Pioneers of the regular mapping paradigm

I've put together a list of who I think were the _major_
players in putting together the 'regular mapping paradigm'
as I see it:

Leonhard Euler (1707-1783)
Hermann von Helmholtz (1821-1894)
Robert Holford Macdowell Bosanquet (1841–1913)
Shohe Tanaka (1862-1945)
Adriaan Daniel Fokker (1887-1972)
Harry Partch (1901-1974)
Erv Wilson (1928- )

I've left two members here off the end to avoid being
tacky.

I'm not sure if Augosto Navarro should be added... I have
no information on him at all, other than hearsay that he
pioneered the tonality diamond.

I'm not sure if Shohe Tanaka should be removed. I have
only his Wikipedia entry (vs. that of Hugo Riemann) to
go on.

Thoughts?

-Carl

Hello Carl`
you can see some of novaro here http://anaphoria.com/novaro27natural.pdf and here http://anaphoria.com/novaro27appox.pdf
--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/index.html>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main/index.asp> 88.9 FM Wed 8-9 pm Los Angeles

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--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>_major_ players in putting together the 'regular mapping paradigm'
>
> Leonhard Euler (1707-1783)
http://www.pricomm.at/files/vortragproftaschner.pdf
"Bereits vor Euler hat der Musiktheoretiker Conrad Henfling dieses
Tonnetz. entworfen."
tr:
'Already before Euler the musicologist Conrad Henfling
had drafted the tonal-(diamond)-grid.'

Pardon, excuse my french:
http://umb-www-01.u-strasbg.fr/lexis/html/cinscription/Leibniz.html
" CORRESPONDANCE AVEC CONRAD HENFLING
Conrad Henfling (1648-1716) a été fonctionnaire à la cour du Margrave
de Ansbach, puis conseiller aulique (Hofrat). Il a été mis en relation
avec Leibniz par la princesse Caroline de Ansbach, plus tard reine
d'Angleterre. L'oeuvre musicologique de Henfling était encore connue
vers 1740, mais visiblement personne ne l'avait réellement lue et elle
finit par tomber dans l'oubli......"

> Hermann von Helmholtz (1821-1894)
> Robert Holford Macdowell Bosanquet (1841–1913)
> Shohe Tanaka (1862-1945)
> Adriaan Daniel Fokker (1887-1972)
> Harry Partch (1901-1974)
> Erv Wilson (1928- )
>
> Thoughts?
How about including some of that guys:

12:
http://en.wikipedia.org/wiki/Zhu_Zaiyu
http://en.wikipedia.org/wiki/Simon_Stevin#Music_theory
1585, simultaneously with, and independently of, Chu Tsai-Yu in China)
to give a mathematically accurate specification for equal temperament.

53:
http://en.wikipedia.org/wiki/Jing_Fang
http://en.wikipedia.org/wiki/53_equal_temperament#History
http://mto.societymusictheory.org/issues/mto.93.0.3/mto.93.0.3.lindley7.gif

612:
http://www.bach-cantatas.com/Topics/Genius.htm
"Newton played pretty well the viola, and invented for instance in
tuning theory the 612-division of the octave.
Lit: new Grove 2nd Ed. Vol.17 p.815-4"

665:
http://de.wikipedia.org/wiki/Moritz_Wilhelm_Drobisch
rediscovered the
http://www.xs4all.nl/~huygensf/doc/measures.html
"Delfi unit: 1/665 part of an octave
Used in Byzantine music theory? Approximately 1/12 part of the
syntonic comma and 1/13 part of the Pythagorean comma."

so far about some of my favourite classical authors
in "regular mappings" that didn't care about

A.S.

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
> Hello Carl`
> you can see some of novaro here
http://anaphoria.com/novaro27natural.pdf

Thanks, I'll look at that!

> and here http://anaphoria.com/novaro27appox.pdf

I'm getting 'not found' for that.

-Carl

I wrote...

> http://anaphoria.com/novaro27natural.pdf
>
> Thanks, I'll look at that!

We basically get the 7-limit diamond, but not quite as
such. He writes it in a strange way, then in the text
says, "Changing the relatives 5/4, 3/2 and 7/4 into
absolutes..." and writes the intervals of the 7-limit
diamond out on a line!

At any rate, he's using it to show all the 7-limit
intervals, not as a template for scales and modulations
as Partch did.

He recommends 53-ET for the 5-limit, and 72-ET for the
7-limit, and the 7-limit generally as having what it takes
for the 'full color' of music.

It's dated 1927. Checking monz's list...

http://tonalsoft.com/enc/e/equal-temperament.aspx

...we see that several people beat him to 53.

Hey monz - I'm a little confused by the standing of some
of the comments at the top of the boxes. For example,
"mercator". Did you mean to eventually put in Gerardus
Mercator (1512-1594)?
Oh, and didn't Gene say Newton was hip to 53?

As another aside, you can put me in for 41 (1998). It's
the first ET that's both consistent and unique in the
11-limit. Probably I'm not the first person to notice
that, but I'm not quite aware of who noticed it earlier.
Also, 46 (2007). It's part of a series of ETs of
decreasing 17-limit TOP badness that goes: 2, 7, 9, 10,
19, 22, 26, 31, 46, 72.

Yes, so to 72. I note the list isn't in chrono order.
There's Haba also in '27! Oh, and there's Novaro in
1951. Well, you can update that.

Overall, I'd like to see more Novaro, and he was definitely
well put-together for 1927. But I'm not seeing him on the
list just yet. I do think I'll add Vicentino, though.

-Carl

Carl Lumma wrote:

> As another aside, you can put me in for 41 (1998). It's
> the first ET that's both consistent and unique in the
> 11-limit. Probably I'm not the first person to notice
> that, but I'm not quite aware of who noticed it earlier.
> Also, 46 (2007). It's part of a series of ETs of
> decreasing 17-limit TOP badness that goes: 2, 7, 9, 10,
> 19, 22, 26, 31, 46, 72.

What about 11:10 and 10:9?

58 is the smallest number of notes that can be unique and properly ordered in the 11-limit (no need to enforce consistency). That's why I put that JI scale on the old wiki.

Graham

I wrote...

> For example,
> "mercator". Did you mean to eventually put in Gerardus
> Mercator (1512-1594)?

D'oh. Got me. It was Nicholas Mercator, (1642-1648).

-C.

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
> Carl Lumma wrote:
>
> > As another aside, you can put me in for 41 (1998). It's
> > the first ET that's both consistent and unique in the
> > 11-limit. Probably I'm not the first person to notice
> > that, but I'm not quite aware of who noticed it earlier.
> > Also, 46 (2007). It's part of a series of ETs of
> > decreasing 17-limit TOP badness that goes: 2, 7, 9, 10,
> > 19, 22, 26, 31, 46, 72.
>
> What about 11:10 and 10:9?

You know, I think I thought as long as it could do 11:10
and 12:11, it was OK. Drat.

Uh monz, don't put me down for 41. :)

> 58 is the smallest number of notes that can be unique and
> properly ordered in the 11-limit

Is there an obvious reason that should be obvious?

-Carl

Carl Lumma wrote:

> You know, I think I thought as long as it could do 11:10
> and 12:11, it was OK. Drat.

That's why Partch needed to add 2 notes to his 41 note periodicity block. And the fact that he nearly had a periodicity block shows you why he was an important pioneer.

> Uh monz, don't put me down for 41. :)

How about the 9-limit?

>>58 is the smallest number of notes that can be unique and >>properly ordered in the 11-limit
> > Is there an obvious reason that should be obvious?

No, not at all. Which is why I like too remind people of it. (It implies a consistent mapping, of course.) If Partch was trying to get a periodicity block, it would have taken him another 15 notes for it to work in a purist fashion.

Graham

Andreas wrote...

> > Leonhard Euler (1707-1783)
> http://www.pricomm.at/files/vortragproftaschner.pdf

I only read English. :(

> 'Already before Euler the musicologist Conrad Henfling
> had drafted the tonal-(diamond)-grid.'
>
> Pardon, excuse my french:

I can only read English. :(:(

> Conrad Henfling (1648-1716)

Can't find info on this cat.

> > Thoughts?
> How about including some of that guys:
>
> 12:
> http://en.wikipedia.org/wiki/Zhu_Zaiyu
> http://en.wikipedia.org/wiki/Simon_Stevin#Music_theory

Interesting. But still too formative to add directly
to the theory.

http://mto.societymusictheory.org/issues/mto.93.0.3/mto.93.0.3.lindley
7.gif

Holy crap! That looks like a horogram! What paper at mto
was this attached to? Wikipedia is currently asking for a
citation for Newton and 53.

> so far about some of my favourite classical authors
> in "regular mappings" that didn't care about

The citations you have haven't convinced me. Calculating
good equal divisions is one thing, but well, here's why I
put each person on the list:

Nicola Vicentino (1511–1576)
....adaptive JI, 31-ET
Leonhard Euler (1707-1783)
....pitch space (rectangular)
Hermann von Helmholtz (1821-1894)
....psychoacoustics
RHM Bosanquet (1841–1913)
....regular mapping, generalized keyboard
Shohe Tanaka (1862-1945)
.....pitch space (triangular)
Adriaan Daniel Fokker (1887-1972)
....periodicity blocks
Harry Partch (1901-1974)
....extended JI (11-limit)
Erv Wilson (1928- )
....pitch space (extended), keyboard mapping, scale theory (MOS)

Maybe Tanaka should even be taken off. I clearly had
forgot Vicentino, though.

-Carl

> > Uh monz, don't put me down for 41. :)
>
> How about the 9-limit?

You know actually, I think you're right. I was really
aware of the problem of fitting Partch's scale, because
I had just read Erv's article.
I'd have to check my notes to be sure, but I think I
tossed the (blue spiral) notebook in '05 after I got
married.
Looking over old e-mails...

June 1999 (from me to you)
"41tET is schismic and is consistent and unique at the 9-limit"

Well whatever, it's not terribly important.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> I've put together a list of who I think were the _major_
> players in putting together the 'regular mapping paradigm'
> as I see it:
>
> Leonhard Euler (1707-1783)
> Hermann von Helmholtz (1821-1894)
> Robert Holford Macdowell Bosanquet (1841–1913)
> Shohe Tanaka (1862-1945)
> Adriaan Daniel Fokker (1887-1972)
> Harry Partch (1901-1974)
> Erv Wilson (1928- )

Nice.

> I've left two members here off the end to avoid being
> tacky.

Probably a good idea, but still it leaves me curious.

> I'm not sure if Augosto Navarro should be added... I have
> no information on him at all, other than hearsay that he
> pioneered the tonality diamond.

I think Kraig might have some stuff up about Navarro (sp?)
but i too have never seen his actual work.

> I'm not sure if Shohe Tanaka should be removed. I have
> only his Wikipedia entry (vs. that of Hugo Riemann) to
> go on.
>
> Thoughts?

I talk about Tanaka's use of the kleisma as a unison-vector
and promo of 53-edo, in this page:

http://tonalsoft.com/enc/k/kleisma.aspx

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

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

> The citations you have haven't convinced me. Calculating
> good equal divisions is one thing, but well, here's why I
> put each person on the list:
>
> Nicola Vicentino (1511–1576)
> ....adaptive JI, 31-ET
> Leonhard Euler (1707-1783)
> ....pitch space (rectangular)
> Hermann von Helmholtz (1821-1894)
> ....psychoacoustics
> RHM Bosanquet (1841–1913)
> ....regular mapping, generalized keyboard
> Shohe Tanaka (1862-1945)
> .....pitch space (triangular)
> Adriaan Daniel Fokker (1887-1972)
> ....periodicity blocks
> Harry Partch (1901-1974)
> ....extended JI (11-limit)
> Erv Wilson (1928- )
> ....pitch space (extended), keyboard mapping, scale theory (MOS)
>
> Maybe Tanaka should even be taken off. I clearly had
> forgot Vicentino, though.

Tanaka is definitely important. His 1890 paper clearly
made use of the unison-vector and periodicity-block
concepts, decades before Fokker.

I'd have to read Bosanquet more deeply to see if
he did the same, before Tanaka. I believe that his
keyboard designs exhibit the same concepts.

Hugo Riemann should also be mentioned. He designed
a lattice in 5-limit prime-space which he called
the _Tonnetz_.

It was Tanaka who showed how unison-vectors of
skhisma and kleisma tile Riemann's space in 53-edo.

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

> > I'm not sure if Shohe Tanaka should be removed. I have
> > only his Wikipedia entry (vs. that of Hugo Riemann) to
> > go on.
> >
> > Thoughts?
>
> I talk about Tanaka's use of the kleisma as a unison-vector
> and promo of 53-edo, in this page:
>
> http://tonalsoft.com/enc/k/kleisma.aspx

Well, it looks like he shouldn't be removed! Do you know of
an earlier instance of a > 1-D periodicity block?

-Carl

monz wrote...

> Tanaka is definitely important. His 1890 paper clearly
> made use of the unison-vector and periodicity-block
> concepts, decades before Fokker.

Apparently.

> I'd have to read Bosanquet more deeply to see if
> he did the same, before Tanaka. I believe that his
> keyboard designs exhibit the same concepts.

My recollection is that he came at it strictly from
a linear point of view. He was big on the schisma,
and therefore on 53-ET. I recall him using multiple
commas to close the 5-limit.

> Hugo Riemann should also be mentioned. He designed
> a lattice in 5-limit prime-space which he called
> the _Tonnetz_.
>
> It was Tanaka who showed how unison-vectors of
> skhisma and kleisma tile Riemann's space in 53-edo.

I thought about Riemann, but it looks like Tanaka made
the bigger contribution and since they were
contemporaries, Riemann has no advantage. At least,
Wikipedia makes it sound like they were working in
isolation, and I assume the only reason Riemann's name
is now so popular is that he was European.

-Carl

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Hi Mikal,

--- In tuning@yahoogroups.com, "mikal haley"
<chipsterthehipster@...> wrote:

> recently i read about a quark in deep space picked
> up by Hubble or something like that

The data was collected by the Chandra X-Ray Observatory
in 2003.

> emanates a tone at fifty-seven octaves below
> middle 'c'.
>
> what would that be in terms of A=440?
>
> i have a terrible math block, this is why I am asking.

The articles refer to it as "a B-flat 57 octaves below
the middle keys of the piano". Using the usual 12-edo
semitone as the measurement from A=440 Hz to B-flat,
the calculation is as follows:

2^(1/12) * 440 = ~233.08 Hz for the B-flat

2^-57 * 233.08 = ~0.00000000000000161733 Hz for the low note

It makes more sense to write that low frequency in
scientific notation: ~1.62 * 10^-15 Hz, or as it would
show up in a spreadsheet or on a calculator: 1.62E-15.

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

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If you're interested in this kinda info. see this page:

http://www.lucytune.com/academic/freq_to_wave.html

and you will find the answer to your query by consulting the diagram.

http://www.lucytune.com/academic/freq_to_wave.html

Charles Lucy lucy@lucytune.com

----- Promoting global harmony through LucyTuning -----

For information on LucyTuning go to: http://www.lucytune.com

LucyTuned Lullabies (from around the world):
http://www.lullabies.co.uk

Skype user = lucytune

http://www.myspace.com/lucytuning

On 25 Sep 2007, at 13:07, mikal haley wrote:

> i am so glad to have to say this: this is basically my first post!
> first topic i have started, because i don't read music. i play by ear
> but have been in this awesome group for three years at least.
>
> recently i read about a quark in deep space picked up by Hubble
> or something like that emanates a tone at fifty-seven octaves below
> middle 'c'.
>
> what would that be in terms of A=440?
>
> i have a terrible math block, this is why I am asking.
>
> thx.
> moderator,
> casiokeyboards@yahoo groups
>
>

On 9/25/07, monz <monz@tonalsoft.com> wrote:
> It makes more sense to write that low frequency in
> scientific notation: ~1.62 * 10^-15 Hz, or as it would
> show up in a spreadsheet or on a calculator: 1.62E-15.

I think it would make more sense to talk about the period, which is

1 / (1.62e-15 Hz) = 19.6 million years

How did they measure this frequency if they can only observe it for a
tiny fraction of one period? I guess they must have observed the
spatial wavelength and either theoretically calculated the propagation
speed or observed it some other way.

Keenan

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

http://mto.societymusictheory.org/issues/mto.93.0.3/mto.93.0.3.lindle
y
> 7.gif
>
> Holy crap! That looks like a horogram! What paper at mto
> was this attached to? Wikipedia is currently asking for a
> citation for Newton and 53.

It was me who responded to Andreas regarding Newton
and 53-edo:

/tuning/topicId_71935.html#72147

The only reference i could dig up was the Lindley article
from which that diagram was taken. I searched the web and
found nothing else substantial about Newton's writings
on music.

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

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> > > I'm not sure if Shohe Tanaka should be removed. I have
> > > only his Wikipedia entry (vs. that of Hugo Riemann) to
> > > go on.
> > >
> > > Thoughts?
> >
> > I talk about Tanaka's use of the kleisma as a unison-vector
> > and promo of 53-edo, in this page:
> >
> > http://tonalsoft.com/enc/k/kleisma.aspx
>
> Well, it looks like he shouldn't be removed! Do you know of
> an earlier instance of a > 1-D periodicity block?

No. As i state in my Encyclopedia page

http://tonalsoft.com/enc/p/periodicity-block.aspx

it appears that Tanaka was the first to describe and
illustrate the concept.

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

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
> > Wikipedia is currently asking for a
> > citation for Newton and 53.
//
> The only reference i could dig up was the Lindley article
> from which that diagram was taken.

Do you know of a URL for this article?

-Carl

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
> > Do you know of
> > an earlier instance of a > 1-D periodicity block?
>
> No. As i state in my Encyclopedia page
>
> http://tonalsoft.com/enc/p/periodicity-block.aspx
>
> it appears that Tanaka was the first to describe and
> illustrate the concept.

Thanks!

Do you know what commas K and S are here?

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
Dear Carl,
> Andreas wrote, when quoting:
> > http://www.pricomm.at/files/vortragproftaschner.pdf
>
> > 'Already before Euler the musicologist Conrad Henfling
> > had drafted the tonal-(diamond)-grid.'
> >
> > Conrad Henfling (1648-1716)
labeled his invention as "Tonnetz"
appearently already before Euler, Helmholtz, Riemann, Tanaka & ct....
Probably J.S.Bach overtook that from Henfling
as he shows that in his 3rd-5th diamond lattice-grid contained his
official heraldic seal:
http://f9g.yahoofs.com/groups/g_16806072/.HomePage/__sr_/d3ca.jpg?grAKv.GBUZJ9e3qx
http://launch.ph.groups.yahoo.com/group/bach_tunings/photos/view/f32e?b=1&m=f&o=0
attend there the german nomenclature of labeling note-names different:

1. in modern engl. the old 'round' B (lat: rotundum) became "Bb"
2. in modern german the old 'quadratic' B[lat: quadratum]became "H".

Werckmeister explains that intrinsically definition as:

H(15/8):= b# = (round_B)-sharp

in order to get rid of the cumbersome former distinction
inbetween 'round' and 'quadratic' two types of Bs.

From that arises sometimes even still today confusion
about the intened pitch as notated by inept translation.
>
> > http://en.wikipedia.org/wiki/Simon_Stevin#Music_theory
>
> Interesting. But still too formative to add directly
> to the theory.
sorry again for the less "formative" original dutch source:
http://www.xs4all.nl/~adcs/stevin/singconst/singconst.html#verlijcking
translation:
http://www.xs4all.nl/~huygensf/doc/singe.html
Fokker's introduction for modern readers:
http://www.xs4all.nl/~huygensf/doc/stevinsp.html
>
>
http://mto.societymusictheory.org/issues/mto.93.0.3/mto.93.0.3.lindley7.gif
>
> Holy crap! That looks like a horogram!
>
Newton's autograph drawing is dated November 1665:
Literature reference:
The original manuscript is located at:
Cambridge Univ.Lib. Signature: Ms.Add.4000,fol.105v

> What paper at mto
> was this attached to? Wikipedia is currently asking for a
> citation for Newton and 53.

It would be nice to have not only his drawing as web-resource
without any further illustrating annotations.
/tuning/topicId_71935.html#72161

All i know hitherto about N's
facsimle reprint originates from:
http://en.wikipedia.org/wiki/Mark_Lindley
's encyclopecic 'tuning & temperature' standard reference article:
"Stimmung und Temperatur" (in F. Zaminer, ed., Geschichte der
Musiktheorie, Vol. 6: Hören, Messen und Rechnen in der Frühen Neuzeit
(Wissenschaftliche Buchgesellschaft, 1987) pp.205-210

Quest:
Does anybody here in that group know about an
compareable text-book on tuning that i could reccomend
to my foreign students as historically introduction?

Lindley's article reproduces also an earlier similar "horogramm" of
http://en.wikipedia.org/wiki/Ren%C3%A9_Descartes
his early:
"1618. Compendium Musicae. A treatise on music theory and the
aesthetics of music written for Descartes's early collaborator Isaac
Beeckman."

> The citations you have haven't convinced me. Calculating
> good equal divisions is one thing, but well....
...fully agreed...
.... i strongly doubt -alike Stevin did also about his own proposal-
that barely theoretically EDO stuff doesn't
make any pratical sense in tuning real pianos or organs:

Hence for contrasting the worn-out 12EDO paradigm:
here comes a dozen absolute-pitchclasses starting from
http://en.wikipedia.org/wiki/A440
as implemented on many pianos that i do coach for the owners:

Circulating in a 5ths-cirlce in absoulte-pitch frequency values:

A 440cps 220 110
E (330>) 329
H 987 = 3E
F# (3B = 2961>) 2960 1480 740 370 185
C# 555 = 3F#
G# (1665>) 1664 832 416 208 104 52 only-on-real-big-organs: 26 13
Eb 39
b 117
F 351
C (3F = 1053>) 1052 526 263
G (3C = 789>) 788 364 197 (>196 98 49=7^2 from Werckmeister VI)
D (3G = 591>) 589 (>588 294 147)
A (441>) 440 cycle ready done!

that's ascending in heptatonic order on the white lower piano keys:

c" 526 Ut vs. 2^0 http://en.wikipedia.org/wiki/Tenor_C
d" 589 Re vs. 2^(1/6)
e" 658 Mi vs. 2^(1/3)
f" 702 Fa vs. 2^(5/12)
g" 788 Sol v. 2^(7/12)
a" 880 La vs. 2^(9/12) or A5
h" 987 Ti vs. 2^(11/12)

Sounds in my ears somehow similar to Werckmeister's #3.
For instance the 3rd C-E beats theortically merely:

1200*ln(E6_1366/1365_5C4)/ln(2) = ~1.26783898...Cents wide sharp

as demanded for example by Werckmeister's colleague and friend:
http://www.arpschnitger.nl/snoordb.html
"The "pulling effect" in the tuning can be heard very clearly in the
long chords and is produced by the mutual influence of the pipes which
are arranged in thirds on the windchest."
The same effect appears also in the above tuning inbetween
piano-strings on the 3rd C-E,
that turns out to be detuned less than an schisma sharp
from an real pure 3rd of 5/4.

another quote illustrates that amazing effect:
http://www.polettipiano.com/Pages/werckengpaul.html
"When one of the tones of a consonance stands a little too high or too
low against the other, this is called "hovering" [see note 1]. This
term originated primarily among organ builders, in that when they are
tuning two pipes together and they are almost pure [against one
another], such pipes, when they are sounding together, make a
trembling or shivering sound. The closer they are to being in tune
with one another,...."

Translator's Notes

1. I have chosen to translate the German word schweben in its
literal meaning: "hovering". In modern German, schweben has come to
mean the regular periodic variation in the loudness of two tones which
are almost identical in frequency caused by phase cancellation, called
"beating" in English (or its literal equivalent in many other
languages). However, it is clear from reading old German texts
(including Praetorius, who disapproved of the term) that in the 17th
century, schweben meant something more than the simple acoustic
phenomenon. For example, there is no doubt that Gleichschweben does
not mean "Equal Beating", nor can an interval "beat by 1/4 comma",
since identical degrees of tempering create different rates of
"beating" depending on the interval's position on the keyboard, as
Werckmeister explains. He makes a subtle distinction between schweben
and schlagen (literally: "beating") when he states that the former
gets "bigger and smaller" while the latter becomes "faster and
slower"; using "beating" for both words would obscure this refinement.
Using modern acoustical terminology, we would say that
schweben/hovering exists in the frequency domain while
schlagen/beating exists in the time domain. When a note of an interval
is "hovering" the result is "beating", and to a certain extend - but
only to a certain extend - the two can be conflated.

"Hovering" is in fact a perfect metaphor for the act of setting
a tempered interval. If we think of one note of an interval having a
right and proper elevation above or below the other note, that is, the
position at which it creates a pure interval with the other note, any
slight deviation from this position can be seen as a sort of
"hovering" slightly above or slightly below this proper position."

Already
http://en.wikipedia.org/wiki/Arnolt_Schlick
reports about that his tuning benefits from that effect.
Why ignoring that over 500years well approved skills of experts?

12ET excludes and fails in meeting that subtle effect for the 3rd C-E
with ~14Cents sharp exludes that subtle effect ~7 times away,
as expected in the Baroque well-tuning paradigm
at least C-E should be less than ~2Cents sharp from pure
then just 5/4.

Just compare the vanishing ~2Cent C-E 3rd:
to it's ugly ~14Cents off counterpart in 12EDO:

http://coba.belmont.edu/fac/tappant/piano.htm
http://en.wikipedia.org/wiki/Piano_key_frequencies

None piano tuner can apply that concrete on real strings
made of copper and steel accurately on an acoustic piano.
Attepting that mess results in an horrible atonal cacophony.
How can human ears tolerate an deviating defect ~14cents on C-E?

The allegdly claimed high fidelity as specificied makes no sense
when trying to obtain an acceptable approximation of 4:5:6 on C:E:G:

4:5*(1366/1365):6*(788/789)

or in Cents approx:

4 : 5 ~+1.3c : 6 ~-2.2c

versus the more than 10 times worser 12EDO 3rd C-E

4: 5 ~+14c : 6 ~-2c

Only heaven knows how long will persist the obsolete 12EDO paradigm
that misleads astray laypersons about alleged human precision
in absurd tuning instructions.

Forget about as soon as possible the still alive fossil:
12ET = a dozen TEars in my ears!

A.S.

Hi Carl,

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "monz" <monz@> wrote:
> > > Wikipedia is currently asking for a
> > > citation for Newton and 53.
> //
> > The only reference i could dig up was the Lindley article
> > from which that diagram was taken.
>
> Do you know of a URL for this article?

Um, yes ... you mean the one that i provided in the
tuning list post, for which i gave the URL in the
message from which the above quote was taken.
OK, here it is again:

http://www.societymusictheory.org/mto/issues/mto.93.0.3/mto.93.0.3.lindley.art

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

Hi Carl,

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "monz" <monz@> wrote:
> > > Do you know of
> > > an earlier instance of a > 1-D periodicity block?
> >
> > No. As i state in my Encyclopedia page
> >
> > http://tonalsoft.com/enc/p/periodicity-block.aspx
> >
> > it appears that Tanaka was the first to describe and
> > illustrate the concept.
>
> Thanks!
>
> Do you know what commas K and S are here?

You must not have been reading my posts in this thread
very carefully. I already said that Tanaka used the
kleisma and skhisma as unison-vectors to create 53-edo
out of the 5-limit Tonnetz.

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

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

Andreas --

May I suggest that "wavering" is an even more apt translation than
"hovering"? Wavering implies oscillation, i.e. your frequency
aspect, while hovering has no such implication.

Daniel Wolf
Frankfurt

outside of being a ring with lines, it doesn't appear to be a horogram to me
--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/index.html>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main/index.asp> 88.9 FM Wed 8-9 pm Los Angeles

Keenan Pepper wrote:
> On 9/25/07, monz <monz@tonalsoft.com> wrote:
>> It makes more sense to write that low frequency in
>> scientific notation: ~1.62 * 10^-15 Hz, or as it would
>> show up in a spreadsheet or on a calculator: 1.62E-15.
>
> I think it would make more sense to talk about the period, which is
>
> 1 / (1.62e-15 Hz) = 19.6 million years
>
> How did they measure this frequency if they can only observe it for a
> tiny fraction of one period? I guess they must have observed the
> spatial wavelength and either theoretically calculated the propagation
> speed or observed it some other way.
>
> Keenan
>

It makes no sense to assign *any* pitch in *any*
tuning/scale/temperament to a "vibration" with a frequency that low.
That's just plain pseudo-scientific bullshit.

On 9/25/07, M. Edward (Ed) Borasky <znmeb@cesmail.net> wrote:
> It makes no sense to assign *any* pitch in *any*
> tuning/scale/temperament to a "vibration" with a frequency that low.
> That's just plain pseudo-scientific bullshit.

Why doesn't it make sense? What do you think should be the cutoff
frequency between sounds that have a pitch and sounds that "it makes
no sense to assign" a pitch?

Also, are you sure "pseudo-scientific" is the word you're looking for?
No one's trying to sell you anything here.

Keenan

monz wrote:

> You must not have been reading my posts in this thread
> very carefully. I already said that Tanaka used the
> kleisma and skhisma as unison-vectors to create 53-edo
> out of the 5-limit Tonnetz.

I've downloaded the paper from Kraig's site:

http://www.anaphoria.com/Shohe.PDF

Already I can see a chapter called "Kleismatische Verwechselung". This clearly describes the temperament class that's in the Middle Path paper as "hanson". So, it's time to change the name back to "kleismatic" in deference to this earlier discovery.

Graham

monz wrote...

> > Do you know what commas K and S are here?
>
> You must not have been reading my posts in this thread
> very carefully. I already said that Tanaka used the
> kleisma and skhisma as unison-vectors to create 53-edo
> out of the 5-limit Tonnetz.

I'm not reading anything very carefully, am I. K and S,
duh. -Carl

monz wrote...

> > --- In tuning@yahoogroups.com, "monz" <monz@> wrote:
> > > > Wikipedia is currently asking for a
> > > > citation for Newton and 53.
> > //
> > > The only reference i could dig up was the Lindley article
> > > from which that diagram was taken.
> >
> > Do you know of a URL for this article?
>
> Um, yes ... you mean the one that i provided in the
> tuning list post, for which i gave the URL in the
> message from which the above quote was taken.
> OK, here it is again:
>
> http://www.societymusictheory.org/mto/issues/mto.93.0.3/
> mto.93.0.3.lindley.art

Yeah, that one. *blush* Though I was hoping for the
original text. -Carl

Andreas wrote...
> http://www.pricomm.at/files/vortragproftaschner.pdf
>
> 'Already before Euler the musicologist Conrad Henfling
> had drafted the tonal-(diamond)-grid.'
>
> Conrad Henfling (1648-1716) labeled his invention as "Tonnetz"
> appearently already before Euler, Helmholtz, Riemann, Tanaka

I'm afraid I can't know the extent of it without an English
source.

> Probably J.S.Bach overtook that from Henfling
> as he shows that in his 3rd-5th diamond lattice-grid contained his
> official heraldic seal:
> http://f9g.yahoofs.com/groups/g_16806072/.HomePage/__sr_/
> d3ca.jpg?grAKv.GBUZJ9e3qx

A bit fanciful from my point of view, I'm afraid.

> > > http://en.wikipedia.org/wiki/Simon_Stevin#Music_theory
> >
> > Interesting. But still too formative to add directly
> > to the theory.
> sorry again for the less "formative" original dutch source:
> http://www.xs4all.nl/~adcs/stevin/singconst/
> singconst.html#verlijcking
>
> translation:
> http://www.xs4all.nl/~huygensf/doc/singe.html
> Fokker's introduction for modern readers:
> http://www.xs4all.nl/~huygensf/doc/stevinsp.html

I'm not seeing a whole lot here. It looks like Vicentino
is the first and furthest researcher of the 16th century.

> Newton's autograph drawing is dated November 1665:
> Literature reference:
> The original manuscript is located at:
> Cambridge Univ.Lib. Signature: Ms.Add.4000,fol.105v

"Stephens! Who do we have in Cambridge??"

> > The citations you have haven't convinced me. Calculating
> > good equal divisions is one thing, but well....
> ...fully agreed...
> .... i strongly doubt -alike Stevin did also about his own
> proposal-that barely theoretically EDO stuff doesn't
> make any pratical sense in tuning real pianos or organs:
>
> Hence for contrasting the worn-out 12EDO paradigm:

On the contrary, I meant that calculating large ETs isn't
impractical enough for my taste! In particular, it doesn't
go far enough toward the regular mapping paradigm. Try:

http://www.google.com/search?q=regular+mapping+paradigm

And choose the first result.

-Carl

Carl Lumma wrote:

> I'm not seeing a whole lot here. It looks like Vicentino
> is the first and furthest researcher of the 16th century.

Vicentino's admirable, but I don't think he had much to do with the regular mapping paradigm. He was mostly pushing the paradigm of his age -- meantone for keyboards. He did mention this keyboard tuning that mixes meantone with just intonation but it's a modern interpretation to say that it was used as adaptive temperament.

Another candidate, though, was his teacher Willaert. I don't think we have any direct treatises. From what Vicentino and Zarlino wrote, however, we can assume he knew all about meantone. There's also an interesting article I have as mto.04.10.1.wibberley1.html which suggests he was teaching adaptive tuning for choirs, but the evidence is indirect.

Graham

Graham wrote...

> > I'm not seeing a whole lot here. It looks like Vicentino
> > is the first and furthest researcher of the 16th century.
>
> Vicentino's admirable, but I don't think he had much to do
> with the regular mapping paradigm.

For me, adaptive tuning is a part of that paradigm. Maybe
you disagree, and you named it, but well... we need a name
for all the good bits.

> He was mostly pushing
> the paradigm of his age -- meantone for keyboards.

He seems like first to 31... probably almost certainly
first to 31 on a real instrument.

But beyond that, his second tuning for the archicemalo
or whatever, with the two manuals a comma apart, was the
real breakthru.

> He did
> mention this keyboard tuning that mixes meantone with just
> intonation but it's a modern interpretation to say that it
> was used as adaptive temperament.

Maybe so.

> Another candidate, though, was his teacher Willaert. I
> don't think we have any direct treatises. From what
> Vicentino and Zarlino wrote, however, we can assume he knew
> all about meantone. There's also an interesting article I
> have as mto.04.10.1.wibberley1.html which suggests he was
> teaching adaptive tuning for choirs, but the evidence is
> indirect.

That is interesting. Whole urls do help.

-Carl

Carl Lumma wrote:
> Graham wrote...
> >>>I'm not seeing a whole lot here. It looks like Vicentino
>>>is the first and furthest researcher of the 16th century.
>>
>>Vicentino's admirable, but I don't think he had much to do >>with the regular mapping paradigm.
> > For me, adaptive tuning is a part of that paradigm. Maybe
> you disagree, and you named it, but well... we need a name
> for all the good bits.

The paradigm is about relationships between tunings or temperaments. As temperamental notation can be used for a piece in adaptive tuning, it's part of the paradigm. But adaptive tuning of choirs and meantone tempering of keyboard instruments was the dominant paradigm of the 16th century. Adaptive tuning of general temperament classes would have been more interesting but also unrealistic.

The main pioneers as I see it are:

Bosanquet -- classification of ETs that's similar to temperament classes and identificition of (his version of) regular temperaments. Generalized keyboards.

Tanaka -- first for kleismatic temperament and periodicity blocks with reference to equal temperament, also mentions schismatic temperament (Helmholtz? Riemann?) and unison vectors (Helmholtz? Ellis?).

Fokker -- unison vectors as vectors, determinants as (by our interpretation) one element of the mapping, 7-limit periodicity blocks.

Wilson -- MOS and scale tree, keyboard mappings for either JI or temperaments, regular temperaments pushed to higher limits, more general ways of classifying temperaments.

>>He was mostly pushing >>the paradigm of his age -- meantone for keyboards.
> > He seems like first to 31... probably almost certainly
> first to 31 on a real instrument.

So he was the first with a rich patron who could afford expensive instruments.

Zarlino was the first to precisely define meantones. Perhaps we should chalk all this up to "the Willaert school".

> But beyond that, his second tuning for the archicemalo
> or whatever, with the two manuals a comma apart, was the
> real breakthru.
>
>>He did >>mention this keyboard tuning that mixes meantone with just >>intonation but it's a modern interpretation to say that it >>was used as adaptive temperament.
> > Maybe so.
> >>Another candidate, though, was his teacher Willaert. I >>don't think we have any direct treatises. From what >>Vicentino and Zarlino wrote, however, we can assume he knew >>all about meantone. There's also an interesting article I >>have as mto.04.10.1.wibberley1.html which suggests he was >>teaching adaptive tuning for choirs, but the evidence is >>indirect.
> > That is interesting. Whole urls do help.

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

There's also a follow up article:

http://www.societymusictheory.org/mto/issues/mto.04.10.1/mto.04.10.1.wibberley2.html

It considers "Syntonic tuning" a synonym for "Just Intonation". Hence a precedent for "syntonic" not referring to meantone.

Graham

Carl Lumma wrote:

>>Newton's autograph drawing is dated November 1665:
>>Literature reference:
>>The original manuscript is located at:
>>Cambridge Univ.Lib. Signature: Ms.Add.4000,fol.105v
> > "Stephens! Who do we have in Cambridge??"

There exists a book, you know. "Music, Science, and Natural Magic in Seventeenth-Century England", Penelope Gouk, Yale UP, 1999. I haven't seen it. The sole, genuine Amazon review says it mentions Newton's interest in music.

http://www.amazon.com/gp/product/customer-reviews/0300073836/ref=cm_cr_dp_all_top/104-5144034-4506308?ie=UTF8&n=283155&s=books#customerReviews

Graham

To be precise, it wasn't Andreas' translation, it is that of Paul Poletti

http://www.polettipiano.com/Pages/werckengpaul.html

which he made in close consulation with me!

The problem with 'wavering' is that it is not a translation of
'schweben(d)' - it is a paraphrase of what you assume Werckmeister to
be saying. But in fact in this text Werckmeister somewhat carefully
uses more than one term: 'schweben' and 'zittern' and 'tremulieren' /
'tremores'. One cannot assume that they are completely interchangeable
and mean the same thing.

What he actually means (in the places where there is a useful
distinction to be made!) is, I believe: 'schweben' is the temperament
of an interval; whereas 'tremulieren' or 'zittern' is the audible
beating which under many circumstances is the physical manifestation
of a tempered interval.

The key passage is at the end where Werckmeister refers to the
difference in theory and practice in traditional meantone-like organ
tuning. He says that the hypothesis of 1/4 comma tempering on the
organ is untrue, and the (major) thirds in traditional tuning actually
'hover' sharp, but because they can't be made to beat (or waver!) -
literally, cannot be brought 'ad tremorem' - the old theorists, and
even the tuners themselves, simply assumed that the tempering must
have been 1/4 comma.

What does this mean? Can one have thirds that don't beat (or 'waver')
despite the tempering being less than 1/4 comma? Well, see note 5 at
the end of the translation. It's because the frequency of an organ
pipe (even at given temperature and humidity) is not an absolutely
fixed quantity, and when two pipes are sounded together they can tend
to approach a pure interval, ie to reduce or eliminate beating which
would otherwise be present.

~~~T~~~

--- In tuning@yahoogroups.com, "djwolf_frankfurt" <djwolf@...> wrote:
>
> --- In tuning@yahoogroups.com, "Andreas Sparschuh" <a_sparschuh@>
>
> Andreas --
>
> May I suggest that "wavering" is an even more apt translation than
> "hovering"? Wavering implies oscillation, i.e. your frequency
> aspect, while hovering has no such implication.
>
> Daniel Wolf
> Frankfurt
>

Because things which are certainly inaudible don't have a pitch?

Because describing a frequency of 1 oscillation per 10 million years
by giving its pitch and the number of octaves below middle C is
useless for any scientific or musical purpose?

Should astrophysicists perhaps start using Helmholtz notation:

Bb,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

- can you be bothered to count the ticks?

Perhaps one should rather describe it as 'pseudo-musical' bullshit.
The science may be correct, but the attempt to connect it with a
musical note through a huge number of powers of 2 is pointless.

Actually, they *are* trying to sell something. They are trying to
obtain or maintain public funding for their research. To achieve this
they need, literally, to sell it to the public.

This is how it works. Research group puts out press release which
makes cute-sounding, but scientifically absurd or pointless,
statements related to music. Reporters love it and it gets into the
world media. Next time research group's funding agency meets they
remember this publicity 'success'. It works because the public, that's
YOU, don't give a damn whether what they read in the science section
is at all useful or meaningful. Science as cute entertainment.

~~~T~~~

--- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
>
> On 9/25/07, M. Edward (Ed) Borasky <znmeb@...> wrote:
> > It makes no sense to assign *any* pitch in *any*
> > tuning/scale/temperament to a "vibration" with a frequency that low.
> > That's just plain pseudo-scientific bullshit.
>
> Why doesn't it make sense? What do you think should be the cutoff
> frequency between sounds that have a pitch and sounds that "it makes
> no sense to assign" a pitch?
>
> Also, are you sure "pseudo-scientific" is the word you're looking for?
> No one's trying to sell you anything here.
>
> Keenan
>

[ Attachment content not displayed ]

[ Attachment content not displayed ]

... I have a very elementary question. On the 'Regular Mapping
Paradigm page' it is explained how JI intervals are mapped to numbers
of steps in a diatonic scale.

In contrast to equal temperaments, what interval you are mapping to
any given number of steps of the diatonic scale clearly depends on
which note you start from. That is, the mapping from JI intervals to
numbers of scale steps is many-to-one, information is lost. Now in the
case of 10/9 and 9/8, we *want* information to be lost, but not in the
case of 6/5 and 5/4, which both map to 2 steps, let alone 9/8 and
16/15. Why then is the starting note not specified, and does this
create problems? (Such as an apparent identity of the diatonic scale
and 7-EDO, at the level of the mapping...)

Of course if one maps to the generators the question evaporates - but
Gene (who I assume to be the author) seems to be making a point of
starting off with scale steps.

The thing is that specifying the generators of a regular but non-equal
temperament *doesn't* tell you what the scale looks like; you have to
say in addition which notes, in the sense of combinations of
generators, are in the scale.
~~~T~~~

> Zarlino was the first to precisely define meantones.
> Perhaps we should chalk all this up to "the Willaert school".

It seems like Vicentino had more students than Willaert.
Maybe not though.

> >>Another candidate, though, was his teacher Willaert. I
> >>don't think we have any direct treatises. From what
> >>Vicentino and Zarlino wrote, however, we can assume he knew
> >>all about meantone. There's also an interesting article I
> >>have as mto.04.10.1.wibberley1.html which suggests he was
> >>teaching adaptive tuning for choirs, but the evidence is
> >>indirect.
> >
> > That is interesting. Whole urls do help.
>
> http://www.societymusictheory.org/mto/issues/mto.04.10.1/
> mto.04.10.1.wibberley1_frames.html

Thanks. That's an awful lot to write about one motet. I
skimmed it quickly and don't see a smoking gun.

> There's also a follow up article:
>
> http://www.societymusictheory.org/mto/issues/mto.04.10.1/
> mto.04.10.1.wibberley2.html

Well he certainly does seem to love Willaert. Makes it
sound like Willaert was first to the 5-limit. Is that
how you're reading it?

-Carl

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
>
>
> ... I have a very elementary question. On the 'Regular Mapping
> Paradigm page' it is explained how JI intervals are mapped to
> numbers of steps in a diatonic scale.
>
> In contrast to equal temperaments, what interval you are mapping to
> any given number of steps of the diatonic scale clearly depends on
> which note you start from. That is, the mapping from JI intervals to
> numbers of scale steps is many-to-one, information is lost. Now in
> the case of 10/9 and 9/8, we *want* information to be lost, but not
> in the case of 6/5 and 5/4, which both map to 2 steps, let alone
> 9/8 and 16/15. Why then is the starting note not specified, and
> does this create problems? (Such as an apparent identity of the
> diatonic scale and 7-EDO, at the level of the mapping...)
>
> Of course if one maps to the generators the question
> evaporates - but Gene (who I assume to be the author) seems to be
> making a point of starting off with scale steps.

Graham is the author, and I'll let him answer. I don't understand
the mapping by steps thing. I've only ever used mapping by
generators. Steps are generators, too, and I'd assume that for
the diatonic scale you'd need to vals in the map (one for the
whole step and one for the half).

> The thing is that specifying the generators of a regular but
> non-equal temperament *doesn't* tell you what the scale looks
> like;

This is precisely why we do not conflate the meaning of
"temperament" and "scale".

> you have to say in addition which notes, in the sense of
> combinations of generators, are in the scale.

Yes.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> > Zarlino was the first to precisely define meantones.
> > Perhaps we should chalk all this up to "the Willaert school".
>
> It seems like Vicentino had more students than Willaert.
> Maybe not though.

I don't know anything about those statistics, but the
Wikipedia page on Vicentino states that "Born in Vicenza,
he may have studied with Adrian Willaert in Venice".
So there's likely a close connection between the two.

> Well he certainly does seem to love Willaert. Makes it
> sound like Willaert was first to the 5-limit. Is that
> how you're reading it?

Willaert may indeed have been one of the first continental
composers to embrace 5-limit as a theoretical ideal.
He was born in 1490, only 8 years after Ramos published
the earliest description of a 5-limit monochord division.

BTW, i just updated the Ramos page to include a Tonescape
Lattice:

http://tonalsoft.com/monzo/ramos/ramos.aspx

5-limit practice most likely took root first in England,
during the 1300s or possibly even a bit earlier. The
Wikipedia page about John Dunstable (c.1390-1453) says:
"The _contenance angloise_, while not defined by Martin
Le Franc, was probably a reference to Dunstaple's stylistic
trait of using full triadic harmony, along with a liking
for the interval of the third." This style was widely
copied shortly afterward on the continent, influencing
especially the Burgundian School.

I wrote some very long posts to this list about this,
back around 2000 or 2001, in response to Margo Schulter.

Also, keep in mind that Marchetto of Padua (obviously
also in Italy) wrote his _Lucidarium_ around 1318, and
his description of the "diesis" and three different
sizes of semitones -- while extremely vague and open to
widely varying interpretations -- obviously implies a
rational tuning ideal with prime-factors beyond 3,
which was the standard pythagorean tuning of the time.

http://tonalsoft.com/monzo/marchetto/marchetto.aspx

(This page also has some recent additions.)

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

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> There exists a book, you know. "Music, Science, and Natural
> Magic in Seventeenth-Century England", Penelope Gouk, Yale
> UP, 1999. I haven't seen it. The sole, genuine Amazon
> review says it mentions Newton's interest in music.

I checked it out. It does mention that, but doesn't add anything to
what little we know here.

Carl Lumma wrote:
>>Zarlino was the first to precisely define meantones. >>Perhaps we should chalk all this up to "the Willaert school".
> > It seems like Vicentino had more students than Willaert.
> Maybe not though.

If they studied together, Willaert was the master. We know Zarlino was a student of Willaert, and had respect for his teacher. Willaert's greatest student was Gabrieli, but he wasn't a theorist and so we don't talk about him so much.

It seems we don't know who Vicentino studied with. We do know he was born in Vicenza, and Willaert was the leading musical figure in nearby Venice. It's difficult to see how Vicentino wouldn't have been familiar with the ideas of Willaert's circle. Wikipedia also links Willaert with the d'Este family. Unfortunately, it's vague about the exact connection with Vicentino (of course we don't trust Wikipedia anyway, but I'm checking my facts with it, so there). The Yale edition of Vicentino's treatise has more details, but I don't have it to hand. (Note that the great Gesualdo married into the same d'Este family, and probably inherited Vicentino's instruments.)

In addition, Vincenzo Galilei was a student of Zarlino. So all these theorists were connected in some way with Willaert.

I'm trying to shift the focus away from individuals and toward schools of thought. 5-limit JI for choirs and meantone for instruments was the emerging paradigm of the day. Willaert seems to have been an important figure behind it (certainly not the originator). We don't have much theory from the time. You can read about this in Jeppesen's Counterpoint book. The culture was still influenced by the medieval guilds, where it was bad to reveal your trade secrets. We have Vicentino's treatise because of an accident -- he lost an argument with Lusitano (still following Pythagorean thinking) and to regain face he wrote down his ideas on the enharmonic to show how clever and learned he was. We can assume that other great musicians and theorists had interesting ideas but didn't publish them. Vicentino was probably in the inner circle and so would have known a lot more than we do about all this.

>>>>Another candidate, though, was his teacher Willaert. I >>>>don't think we have any direct treatises. From what >>>>Vicentino and Zarlino wrote, however, we can assume he knew >>>>all about meantone. There's also an interesting article I >>>>have as mto.04.10.1.wibberley1.html which suggests he was >>>>teaching adaptive tuning for choirs, but the evidence is >>>>indirect.
>>>
>>>That is interesting. Whole urls do help.
>>
>>http://www.societymusictheory.org/mto/issues/mto.04.10.1/
>>mto.04.10.1.wibberley1_frames.html
> > Thanks. That's an awful lot to write about one motet. I
> skimmed it quickly and don't see a smoking gun.

The smoking gun is the it closes the cycle of fifths. That was revolutionary (pun alert!) and whatever the reason was an important milestone in pre-regular mapping thinking. Closing the circle implies "tempering out" (scare quotes because tempering may not have been implied) the Pythagorean comma. Unison vectors are dual to mappings. Whether he meant adaptive tuning, equal temperament, irregular temperament, or strict JI, the fact that he closed the cycle is a regular mapping precursor. It's also interesting that the motet seems to work in JI because you wouldn't expect that for a piece written to demonstrate equal temperament.

>>There's also a follow up article:
>>
>>http://www.societymusictheory.org/mto/issues/mto.04.10.1/
>>mto.04.10.1.wibberley2.html
> > Well he certainly does seem to love Willaert. Makes it
> sound like Willaert was first to the 5-limit. Is that
> how you're reading it?

I think he's overstating his argument, as academics tend to. But what I take from it is that choirs were trying to get 5-limit harmony at the time. And some theorists were dealing with the technicalities.

Graham

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:

> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
> >
> > > Zarlino was the first to precisely define meantones.
> > > Perhaps we should chalk all this up to "the Willaert school".
> >
> > It seems like Vicentino had more students than Willaert.
> > Maybe not though.
>
>
> I don't know anything about those statistics, but the
> Wikipedia page on Vicentino states that "Born in Vicenza,
> he may have studied with Adrian Willaert in Venice".
> So there's likely a close connection between the two.
>
> <snip>
>
> Also, keep in mind that Marchetto of Padua (obviously
> also in Italy) wrote his _Lucidarium_ around 1318, and
> his description of the "diesis" and three different
> sizes of semitones -- while extremely vague and open to
> widely varying interpretations -- obviously implies a
> rational tuning ideal with prime-factors beyond 3,
> which was the standard pythagorean tuning of the time.
>
> http://tonalsoft.com/monzo/marchetto/marchetto.aspx
>
> (This page also has some recent additions.)

In fact, Padua is so close to Venice that it is sometimes
included in the Venice-Padua Metropolitan Area for
census purposes. Hmm ...

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

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:

> > http://tonalsoft.com/monzo/marchetto/marchetto.aspx
> >
> > (This page also has some recent additions.)
>
>
> In fact, Padua is so close to Venice that it is sometimes
> included in the Venice-Padua Metropolitan Area for
> census purposes. Hmm ...

Padua was part of the Republic of Venice from 1405 until
the end of the Republic in 1797, except for one brief
period of a few weeks in 1509. Thus, it seems very likely
to me that Marchetto's ideas would influence the progressive
composers in Venice (*and* Vicenza).

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

Tom Dent wrote:
> ... I have a very elementary question. On the 'Regular Mapping
> Paradigm page' it is explained how JI intervals are mapped to numbers
> of steps in a diatonic scale.
> > In contrast to equal temperaments, what interval you are mapping to
> any given number of steps of the diatonic scale clearly depends on
> which note you start from. That is, the mapping from JI intervals to
> numbers of scale steps is many-to-one, information is lost. Now in the
> case of 10/9 and 9/8, we *want* information to be lost, but not in the
> case of 6/5 and 5/4, which both map to 2 steps, let alone 9/8 and
> 16/15. Why then is the starting note not specified, and does this
> create problems? (Such as an apparent identity of the diatonic scale
> and 7-EDO, at the level of the mapping...)

Many JI intervals map to one generic interval, the same as with an equal temperament. The best analogy is with an irregular temperament. All specific intervals can have different sizes but generic intervals are treated the same way in some senses.

6:5 and 5:4 both map to thirds. A third is an interval of 2 diatonic steps. Whether this is a problem depends on what you want to do. In some cases musica ficta didn't distinguish different thirds in notation. In counterpoint, thirds are interchangeable (but perfect and diminished fifths are not).

> Of course if one maps to the generators the question evaporates - but
> Gene (who I assume to be the author) seems to be making a point of
> starting off with scale steps.

You can check the metadata for authorship.

If you want to distinguish meantone intervals (so 5:4 and 6:5 are different, but 9:8 and 10:9 are not) you can count scale steps on a keyboard (or from 12-equal) along with diatonic scale steps. Most of the time this is what you will do because conventional notation is a 2-dimensional system. But you should be able to see where the diatonic numbers come from. (Also if you happen to know numerical notation...)

You can look at octaves and generators as well and they carry exactly the same information. But "a third" could be generated by either four fifths or three fourths. You need to make the scale unequal to distinguish them.

I start with scale steps because anybody can sit at a keyboard and count them. Generators are a more abstract concept.

> The thing is that specifying the generators of a regular but non-equal
> temperament *doesn't* tell you what the scale looks like; you have to
> say in addition which notes, in the sense of combinations of
> generators, are in the scale. Yes. But you can notate music in terms of the generators, as conventional staff notation does. Some important properties of the music don't depend on the specific tuning you use (regular or otherwise; tempered or otherwise).

Graham

Keenan Pepper wrote:
> On 9/25/07, M. Edward (Ed) Borasky <znmeb@cesmail.net> wrote:
>> It makes no sense to assign *any* pitch in *any*
>> tuning/scale/temperament to a "vibration" with a frequency that low.
>> That's just plain pseudo-scientific bullshit.
>
> Why doesn't it make sense? What do you think should be the cutoff
> frequency between sounds that have a pitch and sounds that "it makes
> no sense to assign" a pitch?

The cutoff frequency between sounds that have a pitch and sounds for
which it makes no sense to assign a pitch is a well defined concept in
psycho-acoustics, although the exact number is open to some debate. I'm
pretty sure Helmholtz came up with some number -- I'd say somewhere in
the range of 2 cycles per second is the absolute lowest frequency that
one could claim was a pitched sound, and I think somewhere between 16
and 32 cycles per second is probably more realistic ... somewhere near
where you'd find the longest organ pipes, for example.

> Also, are you sure "pseudo-scientific" is the word you're looking for?
> No one's trying to sell you anything here.

Perhaps "pseudo-musical bullshit" is a more appropriate designator.
Clearly the fact that they've measured a frequency that low is science,
but the attempt to tie it to music seems like they're trying to impress
rather than enlighten.

[ Attachment content not displayed ]

"mikal haley" wrote...
> [image: frequency to wavelength - the connections]

One should note that EM waves and sound waves are very
different phenomena that do not belong on the same chart
like this.

-Carl

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
> >>Vicentino's admirable, but I don't think he had much to do
> >>with the regular mapping paradigm.
> >
> > For me, adaptive tuning is a part of that paradigm. Maybe
> > you disagree, and you named it, but well... we need a name
> > for all the good bits.
>
> The paradigm is about relationships between tunings or
> temperaments. As temperamental notation can be used for a
> piece in adaptive tuning, it's part of the paradigm.

I would phrase it like, the theory of temperaments doesn't
say anything about what should be tempered: all intervals,
melodic intervals only, etc. And in fact, it should point
this out.

> But adaptive tuning of choirs and meantone tempering of
> keyboard instruments was the dominant paradigm of the 16th
> century.

It was? I see 5-limit JI in Willaert, but not adaptive
tuning. At least, not anything as principled as Vicentino's
2nd archicembalo tuning.

I'm missing the bit where you say Vicentino only made an
instrument because he happenned to find a rich patron.
Well, there is a lot of luck in money, but there's other
stuff too; if Vincentino got funded that might be telling
us something. In any case, building an instrument is a
strong demonstration of understanding that I'm not seeing
the equivalent of in texts by other theorists.

In short, to me, the advocacy of 5-limit JI is one thing,
that I suspect many were doing in the 16th century. Using
two chains of meantone a 1/4-comma apart is much more
unique... in fact I'm unaware of it being suggested in
the period between 1555 and 1999.

-Carl

Tom Dent wrote:

> The problem with 'wavering' is that it is not a translation of
> 'schweben(d)

It is a perfectly good translation of schweben. > ' - it is a paraphrase of what you assume Werckmeister to
> be saying. But in fact in this text Werckmeister somewhat carefully
> uses more than one term: 'schweben' and 'zittern' and 'tremulieren' /
> 'tremores'.
Yes, and Andreas was looking a word to contrast with "beating" for the other terms. In particular, he needed a term without the repercussive aspect of beating. > One cannot assume that they are completely interchangeable
> and mean the same thing.
I do not, the whole purpose here was to distinguish the two phenomena.

>
> What he actually means (in the places where there is a useful
> distinction to be made!) is, I believe: 'schweben' is the temperament
> of an interval;
Then he should have used a nominative form, and namely: Temperatur.

> whereas 'tremulieren' or 'zittern' is the audible
> beating which under many circumstances is the physical manifestation
> of a tempered interval.
No, the beating is one specific attribute of a tempered interval, wavering (schweben) -- like a voter who can't quite make her mind up -- is a perfectly good description of intervals that are intended to have two meanings (to pun): as a third to another tone, and simultaneously, as a a pitch related by a series of perfect fifths.

Just extend the bottom of the chart downwards from C-20 another 37 octaves
A simple copy and paste of the bottom of the chart, line up the C with C-20 TO KEEP THE COLOR PATTERN CONSISTENT and then change the numbers TO CONTINUE THE SERIES.

I assume that you have Photoshop, Graphic Converter or similar and can ADD/SUBTRACT in negative numbers down to 60;-)

Charles Lucy lucy@lucytune.com

----- Promoting global harmony through LucyTuning -----

For information on LucyTuning go to: http://www.lucytune.com

LucyTuned Lullabies (from around the world):
http://www.lullabies.co.uk

Skype user = lucytune

http://www.myspace.com/lucytuning

On 27 Sep 2007, at 06:39, mikal haley wrote:

>
>
> So Charles, where would I find that 57 c's below middle C on this > chart?
> I do "need to know" if you follow me...
>
>
>
> On 9/27/07, M. Edward (Ed) Borasky <znmeb@cesmail.net> wrote:
> Keenan Pepper wrote:
> > On 9/25/07, M. Edward (Ed) Borasky <znmeb@cesmail.net> wrote:
> >> It makes no sense to assign *any* pitch in *any*
> >> tuning/scale/temperament to a "vibration" with a frequency that > low.
> >> That's just plain pseudo-scientific bullshit.
> >
> > Why doesn't it make sense? What do you think should be the cutoff
> > frequency between sounds that have a pitch and sounds that "it makes
> > no sense to assign" a pitch?
>
> The cutoff frequency between sounds that have a pitch and sounds for
> which it makes no sense to assign a pitch is a well defined concept in
> psycho-acoustics, although the exact number is open to some debate. > I'm
> pretty sure Helmholtz came up with some number -- I'd say somewhere in
> the range of 2 cycles per second is the absolute lowest frequency that
> one could claim was a pitched sound, and I think somewhere between 16
> and 32 cycles per second is probably more realistic ... somewhere near
> where you'd find the longest organ pipes, for example.
>
> > Also, are you sure "pseudo-scientific" is the word you're looking > for?
> > No one's trying to sell you anything here.
>
> Perhaps "pseudo-musical bullshit" is a more appropriate designator.
> Clearly the fact that they've measured a frequency that low is > science,
> but the attempt to tie it to music seems like they're trying to > impress
> rather than enlighten.
>
>
>
>
>
> --> very new!
>
> http://particlezen.proboards7.com/index.cgi?
> John 16:33
>
> the*edge*of*everything*
>
>

Carl;

You're being an old prune again;-)

I suggest that you read Guy Murchie if the concept is too "troublesome" for you.

Charles Lucy lucy@lucytune.com

----- Promoting global harmony through LucyTuning -----

For information on LucyTuning go to: http://www.lucytune.com

LucyTuned Lullabies (from around the world):
http://www.lullabies.co.uk

Skype user = lucytune

http://www.myspace.com/lucytuning

On 27 Sep 2007, at 08:23, Carl Lumma wrote:

> "mikal haley" wrote...
> > [image: frequency to wavelength - the connections]
>
> One should note that EM waves and sound waves are very
> different phenomena that do not belong on the same chart
> like this.
>
> -Carl
>
>
>

Hm, it must just be that Gene's and Graham's websites have a similar
layout and aesthetic, so in the absence of a signature I took one for
the other. (However, I should have noticed the different style of
writing...)

I'm still (slightly) confused. If one says 'diatonic scale', it surely
means that the property of having five large steps and two small steps
is important somehow. But the 7-note scale step mapping, which is just
equivalent to saying '2:1 is an octave, 3/2 is a fifth, 5/4 and 6/5
are thirds' (etc.), doesn't necessarily require this. (Are these
statements exclusively characteristic of meantone - in any sense of
the word? Perhaps in schismatic one would rather say '5/4 is a
diminished fourth'??)

If one were to restrict 'meantone' to those tunings generated by
fifths wider than 7-equal but narrower than 5-equal, then the relevant
7-note scale would in fact be diatonic. But in the context of what
you're saying the interval sizes (aka 'tuning') should not play a
determining role.

I think, in context, one has to understand the mapping of meantone
(understood *as* a mapping!) to 7-note 'diatonic' scale steps with a
definition of 'diatonic' that allows the relative sizes of 'semitone'
and 'tone' to take *any* value. (Including negative!) Just 5 of one
and 2 of the other maximally separated.

~~~T~~~

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> You can check the metadata for authorship.
>
> If you want to distinguish meantone intervals (so 5:4 and
> 6:5 are different, but 9:8 and 10:9 are not) you can count
> scale steps on a keyboard (or from 12-equal) along with
> diatonic scale steps. Most of the time this is what you
> will do because conventional notation is a 2-dimensional
> system. But you should be able to see where the diatonic
> numbers come from. (Also if you happen to know numerical
> notation...)
>
> You can look at octaves and generators as well and they
> carry exactly the same information. But "a third" could be
> generated by either four fifths or three fourths. You need
> to make the scale unequal to distinguish them.
>
> I start with scale steps because anybody can sit at a
> keyboard and count them. Generators are a more abstract
> concept.
>

Tom Dent wrote:
> Hm, it must just be that Gene's and Graham's websites have a similar
> layout and aesthetic, so in the absence of a signature I took one for
> the other. (However, I should have noticed the different style of
> writing...)

I use plain HTML for that page. So what you're seeing is the aesthetic your browser gives it.

> I'm still (slightly) confused. If one says 'diatonic scale', it surely
> means that the property of having five large steps and two small steps
> is important somehow. But the 7-note scale step mapping, which is just
> equivalent to saying '2:1 is an octave, 3/2 is a fifth, 5/4 and 6/5
> are thirds' (etc.), doesn't necessarily require this. (Are these
> statements exclusively characteristic of meantone - in any sense of
> the word? Perhaps in schismatic one would rather say '5/4 is a
> diminished fourth'??)

You can have a diatonic scale with different sizes of large steps: 9:8 and 10:9. In fact, this is the original meaning of a diatonic scale. The mapping is the same however you tune it. Also, diatonic steps all look the same on the staff.

But yes, the diatonic scale can usually be taken to have the large and small steps. And you could take "large steps" and "small steps" as the generators. Perhaps that would be clearer. Be free to write your own explanation if you think you can improve on mine :)

In schismatic a 5:4 is indeed a kind of fourth rather than a third. But there are other temperament classes that share the 7 note mapping with meantone. Dicot is the simplest one, where the generator is a neutral third that approximates both 6:5 and 5:4. From Paul's Middle Path paper I also have "mavila", "porcupine", "tetracot", and "amity".

> If one were to restrict 'meantone' to those tunings generated by
> fifths wider than 7-equal but narrower than 5-equal, then the relevant
> 7-note scale would in fact be diatonic. But in the context of what
> you're saying the interval sizes (aka 'tuning') should not play a
> determining role.

If it's a regular tuning you can expect it to be a 5L 2s diatonic. But the exact temperament can vary.

> I think, in context, one has to understand the mapping of meantone
> (understood *as* a mapping!) to 7-note 'diatonic' scale steps with a
> definition of 'diatonic' that allows the relative sizes of 'semitone'
> and 'tone' to take *any* value. (Including negative!) Just 5 of one
> and 2 of the other maximally separated.

You can certainly do that. It isn't clear that the result would be "meantone" though.

Graham

Hi Daniel and Tom,

--- In tuning@yahoogroups.com, Daniel Wolf <djwolf@...> wrote:
>
> Tom Dent wrote:
>
> > The problem with 'wavering' is that it is not a translation of
> > 'schweben(d)
>
> It is a perfectly good translation of schweben.
>
> <snip>
>
> > whereas 'tremulieren' or 'zittern' is the audible
> > beating which under many circumstances is the physical
> > manifestation of a tempered interval.
>
> No, the beating is one specific attribute of a tempered
> interval, wavering (schweben) -- like a voter who can't
> quite make her mind up -- is a perfectly good description
> of intervals that are intended to have two meanings
> (to pun): as a third to another tone, and simultaneously,
> as a a pitch related by a series of perfect fifths.

I agree with Daniel that "wavering" is a good translation
of "schweben" as used in this context.

This is similar to a situation in Roy Carter's English
translation of Schoenberg's _Harmonielehre_. In his
Introduction, Carter discusses the translation of
_Wendepunktgesetze_ as "turning-point tones" or
"pivot tones". Unfortunately i don't have my copy
at hand, but here at least is another discussion of
the term which is available online:

http://books.google.com/books?
id=jBsvaZUNTegC&pg=PA26&lpg=PA26&dq=schoenberg+%22turning+point%
22&source=web&ots=rsYvY4Sgwk&sig=n1sMBFsPa8vqOc-dT6Utu4xxfvk#PPR6,M1

And again speaking of Schoenberg, in my Encyclopedia
page for "gleichschwebende" i note his use of the term
"schwebende" to refer to what is usually translated as
"suspended tonality" ... and i think "wavering tonality"
is a better translation.

http://tonalsoft.com/enc/g/gleichschwebende.aspx

Note also that, as i point out in that page, "schwebend"
historically first referred to equally-beating tempered
intervals, but since then has been taken up by most
German-language theorists to refer to an equal logarithmic
amount of tempering. So apparently it does have both
of the meanings that you two are arguing about.

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

--- In tuning@yahoogroups.com, Charles Lucy <lucy@...> wrote:
> Carl;
>
> You're being an old prune again;-)
>
> I suggest that you read Guy Murchie if the concept is too
> "troublesome" for you.

With sources like that, no wonder you're coming up with stuff
like that chart.

-Carl

Hi Tom,

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

> I'm still (slightly) confused. If one says 'diatonic scale',
> it surely means that the property of having five large steps
> and two small steps is important somehow. But the 7-note
> scale step mapping, which is just equivalent to saying
> '2:1 is an octave, 3/2 is a fifth, 5/4 and 6/5 are thirds'
> (etc.), doesn't necessarily require this.

I've only been skimming many of the posts in this digression
of this thread, but it struck me that no one has mentioned
MOS (or equivalently, DES = distributionally even scales).

The 7MOS=5L+2s structure is one automatic result of
generator=4th (or 5th) and period=octave, if the generator
is anywhere near any form of meantone or pythagorean
in size.

Just thought i should add that into the discussion.

BTW, my Encyclopedia page for DES is blank and i could
use a good definition of it. Any takers? ... I know that
most theorists consider it essentially equivalent to
MOS, so perhaps the definition can be very short and
just link to MOS ... or is there some significant
difference?

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

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
> Hm, it must just be that Gene's and Graham's websites have a similar
> layout and aesthetic, so in the absence of a signature I took one
> for the other. (However, I should have noticed the different style
> of writing...)

Ah, HTML as it was meant to be used -- looking the same. Why
is Wikipedia so awesome? I think it's largely because you don't
have to spend x seconds or minutes learning how to read a page
before you can start reading it. The original www was like this.
You could do serious research from a web search. Now you need a
portal like Wikipedia to protect you. Consumer industry working
its magic again!

-Carl

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
> You can have a diatonic scale with different sizes of large
> steps: 9:8 and 10:9. In fact, this is the original meaning
> of a diatonic scale.

Really? What's the source for that?

-Carl

Greetings,

I don't know if it has been mentioned, but the great American critic and historian, Jacques Barzun celebrates his 100th birthday November 30. He is well, but somewhat frail.

Some may remember that Dr. Barzun was a keen supporter of Harry Partch. He also wrote extensively on Berlioz and a dozen dozen other subjects (like baseball).

I am doing a concert in Calgary (that sadly will not have any microtonal music in it) in his honour.

But I hope that by bringing this to the attention of the group, others who are performing might place some tribute mention in their performances. Also, if any are writers, dare I suggest a brief article written for your favourite journal?

In my estimation, he is one of the greatest minds that North America can claim for itself. Please, let us do something to mark the birthday of the remarkable man!

All best wishes,

Gordon Rumson

Hi Gordon,

Thanks for that info. I've long been an admirer of
his books ... but i had no idea that he was still
alive! Wow.

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

--- In tuning@yahoogroups.com, Gordon Rumson <rumsong@...> wrote:
>
> Greetings,
>
> I don't know if it has been mentioned, but the great
> American critic and historian, Jacques Barzun celebrates
> his 100th birthday November 30. He is well, but
> somewhat frail.
>
> <snip>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, Graham Breed <gbreed@> wrote:
> > You can have a diatonic scale with different sizes of large
> > steps: 9:8 and 10:9. In fact, this is the original meaning
> > of a diatonic scale.
>
> Really? What's the source for that?

The "diatonic scale"

http://tonalsoft.com/enc/d/diatonic-scale.aspx

is a usage which grew out of the "diatonic genus"

http://tonalsoft.com/enc/d/diatonic-genus.aspx

"Diatonic" means "thru tones", and was the term
which Greek theorists used to describe a type of
tetrachord division which is composed of
predominantly "whole-tones".

For most theoretical descriptions, this means that
each tetrachord was to be divided into two large
"tones" and one smaller interval. The exception is
Ptolemy's _diatonon homalon_, which i translate as
"even diatonic" or "equable diatonic" ... Partch
carried over the translation as "smooth diatonic"
from the books he had read. In this division, the
tetrachord is divided into three intervals of
almost equal size.

The ancient greek PIS also provided for a
"tone of disjunction" in two places: one between
the tetrachords diezeugmenon and meson, and
another between tetrachord hypaton and the
"added tone at the bottom", proslambanomenos.

Despite their adherence to tetrachordal similarity,
the Greeks did indeed recognize that notes an
"octave" apart are in some way synonymous. So a
scale which follows these tetrachord divisions
(except for Ptolemy's even diatonic) and recognizes
octave-equivalence, would indeed be composed almost
entirely of "tones" (i.e., "L"), with a couple of
"semitones" ("s"). In fact, any octave-species
would contain 5L+2s.

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

I have discussed again with the author of the translation, Paul
Poletti. (NB! NOT Andreas anybody.) His use of 'frequency domain' to
talk about schweben is somewhat eccentric.

Paul actually agrees with me that schweben is, as Werckmeister
habitually uses the word, a tempering of an interval by a certain
*ratio* or proportion. (Look at the source! -
http://harpsichords.pbwiki.com/Werckmeister_Anmerckungen_1698)

Werckmeister talks of intervals that 'schweben' by 1/4 comma or 1/12
comma, for example. (As does Neidhardt.) In this context it is
absolutely clear that it refers to the amound of tempering, and not to
the beat rate, which in the same context, here, Werckmeister calls
'schlagen' or 'tremuliren'.

Read carefully:

"even when one already knows that one note of a consonance should
hover [schweben] by one-, two-, or three-quarters of a comma against
the other, our poor sense of hearing cannot judge accurately if the
[degree of] hovering [schwebung] is too great or too small, or if they
beat [schlagen] too slowly or too rapidly."

Now, as to the encyclopedia...

The so-called 'equal-beating' temperaments are a pseudo-historical
mirage, brought on by 20th-century writers who, to further their own
purposes, imagined that 17th- and 18th-century authors were incapable
of talking about amounts of temperament using the word 'schweben', and
did not know that the same tempered interval would beat more or less
quickly depending on its pitch.

Whereas if one simply reads the sources it comes out very clearly that
the word was often used to denote the amount of tempering in an
interval, therefore 'gleichschwebend' might, in some contexts, very
easily and logically mean that every interval is tempered by the same
amount.

There is NO evidence or reason to believe that 'schweben', as used by
writers on tuning and temperament, ever meant EXCLUSIVELY the beat
rate. As soon as you realise this, the whole edifice of
'equal-beating' crashes down as a simple mistake of overly
prescriptive translation.

Now I can accept that 'wavering' is, in the dictionary, a correct
translation of 'schweben'. But when Werckmeister talks about a fifth
where the upper note is 'schwebend' by 1/12 comma relative to the
lower note, it would be positively misleading to call it 'wavering'.
What he means is the tempering of the interval.

When people have READ THE SOURCE, which is here:

http://harpsichords.pbwiki.com/Werckmeister_Anmerckungen_1698

we can really start discussing what things mean there. At the moment
it seems as if only Andreas, Paul and I have read it.

~~~T~~~

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
>
> Hi Daniel and Tom,
>
> --- In tuning@yahoogroups.com, Daniel Wolf <djwolf@> wrote:
> >
> > Tom Dent wrote:
> >
> > > The problem with 'wavering' is that it is not a translation of
> > > 'schweben(d)
> >
> > It is a perfectly good translation of schweben.
> >
> > <snip>
> >
> > > whereas 'tremulieren' or 'zittern' is the audible
> > > beating which under many circumstances is the physical
> > > manifestation of a tempered interval.
> >
> > No, the beating is one specific attribute of a tempered
> > interval, wavering (schweben) -- like a voter who can't
> > quite make her mind up -- is a perfectly good description
> > of intervals that are intended to have two meanings
> > (to pun): as a third to another tone, and simultaneously,
> > as a a pitch related by a series of perfect fifths.
>
>
> I agree with Daniel that "wavering" is a good translation
> of "schweben" as used in this context.
>
>
> And again speaking of Schoenberg, in my Encyclopedia
> page for "gleichschwebende" i note his use of the term
> "schwebende" to refer to what is usually translated as
> "suspended tonality" ... and i think "wavering tonality"
> is a better translation.
>
> http://tonalsoft.com/enc/g/gleichschwebende.aspx
>
> Note also that, as i point out in that page, "schwebend"
> historically first referred to equally-beating tempered
> intervals, but since then has been taken up by most
> German-language theorists to refer to an equal logarithmic
> amount of tempering. So apparently it does have both
> of the meanings that you two are arguing about.
>
>
> -monz
>

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
>
>
> I have discussed again with the author of the translation, Paul
> Poletti. (NB! NOT Andreas anybody.) His use of 'frequency domain'
>to
> talk about schweben is somewhat eccentric.
>
> Paul actually agrees with me that schweben is, as Werckmeister
> habitually uses the word, a tempering of an interval by a certain
> *ratio* or proportion. (Look at the source! -
> http://harpsichords.pbwiki.com/Werckmeister_Anmerckungen_1698)
>
> Werckmeister talks of intervals that 'schweben' by 1/4 comma or
1/12
> comma, for example. (As does Neidhardt.) In this context it is
> absolutely clear that it refers to the amound of tempering, and
not to
> the beat rate, which in the same context, here, Werckmeister calls
> 'schlagen' or 'tremuliren'.
>
> Read carefully:
>
> "even when one already knows that one note of a consonance should
> hover [schweben] by one-, two-, or three-quarters of a comma
against
> the other, our poor sense of hearing cannot judge accurately if the
> [degree of] hovering [schwebung] is too great or too small, or if
they
> beat [schlagen] too slowly or too rapidly."
>
> Now, as to the encyclopedia...
>
> The so-called 'equal-beating' temperaments are a pseudo-historical
> mirage, brought on by 20th-century writers who, to further their
own
> purposes, imagined that 17th- and 18th-century authors were
incapable
> of talking about amounts of temperament using the word 'schweben',
and
> did not know that the same tempered interval would beat more or
less
> quickly depending on its pitch.
>
> Whereas if one simply reads the sources it comes out very clearly
that
> the word was often used to denote the amount of tempering in an
> interval, therefore 'gleichschwebend' might, in some contexts, very
> easily and logically mean that every interval is tempered by the
same
> amount.
>
> There is NO evidence or reason to believe that 'schweben', as used
by
> writers on tuning and temperament, ever meant EXCLUSIVELY the beat
> rate. As soon as you realise this, the whole edifice of
> 'equal-beating' crashes down as a simple mistake of overly
> prescriptive translation.
>
> Now I can accept that 'wavering' is, in the dictionary, a correct
> translation of 'schweben'. But when Werckmeister talks about a
fifth
> where the upper note is 'schwebend' by 1/12 comma relative to the
> lower note, it would be positively misleading to call
it 'wavering'.
> What he means is the tempering of the interval.
>
> When people have READ THE SOURCE, which is here:
>
> http://harpsichords.pbwiki.com/Werckmeister_Anmerckungen_1698
>
> we can really start discussing what things mean there. At the
moment
> it seems as if only Andreas, Paul and I have read it.

I have read it, some time ago, and would translate "schweben"
as "floating", not exactly in the usual English sense of
floating "on", but floating "in" or even just "floating...", almost
as in levitating. "Hovering" is excellent, IMO. The concept/effect
should be no mystery at all to anyone who tunes by ear.

Die "Schwebung" clearly refers to the audible effect of tempering
Just intervals. Beating is a subset of that; once in the paper we
might percieve that the tempering and its beating byproduct are
lumped together:

"so würden doch c und c' als 4. und 2. Fuß noch
einmahl so langsam tremuliren / oder schweben / als c' und c'' "

but I think it's clear that this is placing the two effects
in a gradation, for the one interpretation that works throughout
would be "the audible effect/amount of tempering a Just interval".

It is also clear from this paper that our dear Werckchen
wouldn't hesitate to answer my dogged question "does it or
does it not make a difference in the musical character of an
interval whether you temper it up or down, yes or no?"
with a resounding "yes", so you'll forgive me if I choose
to interpret in a most colorful way the absence, on this list,
of a straight answer to that question, LOL.

-Cameron Bobro

Carl Lumma wrote:
> --- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
> >>You can have a diatonic scale with different sizes of large >>steps: 9:8 and 10:9. In fact, this is the original meaning >>of a diatonic scale.
> > > Really? What's the source for that?

Okay, not originally, but JI diatonics have been around a long time.

Graham

Carl Lumma wrote:

> I would phrase it like, the theory of temperaments doesn't
> say anything about what should be tempered: all intervals,
> melodic intervals only, etc. And in fact, it should point
> this out.

I'm not sure what that means, but I can probably agree with it ;) If it's not talking about the tunings it isn't a theory of temperaments. That's why I suggested "regular mapping" as the catch-all.

>>But adaptive tuning of choirs and meantone tempering of
>>keyboard instruments was the dominant paradigm of the 16th
>>century.
> > It was? I see 5-limit JI in Willaert, but not adaptive
> tuning. At least, not anything as principled as Vicentino's
> 2nd archicembalo tuning.

Choirs are naturally going to adapt.

> I'm missing the bit where you say Vicentino only made an
> instrument because he happenned to find a rich patron.
> Well, there is a lot of luck in money, but there's other
> stuff too; if Vincentino got funded that might be telling
> us something. In any case, building an instrument is a
> strong demonstration of understanding that I'm not seeing
> the equivalent of in texts by other theorists.

One thing it tells us is that he probably got a reference from Willaert. But yes, he was a leading member of the 16th Century avant garde. That shouldn't be enough to get on your short list.

> In short, to me, the advocacy of 5-limit JI is one thing,
> that I suspect many were doing in the 16th century. Using
> two chains of meantone a 1/4-comma apart is much more
> unique... in fact I'm unaware of it being suggested in
> the period between 1555 and 1999.

So the fact it was only mentioned as an afterthought, and not picked up on or rediscovered for so long, shows us that it wasn't considered to be important. That's the interesting thing this history tells us. I'd rather give credit to theorists whose ideas got absorbed by the mainstream and ended up leading to the consideration of mappings.

If Vicentino gets a mention, why not the various theorists who proposed dynamic tunings in the 20th Century? Waage may have been the first. And perhaps Sethares is one of your two phantom theorists (I guess not but I think he has a claim).

There's a danger here of opening the floodgates to lots of little theories. You can find all kinds of papers that mention one piece of a puzzle. The real pioneers are those who try to put the puzzle together, or make a considerable advance.

Another thing that deserves mention is this paper:

A Matrix Technique for Analysing Musical Tuning
C Karp - Acustica, 1984

I can't remember exactly what it says. I think it talks about different ways of representing JI and defines quarter comma meantone in matrix terms. By the time I read it, it didn't have anything new to teach me. Maybe it lists mappings as vectors, in which case it could well be the first such example (but what exactly does Helmholtz/Ellis do?)

Anyway, I decided to do a Google search. It's mentioned by my website, a page on lattices by Carter Scholz, and this paper by Robert Kelley:

http://www.robertkelleyphd.com/ReconcilingTonalConflicts.pdf

In fact I already have that PDF, which is good because I can't reach the site today. One interesting thing is that on older, conference paper doesn't mention Karp, so he probably didn't learn anything from it either.

Oh, this is gettable:

http://garnet.acns.fsu.edu/~rtk1218/DT&JI.pdf

Kelley's papers are notable in that they include a matrix of mappings for "diatonic spelling" of 5-limit just intonation. That probably is the first reference for mapping vectors in real music theory. The point of the paper is also that considerations of tuning are still valid even if the tuning concerned was never the composer's intention. That's very much a regular mapping idea, and Kelley would be a regular mapping pioneer if he'd done this before us. As it is, I wonder how much he knows about the new paradigm. Maybe something because he does cite Monzo.

Next, how about a Google Scholar search for the Karp paper? Well, the only citations are from Kelley. Same as for Fokker's periodicity block paper (which Google Scholar doesn't know is online).

Graham

Hi Tom,

Wow, thanks very much for that. In fact, i have not
read any of the sources from c.1600-1800 (my main
interests in tuning lie before and after that period),
so my Encyclopedia entry needs to be corrected.

I appreciate your explanation, and would like to
quote the whole thing on my page, after i edit what's
already there.

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

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

> Now, as to the encyclopedia...
>
> The so-called 'equal-beating' temperaments are a
> pseudo-historical mirage, brought on by 20th-century
> writers who, to further their own purposes, imagined
> that 17th- and 18th-century authors were incapable
> of talking about amounts of temperament using the
> word 'schweben', and did not know that the same tempered
> interval would beat more or less quickly depending on
> its pitch.
>
> Whereas if one simply reads the sources it comes out
> very clearly that the word was often used to denote
> the amount of tempering in an interval, herefore
> 'gleichschwebend' might, in some contexts, very
> easily and logically mean that every interval is
> tempered by the same amount.
>
> There is NO evidence or reason to believe that
> 'schweben', as used by writers on tuning and temperament,
> ever meant EXCLUSIVELY the beat rate. As soon as you
> realise this, the whole edifice of 'equal-beating'
> crashes down as a simple mistake of overly prescriptive
> translation.
>
> Now I can accept that 'wavering' is, in the dictionary,
> a correct translation of 'schweben'. But when Werckmeister
> talks about a fifth where the upper note is 'schwebend'
> by 1/12 comma relative to the lower note, it would be
> positively misleading to call it 'wavering'. What he
> means is the tempering of the interval.
>
> When people have READ THE SOURCE, which is here:
>
> http://harpsichords.pbwiki.com/Werckmeister_Anmerckungen_1698
>
> we can really start discussing what things mean there.
> At the moment it seems as if only Andreas, Paul and I
> have read it.

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
>
> Hi Tom,
>
>
> Wow, thanks very much for that. In fact, i have not
> read any of the sources from c.1600-1800 (my main
> interests in tuning lie before and after that period),

Oops, my bad ... i *have* read Rameau and Fux,
both from that period. But no other sources besides
those two ... and Rameau is in French, and Fux's theory
is based on Palestrina and quasi-5-limit-JI, so neither
of them help with "schwebend".

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

Graham wrote...
> >>But adaptive tuning of choirs and meantone tempering of
> >>keyboard instruments was the dominant paradigm of the 16th
> >>century.
> >
> > It was? I see 5-limit JI in Willaert, but not adaptive
> > tuning. At least, not anything as principled as Vicentino's
> > 2nd archicembalo tuning.
>
> Choirs are naturally going to adapt.

They are, but that's a far cry from a theoretical understanding
of how they do it, which is itself a far cry from building a
keyboard instrument that does the same thing.

> > In short, to me, the advocacy of 5-limit JI is one thing,
> > that I suspect many were doing in the 16th century. Using
> > two chains of meantone a 1/4-comma apart is much more
> > unique... in fact I'm unaware of it being suggested in
> > the period between 1555 and 1999.
>
> So the fact it was only mentioned as an afterthought, and
> not picked up on or rediscovered for so long, shows us that
> it wasn't considered to be important. That's the
> interesting thing this history tells us. I'd rather give
> credit to theorists whose ideas got absorbed by the
> mainstream and ended up leading to the consideration of
> mappings.

If we don't say anything about it, people will assume that
temperament requires the impurity of simultaneous intervals.
But in fact, impurity of melodic intervals only is an
option. Don't you think it's worth saying so, to prevent
wrong assumptions? The fact that it wasn't picked up by
others makes Vicentino's contribution that much more
valuable from my point of view.

-Carl

Carl Lumma wrote:
> Graham wrote...
>>Choirs are naturally going to adapt.

> They are, but that's a far cry from a theoretical understanding
> of how they do it, which is itself a far cry from building a
> keyboard instrument that does the same thing.

Right, and we don't have any theoretical descriptions from the 16th century. Neither do we know why Vicentino built these instruments (but probably it was for the enharmonic scales). We don't even know why Vicentino proposed the second tuning. Probably it was for theoretical purposes -- to compare the sounds of just and tempered chords.

> If we don't say anything about it, people will assume that
> temperament requires the impurity of simultaneous intervals.
> But in fact, impurity of melodic intervals only is an
> option. Don't you think it's worth saying so, to prevent
> wrong assumptions? The fact that it wasn't picked up by
> others makes Vicentino's contribution that much more
> valuable from my point of view.

You can talk about it if you like. You'll probably mention Vicentino in connection with it, but more to the point would be to credit whichever modern theorist proposed the second tuning as a basis for adaptive temperament.

It's strange to say that a contribution's valuable because people found it useless.

Graham

[ Attachment content not displayed ]

Tom --

Thank you for pointing to an online source of the text; it saves me a
trip to the library. Upon review of the text, it's clear -- if the
disctinctions between terms are in fact important, that we need a
term with both an adjective and a noun form. Hoovering, wavering,
floating (the term my wife and son prefer), and hanging won't quite
do, so I cam round to suspended/suspension, despite its use in other
musical contexts, simply because schweben/Schwebungen are always in
relationship to a pure interval, hung or suspended from such.
However, if what we're after is a practical modern translation, the
best might be to simply go with temper/tempering. Werckmeister,
conveniently, reserves the noun _Temperatur_ for the entire keyboard
temperament, so these terms would be available in an English
translation without much ambiguity. But the concern here seems to be
not to give up the connotations of "schweben". So, here goes: a bit
of quick translation of some relevant parts of the text:

"Temperament originated because, in the use of the keyboard, one
cannot have all consonances pure when one moves from one chord to
another, so one must therefore give something to one consonance and
take something away from another so that a tolerable and pleasing
temperament is thereby created. Now when one consonance is somewhat
higher or lower in comparison to the others, one calls this suspended
(Schweben). This name comes principally from organ builders, because
when two pipes are tuned together and the (tunings of the two) are
nearly pure, then such pipes will make -- if they are played
together -- a trembling (Tremoren) or shaking (Zittern). Now, the
closer the tuning (of the two together) is, the slower the trembling
will be, but when they are finally tuned together, the trembling or
the shaking (Beben) allows itself no longer to be heard, and such
pairs whistle as if they were one pipe: this trembling or shaking is
called suspending (Schweben) by the organ builders..."

"When the upper key is too high compared with other, this is called
suspended (Schweben) above; if it is too low one calls it suspended
below or vice versa. If the upper key is too high compared to the
lower, one says that the lower is is suspended below, in the same way
that a tail hangs from a dog and a dog can hang from its tail. These
terms are often the source of misunderstandings, so that one cannot
understand another, and through which quarrels and conflicts between
musicians are created..."

"But when these suspensions (Schwebungen) or tremblings on stringed
instruments -- spinets, clavichords, and the like -- cannot so
clearly be understood..."

"...: this requires experience to know whether these consonances
against the others must be suspended by 1, 2, or 3 quarter commas.
So, indeed, the poor ear cannot accurately know whether these
suspensions are too large or too small, or whether they beat
(schlagen) too slowly or too rapidly. There is also a great
difference if two large or two small pipes are tuned together. For
example, I have tuned the octaves to c and c'...
" ).

djw

> Graham wrote...
Right, and we don't have any theoretical descriptions from
the 16th century. Neither do we know why Vicentino built
these instruments (but probably it was for the enharmonic
scales).

Hi Graham,

Perhaps you never saw Vicentino's book in English translation? Maybe you
never heard any of his musical compositions (which I find damn good). Did you
know Vicentino built keyboards to play upon, as he was a great keyboardist.
BTW, his ideas did catch on...if slowly. (His name is included in Walther's
Lexicon, and probably most any source of music history.)

The enharmonic bit was a result of the Renaissance's late translations of
its musical writings, and their limited absorption. Enharmonic stood for the
use of smaller intervals, indicated with a small dot above the notehead in
order to advance the pitch a dieses. Vicentino wanted to force fit music's
enharmonic genus, as with other "sciences" described in antiquity, into the
Renaissance. In that he "failed."

When I first looked at Carl's list, Vicentino was notable as the only
professional musician. That Vicentino could imagine, design, and perform, only to
gain fame through history for his several accomplishments speaks bounds to
me. We performed one of Vicentino's a cappella works on the AFMM and it was
quite a success. A significant performance error prevents its commercial
release.

best, Johnny

p.s. Vicentino was noted for being the most adventurous of Willaret's
students (in basic contrast to Zarlino)

************************************** See what's new at http://www.aol.com

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, Charles Lucy <lucy@> wrote:
> > Carl;
> >
> > You're being an old prune again;-)
> >
> > I suggest that you read Guy Murchie if the concept is too
> > "troublesome" for you.
>
> With sources like that, no wonder you're coming up with stuff
> like that chart.

I guess we should read that as a neutral statement rather than
a snide remark, as a disparaging tone would have religious
implications, and I would guess we don't want to go there.

Charles Lucy, more power to you if
you're thinking in religious or spiritual or metaphysical ways.
Why the hell not?

-Cameron Bobro

No my thinking is more scientific/engineering than religious "woo-woo"; I consider "religion" (especially the obsessive type) to be a "psychiatric" condition.

I am more sympathetic to Richard Dawkins ideas than to any religious leader; although Dawkins's concept and logic may make sense, his presentation techniques are often severely flawed.

Charles Lucy lucy@lucytune.com

----- Promoting global harmony through LucyTuning -----

For information on LucyTuning go to: http://www.lucytune.com

LucyTuned Lullabies (from around the world):
http://www.lullabies.co.uk

Skype user = lucytune

http://www.myspace.com/lucytuning

On 28 Sep 2007, at 22:51, Cameron Bobro wrote:

> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
> >
> > --- In tuning@yahoogroups.com, Charles Lucy <lucy@> wrote:
> > > Carl;
> > >
> > > You're being an old prune again;-)
> > >
> > > I suggest that you read Guy Murchie if the concept is too
> > > "troublesome" for you.
> >
> > With sources like that, no wonder you're coming up with stuff
> > like that chart.
>
> I guess we should read that as a neutral statement rather than
> a snide remark, as a disparaging tone would have religious
> implications, and I would guess we don't want to go there.
>
> Charles Lucy, more power to you if
> you're thinking in religious or spiritual or metaphysical ways.
> Why the hell not?
>
> -Cameron Bobro
>
>
>

> Charles Lucy, more power to you if you're thinking in
> religious or spiritual or metaphysical ways.
> Why the hell not?
>
> -Cameron Bobro

There's a difference between metaphysics/religion, and
pseudoscience.

I am a bit put out that our moderator did exactly what he
said he'd never allow, by apparently dishing out a medical
diagnosis on the list. However I don't think your
accusations that people change the subject on you are
valid. What I see is people interested in understanding
your language / claims, and you (unintentionally, probably)
making it rather difficult to do so.

-Carl

If you want to insult me due to my "obsession" with Islam, at least desist from doing it on the tuning list Charles!

Oz.

----- Original Message -----
From: Charles Lucy
To: tuning@yahoogroups.com
Sent: 29 Eylül 2007 Cumartesi 2:10
Subject: Re: [tuning] Re: very low note from Perseus cluster - Guy Murchie - Music of The Spheres.

No my thinking is more scientific/engineering than religious "woo-woo"; I consider "religion" (especially the obsessive type) to be a "psychiatric" condition.

I am more sympathetic to Richard Dawkins ideas than to any religious leader; although Dawkins's concept and logic may make sense, his presentation techniques are often severely flawed.

[ Attachment content not displayed ]

Nothing personal Ozan; each to their own; my attitude is just how I
view things.

If you wish to take offence or view my perception as condescending
that is entirely up to you, no offence was intended to anyone. I
support the idea of freedom of speech.

I am actually usually tolerant to all the diverse religious views,
I'll listen (within reason) just don't expect me to necessarily agree
or argue with you or anyone else about religion.

Life's too short;-)

best wishes.

Charles Lucy lucy@lucytune.com

----- Promoting global harmony through LucyTuning -----

For information on LucyTuning go to: http://www.lucytune.com

LucyTuned Lullabies (from around the world):
http://www.lullabies.co.uk

Skype user = lucytune

http://www.myspace.com/lucytuning

On 29 Sep 2007, at 00:24, Ozan Yarman wrote:

>
> If you want to insult me due to my "obsession" with Islam, at least
> desist from doing it on the tuning list Charles!
>
> Oz.
>
> ----- Original Message -----
> From: Charles Lucy
> To: tuning@yahoogroups.com
> Sent: 29 Eylül 2007 Cumartesi 2:10
> Subject: Re: [tuning] Re: very low note from Perseus cluster - Guy
> Murchie - Music of The Spheres.
>
> No my thinking is more scientific/engineering than religious "woo-
> woo"; I consider "religion" (especially the obsessive type) to be a
> "psychiatric" condition.
>
> I am more sympathetic to Richard Dawkins ideas than to any
> religious leader; although Dawkins's concept and logic may make
> sense, his presentation techniques are often severely flawed.
>
>

This is hardly the place to announce one's regard of other people's faiths. If you heard my sincere opinion about your ideas, you would surely be appalled.

So, do not court with disaster.

Oz.
----- Original Message -----
From: Charles Lucy
To: tuning@yahoogroups.com
Sent: 29 Eylül 2007 Cumartesi 3:09
Subject: Re: [tuning] Re: very low note from Perseus cluster - Guy Murchie - Music of The Spheres.

Nothing personal Ozan; each to their own; my attitude is just how I view things.

If you wish to take offence or view my perception as condescending that is entirely up to you, no offence was intended to anyone. I support the idea of freedom of speech.

I am actually usually tolerant to all the diverse religious views, I'll listen (within reason) just don't expect me to necessarily agree or argue with you or anyone else about religion.

Life's too short;-)

best wishes.

Afmmjr@aol.com wrote:
>>Graham wrote...
> > Right, and we don't have any theoretical descriptions from
> the 16th century. Neither do we know why Vicentino built
> these instruments (but probably it was for the enharmonic
> scales).
> > Hi Graham,
> > Perhaps you never saw Vicentino's book in English translation? Maybe you > never heard any of his musical compositions (which I find damn good). Did you > know Vicentino built keyboards to play upon, as he was a great keyboardist. > BTW, his ideas did catch on...if slowly. (His name is included in Walther's > Lexicon, and probably most any source of music history.)

I've not only seen but read the translation. I haven't heard any performances but I've heard good reports of them. Of course I know he built keyboards. I said in the quote that he built keyboards.

Nowhere have I seen him outline a theory of adaptive tuning.

> The enharmonic bit was a result of the Renaissance's late translations of > its musical writings, and their limited absorption. Enharmonic stood for the > use of smaller intervals, indicated with a small dot above the notehead in > order to advance the pitch a dieses. Vicentino wanted to force fit music's > enharmonic genus, as with other "sciences" described in antiquity, into the > Renaissance. In that he "failed."

What does this have to do with adaptive tuning?

> When I first looked at Carl's list, Vicentino was notable as the only > professional musician. That Vicentino could imagine, design, and perform, only to > gain fame through history for his several accomplishments speaks bounds to > me. We performed one of Vicentino's a cappella works on the AFMM and it was > quite a success. A significant performance error prevents its commercial > release.

Ah, so this is about professional solidarity! What was Partch?

> p.s. Vicentino was noted for being the most adventurous of Willaret's > students (in basic contrast to Zarlino)

Do you have evidence that Vicentino was a student of Willaert?

Graham

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

> I am a bit put out that our moderator did exactly what he
> said he'd never allow, by apparently dishing out a medical
> diagnosis on the list. However I don't think your
> accusations that people change the subject on you are
> valid. What I see is people interested in understanding
> your language / claims, and you (unintentionally, probably)
> making it rather difficult to do so.

Well, right there we're off on the wrong foot. Why do you
say "claims"? Having been a newscaster (Slovenian national
radio, two years) and one of these
http://www.usacapoc.army.mil/default.aspx
guys, I am more than aware of the how the word "claim" actually
functions in the big bad world. Good guys and scientists "state"
and "propose", bad guys and kooks "claim". (If you want
to spot an example of a claim being pawned off as a fact,
keep your ears open for "made the case...").

If you were unaware of this, I sure hope you don't vote.
If you were not unaware of this, then using the word
"claim" to describe the things I say is a very transparent
attempt to disparage them from the git-go. Do you
think that such crude old rhetorical tricks are not glaringly
obvious to the very people whose relations
to the ideas you espouse ( :-) ) are most important,
in terms of turning theory into practice and vice versa,
in the longterm?

I state my opinions and experiences as such, and propose
ideas. It's very easy to spot when I'm making a claim, it
goes like this: "I claim..." Not only that, I defy you
to find anyone here as open and shameless about
accepting the artistic validity of things I know to
be of far-fetched or even highly dubious scientific validity:
has anyone else here referred in a postive way to osteomancy?
Anyone else care to stand up and say it's perfectly
okay in the artistic world, and therefore in tuning
music, to consider Pi in terms of Thelema? The irony
is that when I do speak in terms of "fact", I'm extremely
conservative and I believe quite accurate- and certainly
open to straight-up correction on falsifiable issues, which
should be obvious from the times I have written "correct me
if I'm wrong".

> What I see is people interested in understanding
> your language / claims, and you (unintentionally, probably)
> making it rather difficult to do so.

Then once I again, I will go very, very slowly:

Does tempering an interval upward have a different effect
on the musical character of an interval than tempering
the same interval downward?

Anybody?

I am not talking about brute dissonance, beat rates, or
harmonic entropy. I am talking about musical character,
the specific nature of which is subjective of course.

Anybody who cares to understand what I say needs only
to answer that question with a simple "yes" or "no".

-Cameron Bobro

Cameron wrote...
> Then once I again, I will go very, very slowly:
>
> Does tempering an interval upward have a different effect
> on the musical character of an interval than tempering
> the same interval downward?

I've answered this question maybe half a dozen times,
and most recently last week.

> I am not talking about brute dissonance, beat rates, or
> harmonic entropy. I am talking about musical character,
> the specific nature of which is subjective of course.

If it's subjective than what's the point in talking
about it?

> Anybody who cares to understand what I say needs only
> to answer that question with a simple "yes" or "no".

So you only accept answers of the type you specify?

-Carl

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
> > Perhaps you never saw Vicentino's book in English translation?
//
> I've not only seen but read the translation.

Where did you find it?

-Carl

Nicolo Vicentino "Ancient Music Adapted to Modern Practice"
(Translated with Introducton and Notes by Maria Rika Maniates), Yale
University Press, New Haven and London, 1996, pp. 487).

The second paragraph of the Introduction written by Vicentino scholar
Maniates read:

"Vicentino studied in Venice under Adrian Willaert sometime during the
1950s."

Johnny

--- In _tuning@yahoogroups.com_
(/tuning/post?postID=kbG3ADRv6TWTWwRtbKnrFb4tao4MnjIdPefm0fGDPW8wjMA6nlTAbSapgmmkejlyiAW
1CujsrehHmN85f0XV) , Graham Breed <gbreed@...> wrote:
> > Perhaps you never saw Vicentino's book in English translation?
//
> I've not only seen but read the translation.

CL: Where did you find it?

************************************** See what's new at http://www.aol.com

Hi Johnny and Carl,

--- In tuning@yahoogroups.com, Afmmjr@... wrote:
>
>
> Nicolo Vicentino "Ancient Music Adapted to Modern Practice"
> (Translated with Introducton and Notes by Maria Rika Maniates),
Yale
> University Press, New Haven and London, 1996, pp. 487).
>
> The second paragraph of the Introduction written
> by Vicentino scholar Maniates read:
>
> "Vicentino studied in Venice under Adrian Willaert
> sometime during the 1950s."

That date is certainly a typo ... for "1550s"?

Carl, Vicentino's book is readily available. I've had
it on my Amazon wish list for a couple of years now ...
i'd better shell out the money before it's gone.

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

> Carl, Vicentino's book is readily available. I've had
> it on my Amazon wish list for a couple of years now ...
> i'd better shell out the money before it's gone.

If it's not on the web, what good is it?

-Carl

Carl Lumma wrote:
> Cameron wrote...
> >>Then once I again, I will go very, very slowly:
>>
>>Does tempering an interval upward have a different effect
>>on the musical character of an interval than tempering >>the same interval downward? > > I've answered this question maybe half a dozen times,
> and most recently last week.

I do mentions this with regard to dissonance in my detailed and unimportant "prime errors and complexities" PDF. Something like

http://x31eq.com/primerr.pdf

and the latest version (still oh so much work to do!) in the Tuning Math files area.

But, rather than joining the "I refer the honourable gentleman to my prevous answer" game, I'll come out with a "yes".

>>I am not talking about brute dissonance, beat rates, or
>>harmonic entropy. I am talking about musical character,
>>the specific nature of which is subjective of course. > > If it's subjective than what's the point in talking
> about it?

It can often be worth talking about subjective things. Most of what we talk about here is subjective. Dissonance, for example, is subjective. Despite that, a lot of us are in enough agreement about it that we can put numbers to it and construct tuning systems on the basis of those numbers. It looks like most people generally agree with us on it as well (but not its importance). Where people disagree, it's useful to know why and how, even if it's not practical to accommodate them.

Talking about matters of taste has somewhat less value. Arguing about them is something I'll happily label as "pointless". And if somebody tells you to tear up your apple trees because of their taste in orange juice I think you're justified in smiling politely and backing toward the door.

Graham

Carl Lumma wrote:
>>Carl, Vicentino's book is readily available. I've had
>>it on my Amazon wish list for a couple of years now ...
>>i'd better shell out the money before it's gone.
> > > If it's not on the web, what good is it?

It's a good description of a particular microtonal system, in its historical context. Possibly the first European microtonal theory with musical examples. There are also things about 16th century music which are of interest to historians.

For what you were talking about, the second archicembalo tuning, you might find a Google Books search gives you the page you want.

Graham

Hi Graham and Carl,

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> Carl Lumma wrote:
>
> >> [monz:]
> >> Carl, Vicentino's book is readily available. I've had
> >> it on my Amazon wish list for a couple of years now ...
> >> i'd better shell out the money before it's gone.
> >
> >
> > If it's not on the web, what good is it?
>
> It's a good description of a particular microtonal system,
> in its historical context. Possibly the first European
> microtonal theory with musical examples.

I'm pretty sure that award goes to Marchetto of Padua (1318).
However ...

IIRC Marchetto's _Lucidarium_ has only short musical
examples and not complete pieces. I only finally just
ordered Vicentino's book, so i don't know yet, but
i believe that it has at least one complete piece
illustrating Vicentino's version of the enharmonic genus.

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

monz wrote:

> I'm pretty sure that award goes to Marchetto of Padua (1318).
> However ...
> > IIRC Marchetto's _Lucidarium_ has only short musical
> examples and not complete pieces. I only finally just
> ordered Vicentino's book, so i don't know yet, but
> i believe that it has at least one complete piece > illustrating Vicentino's version of the enharmonic genus.

Hope you enjoy Vicentino! Yes, there's a complete piece. I think one for the enharmonic and one for mixed genera. But to Marchetto -- how microtonal are his examples?

Graham

Graham wrote...

> But, rather than joining the "I refer the honourable
> gentleman to my prevous answer" game, I'll come out with a
> "yes".

Cameron and I had a back and forth on this last week.
He replied at least once to my reply, so I know he knows
my answer. He doesn't eloborate any objections to my
answer, just keeps on accusing me of not answering him.

Prior to last week I believe I have even posted spreadsheets
with pretty graphs on this topic.

-Carl

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> Carl Lumma wrote:
> >>Carl, Vicentino's book is readily available. I've had
> >>it on my Amazon wish list for a couple of years now ...
> >>i'd better shell out the money before it's gone.
> >
> >
> > If it's not on the web, what good is it?
//
>
> For what you were talking about, the second archicembalo
> tuning, you might find a Google Books search gives you the
> page you want.

It looks like that section isn't part of the preview. And
for some stupid reason I can't buy it.

-Carl

Hi Graham,

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> monz wrote:
>
> > I'm pretty sure that award goes to Marchetto of Padua (1318).
> > However ...
> >
> > IIRC Marchetto's _Lucidarium_ has only short musical
> > examples and not complete pieces. I only finally just
> > ordered Vicentino's book, so i don't know yet, but
> > i believe that it has at least one complete piece
> > illustrating Vicentino's version of the enharmonic genus.
>
> Hope you enjoy Vicentino!

Me too! ... i'm so busy these days that a stack of copies
of Mahler manuscripts which i've coveted for at least
10 years arrived more than a month ago and i haven't
even unwrapped them yet! Don't know when i'll have time
to read Vicentino ...

> Yes, there's a complete piece. I think one for the
> enharmonic and one for mixed genera. But to Marchetto --
> how microtonal are his examples?

Marchetto's theory calls for the division of the 9:8
whole-tone into 5 parts called "dieses". He then says
that a whole-tone may be divided into two unequal parts
by means of the dieses thus:

* whole-tone (5) = "enharmonic" (2) + "diatonic" (3) semitones

* whole-tone (5) = "chromatic semitone" (4) + diesis (1)

Unfortunately, he does not say whether the dieses are equal
or unequal, and does not indicate whether his division is
logarithmic or arithmetic. And to make matters even more
complicated, he first describes how the whole-tone is divided
into *9* parts, then measures the 5 dieses by means of those 9.

Obviously, his description of the division of the whole-tone
is vague enough to permit a high degree of interpretation.
I've come up with about 4 or 5 of my own, examined here:
(rather sloppily ... i should edit this page someday ...)

http://tonalsoft.com/monzo/marchetto/marchetto.aspx

Jan Herlinger put forward his reading of Marchetto's
division in the early-to-mid 1980s; his English translation
of _Lucidarium_ was published in 1985.

In October 1998, at exactly the same time that published my
webpage with my first interpretation of Marchetto's division,
Jay Rahn published an investigation of it in Music Theory
Online, and his reading has some great similarities to mine:

http://mto.societymusictheory.org/issues/mto.98.4.6/mto.98.4.6.rahn.html

(i've never gotten over how odd it was that two similar
independent examinations of such an esoteric topic would
appear at exactly the same time ...)

I became so absorbed by Marchetto's division of the
whole-tone that i never really finished reading the rest
of the book in a systematic manner, and obviously i need
to do this to clarify my own understanding of his theory.

Margo Schulter knows a lot about Marchetto, but i believe
she is more willing to accept Rahn's reading than any of
mine. Then again, i'm not yet sure which of mine i think
is right ... but at least Margo knows much more about the
rest of Marchetto's theory. You can find her posts on
"Marchettus" in the archive, from 1998 to around 2003 or so.

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

Carl Lumma wrote:

> Cameron and I had a back and forth on this last week.
> He replied at least once to my reply, so I know he knows
> my answer. He doesn't eloborate any objections to my
> answer, just keeps on accusing me of not answering him.

Yes. I saw it.

Graham

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Cameron wrote...
> > Then once I again, I will go very, very slowly:
> >
> > Does tempering an interval upward have a different effect
> > on the musical character of an interval than tempering
> > the same interval downward?
>
> I've answered this question maybe half a dozen times,
> and most recently last week.

You're the only one on this list?
>
> > I am not talking about brute dissonance, beat rates, or
> > harmonic entropy. I am talking about musical character,
> > the specific nature of which is subjective of course.
>
> If it's subjective than what's the point in talking
> about it?

Subjective things aren't discussed on this list?

The specific nature is subjective. Whether or not
the musical character of an interval exists and if it
does, whether the musical character of an interval
is changed differently by tempering an interval up
or down can be answered by anybody- "yes" or "no".

> > Anybody who cares to understand what I say needs only
> > to answer that question with a simple "yes" or "no".
>
> So you only accept answers of the type you specify?

Certainly- I specify that the question be answered, not
deflected by something like harmonic dissonance. Do
you only answer questions in such a way that you can
avoid anything that might question the foundations
of the regular-mapping paradigm as understood on this list?

-Cameron Bobro

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> > Cameron wrote...
> >>
> >>Does tempering an interval upward have a different effect
> >>on the musical character of an interval than tempering
> >>the same interval downward?

>
> I do mentions this with regard to dissonance in my detailed
> and unimportant "prime errors and complexities" PDF.
> Something like
>
> http://x31eq.com/primerr.pdf
>
> and the latest version (still oh so much work to do!) in the
> Tuning Math files area.
>
> But, rather than joining the "I refer the honourable
> gentleman to my prevous answer" game, I'll come out with a
> "yes".

Cool, thanks. I have your paper- it may be obvious
that my question is rhetorical.

Here's the thing- how can we then talk
about approximating just intervals unless we're either
talking about extremely close approximations or specifying
the sign of the error? That is, if we're acknowledge the
existence of, and are concerned with, the musical
character of the intervals.

Lessee, a specific example. What's something audibly different
about 41 and 46 equal, as far as interval character? (We
were discussing these the other day). Why does
41 in general sound "softer" and more limpid to me?

I checked late last night and lucked out, finding a possible
explanation at first try. Lucky because I checked the "neutral
third" first
and then a simple aspect of what I believe is part of an
interval's "character family", which is it's place in
an "o- (u-)tonality".

for 46
Temperings of
11/10 11/9 11/8 11/7 11/6
-8.4825 -8.2775 -3.4919 0.1167 -5.8847

for 41
Temperings of
11/10 11/9 11/8 11/7 11/6
10.6055 3.8116 4.7796 7.7519 4.2956

Both these series are represented quite well, IMO. The total
absolute "error" is very similar between the two. Both are
consistent as far as sign (0.1167 cents difference we can
safely consider as untempered in discussion- differences
that small only matter in private, I believe). Both this
consistency in sign as well as the sheer distances are
part of what makes these good approximations- audibly good.
In the case of 41, a timbre with an 11th partial 6 cents low,
this particular JI series might be so closely represented that
we could ignore sign.

This 11/x series, in both 41 and 46, flies very well
solo, as a coherent whole- with many real-life timbres
and orchestrations it would do "that thing" as far as
being perceived as a JI series, just fine. In a tonal
context, against the 1/1, it loses more purity, and more
importantly IMO the character is different. Better or
worse or more appropriate for which tune, all that
would be subjective, but whatever: they're different.
Different sound. To me, that matters.

>
> >>I am not talking about brute dissonance, beat rates, or
> >>harmonic entropy. I am talking about musical character,
> >>the specific nature of which is subjective of course.
> >
> > If it's subjective than what's the point in talking
> > about it?
>
> It can often be worth talking about subjective things. Most
> of what we talk about here is subjective. Dissonance, for
> example, is subjective. Despite that, a lot of us are in
> enough agreement about it that we can put numbers to it and
> construct tuning systems on the basis of those numbers. It
> looks like most people generally agree with us on it as well
> (but not its importance). Where people disagree, it's
> useful to know why and how, even if it's not practical to
> accommodate them.

I find myself in agreement, in general, with most perceptions
of dissonance that have been expressed around here. That is,
brute dissonance. But I am convinced that consonance and
dissonance aren't opposite poles or even mutually exclusive, take a
gander at 7/6.

-Cameron Bobro

Cameron Bobro wrote:

> Here's the thing- how can we then talk
> about approximating just intervals unless we're either
> talking about extremely close approximations or specifying
> the sign of the error? That is, if we're acknowledge the
> existence of, and are concerned with, the musical > character of the intervals. We can talk about approximating just intervals if we're approximating just intervals. In some contexts I do specify the sign of the error. When it comes to scoring temperaments, sharp and flat are treated the same, which is mainly because I don't know which way to distinguish them.

We can also acknowledge the existence of characteristics of intervals that aren't considered by the simplistic error measures. It doesn't have to be all or nothing.

> Lessee, a specific example. What's something audibly different > about 41 and 46 equal, as far as interval character? (We
> were discussing these the other day). Why does
> 41 in general sound "softer" and more limpid to me?

I can't say much about this because I don't listen to such large ETs as single entities. The general sound will depend on what generators I think in terms of and a whole load of other contextual things.

Usually I think of rank 2 temperaments. That gives me a free parameter (I leave octaves pure) to tweak to get a sound I'm happy with. It's good to have that freedom.

The automated searches, and the theory behind them, are mainly for sorting good temperament classes from bad. When it comes to good approximations (which I find make sense harmonically) and simple mappings (which are also important to me) there aren't usually very many choices left. So there isn't a great need to make the search more intelligent. It can always do with being faster and simpler.

> I checked late last night and lucked out, finding a possible
> explanation at first try. Lucky because I checked the "neutral > third" first > and then a simple aspect of what I believe is part of an > interval's "character family", which is it's place in
> an "o- (u-)tonality". As far as neutral thirds go, the first thing I see (which is different from hearing) is that 41 has a single neutral third: the fifth splits into equal parts. 46, however, has two different candidates for a neutral third.

I find it much simpler to think of tunings that have only one neutral third, like 41. Any temperament where 11:9 and 18:11 will have such a neutral third. For that matter, any interval that equally divides a perfect fifth can be considered an approximation of 11:9 as long as you think in terms of such approximations. Hence tempering out 243:242 and having a single neutral third can be considered identical concepts. Looking for temperament classes that temper out 243:242 is a great way of finding tunings with a single neutral third. Whether you really wanted an approximation of 11:9 or an equal division of a perfect fifth doesn't matter at all because the result is the same.

It also follows that, because 243:242 is quite a simple interval, good 11-limit temperaments with a certain range of error will naturally temper out 243:242 and so have a single neutral third. That means looking for 11-limit temperaments can be a good way of finding tunings with a single neutral third. If all you wanted was neutral thirds there'll be some noise because of the other intervals that get considered but you might find something interesting.

> for 46
> Temperings of > 11/10 11/9 11/8 11/7 11/6
> -8.4825 -8.2775 -3.4919 0.1167 -5.8847
> > for 41
> Temperings of > 11/10 11/9 11/8 11/7 11/6
> 10.6055 3.8116 4.7796 7.7519 4.2956
> > Both these series are represented quite well, IMO. The total > absolute "error" is very similar between the two. Both are
> consistent as far as sign (0.1167 cents difference we can > safely consider as untempered in discussion- differences
> that small only matter in private, I believe). Both this
> consistency in sign as well as the sheer distances are
> part of what makes these good approximations- audibly good.
> In the case of 41, a timbre with an 11th partial 6 cents low,
> this particular JI series might be so closely represented that > we could ignore sign. You're happy with errors of a few cents now?

> This 11/x series, in both 41 and 46, flies very well
> solo, as a coherent whole- with many real-life timbres
> and orchestrations it would do "that thing" as far as
> being perceived as a JI series, just fine. In a tonal
> context, against the 1/1, it loses more purity, and more
> importantly IMO the character is different. Better or
> worse or more appropriate for which tune, all that
> would be subjective, but whatever: they're different.
> Different sound. To me, that matters. Why would $DEITY have created both temperaments if they sounded the same? You can also shrink the octaves of 41 and stretch those of 46 to even out the errors (and get a different sound).

> I find myself in agreement, in general, with most perceptions
> of dissonance that have been expressed around here. That is,
> brute dissonance. But I am convinced that consonance and > dissonance aren't opposite poles or even mutually exclusive, take a > gander at 7/6.

Well, that's it. "Brute dissonance" is what we're talking about.

Graham

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> Cameron Bobro wrote:
>
> > Here's the thing- how can we then talk
> > about approximating just intervals unless we're either
> > talking about extremely close approximations or specifying
> > the sign of the error? That is, if we're acknowledge the
> > existence of, and are concerned with, the musical
> > character of the intervals.
>
> We can talk about approximating just intervals if we're
> approximating just intervals.

For notation and keyboard/instrument mapping, I agree, in
practice as well as theory. For actual music, ignoring the
sign of the error and other factors is too crude, afaic.

>In some contexts I do specify
> the sign of the error. When it comes to scoring
> temperaments, sharp and flat are treated the same, which is
> mainly because I don't know which way to distinguish them.
>
> We can also acknowledge the existence of characteristics of
> intervals that aren't considered by the simplistic error
> measures. It doesn't have to be all or nothing.
>
> > Lessee, a specific example. What's something audibly different
> > about 41 and 46 equal, as far as interval character? (We
> > were discussing these the other day). Why does
> > 41 in general sound "softer" and more limpid to me?
>
> I can't say much about this because I don't listen to such
> large ETs as single entities.

41 or 46 are stretching it as far equal divisions as single
entities, IMO. Anyway I've been talking about a specific
subset in this case (11/x)

>The general sound will depend
> on what generators I think in terms of and a whole load of
> other contextual things.
>
> Usually I think of rank 2 temperaments. That gives me a
> free parameter (I leave octaves pure) to tweak to get a
> sound I'm happy with. It's good to have that freedom.
>
> The automated searches, and the theory behind them, are
> mainly for sorting good temperament classes from bad. When
> it comes to good approximations (which I find make sense
> harmonically) and simple mappings (which are also important
> to me) there aren't usually very many choices left. So
> there isn't a great need to make the search more
> intelligent. It can always do with being faster and simpler.

Of course with subsets of huge EDOs you can have a big pile of
poor approximations and still be in business, who cares if they're
crap if you don't use them?

It seems to me that something MOSs could do very well is just
that- act as a sieve.

>
> > I checked late last night and lucked out, finding a possible
> > explanation at first try. Lucky because I checked the "neutral
> > third" first
> > and then a simple aspect of what I believe is part of an
> > interval's "character family", which is it's place in
> > an "o- (u-)tonality".
>
> As far as neutral thirds go, the first thing I see (which is
> different from hearing) is that 41 has a single neutral
> third: the fifth splits into equal parts. 46, however, has
> two different candidates for a neutral third.

And they're a bit strange, kind of like disjunct 11/9s inside
the fifth.
>
> I find it much simpler to think of tunings that have only
> one neutral third, like 41. Any temperament where 11:9 and
> 18:11 will have such a neutral third. For that matter, any
> interval that equally divides a perfect fifth can be
> considered an approximation of 11:9 as long as you think in
> terms of such approximations. Hence tempering out 243:242
> and having a single neutral third can be considered
> identical concepts. Looking for temperament classes that
> temper out 243:242 is a great way of finding tunings with a
> single neutral third. Whether you really wanted an
> approximation of 11:9 or an equal division of a perfect
> fifth doesn't matter at all because the result is the same.
>
> It also follows that, because 243:242 is quite a simple
> interval, good 11-limit temperaments with a certain range of
> error will naturally temper out 243:242 and so have a single
> neutral third. That means looking for 11-limit temperaments
> can be a good way of finding tunings with a single neutral
> third. If all you wanted was neutral thirds there'll be
> some noise because of the other intervals that get
> considered but you might find something interesting.

I was considering the 11/x series, of which 11/9 is a part,
of course.
>
> > for 46
> > Temperings of
> > 11/10 11/9 11/8 11/7 11/6
> > -8.4825 -8.2775 -3.4919 0.1167 -5.8847
> >
> > for 41
> > Temperings of
> > 11/10 11/9 11/8 11/7 11/6
> > 10.6055 3.8116 4.7796 7.7519 4.2956
> >
> > Both these series are represented quite well, IMO. The total
> > absolute "error" is very similar between the two. Both are
> > consistent as far as sign (0.1167 cents difference we can
> > safely consider as untempered in discussion- differences
> > that small only matter in private, I believe). Both this
> > consistency in sign as well as the sheer distances are
> > part of what makes these good approximations- audibly good.
> > In the case of 41, a timbre with an 11th partial 6 cents low,
> > this particular JI series might be so closely represented that
> > we could ignore sign.
>
> You're happy with errors of a few cents now?

That's not what I said. For the specific 11/x series we're
looking at here, the "errors" are of the same sign, and
close in size, so the 11/x series is cohesive. In the
case of 41, if you play this series without the tonic,
it's damn close to a pure 11/x in sound- against the tonic,
the cohesive nature assures character and it doesn't sound
like a bunch of "approximations". A little high,
a little low, a flier... no I would not be happy with that.

>
> > This 11/x series, in both 41 and 46, flies very well
> > solo, as a coherent whole- with many real-life timbres
> > and orchestrations it would do "that thing" as far as
> > being perceived as a JI series, just fine. In a tonal
> > context, against the 1/1, it loses more purity, and more
> > importantly IMO the character is different. Better or
> > worse or more appropriate for which tune, all that
> > would be subjective, but whatever: they're different.
> > Different sound. To me, that matters.
>
> Why would $DEITY have created both temperaments if they
> sounded the same?

Why, indeed?

>You can also shrink the octaves of 41 and
> stretch those of 46 to even out the errors (and get a
> different sound).

41 would still suck as far as the x/11 series. 34 sidesteps
both series by quite consistently splitting the difference,
so to speak, with other series (7s for example). I think
the "regular mapping paradigm" would benefit from applying
itself to these kinds of series, like the series derived
from an imaginary partial lying at the harmonic mean of
7 and 11, for example. But since I've been waving the flag
for this kind of thing for a long time now, it's sure to be
ignored or pooh-poohed "considering the source", LOL.

> > I find myself in agreement, in general, with most perceptions
> > of dissonance that have been expressed around here. That is,
> > brute dissonance. But I am convinced that consonance and
> > dissonance aren't opposite poles or even mutually exclusive,
>take a
> > gander at 7/6.
>
> Well, that's it. "Brute dissonance" is what we're talking
> about.

Well, I just don't buy brute dissonance as a measure of
a tuning's validity at all, nor do I find that brute
dissonance agrees very much with contextual (musical)
dissonance and consonance. But different strokes...

-Cameron Bobro

Cameron Bobro wrote:
> --- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:

>>We can talk about approximating just intervals if we're >>approximating just intervals. > > For notation and keyboard/instrument mapping, I agree, in
> practice as well as theory. For actual music, ignoring the
> sign of the error and other factors is too crude, afaic.

When it comes to actual music, you can always use your ears to judge the sound. There doesn't have to be a quantitative theory describing each interval. Some people (perhaps even most musicians) would prefer there *not* to be a quantitative theory. And I note you still haven't come up with a quantitative theory. So for now all we have is a crude model for one dimension of the sound.

>>>Lessee, a specific example. What's something audibly different >>>about 41 and 46 equal, as far as interval character? (We
>>>were discussing these the other day). Why does
>>>41 in general sound "softer" and more limpid to me?
>>
>>I can't say much about this because I don't listen to such >>large ETs as single entities. > > 41 or 46 are stretching it as far equal divisions as single
> entities, IMO. Anyway I've been talking about a specific
> subset in this case (11/x)

Yes but you were talking about the general case first. And doesn't the whole 11-limit get dragged in as intervals within this subset?

> Of course with subsets of huge EDOs you can have a big pile of
> poor approximations and still be in business, who cares if they're > crap if you don't use them?

That's what we check complexity for. With a small complexity you end up with most of the intervals you're likely to use meaning something. Or at least pretending to mean something.

> It seems to me that something MOSs could do very well is just
> that- act as a sieve.

>>As far as neutral thirds go, the first thing I see (which is >>different from hearing) is that 41 has a single neutral >>third: the fifth splits into equal parts. 46, however, has >>two different candidates for a neutral third.
> > And they're a bit strange, kind of like disjunct 11/9s inside > the fifth.

One of them will naturally be closer to 16:13 than 11:9.

>>>for 46
>>>Temperings of >>>11/10 11/9 11/8 11/7 11/6
>>>-8.4825 -8.2775 -3.4919 0.1167 -5.8847
>>>
>>>for 41
>>>Temperings of >>>11/10 11/9 11/8 11/7 11/6
>>>10.6055 3.8116 4.7796 7.7519 4.2956

<snip>

> That's not what I said. For the specific 11/x series we're
> looking at here, the "errors" are of the same sign, and > close in size, so the 11/x series is cohesive. In the > case of 41, if you play this series without the tonic,
> it's damn close to a pure 11/x in sound- against the tonic,
> the cohesive nature assures character and it doesn't sound
> like a bunch of "approximations". A little high,
> a little low, a flier... no I would not be happy with that.

You mean they do sound like a bunch of approximations?

From the figures you've got there, the worst interval of 46 does not involve the tonic. And the worst interval from 41 without the tonic is still over 6 cents. To me, these would be "fairly large" errors.

As a benchmark, the worst interval in 11-limit minimax miracle is 3.3 cents from just. 41-equal is about twice as bad by the TOP-RMS measure. And miracle is what I'll normally use in the 11-limit. 41 is outside the range I'd use. So if the argument is that you lose the 11-limit character when you reach 41-equal, I'm in agreement.

There's a graph of the 11-limit approximations here:

http://x31eq.com/miracle.htm#approx

You can see that ever single 11-limit interval improves as you temper away from 41-equal (where the secor is about 117 cents).

Most temperament classes will not get close to this accuracy for equivalent simplicity. If this is the accuracy you want then miracle is the way to go. If you want more simplicity then something has to give. There still won't be many temperament classes to pick through for the other properties you want.

I find that 41-equal does work in the 9-limit. Magic is a good way of mapping it. And the 11/x series without the tonic is really the 9-limit. I don't have a page on magic, but I do have this graph:

http://x31eq.com/pics/magic.png

> 41 would still suck as far as the x/11 series. 34 sidesteps
> both series by quite consistently splitting the difference, > so to speak, with other series (7s for example). I think
> the "regular mapping paradigm" would benefit from applying
> itself to these kinds of series, like the series derived
> from an imaginary partial lying at the harmonic mean of
> 7 and 11, for example. But since I've been waving the flag
> for this kind of thing for a long time now, it's sure to be > ignored or pooh-poohed "considering the source", LOL.

You can use any partials you like with my library. But I don't think you need to in this case. (And wouldn't your specific example give 77:9, not a good match to 34?)

The higher up the harmonic series goes, the closer the partials get. And the closer two notes get, the less difference it makes what kind of mean you take. Each new harmonic is the arithmetic mean of two other harmonics. So you can get similar results by looking at higher prime limits. In the case of 34, you ignore the poor approximation to 7. By my figures, 34-equal is a standout approximation to various limits up to 1.3.5.11.13.17.19.23.

Whether your aim is to look at higher limits with no sevens, or only to split the differences within lower limits, it doesn't matter. The result is the same. And the result partly depends on the approximation to 7:1 being so glaringly bad (and all the signs agreeing).

> Well, I just don't buy brute dissonance as a measure of > a tuning's validity at all, nor do I find that brute > dissonance agrees very much with contextual (musical)
> dissonance and consonance. But different strokes...

I'm happy that tension and release works in 12-equal with 5-limit rules. And the approximation is quite extreme there. The thirds in 12-equal don't sound at all like their just counterparts. Similarly, the "Decatonic Swing" Paul and his band made in 22-equal makes perfect harmonic sense to me. It doesn't matter that 22-equal, under laboratory conditions, doesn't have the same sound as 7-limit JI.

I'm also generally happy with 11-limit harmony in miracle. Although I'm recently finding that I prefer the approximation to 11:8 to that of 7:5. Maybe it's because of the accuracy.

Graham

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> When it comes to actual music, you can always use your ears
> to judge the sound. There doesn't have to be a quantitative
> theory describing each interval. Some people (perhaps even
> most musicians) would prefer there *not* to be a
> quantitative theory.

Now we're getting somewhere- using my ears to judge the sound
is what brought me to questioning the regular mapping
paradigm as practiced here, for what I hear and what's
on paper simply don't agree so far.

>And I note you still haven't come up
> with a quantitative theory. So for now all we have is a
> crude model for one dimension of the sound.

Well there's no law that says you have to read or
understand everything I write, and certainly it's
possible that I have no writing skills.

>
> >>>Lessee, a specific example. What's something audibly different
> >>>about 41 and 46 equal, as far as interval character? (We
> >>>were discussing these the other day). Why does
> >>>41 in general sound "softer" and more limpid to me?
> >>
> >>I can't say much about this because I don't listen to such
> >>large ETs as single entities.
> >
> > 41 or 46 are stretching it as far equal divisions as single
> > entities, IMO. Anyway I've been talking about a specific
> > subset in this case (11/x)
>
> Yes but you were talking about the general case first. And
> doesn't the whole 11-limit get dragged in as intervals
> within this subset?

What do you mean by "11-limit"? (Serious question).
>
> > Of course with subsets of huge EDOs you can have a big pile of
> > poor approximations and still be in business, who cares if
they're
> > crap if you don't use them?
>
> That's what we check complexity for. With a small
> complexity you end up with most of the intervals you're
> likely to use meaning something. Or at least pretending to
> mean something.
>
> > It seems to me that something MOSs could do very well is just
> > that- act as a sieve.
>
> >>As far as neutral thirds go, the first thing I see (which is
> >>different from hearing) is that 41 has a single neutral
> >>third: the fifth splits into equal parts. 46, however, has
> >>two different candidates for a neutral third.
> >
> > And they're a bit strange, kind of like disjunct 11/9s inside
> > the fifth.
>
> One of them will naturally be closer to 16:13 than 11:9.
>
> >>>for 46
> >>>Temperings of
> >>>11/10 11/9 11/8 11/7 11/6
> >>>-8.4825 -8.2775 -3.4919 0.1167 -5.8847
> >>>
> >>>for 41
> >>>Temperings of
> >>>11/10 11/9 11/8 11/7 11/6
> >>>10.6055 3.8116 4.7796 7.7519 4.2956
>
> <snip>
>
> > That's not what I said. For the specific 11/x series we're
> > looking at here, the "errors" are of the same sign, and
> > close in size, so the 11/x series is cohesive. In the
> > case of 41, if you play this series without the tonic,
> > it's damn close to a pure 11/x in sound- against the tonic,
> > the cohesive nature assures character and it doesn't sound
> > like a bunch of "approximations". A little high,
> > a little low, a flier... no I would not be happy with that.
>
> You mean they do sound like a bunch of approximations?

I don't know how you get that out of "it doesn't sound
like a bunch of "approximations".
>
> From the figures you've got there, the worst interval of 46
> does not involve the tonic. And the worst interval from 41
> without the tonic is still over 6 cents. To me, these would
> be "fairly large" errors.

Error from 11/x, in relation to the tonic, yes. Without the
tonic, the intervals are centered around a virtual partial
some 7+ cents lower than 11. The maximum error is <3.4 cents.
In other words, taken on it's own, this particular series
is consistent in proportions with an 11/x series. Against
the tonic, it's consistent with a detuned 11th partial,
therefore consistent in character. It's not great- I just
grabbed 41 as an example- but it doesn't suck.
>
> As a benchmark, the worst interval in 11-limit minimax
> miracle is 3.3 cents from just. 41-equal is about twice as
> bad by the TOP-RMS measure. And miracle is what I'll
> normally use in the 11-limit. 41 is outside the range I'd
> use. So if the argument is that you lose the 11-limit
> character when you reach 41-equal, I'm in agreement.

Can you give me a specific "minimax" example? 3.3 cents maximum
error from a simpler series, even
plus or minus, with no fliers, would be very good indeed. But in
the Scala archive I find for example:

Canasta with Secor's minimax generator of 116.7155941 cents (5:9
exact). XH5, 1976
|
Temperings of
11/10 11/9 11/8 11/7 11/6

-15.4446 2.7388 -0.5840 -16.5104 1.0774

which leaves me scratching my head, for it's a typical
example of "hodge-podge approximations" for this
particular 11s series.

>
> There's a graph of the 11-limit approximations here:
>
> http://x31eq.com/miracle.htm#approx
>
> You can see that ever single 11-limit interval improves as
> you temper away from 41-equal (where the secor is about 117
> cents).
>
> Most temperament classes will not get close to this accuracy
> for equivalent simplicity. If this is the accuracy you want
> then miracle is the way to go. If you want more simplicity
> then something has to give. There still won't be many
> temperament classes to pick through for the other properties
> you want.
>
> I find that 41-equal does work in the 9-limit. Magic is a
> good way of mapping it.

Can you give me a specific example, Scala file preferably,
of 41-equal "magic" mapped?

>And the 11/x series without the
> tonic is really the 9-limit.

Yes, you've got the superparticulars expanding from 10/9.
Surely you see one reason I'm concerned with consistency
in the series itself, not just absolute proximity to
the Just intervals themselves?

>I don't have a page on magic,
> but I do have this graph:
>
> http://x31eq.com/pics/magic.png
>
> > 41 would still suck as far as the x/11 series. 34 sidesteps
> > both series by quite consistently splitting the difference,
> > so to speak, with other series (7s for example). I think
> > the "regular mapping paradigm" would benefit from applying
> > itself to these kinds of series, like the series derived
> > from an imaginary partial lying at the harmonic mean of
> > 7 and 11, for example. But since I've been waving the flag
> > for this kind of thing for a long time now, it's sure to be
> > ignored or pooh-poohed "considering the source", LOL.
>
> You can use any partials you like with my library. But I
> don't think you need to in this case. (And wouldn't your
> specific example give 77:9, not a good match to 34?)

Yeah, sorry for mixing examples (34 has another thing
going IMO, which I'll get around to writing out sometime).
>
> The higher up the harmonic series goes, the closer the
> partials get. And the closer two notes get, the less
> difference it makes what kind of mean you take. Each new
> harmonic is the arithmetic mean of two other harmonics. So
> you can get similar results by looking at higher prime
> limits. In the case of 34, you ignore the poor
> approximation to 7. By my figures, 34-equal is a standout
> approximation to various limits up to 1.3.5.11.13.17.19.23.

Sure- 23d partial "does" 34 remarkably well on it's own, as I've
mentioned several times over the last year.
>
> Whether your aim is to look at higher limits with no sevens,
> or only to split the differences within lower limits, it
> doesn't matter. The result is the same.

More or less, but as far as audibility, I think it's better
to go for harmonic means of audible partials rather than
ratios based on partials of dubious audibility. Also you
steer clear of crap like magic primes.

>And the result
> partly depends on the approximation to 7:1 being so
> glaringly bad (and all the signs agreeing).

Exactly! Vice into virtue. I'd rather have rock-solid
higher-partials/more-complex series in a tuning than
lumpy references to the lowest partials. Easier to
sing, to cut to the chase.

> > Well, I just don't buy brute dissonance as a measure of
> > a tuning's validity at all, nor do I find that brute
> > dissonance agrees very much with contextual (musical)
> > dissonance and consonance. But different strokes...
>
> I'm happy that tension and release works in 12-equal with
> 5-limit rules.

Doesn't work for me- I hear neither true tension nor true
release in 12-equal, and haven't since earliest memory. I
remember cringing when I first heard in school that the M3
was "happy" and "peaceful". What, that theatrical grimace
is supposed to be a smile?

>And the approximation is quite extreme
> there.

It's not an approximation. Different interval.

>The thirds in 12-equal don't sound at all like their
> just counterparts.

They sound quite a bit like 81/64 to me.

> Similarly, the "Decatonic Swing" Paul
> and his band made in 22-equal makes perfect harmonic sense
> to me. It doesn't matter that 22-equal, under laboratory
> conditions, doesn't have the same sound as 7-limit JI.

Have to listen again to that one.
>
> I'm also generally happy with 11-limit harmony in miracle.
> Although I'm recently finding that I prefer the
> approximation to 11:8 to that of 7:5. Maybe it's because of
> the accuracy.

Once again, I'd need a definition of "11-limit" and a Scala
file illustrating this.

-Cameron Bobro

Graham wrote...

> There's a graph of the 11-limit approximations here:
>
> http://x31eq.com/miracle.htm#approx

116.8 looks perfect to me. We should prioritize the
lines with the steepest slope (11 and 9), if anything.
Though the positive error in the the 11 may work against
the negative errors on the other primary intervals in
the secondary intervals. So perhaps 116.75 is better.

> http://x31eq.com/pics/magic.png

The error scale here is larger. But I think 380.75
would be good.

-Carl

> Now we're getting somewhere- using my ears to judge the sound
> is what brought me to questioning the regular mapping
> paradigm as practiced here, for what I hear and what's
> on paper simply don't agree so far.

Did you ever make that list of disagreements I asked for?

-Carl

Sorry- it was Cameron. Cameron who wrote that....

> > Now we're getting somewhere- using my ears to judge the sound
> > is what brought me to questioning the regular mapping
> > paradigm as practiced here, for what I hear and what's
> > on paper simply don't agree so far.
>
> Did you ever make that list of disagreements I asked for?
>
> -Carl

Carl Lumma wrote:
> Graham wrote...
> > >>There's a graph of the 11-limit approximations here:
>>
>>http://x31eq.com/miracle.htm#approx
> > > 116.8 looks perfect to me. We should prioritize the
> lines with the steepest slope (11 and 9), if anything.
> Though the positive error in the the 11 may work against
> the negative errors on the other primary intervals in
> the secondary intervals. So perhaps 116.75 is better.

I've used these before. For the 11-limit, 116.75 cents where 11:8 is just. And for the 9-limit, 116.79 cents where 7:9 is just. Neither is any special optimum because they depend on ignoring the error of 5:1. That makes sense to me because 11:8 and 7:9 are difficult intervals to hear as consonances.

Tunings around here make the quomma (the interval between 10 secors and an octave) a bit smaller than for 72-equal. It means notes differing by a quomma sound more like inflections of each other than distinct notes. That's good for my decimalized thinking but not so good if you want to treat blackjack as a melodic unit.

>>http://x31eq.com/pics/magic.png
> > The error scale here is larger. But I think 380.75
> would be good.

That's close to the TOP-RMS of 380.70 cents. 41-equal is 380.49 cents. With enough notes you might hear the difference. I haven't done much with magic so I don't know if I can.

Graham

Cameron Bobro wrote:
> --- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
> >>When it comes to actual music, you can always use your ears >>to judge the sound. There doesn't have to be a quantitative >>theory describing each interval. Some people (perhaps even >>most musicians) would prefer there *not* to be a >>quantitative theory. > > Now we're getting somewhere- using my ears to judge the sound
> is what brought me to questioning the regular mapping > paradigm as practiced here, for what I hear and what's > on paper simply don't agree so far.

Plenty of the details of what you say you hear do agree.

>>And I note you still haven't come up >>with a quantitative theory. So for now all we have is a >>crude model for one dimension of the sound.
> > Well there's no law that says you have to read or
> understand everything I write, and certainly it's
> possible that I have no writing skills. It's a plain statement of fact that you haven't provided us with a quantitative theory. Why do you have to get confrontational about it?

> What do you mean by "11-limit"? (Serious question).

Ratios of odd numbers no greater than 11 and their octave equivalents.

>>>That's not what I said. For the specific 11/x series we're
>>>looking at here, the "errors" are of the same sign, and >>>close in size, so the 11/x series is cohesive. In the >>>case of 41, if you play this series without the tonic,
>>>it's damn close to a pure 11/x in sound- against the tonic,
>>>the cohesive nature assures character and it doesn't sound
>>>like a bunch of "approximations". A little high,
>>>a little low, a flier... no I would not be happy with that.
>>
>>You mean they do sound like a bunch of approximations?
> > I don't know how you get that out of "it doesn't sound
> like a bunch of "approximations".

Unfortunately, so many people use "do" and "don't" interchangeably that you have to guess what they really mean from the context. If you stand by what you wrote, how does "damn close" not imply "approximation"?

>> From the figures you've got there, the worst interval of 46 >>does not involve the tonic. And the worst interval from 41 >>without the tonic is still over 6 cents. To me, these would >>be "fairly large" errors.
> > Error from 11/x, in relation to the tonic, yes. Without the
> tonic, the intervals are centered around a virtual partial > some 7+ cents lower than 11. The maximum error is <3.4 cents.
> In other words, taken on it's own, this particular series
> is consistent in proportions with an 11/x series. Against > the tonic, it's consistent with a detuned 11th partial,
> therefore consistent in character. It's not great- I just
> grabbed 41 as an example- but it doesn't suck.

This "virtual partial" is similar to the mean implied in calculating the standard deviation. So you seem to agree with my logic in using standard deviations to rate errors where you don't temper the octaves. That's good because nobody else understands it. Note that if you do optimize the octaves, the virtual partial will end up with a near zero error.

>>As a benchmark, the worst interval in 11-limit minimax >>miracle is 3.3 cents from just. 41-equal is about twice as >>bad by the TOP-RMS measure. And miracle is what I'll >>normally use in the 11-limit. 41 is outside the range I'd >>use. So if the argument is that you lose the 11-limit >>character when you reach 41-equal, I'm in agreement.
> > Can you give me a specific "minimax" example? 3.3 cents maximum > error from a simpler series, even
> plus or minus, with no fliers, would be very good indeed. But in
> the Scala archive I find for example:

You bet it's good!

> Canasta with Secor's minimax generator of 116.7155941 cents (5:9 > exact). XH5, 1976

That's exactly what I get. My generator is 116.7155940982074 cents. (May not be accurate to that precision.) I can't find this file in my Scala archive, however, so I can't check it. The mapping by octaves and generators for miracle is

<1, 1, 3, 3, 2]
<0, 6, -7, -2, 15]

> |
> Temperings of > 11/10 11/9 11/8 11/7 11/6
> > -15.4446 2.7388 -0.5840 -16.5104 1.0774
> > which leaves me scratching my head, for it's a typical
> example of "hodge-podge approximations" for this
> particular 11s series. That's plain wrong. The errors in primes should be

2 3 5 7 11
0.0 -1.7 -3.3 -2.3 -0.6

So the errors in your intervals are

11:10 11:9 11:8 11:7 11:6

2.7388 2.7388 -0.5840 1.6731 1.0774

We disagree on 11:10 and 11:7. Furthermore, in both cases our disagreement amounts to 18.1834 +/- 0.0001 cents. Well, 18.1834 cents is the residue you get from closing the cycle of secors at 31 steps. So it looks like you've taken two of the wrong intervals from the 31 note scale. If you're following the convention of / for notes, it means you chose a bad tonic.

>>I find that 41-equal does work in the 9-limit. Magic is a >>good way of mapping it. > > Can you give me a specific example, Scala file preferably,
> of 41-equal "magic" mapped?

A scala file of 41-equal will look the same however it's mapped. What I can give you is a 19 note magic scale

58.9
117.9
203.5
262.5
321.4
380.4
439.3
498.2
583.9
642.8
701.8
760.7
819.6
878.6
964.2
1023.2
1082.1
1141.1
1200.0

It uses some optimal tuning or other. The generator mapping for magic is <0, 5, 1, 12]. For this scale I take 12 thirds up from the 1/1, and 6 down. That means you can find approximations to 9/8, 5/4, 3/2, and 7/4 and also 4/3, etc. But there isn't an 8/7. If you prefer to think utonally, try this:

58.9
117.9
176.8
235.8
321.4
380.4
439.3
498.2
557.2
616.1
701.8
760.7
819.6
878.6
937.5
996.5
1082.1
1141.1
1200.0

>>And the 11/x series without the >>tonic is really the 9-limit. > > Yes, you've got the superparticulars expanding from 10/9.
> Surely you see one reason I'm concerned with consistency
> in the series itself, not just absolute proximity to
> the Just intervals themselves? Um, what's the difference?

>>Whether your aim is to look at higher limits with no sevens, >>or only to split the differences within lower limits, it >>doesn't matter. The result is the same. > > More or less, but as far as audibility, I think it's better
> to go for harmonic means of audible partials rather than
> ratios based on partials of dubious audibility. Also you
> steer clear of crap like magic primes. My online scripts only take subset-prime limits. If you want to use other JI bases you'll have to download the python libraries. The regular mapping paradigm doesn't care either way.

>>And the result >>partly depends on the approximation to 7:1 being so >>glaringly bad (and all the signs agreeing).
> > Exactly! Vice into virtue. I'd rather have rock-solid
> higher-partials/more-complex series in a tuning than
> lumpy references to the lowest partials. Easier to > sing, to cut to the chase.

Have you seen George Secor's paper about his 17 note well temperament? It's based on an incomplete prime limit.

>>>Well, I just don't buy brute dissonance as a measure of >>>a tuning's validity at all, nor do I find that brute >>>dissonance agrees very much with contextual (musical)
>>>dissonance and consonance. But different strokes...
>>
>>I'm happy that tension and release works in 12-equal with >>5-limit rules. > > Doesn't work for me- I hear neither true tension nor true
> release in 12-equal, and haven't since earliest memory. I
> remember cringing when I first heard in school that the M3
> was "happy" and "peaceful". What, that theatrical grimace
> is supposed to be a smile? I wasn't talking about the tuning but music written in it. Many, many people learn music theory in 12-equal and it makes perfect sense to them. You're obviously an exception because you became a microtonalist.

>>And the approximation is quite extreme >>there. > > It's not an approximation. Different interval. I see you're having difficulty with the meanings of words again.

>>The thirds in 12-equal don't sound at all like their >>just counterparts. > > They sound quite a bit like 81/64 to me.

They're nearly 8 cents out, so you're still being generous with the approximations.

>>Similarly, the "Decatonic Swing" Paul >>and his band made in 22-equal makes perfect harmonic sense >>to me. It doesn't matter that 22-equal, under laboratory >>conditions, doesn't have the same sound as 7-limit JI.
> > Have to listen again to that one.

George Secor sent me some pajara examples as well. Not the decatonic scales, but they show the harmony makes sense.

>>I'm also generally happy with 11-limit harmony in miracle. >>Although I'm recently finding that I prefer the >>approximation to 11:8 to that of 7:5. Maybe it's because of >>the accuracy.
> > Once again, I'd need a definition of "11-limit" and a Scala
> file illustrating this.

Try http://x31eq.com/keyboard.html and tell me if I forgot to copy anything over. Or how about csound code?

;;;;;; this at global scope

; low mapping
gikbdSecor ftgen 0, 0, 32, -2, \
0, 0, 1, 1, 1, 2, 2, 3, 3, 4, 4, 4, \
5, 5, 6, 6, 6, 7, 7, 8, 8, 9, 9, 9

gikbdQuomma ftgen 0, 0, 32, -2, \
-1, 0, -1, 0, 1, -1, 0, -1, 0, -1, 0, 1, \
-1, 0, -1, 0, 1, -1, 0, -1, 0, -1, 0, 1

opcode CPSDec, i, iii
inote, iquommas, ituning xin

ioctaves init int(inote)
isecors init frac(inote)*10

; universal decimal tuning
; works for both miracle and negri
isecor_size = 1/(10+ituning)
iquomma_size = abs(ituning/(10+ituning))

xout cpsoct(ioctaves + \
isecors*isecor_size + iquommas*iquomma_size)
endop

;;;;;; and this in the instrument code

ikey init p5
ipitchClass = ikey%24
ioctave = int(ikey/24) + 6
isecors table ipitchClass, gikbdSecor
iquommas table ipitchClass, gikbdQuomma
inote = ioctave + isecors/10
ifreq CPSDec inote, iquommas, 0.2747

Graham

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> Plenty of the details of what you say you hear do agree.

Obviously, why else would I even bother? There are plenty
of tuning "schools" around, like the whole divvying up of
12-tET approach, which would be an utter waste of time for
me.

>
> >>And I note you still haven't come up
> >>with a quantitative theory. So for now all we have is a
> >>crude model for one dimension of the sound.
> >
> > Well there's no law that says you have to read or
> > understand everything I write, and certainly it's
> > possible that I have no writing skills.
>
> It's a plain statement of fact that you haven't provided us
> with a quantitative theory. Why do you have to get
> confrontational about it?

Where do you get "confrontational"?
>
> > What do you mean by "11-limit"? (Serious question).
>
> Ratios of odd numbers no greater than 11 and their octave
> equivalents.

Okay, that's cool by me, because it relates to actual audible
partials. This, from wikipedia:

"The prime limit can be seen as a generalization that does not favor
the number 2. It is defined as the largest prime number in the
factorization of both numerator and denominator. That is, in number
theoretic terms, it measures the smoothness of the numerator and
denominator. The prime limit of the perfect fourth is 3 (the same as
the odd limit), but the minor tone has a prime limit of 5, because 9
can be factorized into 3×3, and 10 into 2×5."

is bogus as far as tuning goes, unless it's qualified with
exponential limit. Except for numerologists, of course, which
is also fine.
>
> >>>That's not what I said. For the specific 11/x series we're
> >>>looking at here, the "errors" are of the same sign, and
> >>>close in size, so the 11/x series is cohesive. In the
> >>>case of 41, if you play this series without the tonic,
> >>>it's damn close to a pure 11/x in sound- against the tonic,
> >>>the cohesive nature assures character and it doesn't sound
> >>>like a bunch of "approximations". A little high,
> >>>a little low, a flier... no I would not be happy with that.
> >>
> >>You mean they do sound like a bunch of approximations?
> >
> > I don't know how you get that out of "it doesn't sound
> > like a bunch of "approximations".
>
> Unfortunately, so many people use "do" and "don't"
> interchangeably that you have to guess what they really mean
> from the context. If you stand by what you wrote, how does
> "damn close" not imply "approximation"?

Sorry- damn close to an 11/x series. Not approximations of
11/x intervals related to the tonic, but the feel of the series
as a whole. I simply cannot
equate this with "approximations" because of the fact that
I hear intervals with similar characteristics which are not
particularly near to each other, and are even seperated by a
zone which does NOT share the characteristics in mind. For
example, a fifth of around 696 cents is more like a fifth
of 702 cents as far as softness than a fifth of 699 cents is.
If "softness" is a red thread in a particular series of
intervals, that can get us "damn close" in feeling with
quite dubious "approximating". If you are actually paying
attention to what I say, you'll see how that apparently
contradicts something else I've been saying, but more on
that later.

> >> From the figures you've got there, the worst interval of 46
> >>does not involve the tonic. And the worst interval from 41
> >>without the tonic is still over 6 cents. To me, these would
> >>be "fairly large" errors.
> >
> > Error from 11/x, in relation to the tonic, yes. Without the
> > tonic, the intervals are centered around a virtual partial
> > some 7+ cents lower than 11. The maximum error is <3.4 cents.
> > In other words, taken on it's own, this particular series
> > is consistent in proportions with an 11/x series. Against
> > the tonic, it's consistent with a detuned 11th partial,
> > therefore consistent in character. It's not great- I just
> > grabbed 41 as an example- but it doesn't suck.
>
> This "virtual partial" is similar to the mean implied in
> calculating the standard deviation. So you seem to agree
> with my logic in using standard deviations to rate errors
> where you don't temper the octaves. That's good because
> nobody else understands it. Note that if you do optimize
> the octaves, the virtual partial will end up with a near
> zero error.

That's great, man! Carl wanted to know my complaints about
the "regular mapping paradigm" and one of them was
the absence, as far as I could tell, of exactly this idea.
"Standard deviations" is good. In my opinion (practice),
series with heavy deviations which have a rhyme and reason
of their own- near-zero in the frame of a tempered octave,
as you put it, or deviating according to partial-oriented
pattern, for example each member of the series "tempered" halfway
toward a member of another interlaced series based on audible
partials, can create both character and consistency, without
what I hear as the spongey character of more-or-less-precisely
approximated low-ratio JI.

>
> >>As a benchmark, the worst interval in 11-limit minimax
> >>miracle is 3.3 cents from just. 41-equal is about twice as
> >>bad by the TOP-RMS measure. And miracle is what I'll
> >>normally use in the 11-limit. 41 is outside the range I'd
> >>use. So if the argument is that you lose the 11-limit
> >>character when you reach 41-equal, I'm in agreement.
> >
> > Can you give me a specific "minimax" example? 3.3 cents maximum
> > error from a simpler series, even
> > plus or minus, with no fliers, would be very good indeed. But in
> > the Scala archive I find for example:
>
> You bet it's good!
>
> > Canasta with Secor's minimax generator of 116.7155941 cents (5:9
> > exact). XH5, 1976
>
> That's exactly what I get. My generator is
> 116.7155940982074 cents. (May not be accurate to that
> precision.) I can't find this file in my Scala archive,
> however, so I can't check it. The mapping by octaves and
> generators for miracle is
>
> <1, 1, 3, 3, 2]
> <0, 6, -7, -2, 15]
>
> > |
> > Temperings of
> > 11/10 11/9 11/8 11/7 11/6
> >
> > -15.4446 2.7388 -0.5840 -16.5104 1.0774
> >
> > which leaves me scratching my head, for it's a typical
> > example of "hodge-podge approximations" for this
> > particular 11s series.
>
> That's plain wrong. The errors in primes should be
>
> 2 3 5 7 11
> 0.0 -1.7 -3.3 -2.3 -0.6

Hmmm, what you have there as errors in primes looks like
temperings of the actual partials to me, which is very
groovy, but I was looking at the deviations from the
actual intervals in question. In order to see if we're
on the same sheet of music here, can you point me to
an explanation of what you mean by errors in primes.

Er, it would also help if I looked at the right
bunch of numbers, LOL-

11/10 2.7388
11/9 2.7388
11/8 -0.5840
11/7 1.6731
11/6 1.0774
>
> So the errors in your intervals are
>
> 11:10 11:9 11:8 11:7 11:6
>
> 2.7388 2.7388 -0.5840 1.6731 1.0774

...as you say. And it certainly
>
> We disagree on 11:10 and 11:7. Furthermore, in both cases
> our disagreement amounts to 18.1834 +/- 0.0001 cents. Well,
> 18.1834 cents is the residue you get from closing the cycle
> of secors at 31 steps. So it looks like you've taken two of
> the wrong intervals from the 31 note scale. If you're
> following the convention of / for notes, it means you chose
> a bad tonic.

Yip, got that straight now. Well that's certainly a
bunch of very very close approximations. And in this
case they certainly qualify as approximations.
Lessee... hmmm, well the Canasta tuning here has done
a fine job with this. It's harder sounding than the
Just, which isn't something I can explain for I shouldn't
be able to hear that, but it's completely coherent.

> >>I find that 41-equal does work in the 9-limit. Magic is a
> >>good way of mapping it.
> >
> > Can you give me a specific example, Scala file preferably,
> > of 41-equal "magic" mapped?
>
> A scala file of 41-equal will look the same however it's
> mapped. What I can give you is a 19 note magic scale

Thanks...
>
> 58.9
> 117.9
> 203.5
> 262.5
> 321.4
> 380.4
> 439.3
> 498.2
> 583.9
> 642.8
> 701.8
> 760.7
> 819.6
> 878.6
> 964.2
> 1023.2
> 1082.1
> 1141.1
> 1200.0
>
> It uses some optimal tuning or other. The generator mapping
> for magic is <0, 5, 1, 12]. For this scale I take 12 thirds
> up from the 1/1, and 6 down. That means you can find
> approximations to 9/8, 5/4, 3/2, and 7/4 and also 4/3, etc.
> But there isn't an 8/7. If you prefer to think utonally,
> try this:
>
> 58.9
> 117.9
> 176.8
> 235.8
> 321.4
> 380.4
> 439.3
> 498.2
> 557.2
> 616.1
> 701.8
> 760.7
> 819.6
> 878.6
> 937.5
> 996.5
> 1082.1
> 1141.1
> 1200.0

These are both very good. I notice in the second one,
which is the first "tuning-group" tuning I've heard
that I could actually use (different strokes, etc.) other
than some of Kraig's tunings and couple of George's
and Margo's (and man, I've plowed through the archives
for a year, are you guys aware of the shear bulk of stuff
you churn out?) the "secor" is a little high. When I got
up at 4 this morning to get in a couple of hours of
monkeying with the secor, I found that tweaking it a
little high or a little low of its ideal value creates
tunings that sound more "as one" to me. So they're actually
lumpier and worse, on paper, by anybody's standards, but I
like them better.

I cannot explain why this is. It seems to me that the secor
generator is hypersensitive, as far as the overall character
of the tunings generated. Which is cool.

>
> >>And the 11/x series without the
> >>tonic is really the 9-limit.
> >
> > Yes, you've got the superparticulars expanding from 10/9.
> > Surely you see one reason I'm concerned with consistency
> > in the series itself, not just absolute proximity to
> > the Just intervals themselves?
>
> Um, what's the difference?

See above, "standard deviation".
>
> >>Whether your aim is to look at higher limits with no sevens,
> >>or only to split the differences within lower limits, it
> >>doesn't matter. The result is the same.
> >
> > More or less, but as far as audibility, I think it's better
> > to go for harmonic means of audible partials rather than
> > ratios based on partials of dubious audibility. Also you
> > steer clear of crap like magic primes.
>
> My online scripts only take subset-prime limits. If you
> want to use other JI bases you'll have to download the
> python libraries. The regular mapping paradigm doesn't care
> either way.
>
> >>And the result
> >>partly depends on the approximation to 7:1 being so
> >>glaringly bad (and all the signs agreeing).
> >
> > Exactly! Vice into virtue. I'd rather have rock-solid
> > higher-partials/more-complex series in a tuning than
> > lumpy references to the lowest partials. Easier to
> > sing, to cut to the chase.
>
> Have you seen George Secor's paper about his 17 note well
> temperament? It's based on an incomplete prime limit.

Certainly- as I've mentioned a number of times before, hunting out
17 by ear is brought me to this list in the first place.
>
> >>>Well, I just don't buy brute dissonance as a measure of
> >>>a tuning's validity at all, nor do I find that brute
> >>>dissonance agrees very much with contextual (musical)
> >>>dissonance and consonance. But different strokes...
> >>
> >>I'm happy that tension and release works in 12-equal with
> >>5-limit rules.
> >
> > Doesn't work for me- I hear neither true tension nor true
> > release in 12-equal, and haven't since earliest memory. I
> > remember cringing when I first heard in school that the M3
> > was "happy" and "peaceful". What, that theatrical grimace
> > is supposed to be a smile?
>
> I wasn't talking about the tuning but music written in it.
> Many, many people learn music theory in 12-equal and it
> makes perfect sense to them.

I also learned music theory in 12-equal and cringed all the
way through.

>
> >>And the approximation is quite extreme
> >>there.
> >
> > It's not an approximation. Different interval.
>
> I see you're having difficulty with the meanings of words again.

If "to approximate" means "to create an object A, which
bears little resemblance to object B, then to insist that
object A resembles object B, for unfathomable reasons", then yes,
I have difficulty with that meaning.
>
> >>The thirds in 12-equal don't sound at all like their
> >>just counterparts.
> >
> > They sound quite a bit like 81/64 to me.
>
> They're nearly 8 cents out, so you're still being generous
> with the approximations.

Sure, but the relationship of an M3 to the fifth is
important- 81/64 pops right out of perfect fifths, and
400 cents right out of near-perfect fifths. Not a great
approximation, but compared to 5/4 vs 400 cents, gimmee
a break. The "5/4" being unlinked to the fifths
is a point of complaint about 34, btw.

>
> >>I'm also generally happy with 11-limit harmony in miracle.
> >>Although I'm recently finding that I prefer the
> >>approximation to 11:8 to that of 7:5. Maybe it's because of
> >>the accuracy.
> >
> > Once again, I'd need a definition of "11-limit" and a Scala
> > file illustrating this.
>
> Try http://x31eq.com/keyboard.html and tell me if I forgot
> to copy anything over. Or how about csound code?

Csound code is great (I'm coding for a show in Berlin at
the moment) and I'll fire it up when I get the chance.

Hmm, nice approach to inegrating the tuning into Csound,
man. My code always looks like the Kabballah, hehe.

>
> ;;;;;; this at global scope
>
> ; low mapping
> gikbdSecor ftgen 0, 0, 32, -2, \
> 0, 0, 1, 1, 1, 2, 2, 3, 3, 4, 4, 4, \
> 5, 5, 6, 6, 6, 7, 7, 8, 8, 9, 9, 9
>
> gikbdQuomma ftgen 0, 0, 32, -2, \
> -1, 0, -1, 0, 1, -1, 0, -1, 0, -1, 0, 1, \
> -1, 0, -1, 0, 1, -1, 0, -1, 0, -1, 0, 1
>
> opcode CPSDec, i, iii
> inote, iquommas, ituning xin
>
> ioctaves init int(inote)
> isecors init frac(inote)*10
>
> ; universal decimal tuning
> ; works for both miracle and negri
> isecor_size = 1/(10+ituning)
> iquomma_size = abs(ituning/(10+ituning))
>
> xout cpsoct(ioctaves + \
> isecors*isecor_size + iquommas*iquomma_size)
> endop
>
> ;;;;;; and this in the instrument code
>
> ikey init p5
> ipitchClass = ikey%24
> ioctave = int(ikey/24) + 6
> isecors table ipitchClass, gikbdSecor
> iquommas table ipitchClass, gikbdQuomma
> inote = ioctave + isecors/10
> ifreq CPSDec inote, iquommas, 0.2747

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Sorry- it was Cameron. Cameron who wrote that....
>
> > > Now we're getting somewhere- using my ears to judge the sound
> > > is what brought me to questioning the regular mapping
> > > paradigm as practiced here, for what I hear and what's
> > > on paper simply don't agree so far.
> >
> > Did you ever make that list of disagreements I asked for?

I think "rarely hearing anything that sounds f-all like JI"
should do for a start, don't you? And "stiff yet spongy
hodge-podge" comes to mind. Nothing personal to anyone here,
anyone is welcome to tell me that my tunings suck, I'll just
laugh.

I'm talking to Graham about the "secor generator" tunings at
the moment, for that's the only "RMP" I've found so far that
might be copasetic, afaict.

Of course I may have overlooked something, and it's all
obviously subjective.

-Cameron Bobro

Cameron Bobro wrote:
> --- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
> >>Plenty of the details of what you say you hear do agree.
> > Obviously, why else would I even bother? There are plenty
> of tuning "schools" around, like the whole divvying up of
> 12-tET approach, which would be an utter waste of time for > me.

Well, it's good to have some ground for agreement.

>>>>And I note you still haven't come up >>>>with a quantitative theory. So for now all we have is a >>>>crude model for one dimension of the sound.
>>>
>>>Well there's no law that says you have to read or
>>>understand everything I write, and certainly it's
>>>possible that I have no writing skills. >>
>>It's a plain statement of fact that you haven't provided us >>with a quantitative theory. Why do you have to get >>confrontational about it?
> > Where do you get "confrontational"? Is this the answering a question with another question game?

>>>What do you mean by "11-limit"? (Serious question).
>>
>>Ratios of odd numbers no greater than 11 and their octave >>equivalents.
> > Okay, that's cool by me, because it relates to actual audible
> partials. This, from wikipedia:

... which does also define the odd limit.

> "The prime limit can be seen as a generalization that does not favor > the number 2. It is defined as the largest prime number in the > factorization of both numerator and denominator. That is, in number > theoretic terms, it measures the smoothness of the numerator and > denominator. The prime limit of the perfect fourth is 3 (the same as > the odd limit), but the minor tone has a prime limit of 5, because 9 > can be factorized into 3�3, and 10 into 2�5."
> > is bogus as far as tuning goes, unless it's qualified with > exponential limit. Except for numerologists, of course, which
> is also fine. Ah, now ... Weighted prime limits are the current fashion for rating temperaments. You have my PDF on them. The important thing is the weighting, which can be thought of statistically as saying how likely an interval is to be used.

Prime limits are obviously not perfect because you have to choose an arbitrary subset instead of setting a weighting over all intervals. But neither do you choose exactly which intervals you're interested in. Choosing certain primes and weighting them falls between two stools, but it's simple and works well enough.

> Sorry- damn close to an 11/x series. Not approximations of
> 11/x intervals related to the tonic, but the feel of the series > as a whole. I simply cannot
> equate this with "approximations" because of the fact that
> I hear intervals with similar characteristics which are not
> particularly near to each other, and are even seperated by a
> zone which does NOT share the characteristics in mind. For > example, a fifth of around 696 cents is more like a fifth
> of 702 cents as far as softness than a fifth of 699 cents is.
> If "softness" is a red thread in a particular series of
> intervals, that can get us "damn close" in feeling with
> quite dubious "approximating". If you are actually paying
> attention to what I say, you'll see how that apparently
> contradicts something else I've been saying, but more on
> that later. This is where you have to set out your ideas in detail. Maybe there's a way we can make use of them. I won't promise anything because it sounds like the pitch-dimension gets too fragmented. But even then maybe better mathematicians can do something with it.

> That's great, man! Carl wanted to know my complaints about
> the "regular mapping paradigm" and one of them was > the absence, as far as I could tell, of exactly this idea.
> "Standard deviations" is good. In my opinion (practice),
> series with heavy deviations which have a rhyme and reason > of their own- near-zero in the frame of a tempered octave,
> as you put it, or deviating according to partial-oriented > pattern, for example each member of the series "tempered" halfway > toward a member of another interlaced series based on audible
> partials, can create both character and consistency, without > what I hear as the spongey character of more-or-less-precisely
> approximated low-ratio JI.

A paradigm doesn't stand or fall by how you measure the errors. Still, we have been thinking about errors over the years.

The first step was to look at complete odd limits. Paul Erlich defined "consistency" to ensure the errors added up correctly. His 22 note paper shows this thinking. This is also why we started looking at the mappings -- to ensure that the errors add up right when you don't have a consistent ET.

The second step was TOP errors. These are what Paul uses in the Middle Path paper. You take the worst weighted error over a prime limit. If you use the right weighting and optimize the scale stretch, it turns out that it doesn't matter much which intervals you look at. The primes themselves are a sufficient subset. Because of that I redefined "TOP" to be more general so that what Paul used is "TOP-max".

The third step is to use standard deviations where you don't optimize the scale stretches. I started writing that PDF to explain why this works, and also why TOP works. Ultimately, we know they're both wrong. Worst or average weighted errors don't work unless you optimize the scale stretch. Standard deviations don't work if you shrink the scales. But still, they're simple, and work well enough to get useful results.

>>That's plain wrong. The errors in primes should be
>>
>> 2 3 5 7 11
>>0.0 -1.7 -3.3 -2.3 -0.6
> > Hmmm, what you have there as errors in primes looks like
> temperings of the actual partials to me, which is very
> groovy, but I was looking at the deviations from the
> actual intervals in question. In order to see if we're
> on the same sheet of music here, can you point me to
> an explanation of what you mean by errors in primes.

Yes. Every composite number is built from prime numbers, and every interval in a regular tuning is built from prime intervals. If they're mapped from an ideal, not only can you get the size of an interval from the size of the primes, but you can also get the error in a composite interval from the errors of the primes that make it up.

> Er, it would also help if I looked at the right
> bunch of numbers, LOL-
> > 11/10 2.7388 > 11/9 2.7388 > 11/8 -0.5840 > 11/7 1.6731 > 11/6 1.0774

11/10 is like 11/5, so take the difference between the errors in 11 and 5. -0.6+3.3=2.7. And so on.

>>We disagree on 11:10 and 11:7. Furthermore, in both cases >>our disagreement amounts to 18.1834 +/- 0.0001 cents. Well, >>18.1834 cents is the residue you get from closing the cycle >>of secors at 31 steps. So it looks like you've taken two of >>the wrong intervals from the 31 note scale. If you're >>following the convention of / for notes, it means you chose >>a bad tonic.
> > Yip, got that straight now. Well that's certainly a > bunch of very very close approximations. And in this
> case they certainly qualify as approximations.
> Lessee... hmmm, well the Canasta tuning here has done
> a fine job with this. It's harder sounding than the
> Just, which isn't something I can explain for I shouldn't
> be able to hear that, but it's completely coherent. The tuning is 11-limit minimax miracle. Canasta is the scale -- 31 generators.

> These are both very good. I notice in the second one,
> which is the first "tuning-group" tuning I've heard
> that I could actually use (different strokes, etc.) other
> than some of Kraig's tunings and couple of George's
> and Margo's (and man, I've plowed through the archives
> for a year, are you guys aware of the shear bulk of stuff
> you churn out?) the "secor" is a little high. When I got
> up at 4 this morning to get in a couple of hours of
> monkeying with the secor, I found that tweaking it a > little high or a little low of its ideal value creates
> tunings that sound more "as one" to me. So they're actually
> lumpier and worse, on paper, by anybody's standards, but I > like them better. Those are magic tunings, so the generator is a major third, not a secor. Secors are for miracle (because George Secor was the first person to descover the temperament class). So which generator are you tweaking?

I don't know how useful the archives are precisely because there's so much in there. Taking part is better (as you're now doing of course). I found some posts from Margo very useful from several years ago but I don't know if you'll get back to them. There are more permanent documents which you're finding but *cough* could maybe study in more detail.

> I cannot explain why this is. It seems to me that the secor > generator is hypersensitive, as far as the overall character > of the tunings generated. Which is cool. A small change in the generator gets magnified in the more complex intervals. And this is more important the better the approximations are to start with. So this makes sense, whichever temperament class it is you're talking about.

>>>>And the 11/x series without the >>>>tonic is really the 9-limit. >>>
>>>Yes, you've got the superparticulars expanding from 10/9.
>>>Surely you see one reason I'm concerned with consistency
>>>in the series itself, not just absolute proximity to
>>>the Just intervals themselves? >>
>>Um, what's the difference?
> > See above, "standard deviation".

It means scoring intervals between partials instead of only the partials measured relative to the fundamental. But intervals either way.

>>Have you seen George Secor's paper about his 17 note well >>temperament? It's based on an incomplete prime limit.
> > Certainly- as I've mentioned a number of times before, hunting out
> 17 by ear is brought me to this list in the first place.

It's sort of on the edge of the regular mapping paradigm. Well temperaments follow mappings but aren't determined by them.

> I also learned music theory in 12-equal and cringed all the > way through. One thing I'll say is that sus4 chords aren't as bad as they should be.

>>>>The thirds in 12-equal don't sound at all like their >>>>just counterparts. >>>
>>>They sound quite a bit like 81/64 to me.
>>
>>They're nearly 8 cents out, so you're still being generous >>with the approximations.
> > Sure, but the relationship of an M3 to the fifth is > important- 81/64 pops right out of perfect fifths, and
> 400 cents right out of near-perfect fifths. Not a great
> approximation, but compared to 5/4 vs 400 cents, gimmee
> a break. The "5/4" being unlinked to the fifths
> is a point of complaint about 34, btw. We were talking about the LucyTuned third before. It gets closer to 5:4 than 400 cents gets to 81:64.

As it happens, the miracle tempered approximation to 81:64 comes out close to 400 cents. Exactly so if you tune to 72-equal. And there's also the approximate 5:4 to compare it with.

Is that unlinking something you complain about?

Graham

> > > What do you mean by "11-limit"? (Serious question).
> >
> > Ratios of odd numbers no greater than 11 and their octave
> > equivalents.
>
> Okay, that's cool by me, because it relates to actual audible
> partials. This, from wikipedia:
>
> "The prime limit can be seen as a generalization that does not
> favor the number 2. It is defined as the largest prime number
> in the factorization of both numerator and denominator. That
> is, in number theoretic terms, it measures the smoothness of
> the numerator and denominator. The prime limit of the perfect
> fourth is 3 (the same as the odd limit), but the minor tone
> has a prime limit of 5, because 9 can be factorized into 3×3,
> and 10 into 2×5."
>
> is bogus as far as tuning goes, unless it's qualified with
> exponential limit.

It's bogus as far as psychoacoustics goes, even if it were
qualified by an exponent limit (the ear doesn't factor).

But primes are useful in tuning theory because their errors
combine in predictable ways -- you can calculate the maximum
error of all the dyads in a tuning, for example, just by
finding the error of the primes.

> I cannot explain why this is. It seems to me that the secor
> generator is hypersensitive, as far as the overall character
> of the tunings generated. Which is cool.

Most generated scales will be very sensitive to the size of
their generator (since the difference in generator size is
compounded as the generator is 'stacked').

-Carl

"Cameron Bobro" <misterbobro@...> wrote:
> > > > Now we're getting somewhere- using my ears to judge the sound
> > > > is what brought me to questioning the regular mapping
> > > > paradigm as practiced here, for what I hear and what's
> > > > on paper simply don't agree so far.
> > >
> > > Did you ever make that list of disagreements I asked for?
>
> I think "rarely hearing anything that sounds f-all like JI"
> should do for a start, don't you?

No, because it doesn't tell us anything about how to improve.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> "Cameron Bobro" <misterbobro@> wrote:
> > > > > Now we're getting somewhere- using my ears to judge the sound
> > > > > is what brought me to questioning the regular mapping
> > > > > paradigm as practiced here, for what I hear and what's
> > > > > on paper simply don't agree so far.
> > > >
> > > > Did you ever make that list of disagreements I asked for?
> >
> > I think "rarely hearing anything that sounds f-all like JI"
> > should do for a start, don't you?
>
> No, because it doesn't tell us anything about how to improve.
>
> -Carl
>

Tell, no, suggest, certainly. If approximating JI either fails,
or succeeds too well (using a bazillion intervals almost
indistinguishable from JI, why not just use a bazillion actual
JI intervals?), then it seems utterly reasonable (not to
mention musically good) to work on the attractiveness,
usability and coherency/integrity of intervals that are NOT
"JI"

This can be done without dumping JI or ignoring the actual
source of JI, which is the harmonic partials. Pretending the
harmonic series doesn't exist or matter would be even more
bogus, by far, than deifying JI. What would be very nice,
I believe, are intervals which are not simple JI but function
well in terms of the audible partials.

One way of doing this is already done in practice here-
splitting the difference between 7/4 and 9/5 for example
works functionally and, to my ears, in terms of physical
sound.

One reason I keep harping on about 400 cents vs. 5/4 is that
400 cents is actually an example of what I'm talking about.
It splits the difference, not tragically badly, between 5/4
and 9/7 and in flexibly pitched music it can be tweaked on
the fly to work as either (leading tone, sing high! I'm sure
others have heard similiar instructions) as well as a
product-of-fifths Pyth. 3d. The one-dimensional interpretation
as 400-cents as 5/4 bugs me not just because it can make me cringe
on fixed-pitch instruments, but because it hampers thoughts on
these "other families" of intervals.

Another way is examining heavy "temperings" in terms of
what Graham calls "standard deviation". And so on. Any
approach requires losing "near simple JI good, far from simple
JI bad" of course. And seriously unhooking concordance from
consonance.

Also, IMO, it is imperative to address issues like
tetrachordal/scalar harmony and harmonic-series harmony,
for what is perfect in one can be a disaster in others.
This issue can be described in shorthand as the 7/4 "vs" 9/5
issue.

There is no way, absolutely no way, that I'm the first or only
to say "lets make virtues from vices" on this tuning list.
It should also be obvious that I'm advocating the
preservation of JI by not watering it down. Not everyone
may agree with the idea of let just be just and not-just be
not-just, avoiding neither-fish-nor-fowl.

-Cameron Bobro

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> Cameron Bobro wrote:
> > --- In tuning@yahoogroups.com, Graham Breed <gbreed@> wrote:
> >
> >>Plenty of the details of what you say you hear do agree.
> >
> > Obviously, why else would I even bother? There are plenty
> > of tuning "schools" around, like the whole divvying up of
> > 12-tET approach, which would be an utter waste of time for
> > me.
>
> Well, it's good to have some ground for agreement.
>
> >>>>And I note you still haven't come up
> >>>>with a quantitative theory. So for now all we have is a
> >>>>crude model for one dimension of the sound.
> >>>
> >>>Well there's no law that says you have to read or
> >>>understand everything I write, and certainly it's
> >>>possible that I have no writing skills.
> >>
> >>It's a plain statement of fact that you haven't provided us
> >>with a quantitative theory. Why do you have to get
> >>confrontational about it?
> >
> > Where do you get "confrontational"?
>
> Is this the answering a question with another question game?

I don't know- is it? Hahaha! Nah, Carl is the indisputed
master of that game. If you look closely at your "question",
you'll find that it is actually a statement. Consider
the classic example: "Have you stopped beating your
wife, yes or no?"
>
> >>>What do you mean by "11-limit"? (Serious question).
> >>
> >>Ratios of odd numbers no greater than 11 and their octave
> >>equivalents.
> >
> > Okay, that's cool by me, because it relates to actual audible
> > partials. This, from wikipedia:
>
> ... which does also define the odd limit.

The limit I use is "highest partial referred to".
>
> > "The prime limit can be seen as a generalization that does not
favor
> > the number 2. It is defined as the largest prime number in the
> > factorization of both numerator and denominator. That is, in
number
> > theoretic terms, it measures the smoothness of the numerator and
> > denominator. The prime limit of the perfect fourth is 3 (the
same as
> > the odd limit), but the minor tone has a prime limit of 5,
because 9
> > can be factorized into 3×3, and 10 into 2×5."
> >
> > is bogus as far as tuning goes, unless it's qualified with
> > exponential limit. Except for numerologists, of course, which
> > is also fine.
>
> Ah, now ... Weighted prime limits are the current fashion
> for rating temperaments. You have my PDF on them. The
> important thing is the weighting, which can be thought of
> statistically as saying how likely an interval is to be used.

I have yet to find any concrete connection between this and the
audible, but we'll see.
>
> Prime limits are obviously not perfect because you have to
> choose an arbitrary subset instead of setting a weighting
> over all intervals. But neither do you choose exactly which
> intervals you're interested in. Choosing certain primes and
> weighting them falls between two stools, but it's simple and
> works well enough.
>
> > Sorry- damn close to an 11/x series. Not approximations of
> > 11/x intervals related to the tonic, but the feel of the series
> > as a whole. I simply cannot
> > equate this with "approximations" because of the fact that
> > I hear intervals with similar characteristics which are not
> > particularly near to each other, and are even seperated by a
> > zone which does NOT share the characteristics in mind. For
> > example, a fifth of around 696 cents is more like a fifth
> > of 702 cents as far as softness than a fifth of 699 cents is.
> > If "softness" is a red thread in a particular series of
> > intervals, that can get us "damn close" in feeling with
> > quite dubious "approximating". If you are actually paying
> > attention to what I say, you'll see how that apparently
> > contradicts something else I've been saying, but more on
> > that later.
>
> This is where you have to set out your ideas in detail.
> Maybe there's a way we can make use of them. I won't
> promise anything because it sounds like the pitch-dimension
> gets too fragmented. But even then maybe better
> mathematicians can do something with it.
>
> > That's great, man! Carl wanted to know my complaints about
> > the "regular mapping paradigm" and one of them was
> > the absence, as far as I could tell, of exactly this idea.
> > "Standard deviations" is good. In my opinion (practice),
> > series with heavy deviations which have a rhyme and reason
> > of their own- near-zero in the frame of a tempered octave,
> > as you put it, or deviating according to partial-oriented
> > pattern, for example each member of the series "tempered"
halfway
> > toward a member of another interlaced series based on audible
> > partials, can create both character and consistency, without
> > what I hear as the spongey character of more-or-less-precisely
> > approximated low-ratio JI.
>
> A paradigm doesn't stand or fall by how you measure the
> errors.

Well I think measuring "errors" by ear is a pretty good
ruler.

>Still, we have been thinking about errors over the
> years.

Yes. It seems to me that once you've got a certain amount
of deviation from whatever model, it's time to ask whether
you're "approximating" or actually measuring something else.
I read about dinosaurs with my little boy- it's interesting
when the books mention reclassifications and new
descriptions of dinosaurs (lots of revision in the last
25 years or so).
>
> The first step was to look at complete odd limits. Paul
> Erlich defined "consistency" to ensure the errors added up
> correctly. His 22 note paper shows this thinking. This is
> also why we started looking at the mappings -- to ensure
> that the errors add up right when you don't have a
> consistent ET.
>
> The second step was TOP errors. These are what Paul uses in
> the Middle Path paper. You take the worst weighted error
> over a prime limit. If you use the right weighting and
> optimize the scale stretch, it turns out that it doesn't
> matter much which intervals you look at. The primes
> themselves are a sufficient subset. Because of that I
> redefined "TOP" to be more general so that what Paul used is
> "TOP-max".
>
> The third step is to use standard deviations where you don't
> optimize the scale stretches. I started writing that PDF to
> explain why this works, and also why TOP works. Ultimately,
> we know they're both wrong. Worst or average weighted
> errors don't work unless you optimize the scale stretch.
> Standard deviations don't work if you shrink the scales.
> But still, they're simple, and work well enough to get
> useful results.
>
> >>That's plain wrong. The errors in primes should be
> >>
> >> 2 3 5 7 11
> >>0.0 -1.7 -3.3 -2.3 -0.6
> >
> > Hmmm, what you have there as errors in primes looks like
> > temperings of the actual partials to me, which is very
> > groovy, but I was looking at the deviations from the
> > actual intervals in question. In order to see if we're
> > on the same sheet of music here, can you point me to
> > an explanation of what you mean by errors in primes.
>
> Yes. Every composite number is built from prime numbers,
> and every interval in a regular tuning is built from prime
> intervals. If they're mapped from an ideal, not only can
> you get the size of an interval from the size of the primes,
> but you can also get the error in a composite interval from
> the errors of the primes that make it up.
>
> > Er, it would also help if I looked at the right
> > bunch of numbers, LOL-
> >
> > 11/10 2.7388
> > 11/9 2.7388
> > 11/8 -0.5840
> > 11/7 1.6731
> > 11/6 1.0774
>
> 11/10 is like 11/5, so take the difference between the
> errors in 11 and 5. -0.6+3.3=2.7. And so on.
>
> >>We disagree on 11:10 and 11:7. Furthermore, in both cases
> >>our disagreement amounts to 18.1834 +/- 0.0001 cents. Well,
> >>18.1834 cents is the residue you get from closing the cycle
> >>of secors at 31 steps. So it looks like you've taken two of
> >>the wrong intervals from the 31 note scale. If you're
> >>following the convention of / for notes, it means you chose
> >>a bad tonic.
> >
> > Yip, got that straight now. Well that's certainly a
> > bunch of very very close approximations. And in this
> > case they certainly qualify as approximations.
> > Lessee... hmmm, well the Canasta tuning here has done
> > a fine job with this. It's harder sounding than the
> > Just, which isn't something I can explain for I shouldn't
> > be able to hear that, but it's completely coherent.
>
> The tuning is 11-limit minimax miracle. Canasta is the
> scale -- 31 generators.
>
> > These are both very good. I notice in the second one,
> > which is the first "tuning-group" tuning I've heard
> > that I could actually use (different strokes, etc.) other
> > than some of Kraig's tunings and couple of George's
> > and Margo's (and man, I've plowed through the archives
> > for a year, are you guys aware of the shear bulk of stuff
> > you churn out?) the "secor" is a little high. When I got
> > up at 4 this morning to get in a couple of hours of
> > monkeying with the secor, I found that tweaking it a
> > little high or a little low of its ideal value creates
> > tunings that sound more "as one" to me. So they're actually
> > lumpier and worse, on paper, by anybody's standards, but I
> > like them better.
>
> Those are magic tunings, so the generator is a major third,
> not a secor. Secors are for miracle (because George Secor
> was the first person to descover the temperament class). So
> which generator are you tweaking?

The secor, so I guess I'm only talking about "miracle" tunings.
>
> I don't know how useful the archives are precisely because
> there's so much in there. Taking part is better (as you're
> now doing of course). I found some posts from Margo very
> useful from several years ago but I don't know if you'll get
> back to them. There are more permanent documents which
> you're finding but *cough* could maybe study in more detail.
>
> > I cannot explain why this is. It seems to me that the secor
> > generator is hypersensitive, as far as the overall character
> > of the tunings generated. Which is cool.
>
> A small change in the generator gets magnified in the more
> complex intervals. And this is more important the better
> the approximations are to start with. So this makes sense,
> whichever temperament class it is you're talking about.
>
> >>>>And the 11/x series without the
> >>>>tonic is really the 9-limit.
> >>>
> >>>Yes, you've got the superparticulars expanding from 10/9.
> >>>Surely you see one reason I'm concerned with consistency
> >>>in the series itself, not just absolute proximity to
> >>>the Just intervals themselves?
> >>
> >>Um, what's the difference?
> >
> > See above, "standard deviation".
>
> It means scoring intervals between partials instead of only
> the partials measured relative to the fundamental. But
> intervals either way.

But when you're scoring intervals "between partials" you're
not approximating intervals derived directly from the
partials (in relation to the tonic). Losing the idea that
these are approximations and treating them as something new
is something I'm "advocating". Go ahead and make them
"worse" as far as intervals above a tonic, but better
on their own terms. For example, accept your idea
about "standard deviations" with gusto- apply it to
series that just plain sucks as far approximating
simple-JI relationships with the tonic.

>
> >>Have you seen George Secor's paper about his 17 note well
> >>temperament? It's based on an incomplete prime limit.
> >
> > Certainly- as I've mentioned a number of times before, hunting
>out
> > 17 by ear is brought me to this list in the first place.
>
> It's sort of on the edge of the regular mapping paradigm.
> Well temperaments follow mappings but aren't determined by them.
>
> > I also learned music theory in 12-equal and cringed all the
> > way through.
>
> One thing I'll say is that sus4 chords aren't as bad as they
> should be.

According to my view of 12-tET ("ultimate evolution of
Pythagorean), sus4 chords should and are good in 12-tET. IIRC
Stockhausen of all people wrote a paper on what are called "sus4"
chords as structural units, LOL.

>
> >>>>The thirds in 12-equal don't sound at all like their
> >>>>just counterparts.
> >>>
> >>>They sound quite a bit like 81/64 to me.
> >>
> >>They're nearly 8 cents out, so you're still being generous
> >>with the approximations.
> >
> > Sure, but the relationship of an M3 to the fifth is
> > important- 81/64 pops right out of perfect fifths, and
> > 400 cents right out of near-perfect fifths. Not a great
> > approximation, but compared to 5/4 vs 400 cents, gimmee
> > a break. The "5/4" being unlinked to the fifths
> > is a point of complaint about 34, btw.
>
> We were talking about the LucyTuned third before. It gets
> closer to 5:4 than 400 cents gets to 81:64.

Better save that for another day...
>
> As it happens, the miracle tempered approximation to 81:64
> comes out close to 400 cents. Exactly so if you tune to
> 72-equal. And there's also the approximate 5:4 to compare
> it with.
>
Graham, referring to the M3ds and P5ths in 34:
> Is that unlinking something you complain about?

Well, with triads and common tones it's bound to create
big problems, but you probably analized for example Liszt
in school- with nice sleazy Romantic-stylee voice leading,
I think 34 is pretty ideal for oozing around.

-Cameron Bobro

Cameron Bobro wrote:

> The limit I use is "highest partial referred to".

That'd be called an "integer limit". You have different intervals in different octaves which may be harder to think of. But it's perfectly valid. Normally we use weighted prime limits for octave-specific cases. That's the fashion.

Note that odd limits and integer limits are very similar.

>>A paradigm doesn't stand or fall by how you measure the >>errors. > > Well I think measuring "errors" by ear is a pretty good
> ruler. Yes, but you aren't sorting through millions of regular temperament classes to find the ones worth listening to.

>>Still, we have been thinking about errors over the >>years.
> > Yes. It seems to me that once you've got a certain amount
> of deviation from whatever model, it's time to ask whether
> you're "approximating" or actually measuring something else.
> I read about dinosaurs with my little boy- it's interesting
> when the books mention reclassifications and new > descriptions of dinosaurs (lots of revision in the last
> 25 years or so).

You can ask, and then shrug your shoulders and get on with your life.

>>It means scoring intervals between partials instead of only >>the partials measured relative to the fundamental. But >>intervals either way.
> > But when you're scoring intervals "between partials" you're
> not approximating intervals derived directly from the > partials (in relation to the tonic). Losing the idea that > these are approximations and treating them as something new > is something I'm "advocating". Go ahead and make them > "worse" as far as intervals above a tonic, but better
> on their own terms. For example, accept your idea
> about "standard deviations" with gusto- apply it to > series that just plain sucks as far approximating
> simple-JI relationships with the tonic. You're hardly advocating anything original. What happened to the critical flaw that was going to bring the regular mapping paradigm falling down around our ears?

You've got my prime errors PDF, haven't you? There's a table, Table 3 no less, on page 8. In that I give the standard deviation for a range of equal temperaments in the 7-limit, including 6-equal which pretty much sucks to my ears. What enlightenment am I supposed to gain from the resulting number? Note that the STD errors agree with the TOP-RMS errors to around 2 figures in all cases.

> According to my view of 12-tET ("ultimate evolution of > Pythagorean), sus4 chords should and are good in 12-tET. IIRC > Stockhausen of all people wrote a paper on what are called "sus4" > chords as structural units, LOL.

Schoenberg mentioned them as a basis for harmony as well.

> Graham, referring to the M3ds and P5ths in 34:
> >>Is that unlinking something you complain about?
> > Well, with triads and common tones it's bound to create > big problems, but you probably analized for example Liszt
> in school- with nice sleazy Romantic-stylee voice leading, > I think 34 is pretty ideal for oozing around. No, I don't remember any Liszt at school. Did he use 34-equal?

Graham

Cameron Bobro wrote:

> Tell, no, suggest, certainly. If approximating JI either fails,
> or succeeds too well (using a bazillion intervals almost
> indistinguishable from JI, why not just use a bazillion actual
> JI intervals?), then it seems utterly reasonable (not to > mention musically good) to work on the attractiveness, > usability and coherency/integrity of intervals that are NOT
> "JI" Or you could look for temperament classes with middling-low errors and low complexity. You don't have to fall into a false dichotomy.

> This can be done without dumping JI or ignoring the actual
> source of JI, which is the harmonic partials. Pretending the
> harmonic series doesn't exist or matter would be even more
> bogus, by far, than deifying JI. What would be very nice, > I believe, are intervals which are not simple JI but function > well in terms of the audible partials. > > One way of doing this is already done in practice here- > splitting the difference between 7/4 and 9/5 for example > works functionally and, to my ears, in terms of physical > sound. To get 16/9?

> One reason I keep harping on about 400 cents vs. 5/4 is that
> 400 cents is actually an example of what I'm talking about.
> It splits the difference, not tragically badly, between 5/4
> and 9/7 and in flexibly pitched music it can be tweaked on
> the fly to work as either (leading tone, sing high! I'm sure
> others have heard similiar instructions) as well as a > product-of-fifths Pyth. 3d. The one-dimensional interpretation
> as 400-cents as 5/4 bugs me not just because it can make me cringe
> on fixed-pitch instruments, but because it hampers thoughts on
> these "other families" of intervals.

Is that all you were getting at? Dominant temperament does it. It's in the Middle Path paper. So now you've gone from saying only small errors are meaningful to the other end of the spectrum.

> Another way is examining heavy "temperings" in terms of
> what Graham calls "standard deviation". And so on. Any > approach requires losing "near simple JI good, far from simple
> JI bad" of course. And seriously unhooking concordance from
> consonance. I call them standard deviations because that's what they're called. Microsoft Excel can do them, and the Windows calculator, and a middling good desktop calculator. They still tell you how close you get to JI. (Unless you get silly about the octave stretch.)

> Also, IMO, it is imperative to address issues like > tetrachordal/scalar harmony and harmonic-series harmony,
> for what is perfect in one can be a disaster in others. > This issue can be described in shorthand as the 7/4 "vs" 9/5 > issue. How do you address that?

> There is no way, absolutely no way, that I'm the first or only > to say "lets make virtues from vices" on this tuning list. > It should also be obvious that I'm advocating the
> preservation of JI by not watering it down. Not everyone
> may agree with the idea of let just be just and not-just be
> not-just, avoiding neither-fish-nor-fowl.

It isn't obvious because you dart around saying this and that and not getting to the point. In this case, de gustibus non est disputandum.

Graham

Honestly Cameron, I can't figure out what you're on about.

> Tell, no, suggest, certainly. If approximating JI either fails,
> or succeeds too well (using a bazillion intervals almost
> indistinguishable from JI, why not just use a bazillion actual
> JI intervals?), then it seems utterly reasonable (not to
> mention musically good) to work on the attractiveness,
> usability and coherency/integrity of intervals that are NOT
> "JI"

???

> This can be done without dumping JI or ignoring the actual
> source of JI, which is the harmonic partials. Pretending the
> harmonic series doesn't exist or matter would be even more
> bogus, by far, than deifying JI. What would be very nice,
> I believe, are intervals which are not simple JI but function
> well in terms of the audible partials.

Such as....

> One way of doing this is already done in practice here-
> splitting the difference between 7/4 and 9/5 for example
> works functionally and, to my ears, in terms of physical
> sound.

Sounds like you're recommending 12-ET!

> One reason I keep harping on about 400 cents vs. 5/4 is that
> 400 cents is actually an example of what I'm talking about.
> It splits the difference, not tragically badly, between 5/4
> and 9/7 and in flexibly pitched music it can be tweaked on
> the fly to work as either (leading tone, sing high! I'm sure
> others have heard similiar instructions) as well as a
> product-of-fifths Pyth. 3d. The one-dimensional interpretation
> as 400-cents as 5/4 bugs me not just because it can make me
> cringe on fixed-pitch instruments, but because it hampers
> thoughts on these "other families" of intervals.

Where has anybody said it can't also function as 9:7?

> Another way

...to "suggest" how to improve the regular mapping
paradigm is...

> is examining heavy "temperings" in terms of
> what Graham calls "standard deviation". And so on. Any
> approach requires losing "near simple JI good, far from simple
> JI bad" of course. And seriously unhooking concordance from
> consonance.

[At this point, blood starts pouring out of reader's nose.]

> Also, IMO, it is imperative to address issues like
> tetrachordal/scalar harmony and harmonic-series harmony,
> for what is perfect in one can be a disaster in others.
> This issue can be described in shorthand as the 7/4 "vs" 9/5
> issue.

Lots of scale theory has been done around here. I couldn't
tell you if any of it addresses whatever issue you think
you've identified.

> It should also be obvious that I'm advocating the
> preservation of JI by not watering it down. Not everyone
> may agree with the idea of let just be just and not-just be
> not-just, avoiding neither-fish-nor-fowl.

Uh, I certainly have no problem with JI.

-Carl

Cameron wrote...
> The limit I use is "highest partial referred to".

That's been called "integer limit" around here. It's a
perfectly valid thing.

> > Ah, now ... Weighted prime limits are the current fashion
> > for rating temperaments. You have my PDF on them. The
> > important thing is the weighting, which can be thought of
> > statistically as saying how likely an interval is to be used.
>
> I have yet to find any concrete connection between this and
> the audible, but we'll see.

How many weighted-prime-optimized tunings have you tested
against other kinds of tunings?

> > Still, we have been thinking about errors over the
> > years.
>
> Yes. It seems to me that once you've got a certain amount
> of deviation from whatever model, it's time to ask whether
> you're "approximating" or actually measuring something else.
> I read about dinosaurs with my little boy- it's interesting
> when the books mention reclassifications and new
> descriptions of dinosaurs (lots of revision in the last
> 25 years or so).

'Approximations' in music will be subservient to musical context
and the musical background of the listener.

> But when you're scoring intervals "between partials" you're
> not approximating intervals derived directly from the
> partials (in relation to the tonic). Losing the idea that
> these are approximations and treating them as something new
> is something I'm "advocating".

If you want to advocate something, you should be able to
give an example. Can you please give an example of what
you're advocating? Hopefully something that refers directly
to the intonation of pitches?

-Carl

http://www.versiontracker.com/dyn/moreinfo/macosx/32491

I wonder if it will work as well as Melodyne.

At least the price is less.

Having downloaded the trial we'll see ;-)

Charles Lucy lucy@lucytune.com

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