal),
> this must be the Egyptian version then ! ;)
>
> > BTW, I know I've asked you before, but where did you get the
> > name "Mohajira" and its association to particular scales?
> Mohajira is an arabian word that means "migratory". That was the
first
> feeling I got when I discovered it,

So Manuel got it from you then . . . very interesting.

> trying the different modes of a 13-limit
> Mohajira and finding they were reminding me, in my sense, of many
different
> cultures : arabian, persian, malagasy, even celtic sometimes ...
> Mohajira is basically formed by the succession of six neutral
thirds.

Yes; the scale has come up many times on this list; for example
Blackjack is composed of 3 interlaced Mohajira scales (the Blackjack
generator being 1/3 of a neutral third and the period being an
octave). In the paper I'm sending you, you'll see a few different
scales fitting this description: Beatles[7] =
133+222+133+222+133+222+133, and Dicot[7] =
148+205+148+205+148+205+148. These are derived as temperaments of 7-
limit and 5-limit JI, respectively . . . the 11-limit and 13-limit
cases are going to have to wait for future parts of the paper (what
I'm sending you is actually only part 1). On the tuning-math list,
there has been some discussion of 11-limit and 13-limit temperaments,
and I believe a few of the significant ones had a neutral third
generator and octave period. Any of these temperaments can be taken
as a starting point for deriving a whole host of JI scales,
via "detempering" . . . so I believe the math has value even for
those who wish to stick with a pure form of JI.

🔗Jacques Dudon (AEH) <fotosonix@wanadoo.fr>

le 1/08/05 21:26, wallyesterpaulrus à wallyesterpaulrus@yahoo.com a écrit :

>>> BTW, I know I've asked you before, but where did you get the
>>> name "Mohajira" and its association to particular scales?
>> Mohajira is an arabian word that means "migratory". That was the
>> first feeling I got when I discovered it,
>
> So Manuel got it from you then . . .
May be not directly. If I remember well (that was many years ago),
I was searching for eventual traditionnal sources for that that type of
scale, and had a correspondance on that subject only with John Chalmers, and
several arabian and persian music masters. But may be Manuel got it from my
CD "Lumières Audibles" (1996), where the wole thing is explained.

> In the paper I'm sending you, you'll see a few different
> scales fitting this description: Beatles[7] =
> 133+222+133+222+133+222+133, and Dicot[7] =
> 148+205+148+205+148+205+148. These are derived as temperaments of 7-
> limit and 5-limit JI, respectively

It seems strange that the last one would be derived from 5 limit, but that's
not impossible.
This second one I would definively class in the Mohajira family, but the
first one would fit better in what I call the "Buzurg" family, because of
133 being in between 14/13 and 13/12, the "neutral-minor" seconds ofthe
Buzurg (medieval persian) tetrachord, composed of 14/13 8/7 13/12
Of course this would normally leave an 9/8 after, but who knows.
I composed in 1998 for the Ensemble de Musique Microtonale du Thoronet
several pieces that were intitled "Estrangetés & arabesques", in a decaphone
including such a scale and I love it.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning@yahoogroups.com, "Jacques Dudon (AEH)" <fotosonix@w...>
wrote:
> - the extreme in the other direction, only from the point of wiew of
> this precise representation (Chalmer's) ;

Yes. But I don't see that there is any better way to represent audible
similarity of tetrachords by nearness on a diagram.

> I did draw some other representations
> where enharmonic, chromatic, 3 equal parts, diatonic are also in
> continuity and followed by these "diatonics with semitones ranging
> from 17/16 to a comma" I call hyperdiatonics - may be an improper
> name, but then, on the other side, "semi-diatonics" also...

I can see that this is possible, even on Chalmer's diagram, when you
take specific orderings of the intervals within the tetrachord. But I
usually think of that as involving also what "mode" it is. I take
diatonic, chromatic, enharmonic to be determined purely by the size of
the largest interval, no matter where it is. But I am happy to learn
from you.

I don't know what is meant by "semidiatonic". Please explain.

> Consider the following representation :
> a square with a basis = Log (4/3)
> (and appropriate scaling for cents if you wish)
> The square's diagonal passing by zero,
> Two other lines passing by zero and points Log (8/7) and Log (7/6)
on the
> upper side of the square.
> parralal lines to the basis cut the square for many famous tetrachord
> divisions,
> and showing a total continuity between (from bottom to top)
enharmonics to
> different slendros, passing by "hyperchromatics", chromatics, Hijaz,
Buzurg,
> Bayati, Homalon, semidiatonics, diatonics, "hyperdiatonics", slendros...
> Sometimes you have different modes, but all main greek and arab and more
> tetrachords are there.
> The Log (8/7) line shows the aliquote division of interval a + b
> The Log (7/6) line shows the arithmetic division of interval a + b

I'm sorry I couldn't decode this, but I would dearly love to
understand, and to learn the meaning of these terms. Can you upload a
file of such a diagram to the tuning_files Yahoo group?

> > The kind you are referring to, differ from
> > more common diatonics in a dimension "at right angles" to the
> > diatonic-chromatic-enharmonic dimension.
> Sorry, I don't understand this sentance. Can you develop ? Where are the
> right angles you're talking about ?

I should have said, in a dimension _orthogonal_ to the
diatonic-chromatic-enharmonic dimension, since it is actually shown at
60 degrees on Chalmers' diagram. I understand Aristoxenos'
diatonic-chromatic-enharmonic dimension to be determined purely by the
size of the _largest_ division. The other dimension I speak of is
based on the size of the _smallest_ division.

I have drawn a diagram and uploaded it here.
/tuning/files/Keenan/TetrachordChart.doc

I am sorry it is a Microsoft Word document.

It is effectively a zoom-in on one sixth of Chalmers' diagram. I began
drawing this when I was investigating the classification of Byzantine
tetrachords by its own practitioners, which is why it is based on
72-EDO. The numbers are the sizes, in steps of 72-EDO, for each
interval making up the tetrachord. They should be considered as not
being in any particular order. Here's a conversion table for these steps.

No. Cents Interval Approximated
steps name ratios
-----------------------------------------------------------
0 0 unison 1:1
1 17 twefthtone
2 33 sixthtone
3 50 quartertone
4 67 narrow subminor second 24:25 (27:28 25:26)
5 83 subminor second 20:21 (21:22)
6 100 narrow minor second 16:17 (17:18)
7 117 minor second 15:16 (14:15)
8 133 wide minor second 12:13 (13:14)
9 150 neutral second 11:12
10 167 wide neutral second 10:11
11 183 narrow major second 9:10
12 200 major second 8:9
13 217 narrow supermajor second 15:17 (22:25)
14 233 supermajor second 7:8
15 250 wide supermajor second 13:15
16 267 subminor third 6:7
17 283 wide subminor third 15:17 (11:13)
18 300 narrow minor third 27:32 (21:25 16:19)
19 317 minor third 5:6
20 333 narrow neutral third 14:17
21 350 neutral third 9:11 (22:27)
22 367 wide neutral third 13:16
23 383 major third 4:5
24 400 wide major third 19:24 (27:34)
25 417 narrow supermajor third 11:14
26 433 supermajor third 7:9 (25:32)
27 450 narrow subfourth 10:13 (17:22)
28 467 subfourth 16:21
29 483 wide subfourth
30 500 perfect fourth 3:4

By using 72-EDO, every tetrachord ever used can be represented to
sufficient accuracy for melodic purposes, by one or more of these
circles. And because of 72-EDO's good approximations to sensible just
intonation, it is usually sufficiently accurate for harmonic purposes too.

I would be very interested to see your names for tetrachords (or the
scales or modes containing them) mapped out on this diagram.

> > John Chalmers calls them "improper diatonics" here
> > http://sonic-arts.org/chalmers/diagrams.htm
> right ; improper diatonics, and neither enharmonic nor chromatic -
that's
> why they deserve to be better defined than only "improper diatonics"...

"Improper" may be an unfortunate term. It applies not to the scale's
diatonicity, but to its "scale-ness" or its distinctness of interval
categories. It applies to all scales, not merely tetrachordal ones. It
has a specific meaning from David Rothenberg, which you can find at
http://www.tonalsoft.com/enc

> example : 9/8 34/27 4/3 = because of a semitone smaller than 100
cents you
> would call this "improper diatonic" ?

Yes. And there are proper and improper chromatics too as you will see
from my diagram. All enharmonics are improper, which makes it a little
easier to see how Byzantine choirs could have made that transference
of meaning with the term enharmonic. Although they can't have thought
that all improper scales are enharmonic, since their "hard chromatics"
are also improper.

-- Dave Keenan

🔗monz <monz@tonalsoft.com>

Hi Dave,

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> I have drawn a diagram and uploaded it here.
> /tuning/files/Keenan/
TetrachordChart.doc
>
> I am sorry it is a Microsoft Word document.
>
> It is effectively a zoom-in on one sixth of Chalmers'
> diagram. I began drawing this when I was investigating
> the classification of Byzantine tetrachords by its own
> practitioners, which is why it is based on 72-EDO.
> The numbers are the sizes, in steps of 72-EDO, for each
> interval making up the tetrachord. They should be
> considered as not being in any particular order. ...

Kudos to you for a great diagram! ... something *i* in
particular can really appreciate! ;-)

Please, i beg you, discard the use of "hard" and "soft"
as the English translations of the Greek "syntonon"
and "malakon" respectively -- "tense" and "relaxed"
are far preferable, as these terms stem directly from
the theories of Aristoxenus, and his particular idiosyncracy
was to eschew the use of any string *length* measurements,
and to focus instead on the *tension* of strings to produce
higher or lower pitches.

Aristoxenus did this so that he could therefore avoid
the use of ratios, and instead focus on the *audible
perception* of concordance of 4ths and 5ths in building
a closed tempered 12-tone scale from chains of concordant
4ths and 5ths. This makes him the first fully documentable
advocate of 12-tET, or at least something close to it.

Partch used "hard" and "soft" in _Genesis of a Music_,
transmitted (if i'm not mistaken) from Shirlaw's survey
of harmony published in the 1700's. Because Partch is
such an important microtonal pioneer, his terminology
tends to be repeated by more recent writers. But "tense"
and "relaxed" are the perfect English words to convey
the meanings intended by Aristoxenus, and those genera
were first named by him.

See my webpages:

http://tonalsoft.com/enc/m/malakon.aspx

http://tonalsoft.com/enc/s/syntonon.aspx

http://tonalsoft.com/enc/t/tuning-by-concords.aspx

http://tonalsoft.com/monzo/aristoxenus/318tet.htm

(the last one is long and still very sloppy, in need
of editing ... commit to reading at your own risk ...)

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

Dave Keenan wrote, in reply to Jacques Dudon:
> > Consider the following representation :
> > a square with a basis = Log (4/3)
> > (and appropriate scaling for cents if you wish)
> > The square's diagonal passing by zero,
> > Two other lines passing by zero and points Log (8/7) and Log (7/6)
> > on the
> > upper side of the square.
> > parralal lines to the basis cut the square for many famous tetrachord
> > divisions,
> > and showing a total continuity between (from bottom to top)
> > enharmonics to
> > different slendros, passing by "hyperchromatics", chromatics, Hijaz,
> > Buzurg,
> > Bayati, Homalon, semidiatonics, diatonics, "hyperdiatonics", slendros...
> > Sometimes you have different modes, but all main greek and arab and more
> > tetrachords are there.
> > The Log (8/7) line shows the aliquote division of interval a + b
> > The Log (7/6) line shows the arithmetic division of interval a + b
>
> I'm sorry I couldn't decode this, but I would dearly love to
> understand, and to learn the meaning of these terms. Can you upload a
> file of such a diagram to the tuning_files Yahoo group?

Jacques, may I second this request? I felt there was _almost_
enough information there to draw the diagram, but not quite!
When you say "the square's diagonal passing by zero", do you
mean that the square is cented on the origin?

> > > The kind you are referring to, differ from
> > > more common diatonics in a dimension "at right angles" to the
> > > diatonic-chromatic-enharmonic dimension.
> > Sorry, I don't understand this sentance. Can you develop ? Where are the
> > right angles you're talking about ?
>
> I should have said, in a dimension _orthogonal_ to the
> diatonic-chromatic-enharmonic dimension, since it is actually shown at
> 60 degrees on Chalmers' diagram. I understand Aristoxenos'
> diatonic-chromatic-enharmonic dimension to be determined purely by the
> size of the _largest_ division. The other dimension I speak of is
> based on the size of the _smallest_ division.
>
> I have drawn a diagram and uploaded it here.
>
/tuning/files/Keenan/TetrachordCha
rt.doc
>
> I am sorry it is a Microsoft Word document.

Dave,
Don't be sorry! That's a masterly way of representing the
Byzantine tetrachords - a lot of information in very compact form.

> It is effectively a zoom-in on one sixth of Chalmers' diagram. I began
> drawing this when I was investigating the classification of Byzantine
> tetrachords by its own practitioners, which is why it is based on
> 72-EDO. The numbers are the sizes, in steps of 72-EDO, for each
> interval making up the tetrachord. They should be considered as not
> being in any particular order. Here's a conversion table for these steps.
>
> No. Cents Interval Approximated
> steps name ratios
> -----------------------------------------------------------
> 0 0 unison 1:1
> 1 17 twefthtone
> 2 33 sixthtone
> 3 50 quartertone
> 4 67 narrow subminor second 24:25 (27:28 25:26)
> 5 83 subminor second 20:21 (21:22)
> 6 100 narrow minor second 16:17 (17:18)
> 7 117 minor second 15:16 (14:15)

etc...

Dave, why wouldn't you call the 4-step interval a "thirdtone",
by analogy with the names of the three preceding intervals?
Perhaps the Byzantine musicians themselves spoke of a "narrow
subminor second"? Seems unlikely ...

...

> > example : 9/8 34/27 4/3 = because of a semitone smaller than 100
> > cents you
> > would call this "improper diatonic" ?
>
> Yes. And there are proper and improper chromatics too as you will see
> from my diagram. All enharmonics are improper, which makes it a little
> easier to see how Byzantine choirs could have made that transference
> of meaning with the term enharmonic. Although they can't have thought
> that all improper scales are enharmonic, since their "hard chromatics"
> are also improper.

The diagram certainly makes it easy to see that the names employ
categories on two dimensions.

--
No virus found in this outgoing message.
Checked by AVG Anti-Virus.
Version: 7.0.338 / Virus Database: 267.10.1/64 - Release Date: 4/8/05

🔗Jacques Dudon (AEH) <fotosonix@wanadoo.fr>

le 6/08/05 1:37, Dave Keenan à d.keenan@bigpond.net.au a écrit :

> --- In tuning@yahoogroups.com, "Jacques Dudon (AEH)" <fotosonix@w...>
> wrote:
>> - the extreme in the other direction, only from the point of wiew of
>> this precise representation (Chalmer's) ;
>
> Yes. But I don't see that there is any better way to represent audible
> similarity of tetrachords by nearness on a diagram.

Similarity perhaps, but my order is different, let me explain why :

To arrive to this - most logical - order, I reversed the usual greek order
of the diatonic tetrachord intervals, to place the diatonic semitone at the
end (C D E F instead of E F G A).
Then appears a simple rule that applies the same way to enharmonic,
chromatic, and diatonic genera :

A tetrachord is formed by one first division of a 4/3 interval into two
intervals any possible way, then by a second division, of the first of those
two intervals into two other sub-intervals.
When this second division is done into two sensibly equal intervals, this
defines most classical greek tetrachords (or their permutations).

We know it made sense, for the greeks and also the arabs and the persians,
to divide the distance between two ligatures of a string instrument in two
equal parts to obtain a note in the the middle.
That's how came the idea of my diagramm.

>> where enharmonic, chromatic, 3 equal parts, diatonic are also in
>> continuity and followed by these "diatonics with semitones ranging
>> from 17/16 to a comma" I call hyperdiatonics

> I can see that this is possible, even on Chalmer's diagram, when you
> take specific orderings of the intervals within the tetrachord. But I
> usually think of that as involving also what "mode" it is. I take
> diatonic, chromatic, enharmonic to be determined purely by the size of
> the largest interval, no matter where it is. But I am happy to learn
> from you.
I do to, the scales englobe the modes.

> I don't know what is meant by "semidiatonic". Please explain.

Chalmers refers to those, I think, as "neutral diatonics", and the three
typical examples are Rast, Mohajira, and Bayati, respectively using
4 3 3, 3 4 3, and 3 3 4 quartertones.

>> Consider the following representation :
>> a square with a basis = Log (4/3)
>> (and appropriate scaling for cents if you wish)
>> The square's diagonal passing by zero,
>> Two other lines passing by zero and points Log (8/7) and Log (7/6)
>> on the upper side of the square.
>> parralal lines to the basis cut the square for many famous tetrachord
>> divisions,
>> and showing a total continuity between (from bottom to top) enharmonics to
>> different slendros, passing by "hyperchromatics", chromatics, Hijaz,
>> Buzurg,
>> Bayati, Homalon, semidiatonics, diatonics, "hyperdiatonics", slendros...
>> Sometimes you have different modes, but all main greek and arab and more
>> tetrachords are there.
>> The Log (8/7) line shows the aliquote division of interval a + b
>> The Log (7/6) line shows the arithmetic division of interval a + b
We could also add : the line in the middle shows the geometric division of
interval a + b, where a = b

> I'm sorry I couldn't decode this, but I would dearly love to
> understand, and to learn the meaning of these terms. Can you upload a
> file of such a diagram to the tuning_files Yahoo group?

Sorry, I did'nt signed, and I don't have the time to do that at present.
This diagramm is part of a conference I gave last year that was intitled :
"The confusion of genera : attempts of resolutions", unfortunately in
french.
I know I should put it on our website, it's like that.
Our 5th "Noces harmoniques" festival approaches... later on may be.

----------------------------------------------------------------
Jacques Dudon
Atelier d'Exploration Harmonique - Les Camails 83340 LE THORONET - France
tel & fax 33 4 94 73 87 78
http://aeh.free.fr

🔗Jacques Dudon (AEH) <fotosonix@wanadoo.fr>

le 7/08/05 13:30, Yahya Abdal-Aziz à yahya@melbpc.org.au a écrit :

> Dave Keenan wrote, in reply to Jacques Dudon:
>>> Consider the following representation :
>>> a square with a basis = Log (4/3)
>>> (and appropriate scaling for cents if you wish)
>>> The square's diagonal passing by zero,
>>> Two other lines passing by zero and points Log (8/7) and Log (7/6)
>>> on the upper side of the square.
>>> parralal lines to the basis cut the square for many famous tetrachord
>>> divisions,
>>> and showing a total continuity between (from bottom to top)
>>> enharmonics to different slendros, passing by "hyperchromatics", chromatics,
>>> Hijaz, Buzurg, Bayati, Homalon, semidiatonics, diatonics, "hyperdiatonics",
slendros...
>>> Sometimes you have different modes, but all main greek and arab and more
>>> tetrachords are there.
>>> The Log (8/7) line shows the aliquote division of interval a + b
>>> The Log (7/6) line shows the arithmetic division of interval a + b
>>
>> I'm sorry I couldn't decode this, but I would dearly love to
>> understand, and to learn the meaning of these terms. Can you upload a
>> file of such a diagram to the tuning_files Yahoo group?
>
> Jacques, may I second this request? I felt there was _almost_
> enough information there to draw the diagram, but not quite!
> When you say "the square's diagonal passing by zero", do you
> mean that the square is centered on the origin?

No - sorry for the rough explanation, by zero (cents...) I meant the lower
corner, left side, of the square.
Now, you may cut the square by as many horizontal lines as you wish, giving
you logarithmic representations of many classical tetrachords
(If you do that with a 498 x 498 mm square, then 1 mm = 1 cent...)
Start up from the "zero point", with the logs of :
28/27 : Wilson hyperenharmonic
16/15 : Didymus enharmonic
10/9 : Erathosthenes chromatic
9/8 : Quintilianus chromatic
8/7 : Al-Farabi Hijaz (from 15/14)
7/6 : Safi al-dîn Buzurg
32/27 : Ibn Sina Bayati
6/5 : Ptolemee Homalon
11/9 : Vali scale (and Homalon transposition)
5/4 : Ptolemee Syntonon
4/3 : M & N slendros

you may find some more...

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:
/tuning/files/Keenan/TetrachordCha
> rt.doc
> >
> > I am sorry it is a Microsoft Word document.
>
> Dave,
> Don't be sorry! That's a masterly way of representing the
> Byzantine tetrachords - a lot of information in very compact form.

Thanks. But it's really John Chalmers' idea.

I haven't actually shown which ones are typical representatives of
Byzantine chant.

Rami Vitale's Byzantine superset scale was designed to contain the
following tetrachords (although he gave them in strictly rational
terms). I've given their intervals in descending order of size (in
steps of 72-EDO), for easy mapping to the diagram, but this is not
necessarily a permutation that is actually used.

12 11 7 diatonic (proper)
14 11 5 improper diatonic (Byzantine "enharmonic")
14 12 4 improper diatonic (Byzantine "enharmonic")
16 7 7 soft chromatic
19 7 4 hard chromatic

> > No. Cents Interval Approximated
> > steps name ratios
> > -----------------------------------------------------------
> > 0 0 unison 1:1
> > 1 17 twefthtone
> > 2 33 sixthtone
> > 3 50 quartertone
> > 4 67 narrow subminor second 24:25 (27:28 25:26)
> > 5 83 subminor second 20:21 (21:22)
> > 6 100 narrow minor second 16:17 (17:18)
> > 7 117 minor second 15:16 (14:15)
>
> etc...
>
> Dave, why wouldn't you call the 4-step interval a "thirdtone",
> by analogy with the names of the three preceding intervals?

You're absolutely right. I should have. Particularly in this context.
That goes back to at least Aristoxenos. You can call the 6 step
interval a semitone too.

> Perhaps the Byzantine musicians themselves spoke of a "narrow
> subminor second"? Seems unlikely ...

You're right, they did not. :-) I copied and pasted this table from
another document of mine, where the emphasis was on a consistent
naming system.

-- Dave Keenan

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
> Please, i beg you, discard the use of "hard" and "soft"
> as the English translations of the Greek "syntonon"
> and "malakon" respectively -- "tense" and "relaxed"
> are far preferable, as these terms stem directly from
> the theories of Aristoxenus, and his particular idiosyncracy
> was to eschew the use of any string *length* measurements,
> and to focus instead on the *tension* of strings to produce
> higher or lower pitches.

It is clear that you are correct in saying that these terms refer to
string tension, but cannot the terms "hard" and "soft" also refer to
string tension? I any case, I think it is a little too late. All the
literature in English I could find on Byzantine chant uses the terms
hard and soft for the two kinds of chromatic mode. It's all through
the Scala archive and modenam.par. I suspect you're flogging a dead
horse here. Sorry.

The other problem is that, in English at least, "tense" and "relaxed"
are more likely to be used about the emotional state of _people_ and I
wouldn't want people to get the idea that these modes are named after
the emotional state they induce. "hard" and "soft" seem neutral in
this regard.

> See my webpages:
>
> http://tonalsoft.com/enc/m/malakon.aspx
>
> http://tonalsoft.com/enc/s/syntonon.aspx
>
> http://tonalsoft.com/enc/t/tuning-by-concords.aspx
>
> http://tonalsoft.com/monzo/aristoxenus/318tet.htm
>
> (the last one is long and still very sloppy, in need
> of editing ... commit to reading at your own risk ...)

That last one is by far the most interesting to me. I'm still working
carefully through it. Thanks!

-- Dave Keenan

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning@yahoogroups.com, "Jacques Dudon (AEH)" <fotosonix@w...>
wrote:
> A tetrachord is formed by one first division of a 4/3 interval into two
> intervals any possible way, then by a second division, of the first
of those
> two intervals into two other sub-intervals.
> When this second division is done into two sensibly equal intervals,
this
> defines most classical greek tetrachords (or their permutations).
>
> We know it made sense, for the greeks and also the arabs and the
persians,
> to divide the distance between two ligatures of a string instrument
in two
> equal parts to obtain a note in the the middle.
> That's how came the idea of my diagramm.

OK. So yours is one-dimensional, and only classifies those tetrachords
that have two equal intervals? -- only 11 of the 48 sensibly
heptatonic ones on my diagram? I'll admit these are by far the most
common, but they are not all.

If your classification scheme is to be applied to tetrachords such as
6 10 14, 5 10 15, 4 10 16 (in twelfthtones) it appears it will give
paradoxical results.

Consider 6 10 14. It is equally distant from 7 7 16 (a soft chromatic)
and 4 13 13 (a "hyperdiatonic") so which category would you put it in
given that these two categories are not even adjacenty to one another,
having the category "diatonic" in between?

In Chalmer's system it would clearly be an improper soft chromatic.

But you do have a point here. The _typical_ diatonics are not the
neutral 9 9 12 or equal 10 10 10, but those which on my diagram are
"around the corner", and bordering on improper, 12 11 7, 12 12 6.

> > I don't know what is meant by "semidiatonic". Please explain.
>
> Chalmers refers to those, I think, as "neutral diatonics", and the three
> typical examples are Rast, Mohajira, and Bayati, respectively using
> 4 3 3, 3 4 3, and 3 3 4 quartertones.

Ah yes. I am familiar with them by the name neutral diatonics.

Jacques,

Could you possibly find time to give typical interval sizes in
approximate twelftones for the following, correcting those I've listed
as suggestions

enharmonics 3 3 24
slendros
"hyperchromatics" 4 4 22
chromatics 6 6 18
Hijaz
Buzurg
Bayati
Homalon
semidiatonics 9 9 12
diatonics 12 12 6
"hyperdiatonics" 13 13 4
slendros

I find it interesting that you have slendros in two places here.

-- Dave Keenan

🔗Dave Keenan <d.keenan@bigpond.net.au>

🔗Jacques Dudon (AEH) <fotosonix@wanadoo.fr>

le 8/08/05 3:18, Dave Keenan à d.keenan@bigpond.net.au a écrit :

> --- In tuning@yahoogroups.com, "Jacques Dudon (AEH)" <fotosonix@w...>
> wrote:
>> A tetrachord is formed by one first division of a 4/3 interval into two
>> intervals any possible way, then by a second division, of the first
>> of those two intervals into two other sub-intervals.
>> When this second division is done into two sensibly equal intervals,
>> this
>> defines most classical greek tetrachords (or their permutations).
>>
>> We know it made sense, for the greeks and also the arabs and the
>> persians,
>> to divide the distance between two ligatures of a string instrument
>> in two equal parts to obtain a note in the the middle.
>> That's how came the idea of my diagramm.
>
> OK. So yours is one-dimensional, and only classifies those tetrachords
> that have two equal intervals ?

well, not exactly equal - you have a collection of commas here...
And each horizontal line shows TWO tetrachords, depending on the choice
of the division, between "aliquote" or "arithmetic".

> only 11 of the 48 sensibly heptatonic ones on my diagram?
> I'll admit these are by far the most common, but they are not all.

Of course, I never thought of replacing Chalmers 2-D system anyway.
As I said, it is a proposition of classification. I should add a
one-dimensional classification, that I found very useful.

> If your classification scheme is to be applied to tetrachords such as
> 6 10 14, 5 10 15, 4 10 16 (in twelfthtones) it appears it will give
> paradoxical results.

Wait a minute, you need at least a second dimension here !
But you can also easily use my diagramm in a 2-D mode, simply by marking
with a dot where you want the division of (a + b) to be...

> Consider 6 10 14. It is equally distant from 7 7 16 (a soft chromatic)
> and 4 13 13 (a "hyperdiatonic") so which category would you put it in
> given that these two categories are not even adjacenty to one another,
> having the category "diatonic" in between?
>
> In Chalmer's system it would clearly be an improper soft chromatic.
>
> But you do have a point here. The _typical_ diatonics are not the
> neutral 9 9 12 or equal 10 10 10, but those which on my diagram are
> "around the corner", and bordering on improper, 12 11 7, 12 12 6.

We all love what's improper, don't we ?

> Jacques,
>
> Could you possibly find time to give typical interval sizes in
> approximate twelftones for the following, correcting those I've listed
> as suggestions
>
> enharmonics 3 3 24
> slendros
> "hyperchromatics" 4 4 22
> chromatics 6 6 18
> Hijaz
> Buzurg
> Bayati
> Homalon
> semidiatonics 9 9 12
> diatonics 12 12 6
> "hyperdiatonics" 13 13 4
> slendros
>
> I find it interesting that you have slendros in two places here.

A relation is well possible, but I did not place the slendros after
enharmonics,
only at the last end.

🔗Jacques Dudon (AEH) <fotosonix@wanadoo.fr>

le 8/08/05 3:33, Dave Keenan à d.keenan@bigpond.net.au a écrit :

> --- In tuning@yahoogroups.com, "Jacques Dudon (AEH)" <fotosonix@w...>
> wrote:
>> Now, you may cut the square by as many horizontal lines as you wish,
> giving
>> you logarithmic representations of many classical tetrachords
>> (If you do that with a 498 x 498 mm square, then 1 mm = 1 cent...)
>> Start up from the "zero point", with the logs of :
>> 28/27 : Wilson hyperenharmonic
>> 16/15 : Didymus enharmonic
>> 10/9 : Erathosthenes chromatic
>> 9/8 : Quintilianus chromatic
>> 8/7 : Al-Farabi Hijaz (from 15/14)
>> 7/6 : Safi al-dîn Buzurg
>> 32/27 : Ibn Sina Bayati
>> 6/5 : Ptolemee Homalon
>> 11/9 : Vali scale (and Homalon transposition)
>> 5/4 : Ptolemee Syntonon
>> 4/3 : M & N slendros
>>
>> you may find some more...
>
> Thanks for that. Please ignore my request re approximate twelfthtones.
> I'll translate below. Please correct me if I'm wrong.
>
> 2 2 26 : Wilson hyperenharmonic
> 3.5 3.5 23 : Didymus enharmonic
> 5.5 5.5 19 : Erathosthenes chromatic
> 6 6 18 : Quintilianus chromatic
> 7 7 16 : Al-Farabi Hijaz (from 15/14)
> 8 8 14 : Safi al-dîn Buzurg
> 9 9 12 : Ibn Sina Bayati
> 9.5 9.5 11 : Ptolemee Homalon
> 10.5 10.5 9 : Vali scale (and Homalon transposition)
> 11.5 11.5 7 : Ptolemee Syntonon
> ? ("hyperdiatonic")
> 15 15 0 : M & N slendros

I trust you to be right, but as I don't use equal divisions, I don't qualify
here to say if these would be good approximations.
It seems to me that Ptolemy Homalon (Chalmers 474) would rather be 9 10 11
and Ptolemy Syntonon 12 11 7 , etc.
But I have not seen your diagramm, which I am sure must be very interesting.
One suggestion I can make to join these two worlds, is on my diagramm, to
draw a grid of 30 (or 60) parralal vertical lines between zero and 498
cents, that is, every 16,6 or 8,3 cents - then look at what matches.

I see that in my list I didn't mention any "hyperdiatonic", as I call them
(properly or improperly ...) - line 9/7 up from 1/1 seems typical enough,
giving both
8/7 + 9/8 + 28/27 and 9/8 + 8/7 + 28/27 - which would be approached by
14 12 4 and 12 14 4

🔗Dave Keenan <d.keenan@bigpond.net.au>

🔗Jacques Dudon (AEH) <fotosonix@wanadoo.fr>

le 10/08/05 11:24, Dave Keenan à d.keenan@bigpond.net.au a écrit :

> Jacques,
>
> Here's my tetrachord diagram as a .GIF file on my own website, so you
> don't have to join the tuning_files yahoo group, or use Microsoft Word.
>
> http://dkeenan.com/Music/TetrachordChart1.GIF
>
> I believe I have understood your description of your diagram and I
> show what seems like the obvious generalisation to two dimensions, and
> why it fails, here:
>
> http://dkeenan.com/Music/TetrachordChart3.GIF
>
> And here I show a mapping, of the categories in two dimensions, that
> works:
>
> http://dkeenan.com/Music/TetrachordChart4.GIF
>
> As you will see, it corresponds to a simple geometric transformation
> of John Chalmers' diagram.
>
> -- Dave Keenan

Thanks Dave, now I can look at your work !

I already replied on your commentary on "TetrachordChart3" -
May be I can add : is there any better *uncomplete* ONE-dimensionnal way of
classification than mine ? ...
Anyway, the main interest of my diagramm, besides proposing a linear
classification (that can be completed), is in pointing the
aliquot/arithmetic divisions of many classical tetrachords. Thus it aplies
rather to a JI concept than to equal-divisions.
In the example you give of a non-fitting tetrachord (4+10+16 and
permutations), what are the three intervals that define 4, 10, and 16 ?
25/24, 11/10 and 7/6 ? then the product is not exactly 4/3 and I don't know
why I would want to fit this tetrachord in this aliquot/arithmetic system.
The only thing we can say is that the upper order (10+16+4 or 16+10+4) seems
closer to the aliquot/arithmetic divisions, so could be related in priority
to line 32/25 - may be.
I have to plunge deeper in "TetrachordChart4", but at a first glance I don't
understand why you place chromatics and enharmonics near the top of the
square, because of permutations may be ?

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning@yahoogroups.com, "Jacques Dudon (AEH)" <fotosonix@w...>
wrote:
> Thanks Dave, now I can look at your work !
>
> I already replied on your commentary on "TetrachordChart3" -
> May be I can add : is there any better *uncomplete* ONE-dimensionnal
way of
> classification than mine ? ...

Not that I know of. It is kind of neat the way it shows diatonics with
an LLs (Large Large small) permutation and chromatics and enharmonics
with an ssL permutation (the common forms of each) by keeping them in
a straight line. But other permutations are certainly used. For
example the Byzantine soft and hard chromatics are sLs, and their
"hyperdiatonics" occur as both LsL and sLL and their diatonics as LLs
and sLL.

> Anyway, the main interest of my diagramm, besides proposing a linear
> classification (that can be completed), is in pointing the
> aliquot/arithmetic divisions of many classical tetrachords. Thus it
aplies
> rather to a JI concept than to equal-divisions.

OK. But how much might this be due simply to the use of rational
arithmetic (which has been much simpler than using logarithmic
measures until fairly recently) in specifying what is intended merely
as a division into two approximately equal parts for purely melodic
(not harmonic) reasons?

> In the example you give of a non-fitting tetrachord (4+10+16 and
> permutations), what are the three intervals that define 4, 10, and 16 ?
> 25/24, 11/10 and 7/6 ? then the product is not exactly 4/3 and I
don't know
> why I would want to fit this tetrachord in this aliquot/arithmetic
system.

If you insist that it be specified with rational arithmetic, then it
could be 240/231, 11/10, 7/6.

25/24 is only one ratio that is approximated by 4 twelfthtones. 28/27
is another common one, but I decided to only list one representative
ratio for each step, on the diagram.

You could also consider the permutations of the examples
5 10 15 (104/99 11/10 15/13) or
3 10 16 (40/39 11/10 13/11)
which cause similar difficulties for a simple-minded extension of your
system to 2 dimensions.

I note that 104/99 is very close to 21/20.

> The only thing we can say is that the upper order (10+16+4 or
16+10+4) seems
> closer to the aliquot/arithmetic divisions, so could be related in
priority
> to line 32/25 - may be.

I'm not sure I understand the meaning of "aliquot" and "arithmetic" in
this context. Or the significance of them in classifying tetrachords.

> I have to plunge deeper in "TetrachordChart4", but at a first glance
I don't
> understand why you place chromatics and enharmonics near the top of the
> square, because of permutations may be ?

Yes. It is so that if any tetrachord is classified as chromatic, then
so are all its permutations. The same for enharmonic or diatonic. The
same for proper, improper or pentatonic. Presumably one then uses mode
names like Lydian, Myxolydian, Phrygian etc. to differentiate the
permutations, although I haven't got that part worked out yet. I
probably should just buy a copy of John Chalmers' book. :-)

-- Dave Keenan

🔗Dave Keenan <d.keenan@bigpond.net.au>

Jacques,

This may be of interest to you, if you don't already have it. I found
it while searching for something else.

Date: Fri, 2 Jun 95 4:41:30 PDT
From: "John H. Chalmers" <non12@cyber.net>
To: tuning
Subject: Superparticular tetrachord list
Message-ID: <9506020441.aa02376@cyber.cyber.net>

Here is the complete list of the 26 possible tetrachords
composed of superparticular ratios. The only intervals less
than 81/80 that occur are 256/255, 136/135, 100/99 and
96/95. This list was originally computed by I.E. Hoffman
and published by Prof. Vogel in Bonn.

1 5 / 4 x 17 / 16 x 256 / 255
2 5 / 4 x 18 / 17 x 136 / 135
3 5 / 4 x 19 / 18 x 96 / 95
4 5 / 4 x 20 / 19 x 76 / 75
5 5 / 4 x 21 / 20 x 64 / 63
6 5 / 4 x 22 / 21 x 56 / 55
7 5 / 4 x 24 / 23 x 46 / 45
8 5 / 4 x 26 / 25 x 40 / 39
9 5 / 4 x 28 / 27 x 36 / 35
10 5 / 4 x 31 / 30 x 32 / 31
11 6 / 5 x 11 / 10 x 100 / 99
12 6 / 5 x 12 / 11 x 55 / 54
13 6 / 5 x 13 / 12 x 40 / 39
14 6 / 5 x 15 / 14 x 28 / 27
15 6 / 5 x 16 / 15 x 25 / 24
16 6 / 5 x 19 / 18 x 20 / 19
17 7 / 6 x 9 / 8 x 64 / 63
18 7 / 6 x 10 / 9 x 36 / 35
19 7 / 6 x 12 / 11 x 22 / 21
20 7 / 6 x 15 / 14 x 16 / 15
21 8 / 7 x 8 / 7 x 49 / 48
22 8 / 7 x 9 / 8 x 28 / 27
23 8 / 7 x 10 / 9 x 21 / 20
24 8 / 7 x 13 / 12 x 14 / 13
25 9 / 8 x 10 / 9 x 16 / 15
26 10 / 9 x 11 / 10 x 12 / 11

--John

> > rather to a JI concept than to equal-divisions.
>
> OK. But how much might this be due simply to the use of rational
> arithmetic (which has been much simpler than using logarithmic
> measures until fairly recently) in specifying what is intended merely
> as a division into two approximately equal parts for purely melodic
> (not harmonic) reasons?
>
> > In the example you give of a non-fitting tetrachord (4+10+16 and
> > permutations), what are the three intervals that define 4, 10, and
16 ?
> > 25/24, 11/10 and 7/6 ? then the product is not exactly 4/3 and I
> don't know
> > why I would want to fit this tetrachord in this aliquot/arithmetic
> system.
>
> If you insist that it be specified with rational arithmetic, then it
> could be 240/231, 11/10, 7/6.
>
> 25/24 is only one ratio that is approximated by 4 twelfthtones. 28/27
> is another common one, but I decided to only list one representative
> ratio for each step, on the diagram.
>
> You could also consider the permutations of the examples
> 5 10 15 (104/99 11/10 15/13) or
> 3 10 16 (40/39 11/10 13/11)
> which cause similar difficulties for a simple-minded extension of your
> system to 2 dimensions.
>
> I note that 104/99 is very close to 21/20.
>
> > The only thing we can say is that the upper order (10+16+4 or
> 16+10+4) seems
> > closer to the aliquot/arithmetic divisions, so could be related in
> priority
> > to line 32/25 - may be.
>
> I'm not sure I understand the meaning of "aliquot" and "arithmetic" in
> this context. Or the significance of them in classifying tetrachords.
>
> > I have to plunge deeper in "TetrachordChart4", but at a first glance
> I don't
> > understand why you place chromatics and enharmonics near the top
of the
> > square, because of permutations may be ?
>
> Yes. It is so that if any tetrachord is classified as chromatic, then
> so are all its permutations. The same for enharmonic or diatonic. The
> same for proper, improper or pentatonic. Presumably one then uses mode
> names like Lydian, Myxolydian, Phrygian etc. to differentiate the
> permutations, although I haven't got that part worked out yet. I
> probably should just buy a copy of John Chalmers' book. :-)
>
> -- Dave Keenan

🔗Jacques Dudon (AEH) <fotosonix@wanadoo.fr>

First I want to apologize for an enormous stupidity I wrote
in the description of my tetrachords diagramm :

>> Two other lines passing by zero and points Log (8/7) and Log (7/6) on the
>> upper side of the square.

I just forgot to say that these were not straight lines !!...
The central comma of the a*b interval has a smaller proportion, in interval
dimension, with a*b 's dimension when it goes down towards the base of the
square, when a*b gets smaller.
ex: 49/48, the largest comma, measures about 1/14 of 4/3
while 64/63 (Diatonon Malakon) measures about 1/16 of 9/7
while 121/120 (Ptolemy Homalon) measures about 1/22 of 6/5 ... and so on
As a result the aliquot and arithmetic division lines look slightly
incurved, like the begining of a horn.
I could have done it non-logarithmic, for example showing the real ligature
divisions for one same string length - then the aliquote line would have be
a straigth line ; or in hertz, then the arithmetical division line would
have be a straigth line - but in a logarithmic representation, both are
slighty curved.
My excuses for those who did a drawing out of my explanations !
May be calculations need not to be done at every line, but at least at 3 or
4 heights (especially in the middle and higher parts) to give an idea of the
curves. Sorry about that.
My thanks to Dave, who did make me think more precisely of it...

le 10/08/05 16:15, Dave Keenan à d.keenan@bigpond.net.au a écrit :

> --- In tuning@yahoogroups.com, "Jacques Dudon (AEH)" <fotosonix@w...>
> wrote:
>> Thanks Dave, now I can look at your work !
>>
>> I already replied on your commentary on "TetrachordChart3" -
>> May be I can add : is there any better *uncomplete* ONE-dimensionnal
>> way of classification than mine ? ...
>
> Not that I know of. It is kind of neat the way it shows diatonics with
> an LLs (Large Large small) permutation and chromatics and enharmonics
> with an ssL permutation (the common forms of each) by keeping them in
> a straight line. But other permutations are certainly used. For
> example the Byzantine soft and hard chromatics are sLs, and their
> "hyperdiatonics" occur as both LsL and sLL and their diatonics as LLs
> and sLL.
definitively.

>> Anyway, the main interest of my diagramm, besides proposing a linear
>> classification (that can be completed), is in pointing the
>> aliquot/arithmetic divisions of many classical tetrachords. Thus it
>> aplies rather to a JI concept than to equal-divisions.
>
> OK. But how much might this be due simply to the use of rational
> arithmetic (which has been much simpler than using logarithmic
> measures until fairly recently) in specifying what is intended merely
> as a division into two approximately equal parts for purely melodic
> (not harmonic) reasons ?

Who knows, but it has been pointed that scholars were only trying, with such
methods as ratios or aliquot divisions on a monochord, to reproduce what
they were hearing in popular music. Personnally, I don't see why melodic
lines would be "approximate" or of "approximately equal" intonations either.

>> In the example you give of a non-fitting tetrachord (4+10+16 and
>> permutations), what are the three intervals that define 4, 10, and 16 ?
>> 25/24, 11/10 and 7/6 ? then the product is not exactly 4/3 and I
>> don't know
>> why I would want to fit this tetrachord in this aliquot/arithmetic
>> system.
>
> If you insist that it be specified with rational arithmetic, then it
> could be 240/231, 11/10, 7/6.
ahhhh ! my *archaïc* rational instincts are satisfied now... :-)

> 25/24 is only one ratio that is approximated by 4 twelfthtones. 28/27
> is another common one, but I decided to only list one representative
> ratio for each step, on the diagram.
sure.
I noticed degrees 1, 2, 3, 29 are left blank, may be by lack of space for
the digits, or because of too much choices ? but could not you risk to
choose the more common intervals you find between all others, or was it too
complex ?

> You could also consider the permutations of the examples
> 5 10 15 (104/99 11/10 15/13) or
> 3 10 16 (40/39 11/10 13/11)
> which cause similar difficulties for a simple-minded extension of your
> system to 2 dimensions.

absolutely.

> I note that 104/99 is very close to 21/20.
a lovely 2080/2079 13-limit skisma

>> The only thing we can say is that the upper order (10+16+4 or
>> 16+10+4) seems
>> closer to the aliquot/arithmetic divisions, so could be related in
>> priority to line 32/25 - may be.
>
> I'm not sure I understand the meaning of "aliquot" and "arithmetic" in
> this context. Or the significance of them in classifying tetrachords.
In this context it has not much meaning. I was just figuring out the shorter
distance to the aliquot or arithmetic lines, since this is the logic of my
diagramm - for each of your three solutions. So if a choice had to be done,
this could have been one.
For rather harmonic reasons, if you had said the tetrachord was 25/24,
192/175, 7/6, I would have ranged it also in line 32/25. Now if you say that
the three intervals are 240/231, 11/10, 7/6, I am even more embarrassed...
And I will not try to range a contrabass in a violin box.

The significance of these aliquot and arithmetic divisions in classifying
tetrachords, is only based on the observation that a significant number of
tetrachords mentionned by the ancients share this property to show an
aliquot or arithmetic division somewhere between consecutive intervals.
What I just did, for pedagogical purpose, is to put those in order along a
vertical axis (the vertical sides of the square do not intend to be 1/1 and
all four notes can serve as reference notes).
Now I am quite satisfied to have BOTH aliquot and arithmetic divisions and
the resulting commas on each tetrachord line, because these precise
permutations and the modulations they offer have strong musical sense.
This is how you pass "through the mirror" from harmonics to subharmonics, or
from a minor to a major mode, see for example how a simple 9/8 > 10/9
(81/80) mutation on the 5/4 line reverses the scale, and changes the tonic
from C major to A minor, etc.

>> I have to plunge deeper in "TetrachordChart4", but at a first glance
>> I don't
>> understand why you place chromatics and enharmonics near the top of the
>> square, because of permutations may be ?
>
> Yes. It is so that if any tetrachord is classified as chromatic, then
> so are all its permutations. The same for enharmonic or diatonic. The
> same for proper, improper or pentatonic. Presumably one then uses mode
> names like Lydian, Myxolydian, Phrygian etc. to differentiate the
> permutations, although I haven't got that part worked out yet. I
> probably should just buy a copy of John Chalmers' book. :-)
>
> -- Dave Keenan

Congratulations, if you don't even have John Chalmers' book, to be able to
explain the extension of my diagramm into his system like you did. Certainly
everything is already there, in the "Divisions of the tetrachord", an
incredible research work and sum of ressources.

🔗Jacques Dudon (AEH) <fotosonix@wanadoo.fr>

le 10/08/05 16:29, Dave Keenan à d.keenan@bigpond.net.au a écrit :

> Jacques,
>
> This may be of interest to you, if you don't already have it. I found
> it while searching for something else.
>
> Date: Fri, 2 Jun 95 4:41:30 PDT
> From: "John H. Chalmers" <non12@cyber.net>
> To: tuning
> Subject: Superparticular tetrachord list
> Message-ID: <9506020441.aa02376@cyber.cyber.net>
>
> Here is the complete list of the 26 possible tetrachords
> composed of superparticular ratios. The only intervals less
> than 81/80 that occur are 256/255, 136/135, 100/99 and
> 96/95. This list was originally computed by I.E. Hoffman
> and published by Prof. Vogel in Bonn.
>
> 1 5 / 4 x 17 / 16 x 256 / 255
> 2 5 / 4 x 18 / 17 x 136 / 135
> 3 5 / 4 x 19 / 18 x 96 / 95
> 4 5 / 4 x 20 / 19 x 76 / 75
> 5 5 / 4 x 21 / 20 x 64 / 63
> 6 5 / 4 x 22 / 21 x 56 / 55
> 7 5 / 4 x 24 / 23 x 46 / 45
> 8 5 / 4 x 26 / 25 x 40 / 39
> 9 5 / 4 x 28 / 27 x 36 / 35
> 10 5 / 4 x 31 / 30 x 32 / 31
> 11 6 / 5 x 11 / 10 x 100 / 99
> 12 6 / 5 x 12 / 11 x 55 / 54
> 13 6 / 5 x 13 / 12 x 40 / 39
> 14 6 / 5 x 15 / 14 x 28 / 27
> 15 6 / 5 x 16 / 15 x 25 / 24
> 16 6 / 5 x 19 / 18 x 20 / 19
> 17 7 / 6 x 9 / 8 x 64 / 63
> 18 7 / 6 x 10 / 9 x 36 / 35
> 19 7 / 6 x 12 / 11 x 22 / 21
> 20 7 / 6 x 15 / 14 x 16 / 15
> 21 8 / 7 x 8 / 7 x 49 / 48
> 22 8 / 7 x 9 / 8 x 28 / 27
> 23 8 / 7 x 10 / 9 x 21 / 20
> 24 8 / 7 x 13 / 12 x 14 / 13
> 25 9 / 8 x 10 / 9 x 16 / 15
> 26 10 / 9 x 11 / 10 x 12 / 11
>
> --John

Oh yes, I love that ! thanks. Of course many arithmetic divisions -
Glad also to see a few high primes (17, 19, 23, 31) are present.

Microtonal meeting in France

🔗Jacques Dudon (AEH) <fotosonix@wanadoo.fr>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

🔗Jacques Dudon (AEH) <fotosonix@wanadoo.fr>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗Jacques Dudon (AEH) <fotosonix@wanadoo.fr>

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

🔗Dave Keenan <d.keenan@bigpond.net.au>

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

🔗Dave Keenan <d.keenan@bigpond.net.au>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗Dave Keenan <d.keenan@bigpond.net.au>

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

🔗Jacques Dudon (AEH) <fotosonix@wanadoo.fr>

🔗Dave Keenan <d.keenan@bigpond.net.au>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗Afmmjr@aol.com

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Kurt Bigler <kkb@breathsense.com>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Kurt Bigler <kkb@breathsense.com>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Afmmjr@aol.com

🔗Ozan Yarman <ozanyarman@superonline.com>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗Jacques Dudon (AEH) <fotosonix@wanadoo.fr>

🔗Dave Keenan <d.keenan@bigpond.net.au>

🔗Jacques Dudon (AEH) <fotosonix@wanadoo.fr>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗Jacques Dudon (AEH) <fotosonix@wanadoo.fr>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗Dave Keenan <d.keenan@bigpond.net.au>

🔗monz <monz@tonalsoft.com>

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

🔗Jacques Dudon (AEH) <fotosonix@wanadoo.fr>

🔗Jacques Dudon (AEH) <fotosonix@wanadoo.fr>

🔗Dave Keenan <d.keenan@bigpond.net.au>

🔗Dave Keenan <d.keenan@bigpond.net.au>

🔗Dave Keenan <d.keenan@bigpond.net.au>

🔗Dave Keenan <d.keenan@bigpond.net.au>

🔗Jacques Dudon (AEH) <fotosonix@wanadoo.fr>

🔗Jacques Dudon (AEH) <fotosonix@wanadoo.fr>

🔗Dave Keenan <d.keenan@bigpond.net.au>

🔗Dave Keenan <d.keenan@bigpond.net.au>

🔗Jacques Dudon (AEH) <fotosonix@wanadoo.fr>

🔗Jacques Dudon (AEH) <fotosonix@wanadoo.fr>

🔗Dave Keenan <d.keenan@bigpond.net.au>

🔗Dave Keenan <d.keenan@bigpond.net.au>

🔗Jacques Dudon (AEH) <fotosonix@wanadoo.fr>

🔗Jacques Dudon (AEH) <fotosonix@wanadoo.fr>

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

🔗Keenan Pepper <keenanpepper@gmail.com>

🔗Carl Lumma <clumma@yahoo.com>