Otonal / Utonal

One way that has helped me understand the utonal and otonal
difference is as follows

Otonal - a collection of pitches that can be construed as the
overtones of a common fundamental:

5/4 3/2 9/8 for example

Utonal - a collection of pitches that can be construed as
fundamentals having a common overtone:

4/3 8/5 8/7. These having an overtone of 1/1 (1/1 is the 3/2 of 4/3,
1/1 is the 5/4 of 8/5 and 1/1 is the 7/4 of 8/7).

An otonal group share a common udentity and a utonal group share a
common odentity. u for undernumber and o for overnumber.

Here 1/1 refers to a reference pitch. Any pitch can be construed
either as a new fundamental (the overtone series being the
overarching example), or as the overtone of a new fundamental.

Partch's tonality diamond takes 1 3 5 7 9 and 11 'identities' and
treats them each way, as overtone (odentity) and fundamental
(udentity).

Mark

This looks like a good explanation, but what notes (indicated as
scale notes, such as c4-e4-g4) do 5/4 3/2 9/8, and 4/3 8/5 8/7
actually represent? (are they stacked, or hinged at the bottom,
etc?...)

thanks, Kelly

--- In tuning@yahoogroups.com, "microtonalist" <mark@e...> wrote:
>
> One way that has helped me understand the utonal and otonal
> difference is as follows
>
> Otonal - a collection of pitches that can be construed as the
> overtones of a common fundamental:
>
> 5/4 3/2 9/8 for example
>
> Utonal - a collection of pitches that can be construed as
> fundamentals having a common overtone:
>
> 4/3 8/5 8/7. These having an overtone of 1/1 (1/1 is the 3/2 of
4/3,
> 1/1 is the 5/4 of 8/5 and 1/1 is the 7/4 of 8/7).
>
> An otonal group share a common udentity and a utonal group share a
> common odentity. u for undernumber and o for overnumber.
>
> Here 1/1 refers to a reference pitch. Any pitch can be construed
> either as a new fundamental (the overtone series being the
> overarching example), or as the overtone of a new fundamental.
>
> Partch's tonality diamond takes 1 3 5 7 9 and 11 'identities' and
> treats them each way, as overtone (odentity) and fundamental
> (udentity).
>
> Mark
>

Hi Kelly,

--- In tuning@yahoogroups.com, "traktus5" <kj4321@h...> wrote:
>
> This looks like a good explanation, but what notes
> (indicated as scale notes, such as c4-e4-g4) do 5/4 3/2 9/8,
> and 4/3 8/5 8/7 actually represent? (are they stacked,
> or hinged at the bottom, etc?...)

I have no idea what you mean by "hinged at the bottom",
but hopefully i can help a little ...

Partch used G-392 Hz as his 1/1. This is the G above
middle-C, so it would usually be called "G4".

Now, the problem with using regular musical letter-name
nominals to describe Partch's theories is that in addition
to the standard letter-names, you also have to employ either
additional accidentals, or cents deviations, or some such.

I like to use a notation i devised called "HEWM", which
in its simplified form uses 72-edo (72-tone equal-temperament)
as the basis for a small set of accidentals, thus:

lower raise inflection cents

b # semitone 100
v ^ 1/4-tone 50
< > 1/6-tone 331/3
- + 1/12-tone 162/3

There is a more complicated version which gives an exact
representation of JI, but the 72-edo version does a fairly
good job of representing Partch's 11-limit JI, so i prefer
to use it.

Essentially:

* the plain letter-names represent 3-limit pythagorean
diatonic tuning;

* the sharp (#) and flat (b) represent pythagorean
chromatic deviation from the diatonic (~114 cents);

* the plus (+) and minus (-) represent deviations from
pythagorean by a syntonic-comma (~22 cents);

* the greater-than (>) and less-than (<) represent
deviations from pythagorean by a septimal-comma (~27 cents);

* the up-arrow (^) and down-arrow (v) represent
deviations from pythagorean by an undecimal-diesis (~53 cents).

You can read a lot more about HEWM here:

http://tonalsoft.com/enc/h/hewm.aspx

So, assuming G4 as the 1/1 ... 5/4 3/2 9/8 can be notated
as B-4 D5 A5, and 4/3 8/5 8/7 is C5 Eb+5 A>4.

If you prefer to use C4 (middle-C) as your 1/1, then you get:
5/4 3/2 9/8 = E-4 G4 D4, and 4/3 8/5 8/7 = F4 Ab+4 D>4.

Hope that helps.

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

All ratios are expressed in relation to the 1/1, i.e. 5/4 is a just
major third above the 1/1. This is exactly Partch's stated position.
All you have to do is choose which pitch represents the 1/1.

Therefore 1/1 5/4 3/2 represents the JI major chord.
Therefore 1/1 8/5 4/3 represents the JI minor chord.

In this notation 9/8 is always 9/8. Some microtonalists use a:b to
indicate an interval, rather than a pitch. Whether a ratio represents
a pitch (relative to a fixed 1/1) or an interval, as in the 6/5 from
5/4 up to 3/2, can lead to confusion. Partch himself made this

No stacking or hinging required.

--- In tuning@yahoogroups.com, "traktus5" <kj4321@h...> wrote:
>
> This looks like a good explanation, but what notes (indicated as
> scale notes, such as c4-e4-g4) do 5/4 3/2 9/8, and 4/3 8/5 8/7
> actually represent? (are they stacked, or hinged at the bottom,
> etc?...)
>
> thanks, Kelly
>
>
>
> --- In tuning@yahoogroups.com, "microtonalist" <mark@e...> wrote:
> >
> > One way that has helped me understand the utonal and otonal
> > difference is as follows
> >
> > Otonal - a collection of pitches that can be construed as the
> > overtones of a common fundamental:
> >
> > 5/4 3/2 9/8 for example
> >
> > Utonal - a collection of pitches that can be construed as
> > fundamentals having a common overtone:
> >
> > 4/3 8/5 8/7. These having an overtone of 1/1 (1/1 is the 3/2 of
> 4/3,
> > 1/1 is the 5/4 of 8/5 and 1/1 is the 7/4 of 8/7).
> >
> > An otonal group share a common udentity and a utonal group share
a
> > common odentity. u for undernumber and o for overnumber.
> >
> > Here 1/1 refers to a reference pitch. Any pitch can be construed
> > either as a new fundamental (the overtone series being the
> > overarching example), or as the overtone of a new fundamental.
> >
> > Partch's tonality diamond takes 1 3 5 7 9 and 11 'identities' and
> > treats them each way, as overtone (odentity) and fundamental
> > (udentity).
> >
> > Mark
> >
>

Dear microtonalist

As u wrote : .....An otonal group share a common udentity and a utonal
group share a
common odentity. u for undernumber and o for overnumber.....

This is the same as I told :

1- "" utonal systems which I think it is better for me to call them ""
constant/decreasing rational system "" the nominator is constant and
denominator is decreasing """" the nominator here is the udentity.

2- utonal systems which I think it is better for me to call them ""
constant/decreasing rational system "" the nominator is constant
(odentity)and denominator is decreasing

3- you tell that :""Utonal - a collection of pitches that can be
construed as
fundamentals having a common overtone:
4/3 8/5 8/7. These having an overtone of 1/1 (1/1 is the 3/2 of 4/3,
1/1 is the 5/4 of 8/5 and 1/1 is the 7/4 of 8/7).""

I found that each degree of utonal and otonal systems are inverse of
each other due to 2/1(3/2 & 4/3 are inverse of each other so , . so may
be the musical sense of major tonality is inverse of minor tonality
because of that !!!!!

1200-each degree of otonal systems=each degree of utonal system and vice
versa.

4-in otonal and utonal systems with same number of degrees ,
udentity=2*odentity.

Dear monzo

1- although I see my mails in microsoft outlook , but I can also see the
tables inserted in the mail in internet explorer.

2- as u wrote: "A perfect example of the latter is a JI minor-7th chord
with ratios 1/1 - 6/5 - 3/2 - 9/5"

In my opinion it is a hybrid chord made from two utonal and otonal
sytems which have some degrees in common :

1/1... 6/5...3/2...2/1

1/1..6/5..7/5..8/5..9/5..2/1

U can also construct this chord in this i/c rational system:

10/10......20/10(1/1...2/1). May be we can divide one rational otonal
system to two or more otonal and utonal systems.

________________________________

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf
Of microtonalist
Sent: Monday, October 10, 2005 5:38 PM
To: tuning@yahoogroups.com
Subject: [tuning] Otonal / Utonal

One way that has helped me understand the utonal and otonal
difference is as follows

Otonal - a collection of pitches that can be construed as the
overtones of a common fundamental:

5/4 3/2 9/8 for example

Utonal - a collection of pitches that can be construed as
fundamentals having a common overtone:

4/3 8/5 8/7. These having an overtone of 1/1 (1/1 is the 3/2 of 4/3,
1/1 is the 5/4 of 8/5 and 1/1 is the 7/4 of 8/7).

An otonal group share a common udentity and a utonal group share a
common odentity. u for undernumber and o for overnumber.

Here 1/1 refers to a reference pitch. Any pitch can be construed
either as a new fundamental (the overtone series being the
overarching example), or as the overtone of a new fundamental.

Partch's tonality diamond takes 1 3 5 7 9 and 11 'identities' and
treats them each way, as overtone (odentity) and fundamental
(udentity).

Mark

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hello Microtonalists! "microtonalist" <mark@e...> wrote:

> One way that has helped me understand the utonal and otonal
> difference is as follows> Otonal - a collection of pitches that
>can be construed as the ...

As one ascends the harmonic series, when can one start considering
the chords utonally? Is it the first chord of the harmonic series
(8:10:13) which, in the subharmonic series, is the first type of
minor triad (ie, 1/5:4:3)?

(And as the secondary issue, why do you choose 12:15:20 over
8:10:13? Is it because of the purportedly discordant quality of the
natural 13th harmonic? Or is 12-15-20 favored because it reduces
down to the constituent intervals 5/4 and 4/3? And 12-15-20 is "5-
limit", correct?!)

And, how does one reckon utonality theory with theories which see
chords as having mutliple root associations? Is there a good and
fair critique of utonality, along these or other lines, that's been
published anywhere?

thanks, Kelly

>
> An otonal group share a common udentity and a utonal group share a
> common odentity. u for undernumber and o for overnumber.
>
> Here 1/1 refers to a reference pitch. Any pitch can be construed
> either as a new fundamental (the overtone series being the
> overarching example), or as the overtone of a new fundamental.
>
> Partch's tonality diamond takes 1 3 5 7 9 and 11 'identities' and
> treats them each way, as overtone (odentity) and fundamental
> (udentity).
>
> Mark
>

--- In tuning@yahoogroups.com, "traktus5" <kj4321@h...> wrote:
>
> hello Microtonalists! "microtonalist" <mark@e...> wrote:
>
> > One way that has helped me understand the utonal and otonal
> > difference is as follows> Otonal - a collection of pitches that
> >can be construed as the ...
>
> As one ascends the harmonic series, when can one start considering
> the chords utonally? Is it the first chord of the harmonic series
> (8:10:13) which, in the subharmonic series, is the first type of
> minor triad (ie, 1/5:4:3)?
Chords ascending in the harmonic series are never utonal.
>
> (And as the secondary issue, why do you choose 12:15:20 over
> 8:10:13? Is it because of the purportedly discordant quality of
the
> natural 13th harmonic? Or is 12-15-20 favored because it reduces
> down to the constituent intervals 5/4 and 4/3? And 12-15-20 is "5-
> limit", correct?!)
None of these. Simply that the components of one chord are the ratio
complements of the other. The otonal chord does not contain 16/13, so
the utonal one does not contain 13/8.
>
> And, how does one reckon utonality theory with theories which see
> chords as having mutliple root associations? Is there a good and
> fair critique of utonality, along these or other lines, that's been
> published anywhere?
None.

However, for interest, Erno Lendvai has written about the concept of
untonality (though he prefers to separate his tonal universes into
diatonic and pentatonic, which is an interesting insight) Consider

1/1 5/4 3/2 7/4 these are otonal, and are found in diatonic music
1/1 8/5 4/3 8/7 these are utonal, and are found in pentatonic
music. (From Lendvai - not me)

Lendvai also demonstrates the complementary quality of the chromatic

C D E Fsharp (as 11/8) G Bflat(as 7/4)
C Eflat F Aflat (8/5)

D in the above contexts would either be 9/8 or 8/7 respectively
(these are my interpretations of Lendvai's chromatic scale form)

Note that to Lendvai it is the form 1/1 4/3 8/5 which is the primary
mode of presentation of the pentatonic tonic.

But post Partch I am not aware of any significant addition to his
concept of tonality

>

On Mon, 10 Oct 2005, "microtonalist" <mark@...> wrote:
>
> One way that has helped me understand the utonal and otonal
> difference is as follows
>
> Otonal - a collection of pitches that can be construed as the
> overtones of a common fundamental:
>
> 5/4 3/2 9/8 for example
>
> Utonal - a collection of pitches that can be construed as
> fundamentals having a common overtone:
>
> 4/3 8/5 8/7. These having an overtone of 1/1 (1/1 is the 3/2 of 4/3,
> 1/1 is the 5/4 of 8/5 and 1/1 is the 7/4 of 8/7).
>
> An otonal group share a common udentity and a utonal group share a
> common odentity. u for undernumber and o for overnumber.
>
> Here 1/1 refers to a reference pitch. Any pitch can be construed
> either as a new fundamental (the overtone series being the
> overarching example), or as the overtone of a new fundamental.
>
> Partch's tonality diamond takes 1 3 5 7 9 and 11 'identities' and
> treats them each way, as overtone (odentity) and fundamental
> (udentity).
>
> Mark

Hi Mark,

This very neat way of putting things does point up
the fundamental duality of the two notions. As such,
it's a valuable reminder that otonality and utonality
fit together in a larger scheme of things, each with
[expressive] possibilities the other necessarily lacks.

Thank you!

Apart from their suggestive power, is there a concise
definition of the terms "odentity" and "udentity"?

Regards,
Yahya

--
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Hi Yahya,

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

> This very neat way of putting things does point up
> the fundamental duality of the two notions. As such,
> it's a valuable reminder that otonality and utonality
> fit together in a larger scheme of things, each with
> [expressive] possibilities the other necessarily lacks.
>
> Thank you!

Otonality/Utonality are indeed dualistic ... this was a
fundamental (pardon the pun) aspect of Partch's theory.

But note that they are really only two sides of the
same coin. I just wrote here yesterday explaining how
the same chord may be interpreted as either an Otonality
or Utonality:

/tuning/topicId_61291.html#61301

> Apart from their suggestive power, is there a concise
> definition of the terms "odentity" and "udentity"?

odentity = an identity of an Otonality,
found among the numerators of the ratios

udentity = an identity of a Utonality,
found among the denominators of the ratios

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

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> Hi Yahya,
>
> --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:
>
> > This very neat way of putting things does point up
> > the fundamental duality of the two notions. As such,
> > it's a valuable reminder that otonality and utonality
> > fit together in a larger scheme of things, each with
> > [expressive] possibilities the other necessarily lacks.
> >
> > Thank you!
>
>
> Otonality/Utonality are indeed dualistic ... this was a
> fundamental (pardon the pun) aspect of Partch's theory.
>
> But note that they are really only two sides of the
> same coin. I just wrote here yesterday explaining how
> the same chord may be interpreted as either an Otonality
> or Utonality:
>
> /tuning/topicId_61291.html#61301
>
>
> > Apart from their suggestive power, is there a concise
> > definition of the terms "odentity" and "udentity"?
>
>
> odentity = an identity of an Otonality,
> found among the numerators of the ratios
>
> udentity = an identity of a Utonality,
> found among the denominators of the ratios

http://tonalsoft.com/enc/i/identity.aspx

http://tonalsoft.com/enc/o/odentity.aspx

http://tonalsoft.com/enc/u/udentity.aspx

Those definitions are perhaps a little too concise.
But at least the first part of each comes from Partch,
who invented the terms.

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

--- In tuning@yahoogroups.com, "Mohajeri Shahin" <shahinm@k...> wrote:

> 4-in otonal and utonal systems with same number of degrees ,
> udentity=2*odentity.

How did you come to that conclusion?

--- In tuning@yahoogroups.com, "microtonalist" <mark@e...> wrote:

>
> However, for interest, Erno Lendvai has written about the concept of
> untonality (though he prefers to separate his tonal universes into
> diatonic and pentatonic, which is an interesting insight)

I do believe many academics have neglected the possibility and reality of pentatonic
hearing.

> Consider
>
> 1/1 5/4 3/2 7/4 these are otonal, and are found in diatonic music

Isn't 1/1 8/5 4/3 8/7 then also (equally, as much) found in diatonic music?

> 1/1 8/5 4/3 8/7 these are utonal, and are found in pentatonic
> music. (From Lendvai - not me)

Since the pentatonic and diatonic scales are symmetrical about the otonal/utonal "axis", I
don't see how either could be more one than the other. But I'd love to hear the thinking
behind this idea.

> But post Partch I am not aware of any significant addition to his
> concept of tonality

Whose concept? Partch's or Lendvai's?

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

> I do believe many academics have neglected the possibility and
reality of pentatonic
> hearing.

What does "pentatonic hearing" mean?

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
>
> --- In tuning@yahoogroups.com, "microtonalist" <mark@e...> wrote:
>
> >
> > However, for interest, Erno Lendvai has written about the concept
of
> > untonality (though he prefers to separate his tonal universes
into
> > diatonic and pentatonic, which is an interesting insight)
>
> I do believe many academics have neglected the possibility and
reality of pentatonic
> hearing.
>
> > Consider
> >
> > 1/1 5/4 3/2 7/4 these are otonal, and are found in diatonic
music
>
> Isn't 1/1 8/5 4/3 8/7 then also (equally, as much) found in
diatonic music?
>
> > 1/1 8/5 4/3 8/7 these are utonal, and are found in pentatonic
> > music. (From Lendvai - not me)
>

Lendvai states that all intervals can be found in all systems. What
is significant is that the dissonant intervals of one are the
consonant intervals in the other.

Thus 3/2 is consonant in diatony but dissonant in pentatony.
Conversely 4/3 is consonant in pentatony but dissonant in diatony.

similarly for other otonal and utonal ratios. Characteristically
Lendvai makes the six-four minor chord the tonic of a pentatonic
system i.e. C F Aflat (reading up from C).

There are intervals that for a given 1/1 are neither utonal or otonal:

6/5 5/3 7/5 9/7 etc, as neither odentity or udentity contain powers
of 2 only. What should we call them - otonal or utonal? - or neither?

Finally, it should be of interest to those who don't already know
that although a string or a pipe vibrates according to the harmonic
series, if a string is equally divided or a pipe has holes made at
equal points along it then a subharmonic series results from the
resulting scales. It may be interesting to conjecture that perhaps
for early humans, equal division of the physical object (string or
bone flute) was easier than logarithmic.

Mark

> Since the pentatonic and diatonic scales are symmetrical about the
otonal/utonal "axis", I
> don't see how either could be more one than the other. But I'd love
to hear the thinking
> behind this idea.
>
> > But post Partch I am not aware of any significant addition to his
> > concept of tonality
>
> Whose concept? Partch's or Lendvai's?
>
Partch's concept of tonality

tuning@yahoogroups.com wrote:

>
>
>--- In tuning@yahoogroups.com, "microtonalist" <mark@e...> wrote:
>
> >
>
> >
>>But post Partch I am not aware of any significant addition to his >>concept of tonality
>> >>
I think that the Eikosany and the othe CPS structures which are like completments to the diamonds are quite a step in this direction.
I wrote an article on this back when which is reproduced here
http://anaphoria.com/cps.PDF

also the scales of Mt Meru are constructed along the line of difference and combination tones reinforcing a series of self contained series of tones allowing
--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

> Finally, it should be of interest to those who don't already know
> that although a string or a pipe vibrates according to the harmonic
> series, if a string is equally divided or a pipe has holes made at
> equal points

divided how, exactly? any division of equal spacing?...

along it then a subharmonic series results from the
> resulting scales. It may be interesting to conjecture that perhaps
> for early humans, equal division of the physical object (string or
> bone flute) was easier than logarithmic.

Can someone elaborate?

thanks.

Hi Monz,

On Wed, 12 Oct 2005, you wrote:
> Hi Yahya,
> --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz"
> <yahya@m...> wrote:
>
> > This very neat way of putting things does point up
> > the fundamental duality of the two notions. As such,
> > it's a valuable reminder that otonality and utonality
> > fit together in a larger scheme of things, each with
> > [expressive] possibilities the other necessarily lacks.
> >
> > Thank you!
>
>
> Otonality/Utonality are indeed dualistic ... this was a
> fundamental (pardon the pun) aspect of Partch's theory.
>
> But note that they are really only two sides of the
> same coin.
Which is pretty much what dual concepts always are.
An explanation of one that fails to note the other often
lacks something in depth.

> ... I just wrote here yesterday explaining how
> the same chord may be interpreted as either an Otonality
> or Utonality:
>
> /tuning/topicId_61291.html#61301

Yep, read it, thank you.

> > Apart from their suggestive power, is there a concise
> > definition of the terms "odentity" and "udentity"?
>
> odentity = an identity of an Otonality,
> found among the numerators of the ratios
>
> udentity = an identity of a Utonality,
> found among the denominators of the ratios
>
Too concise! :-) As you realised, for you wrote later -

> http://tonalsoft.com/enc/i/identity.aspx
>
> http://tonalsoft.com/enc/o/odentity.aspx
>
> http://tonalsoft.com/enc/u/udentity.aspx
>
>
> Those definitions are perhaps a little too concise.
> But at least the first part of each comes from Partch,
> who invented the terms.

These definitions from Tonalsoft were more informative,
thank you. The quotes from Partch seemed more like some
kind of mystic mumbo-jumbo than either definition or
explanation. Contrariwise, your words shed more light.

Still there seems to be some circularity in the discussion.
Given that there is a natural hierarchy of concepts involved
here, wouldn't it be best if any explanation started with
the most fundamental and built on that?

The way the term "identity" is used by Partch is confusing,
too. It's quite unlike the uses I have encountered in
abstract mathematics, in philosophy and in psychology.
In the "mathematical calculus of frequency ratios" (to
dignify our simple calculations for chords), the only true
identity is that ratio, which, when multiplied by any other,
leaves it unchanged, namely the ratio 1:1. And there is only
one identity possible in such a system, as there is in a
mathematical group. But this is not what "identity" means
in your definitions, is it?

In fact, I suspect that there's a better term waiting to
replace it. Your expression "pole of tonality" is suggestive,
when you describe it as -

"one of the odd-number ingredients, one or several or all
of which act as a pole of tonality [for example, 1-3-5-7-9-11
in Partch's theory]."

What "pole" brings to mind is firstly, the diametrically
opposite North and South Poles of a globe or of magnetism;
and secondly, a fixed structure, point or pivot around which
satellites spin, like dancers round a Maypole.

To use your example 1-3-5-7-9-11, and take "one or several
or all ... as a pole of tonality", we might take -
a) just the 5; or
b) the 5, 7 and 11; or
c) the 1, 3, 5, 7, 9 and 11
as our "poles of tonality". Only in case a) do we have a single
"pole" like a Maypole; and in none of these cases do we have
diametrically opposite poles like the North and South poles.

Perhaps the term "pivot" might suitable generalise the notion
of "poles of tonality". But I suppose the term "pivot" already
has too many other meanings in music?

Regards,
Yahya

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--- In tuning@yahoogroups.com, "microtonalist" <mark@e...> wrote:

> Thus 3/2 is consonant in diatony but dissonant in pentatony.
> Conversely 4/3 is consonant in pentatony but dissonant in diatony.

Is there any evidence this is so?

Hi Yahya,

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

> The way the term "identity" is used by Partch is confusing,
> too. It's quite unlike the uses I have encountered in
> abstract mathematics, in philosophy and in psychology.
> In the "mathematical calculus of frequency ratios" (to
> dignify our simple calculations for chords), the only true
> identity is that ratio, which, when multiplied by any other,
> leaves it unchanged, namely the ratio 1:1. And there is only
> one identity possible in such a system, as there is in a
> mathematical group. But this is not what "identity" means
> in your definitions, is it?
>
> In fact, I suspect that there's a better term waiting to
> replace it. Your expression "pole of tonality" is suggestive,
> when you describe it as -
>
> "one of the odd-number ingredients, one or several or all
> of which act as a pole of tonality [for example, 1-3-5-7-9-11
> in Partch's theory]."
>
> What "pole" brings to mind is firstly, the diametrically
> opposite North and South Poles of a globe or of magnetism;
> and secondly, a fixed structure, point or pivot around which
> satellites spin, like dancers round a Maypole.
>
> To use your example 1-3-5-7-9-11, and take "one or several
> or all ... as a pole of tonality", we might take -
> a) just the 5; or
> b) the 5, 7 and 11; or
> c) the 1, 3, 5, 7, 9 and 11
> as our "poles of tonality". Only in case a) do we have a single
> "pole" like a Maypole; and in none of these cases do we have
> diametrically opposite poles like the North and South poles.
>
> Perhaps the term "pivot" might suitable generalise the notion
> of "poles of tonality". But I suppose the term "pivot" already
> has too many other meanings in music?

Partch's theory is based on the idea of a "numerary nexus",
which you should also look up in the Tonalsoft Encyclopedia.

In a nutshell, Partch's "nexus" simply means that all the
notes of a Utonality or Otonality share a common factor in
either the numerator or denominator of the ratios.

If the nexus is in the denominator, then the chord is best
described as an Otonality, and vice versa: if the nexus is
in the numerator, then the chord is best described as a
Utonality.

"Identity" is the generic word, which can refer to either
Odentities or Udentities. If the nexus is in the denominator,
then the numerators of the ratios give the Odentities, and
vice versa: if the nexus is in the numerator, then the
denominators give the Udentities.

That's the best explanation i can give.

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

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In tuning@yahoogroups.com, "microtonalist" <mark@e...> wrote:
>
> > Thus 3/2 is consonant in diatony but dissonant in pentatony.
> > Conversely 4/3 is consonant in pentatony but dissonant in diatony.
>
> Is there any evidence this is so?
>
Lendvai provides some examples, which sonically are suggestive, but
otherwise it's just an assertion of his.

What I will say from my own experience is that consonance and
dissonance are relative concepts, and can often only be defined in
the context of the music. Absolute consonance and dissonance are
chimerae, IMHO.

One of the best examples of this is when an accidental 'consonance'
creeps into a performance of Webern or similar 'squeaky gate' music -
it sounds jarring and disturbs the flow.

Mark

PS 'squeaky gate' is used here not in the pejorative sense, but as a
descriptive term.

(IF you know all this, please ignore it: else read on)

Suppose one started with a string, and divided it into say twelve equal
sized parts. The full string would then have (for a given thickness and
tension of string) a frequency, which we will call f.

Stopping the string 1/12th of the way from one end (the first division)
will result in a string that is 11/12ths of the original length. If we
make the assumption that tension does not change as a result of
stopping the string, then the new frequency will be 12/11ths of the old
frequency.

similarly stopping one more 12th along, 10/12ths of the string remain,
giving a frequency that is 12/10ths of the unstopped string frequency.

And so it goes. With each new step the interval size increases:

Open String = 1/1
1st stop/fret = 12/11
2nd = 12/10 = 6/5
3rd = 12/9 = 4/3
4th = 12/8 = 3/2
5th = 12/7
6th = 12/6 = 2/1 (octave)
etc.

The intervals between adjacent ascending notes is 12/11 11/10 10/9 9/8
8/7 ... are the same intervals between adjacent notes of the harmonic
series, but in reverse order.

Of course, any equal divisions along a string produce a subharmonic
series or a partition of it.

All of the above is true for pipes, with the caveat that pipes suffer
from end effects, the helmholtz effect (mass of the air where the holes
are) and other perturbations away from the pure solution.

As an example of a partition of a subharmonic series, suppose the first
stopping point is 2/13ths of the length of the string. The first note
is then 13/11. Subsequent stopping points are spaced as 1/8th of the
unstopped string length. The scale is then

2/13th +1/8th +2/8th +3/8th +4/8th ...
13/11 104/75 52/31 52/27 26/17 ...

hopefully this makes some sense.

Mark

--- In tuning@yahoogroups.com, "traktus5" <kj4321@h...> wrote:
>
> > Finally, it should be of interest to those who don't already know
> > that although a string or a pipe vibrates according to the harmonic
> > series, if a string is equally divided or a pipe has holes made at
> > equal points
>
> divided how, exactly? any division of equal spacing?...
>
> along it then a subharmonic series results from the
> > resulting scales. It may be interesting to conjecture that perhaps
> > for early humans, equal division of the physical object (string or
> > bone flute) was easier than logarithmic.
>
> Can someone elaborate?
>
> thanks.
>

Hi Mark,

--- In tuning@yahoogroups.com, "microtonalist" <mark@e...> wrote:

> What I will say from my own experience is that consonance
> and dissonance are relative concepts, and can often only be
> defined in the context of the music. Absolute consonance and
> dissonance are chimerae, IMHO.
>
> One of the best examples of this is when an accidental
> 'consonance' creeps into a performance of Webern or similar
> 'squeaky gate' music - it sounds jarring and disturbs
> the flow.

This has come up here several times before.

First of all, it's been a part of my theory from the
beginning -- following Helmholtz, Schoenberg, and Partch
-- that consonance and dissonance are not opposed in kind
but only in degree. That is, they're not two different
things, but rather two opposite ends of a continuum.

Thus i chose to use the term "sonance" to represent
the whole continuum, instead of always having to say
"relative consonance/dissonance" as Partch did. Thinking
about it this way also greatly clarifies Schoenberg's
thoughts about the subject.

Then, several years ago, it was recognized by a bunch
of us here on this list that sonance is context dependant,
exactly as you say in what i quoted above. So several
of us decided:

* to use the term "sonance" (and the related terms
"consonance" and "dissonance") specifically to refer
to cases where musical context determined the relative
consonance or dissonance of a particular sonority, and

* to employ the term "accordance" (and the related
"concord" and "discord") specifically to refer to the
psychoacoustical perception of a sonority, independent
of its musical context.

I wrote about this here to someone else recently (a couple
of months ago) and it caused a big discussion, with most
people saying that the terms should be used the other
way around: "concord/discord" for the context-dependent
cases and "consonance/dissonance" for the psychoacoustical
perception.

The terms have been defined in the first way in my
Encyclopedia for 7 years now, and obviously are a part
of many Encyclopedia entries, so i'm not about to
change them. So i argued a bit in the recent discussion,
but didn't give it much more thought and don't remember
if there was a consensus, and if so, what it was.

I'll be continuing to use "accordance/concord/discord" for the
psychoacoustical perception, and "sonance/consonance/dissonance"
for the musical-context-dependent perception.

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

This is a very enlightening explanation.
----- Original Message -----
From: microtonalist
To: tuning@yahoogroups.com
Sent: 10 Ekim 2005 Pazartesi 17:08
Subject: [tuning] Otonal / Utonal

One way that has helped me understand the utonal and otonal
difference is as follows

Otonal - a collection of pitches that can be construed as the
overtones of a common fundamental:

5/4 3/2 9/8 for example

Utonal - a collection of pitches that can be construed as
fundamentals having a common overtone:

4/3 8/5 8/7. These having an overtone of 1/1 (1/1 is the 3/2 of 4/3,
1/1 is the 5/4 of 8/5 and 1/1 is the 7/4 of 8/7).

An otonal group share a common udentity and a utonal group share a
common odentity. u for undernumber and o for overnumber.

Here 1/1 refers to a reference pitch. Any pitch can be construed
either as a new fundamental (the overtone series being the
overarching example), or as the overtone of a new fundamental.

Partch's tonality diamond takes 1 3 5 7 9 and 11 'identities' and
treats them each way, as overtone (odentity) and fundamental
(udentity).

Mark

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
>
> > I do believe many academics have neglected the possibility and
> reality of pentatonic
> > hearing.
>
> What does "pentatonic hearing" mean?

Many academics in a number of different music theory fields take it
that when we hear music, we hear partly in terms of generic diatonic
interval classes -- thus, a 300-cent interval will be heard as an
augmented second or a minor third (depending on context), which
correspond to 1 or 2 steps, respectively, in the 7-note diatonic
scale. In other words, everything is heard in terms of a presumed
underlying (though flexible) 7-note framework. I'd call this an
assumption of "diatonic hearing", and it may apply to much Western
music, but even in the blues I'm pretty sure that quite a bit
of "pentatonic hearing" happens instead, with 5 rather than 7 generic
interval sizes per octave.

--- In tuning@yahoogroups.com, "microtonalist" <mark@e...> wrote:

> Thus 3/2 is consonant in diatony but dissonant in pentatony.
> Conversely 4/3 is consonant in pentatony but dissonant in diatony.

Doesn't jibe with my experience at all, but is reminiscent of Yasser's
theory.

> similarly for other otonal and utonal ratios. Characteristically
> Lendvai makes the six-four minor chord the tonic of a pentatonic
> system i.e. C F Aflat (reading up from C).

What kind of pentatonic system uses that as a tonic?

> There are intervals that for a given 1/1 are neither utonal or otonal:

No interval is either utonal or otonal; you need at least three notes
for that. Do you mean for 1/1 to be the third note?

> 6/5 5/3 7/5 9/7 etc, as neither odentity or udentity contain powers
> of 2 only.

Even 5/4 or 8/5 are neither otonal nor utonal: they are just as easily
expressed as 1/(4/5) and 1/(5/8).

> What should we call them - otonal or utonal? - or neither?

It seems you may be misunderstanding the concept. It takes at least
three notes to be able to say whether the structure is more like a
harmonic series or more like a subharmonic series (otonal or utonal).

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> Hi Mark,
>
>
> --- In tuning@yahoogroups.com, "microtonalist" <mark@e...> wrote:
>
> > What I will say from my own experience is that consonance
> > and dissonance are relative concepts, and can often only be
> > defined in the context of the music. Absolute consonance and
> > dissonance are chimerae, IMHO.
> >
> > One of the best examples of this is when an accidental
> > 'consonance' creeps into a performance of Webern or similar
> > 'squeaky gate' music - it sounds jarring and disturbs
> > the flow.

With regard to what Monz wrote below, I'd say that the explanation is
that there are indeed context-independent things we can
call 'concordance' and 'discordance' (though these are still relative
terms, opposite ends of a continuum). And this is the very
distinction that's needed to explain these disturbances in Webern's
music -- a concordance stands out like a sore thumb in the context of
lots of discordance, because it is *sonically* different from a
discordance, whatever the context.

> This has come up here several times before.
>
> First of all, it's been a part of my theory from the
> beginning -- following Helmholtz, Schoenberg, and Partch
> -- that consonance and dissonance are not opposed in kind
> but only in degree. That is, they're not two different
> things, but rather two opposite ends of a continuum.
>
> Thus i chose to use the term "sonance" to represent
> the whole continuum, instead of always having to say
> "relative consonance/dissonance" as Partch did. Thinking
> about it this way also greatly clarifies Schoenberg's
> thoughts about the subject.
>
> Then, several years ago, it was recognized by a bunch
> of us here on this list that sonance is context dependant,
> exactly as you say in what i quoted above. So several
> of us decided:
>
> * to use the term "sonance" (and the related terms
> "consonance" and "dissonance") specifically to refer
> to cases where musical context determined the relative
> consonance or dissonance of a particular sonority, and
>
> * to employ the term "accordance" (and the related
> "concord" and "discord") specifically to refer to the
> psychoacoustical perception of a sonority, independent
> of its musical context.
>
>
> I wrote about this here to someone else recently (a couple
> of months ago) and it caused a big discussion, with most
> people saying that the terms should be used the other
> way around: "concord/discord" for the context-dependent
> cases and "consonance/dissonance" for the psychoacoustical
> perception.
>
> The terms have been defined in the first way in my
> Encyclopedia for 7 years now, and obviously are a part
> of many Encyclopedia entries, so i'm not about to
> change them. So i argued a bit in the recent discussion,
> but didn't give it much more thought and don't remember
> if there was a consensus, and if so, what it was.
>
> I'll be continuing to use "accordance/concord/discord" for the
> psychoacoustical perception, and "sonance/consonance/dissonance"
> for the musical-context-dependent perception.
>
>
>
> -monz
> http://tonalsoft.com
> Tonescape microtonal music software
>

Hi Paul and Gene,

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
>
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> >
> > --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> > <wallyesterpaulrus@y...> wrote:
> >
> > > I do believe many academics have neglected the
> > > possibility and reality of pentatonic hearing.
> >
> > What does "pentatonic hearing" mean?
>
> Many academics in a number of different music theory fields
> take it that when we hear music, we hear partly in terms
> of generic diatonic interval classes -- thus, a 300-cent
> interval will be heard as an augmented second or a minor
> third (depending on context), which correspond to 1 or 2
> steps, respectively, in the 7-note diatonic scale.
> In other words, everything is heard in terms of a presumed
> underlying (though flexible) 7-note framework. I'd call
> this an assumption of "diatonic hearing", and it may apply
> to much Western music, but even in the blues I'm pretty
> sure that quite a bit of "pentatonic hearing" happens
> instead, with 5 rather than 7 generic interval sizes per
> octave.

I agree with Paul 100%, about both the academic neglect
of "pentatonic hearing", and its relevance to the blues.

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

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

> I agree with Paul 100%, about both the academic neglect
> of "pentatonic hearing", and its relevance to the blues.

Blues sounds decidedly un-pentatonic to me. I'd say a good scale for
producing a vaguely bluesy sound is Orwell[9]; not that it's a blues
scale, but all those 7/6s give it a sort of blue note feel. Isn't that
what you need for blues--a few blue notes?

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

> Many academics in a number of different music theory fields take it
> that when we hear music, we hear partly in terms of generic diatonic
> interval classes -- thus, a 300-cent interval will be heard as an
> augmented second or a minor third (depending on context), which
> correspond to 1 or 2 steps, respectively, in the 7-note diatonic
> scale. In other words, everything is heard in terms of a presumed
> underlying (though flexible) 7-note framework. I'd call this an
> assumption of "diatonic hearing", and it may apply to much Western
> music, but even in the blues I'm pretty sure that quite a bit
> of "pentatonic hearing" happens instead, with 5 rather than 7 generic
> interval sizes per octave.

From your language it seems the underlying assumption is some kind of
meantone system (since we have augmented seconds vs minor thirds,
etc.) Is this another of those Eytan Agmon deals, where people assume
meantone without, apparently, realizing they are doing so?

> I agree with Paul 100%, about both the academic neglect
> of "pentatonic hearing", and its relevance to the blues.

I don't know about its neglect (not that I doubt it), but
I certainly agree about its import.

-Carl

Gene Ward Smith wrote:

> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
> > >>I agree with Paul 100%, about both the academic neglect
>>of "pentatonic hearing", and its relevance to the blues.
> > > Blues sounds decidedly un-pentatonic to me. I'd say a good scale for
> producing a vaguely bluesy sound is Orwell[9]; not that it's a blues
> scale, but all those 7/6s give it a sort of blue note feel. Isn't that
> what you need for blues--a few blue notes?

I play 7/6, which raises a few eyebrows but is generally accepted. Monz found 19/16 used as a blue note. But the general definition in practically all the literature (I don't know any exceptions) speaks about neutral thirds and sevenths. Guitarists play it like that.

I've asked before and do it again: Who brought up the notion that blue notes are septimal intervals, or are extra low thirds?

klaus

Hi Gene and klaus,

--- In tuning@yahoogroups.com, klaus schmirler <KSchmir@o...> wrote:
>
> Gene Ward Smith wrote:
>
> > --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
> >
> >
> >> I agree with Paul 100%, about both the academic neglect
> >> of "pentatonic hearing", and its relevance to the blues.
> >
> >
> > Blues sounds decidedly un-pentatonic to me. I'd say a
> > good scale for producing a vaguely bluesy sound is
> > Orwell[9]; not that it's a blues scale, but all those
> > 7/6s give it a sort of blue note feel. Isn't that
> > what you need for blues--a few blue notes?

Blues is founded essentially on the minor pentatonic scale
-- to take the simplest example: A C D E G.

Just try a little improvising with that scale, making sure
that you keep the tonic on A, and not C or another note,
and what you play will probably already sound quite bluesy,
even in 12-edo.

The way blues guitartists play the blues, they basically
fret those 5 notes, and then apply strategic pitch-bending
wherever they feel that it belongs, most usually bending
up the minor-3rd and the 4th ... and remember that on the
guitar, the bending only goes in one direction: up.

And the blues was originally (back in the late 1800s)
a strictly vocal music, so the guitar style evolved
after the manner of the vocals, which we know is free
to roam wherever the singer wants it to.

A blues scale typically played by keyboard players
would be: A C D D#/Eb E G G#/Ab. Those chromatic notes
are simply an analgoue of what the guitarists do,
since you can't bend notes on a regular piano.

> I play 7/6, which raises a few eyebrows but is generally
> accepted. Monz found 19/16 used as a blue note. But the
> general definition in practically all the literature
> (I don't know any exceptions) speaks about neutral thirds
> and sevenths. Guitarists play it like that.
>
> I've asked before and do it again: Who brought up the
> notion that blue notes are septimal intervals, or are
> extra low thirds?

I'm certain that the idea that blue notes are septimal
ratios comes from the fact that in the blues, *every*
chord is a dominant-7th chord.

This is fundamentally different from Euro-style
"common-practice" usage, where the dominant-7th chord
is a dissonance which must be resolved, and typically
only appears as the "dominant" (V), with the root of
the chord on the 5th degree of the diatonic scale ...
or it functions exactly like that in cases where a
different chord has chromatic substitutions.

In blues this is not the case. *Every* chord is a
dominant-7th chord, and none of them set up the kind
of tension-->resolution expectation as they do in
Euro-style music.

Thus, in blues the dominant-7th chord is the funadamental
consonance. This IMO encourages the listener's ears to
interpret *every* chord as a 4:5:6:7 proportion.

So let's say we have a vocalist or guitarist jamming
away on the A C D E G pentatonic scale with pitch-bends.
The chords which go with that scale in the blues
typically follow the 12-measure progression:
A7 / / / D7 / A7 / E7 D7 A7 E7.

If the listener's ear is already hearing the melodic G
as a 7/4 above A, the melodic C as a 7/4 above D, and
the melodic D as a 7/4 above E, and with all the bending
going on, the guitarist and vocalist may indeed be
*playing* those notes lower in pitch so that they
really *are* septimal intervals.

If the D is approximately a 4/3 above the tonic A, and
the C is approximately a 7/4 above the D, then when the
A7 comes back around, that C will be heard as a 7/6.

So that's my theory on why theorists persist in claiming
that septimal intervals are important to the blues.

Of course the real situation is much more complicated
than that. Old-school (i.e., Delta) blues musicians
didn't know anything about Western music-theory, and
played and sang whatever damn notes they felt. For an
example of the richness of notes available to them, see:

http://sonic-arts.org/monzo/rjohnson/drunken.htm

The tonic of this piece is D. The scale Johnson used
in his vocals in this song is interesting in that,
if you try to describe it in terms of 12-edo or meantone,
you basically get: D F# G G#/Ab A A#/Bb B C C#.
So it's essentially a chromatic scale, but with a
big gap between the D and F#. But it could still,
with less accuracy, be notated and performed simply
with D F G A C, and it wouldn't really sound "wrong".

And keep in mind that this tune is one of Robert Johnson's
*simplest* performances. Given all the time i spent on
analyzing this, i wish i had done _Hellhound On My Trail_
instead, which is mind-boggling in its pitch complexity,
considering that it's just one guy playing a guitar and
singing.

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

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

> In blues this is not the case. *Every* chord is a
> dominant-7th chord, and none of them set up the kind
> of tension-->resolution expectation as they do in
> Euro-style music.

It seems to me there is a contradiction here. You can't very well have
a lot of dominant seventh chords *and* a scale with only five notes in it.

> Thus, in blues the dominant-7th chord is the funadamental
> consonance. This IMO encourages the listener's ears to
> interpret *every* chord as a 4:5:6:7 proportion.

That makes more sense if in fact the chords are not dominant seventh
chords in a strict sense, but chords like CEGA#, with augmented
seconds; but then you might be getting blue notes in the form
GA#, etc.

> If the listener's ear is already hearing the melodic G
> as a 7/4 above A, the melodic C as a 7/4 above D, and
> the melodic D as a 7/4 above E, and with all the bending
> going on, the guitarist and vocalist may indeed be
> *playing* those notes lower in pitch so that they
> really *are* septimal intervals.

It makes sense to me, but do they?

> The tonic of this piece is D. The scale Johnson used
> in his vocals in this song is interesting in that,
> if you try to describe it in terms of 12-edo or meantone,
> you basically get: D F# G G#/Ab A A#/Bb B C C#.

What does G#/Ab signify?

Hi Gene,

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > In blues this is not the case. *Every* chord is a
> > dominant-7th chord, and none of them set up the kind
> > of tension-->resolution expectation as they do in
> > Euro-style music.
>
> It seems to me there is a contradiction here. You can't
> very well have a lot of dominant seventh chords *and* a
> scale with only five notes in it.

Yes, absolutely correct -- i should have gone into more
detail in my explanation.

In effect, blues actually uses two different pitch-class sets
simultaneously -- one melodically and another harmonically.

Melodically, the scale basis is the minor pentatonic, and
harmonically, the chords are all dominant-7th chords.

Thus, for the example i used where the tonic is "A", the
melodic basis is the A C D E G minor pentatonic, with lots
of pitch-bending and sliding around allowed. The harmonic
basis is the 9-note scale A B C C# D E F# G G#, which gives
all the notes necessary for the A7, D7, and E7 chords.

This 9-note scale is interesting in itself, because it
exhibits tetrachordal similarity: A B C C# D and E F# G G# A
both have the same tone(t) and semitone(s) pattern: t s s s,
with a "tone of disjunction" between D and E which mirrors
the tone between A and B so that one might also recognize
pentachordal similarity. But that scale is really only an
artifact of the chords. Notice also that the A C D E G
minor pentatonic is included as a subset within it.

> > Thus, in blues the dominant-7th chord is the funadamental
> > consonance. This IMO encourages the listener's ears to
> > interpret *every* chord as a 4:5:6:7 proportion.
>
> That makes more sense if in fact the chords are not
> dominant seventh chords in a strict sense, but chords
> like CEGA#, with augmented seconds; but then you might
> be getting blue notes in the form GA#, etc.

What you're saying is correct, and in fact the G A#
augmented-2nd is quite close to the 7/6 ratio i mentioned
later. But i doubt very strongly that any blues musician,
at least from the first half of the 20th century, was
using meantone either in theory or practice in any form.
The basic blues instrument has always been the 12-edo guitar.

> > If the listener's ear is already hearing the melodic G
> > as a 7/4 above A, the melodic C as a 7/4 above D, and
> > the melodic D as a 7/4 above E, and with all the bending
> > going on, the guitarist and vocalist may indeed be
> > *playing* those notes lower in pitch so that they
> > really *are* septimal intervals.
>
> It makes sense to me, but do they?

As i said, blues musicians and vocalists bend and slide
the pitches of the notes all over the place ... unless
they simply can't, as on a piano -- in which case, they
actually often resort to playing two notes a semitone
apart simultaneously, in order to sort of trick the ear
into accepting that it is hearing a note that lies between
those two.

This is a standard part of blues piano technique, and
something that the visionary jazz pianist/composer
Thelonius Monk made into a distinctive part of his
harmonic style.

> > The tonic of this piece is D. The scale Johnson used
> > in his vocals in this song is interesting in that,
> > if you try to describe it in terms of 12-edo or meantone,
> > you basically get: D F# G G#/Ab A A#/Bb B C C#.
>
> What does G#/Ab signify?

* That if one insists on labeling the notes in Johnson's
vocal with the standard meantone names, then one would
have to call it G# when ascending and Ab when descending.

* That if one insists on labeling the notes in Johnson's
vocal as 12-edo notes, then it is equally valid to call
that 12-edo note by either name.

Same thing goes for A#/Bb.

But my point really is that Johnson's vocal is so rich in
microtonal subtlety that it's pointless to force it into
these notational straight-jackets.

I've written here many times before about how hard it was
for me, and how long it took, to recognize what was really
going on pitch-wise in the blues, because of the fact that
i had gotten so firmly indoctrinated into standard Western
(Euro) music-theory.

It wasn't until i finally got a firm grasp on Partch's
theories and on microtonality in general that i finally was
able to really recognize and copy what blues musicians do.
Before that, my attempts to make blues music were truly
laughable.

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

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

> > > Thus, in blues the dominant-7th chord is the funadamental
> > > consonance. This IMO encourages the listener's ears to
> > > interpret *every* chord as a 4:5:6:7 proportion.
> >
> > That makes more sense if in fact the chords are not
> > dominant seventh chords in a strict sense, but chords
> > like CEGA#, with augmented seconds; but then you might
> > be getting blue notes in the form GA#, etc.
>
>
> What you're saying is correct, and in fact the G A#
> augmented-2nd is quite close to the 7/6 ratio i mentioned
> later. But i doubt very strongly that any blues musician,
> at least from the first half of the 20th century, was
> using meantone either in theory or practice in any form.
> The basic blues instrument has always been the 12-edo guitar.

Of course, 12-edo is also a meantone, so i should have
been more accurate and said that "i doubt very strongly that
any blues musician ... was using a non-12edo meantone ...".

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

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

> This 9-note scale is interesting in itself, because it
> exhibits tetrachordal similarity: A B C C# D and E F# G G# A
> both have the same tone(t) and semitone(s) pattern: t s s s,
> with a "tone of disjunction" between D and E which mirrors
> the tone between A and B so that one might also recognize
> pentachordal similarity. But that scale is really only an
> artifact of the chords. Notice also that the A C D E G
> minor pentatonic is included as a subset within it.

I note that Orwell[9] is also a 9-note scale with a lot of septimal
harmony, though no tetrads. The pattern there is SLSLSLSLS.

> It wasn't until i finally got a firm grasp on Partch's
> theories and on microtonality in general that i finally was
> able to really recognize and copy what blues musicians do.
> Before that, my attempts to make blues music were truly
> laughable.

Which scales or tunings would you regard as suitable for making a more
or less bluesy sound? Would you consider Orwell[9], for instance, as
too far from the requirements?

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > This 9-note scale is interesting in itself, because it
> > exhibits tetrachordal similarity: A B C C# D and E F# G G# A
> > both have the same tone(t) and semitone(s) pattern: t s s s,
> > with a "tone of disjunction" between D and E which mirrors
> > the tone between A and B so that one might also recognize
> > pentachordal similarity. But that scale is really only an
> > artifact of the chords. Notice also that the A C D E G
> > minor pentatonic is included as a subset within it.
>
> I note that Orwell[9] is also a 9-note scale with a lot of
> septimal harmony, though no tetrads. The pattern there
> is SLSLSLSLS.
>
> > It wasn't until i finally got a firm grasp on Partch's
> > theories and on microtonality in general that i finally was
> > able to really recognize and copy what blues musicians do.
> > Before that, my attempts to make blues music were truly
> > laughable.
>
> Which scales or tunings would you regard as suitable for
> making a more or less bluesy sound? Would you consider
> Orwell[9], for instance, as too far from the requirements?
>

Honestly, i never tried setting up an actual tuning and
composing a blues in it until just now. I pretty much
just did it "on the fly" before, bending notes on either
the guitar or synthesizer.

Do you mean the 9-note orwell MOS which i show on the
graphs at the bottom of the Encyclopedia page, with a
generator of ~271.304 cents?

http://tonalsoft.com/enc/o/orwell.aspx

I set it up in Tonescape and started composing a blues
in it, but i made A the tonic and it lacks anything
close to a G.

Recommend an orwell[9] for me which would be good with
A as tonic. Cents values are fine ... if you can give
enough info for me to set up a lattice of at least
2 dimensions that would be great.

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

No wonder Erkan Ogur seems to incorporate Maqam Music so easily with the Blues style.
----- Original Message -----
From: klaus schmirler
To: tuning@yahoogroups.com
Sent: 15 Ekim 2005 Cumartesi 12:29
Subject: Re: [tuning] Re: "pentatonic hearing" (was: Otonal / Utonal)

Gene Ward Smith wrote:

> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
>
>>I agree with Paul 100%, about both the academic neglect
>>of "pentatonic hearing", and its relevance to the blues.
>
>
> Blues sounds decidedly un-pentatonic to me. I'd say a good scale for
> producing a vaguely bluesy sound is Orwell[9]; not that it's a blues
> scale, but all those 7/6s give it a sort of blue note feel. Isn't that
> what you need for blues--a few blue notes?

I play 7/6, which raises a few eyebrows but is generally accepted.
Monz found 19/16 used as a blue note. But the general definition in
practically all the literature (I don't know any exceptions) speaks
about neutral thirds and sevenths. Guitarists play it like that.

I've asked before and do it again: Who brought up the notion that blue
notes are septimal intervals, or are extra low thirds?

klaus

HI

AS I TOLD BEFORE THIS SYSTEM WAS FIRST STUDIED BY IBN SINA IN TANBOUR
BAGHDAD.

THANKS

________________________________

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf
Of microtonalist
Sent: Friday, October 14, 2005 4:35 PM
To: tuning@yahoogroups.com
Subject: [tuning] Re: equal divisions, subharmonic?

(IF you know all this, please ignore it: else read on)

Suppose one started with a string, and divided it into say twelve equal
sized parts. The full string would then have (for a given thickness and
tension of string) a frequency, which we will call f.

Stopping the string 1/12th of the way from one end (the first division)
will result in a string that is 11/12ths of the original length. If we
make the assumption that tension does not change as a result of
stopping the string, then the new frequency will be 12/11ths of the old
frequency.

similarly stopping one more 12th along, 10/12ths of the string remain,
giving a frequency that is 12/10ths of the unstopped string frequency.

And so it goes. With each new step the interval size increases:

Open String = 1/1
1st stop/fret = 12/11
2nd = 12/10 = 6/5
3rd = 12/9 = 4/3
4th = 12/8 = 3/2
5th = 12/7
6th = 12/6 = 2/1 (octave)
etc.

The intervals between adjacent ascending notes is 12/11 11/10 10/9 9/8
8/7 ... are the same intervals between adjacent notes of the harmonic
series, but in reverse order.

Of course, any equal divisions along a string produce a subharmonic
series or a partition of it.

All of the above is true for pipes, with the caveat that pipes suffer
from end effects, the helmholtz effect (mass of the air where the holes
are) and other perturbations away from the pure solution.

As an example of a partition of a subharmonic series, suppose the first
stopping point is 2/13ths of the length of the string. The first note
is then 13/11. Subsequent stopping points are spaced as 1/8th of the
unstopped string length. The scale is then

2/13th +1/8th +2/8th +3/8th +4/8th ...
13/11 104/75 52/31 52/27 26/17 ...

hopefully this makes some sense.

Mark

--- In tuning@yahoogroups.com, "traktus5" <kj4321@h...> wrote:
>
> > Finally, it should be of interest to those who don't already know
> > that although a string or a pipe vibrates according to the harmonic
> > series, if a string is equally divided or a pipe has holes made at
> > equal points
>
> divided how, exactly? any division of equal spacing?...
>
> along it then a subharmonic series results from the
> > resulting scales. It may be interesting to conjecture that perhaps
> > for early humans, equal division of the physical object (string or
> > bone flute) was easier than logarithmic.
>
> Can someone elaborate?
>
> thanks.
>

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> (If you know all this, please ignore it: else read on)

Also, not everyone knows what you know, and sometimes people like to
know without having to go find some theorist's work somewhere. What
is obvious to you is not obvious to another person.

and there's no need to shout, either. See nettiquette rules, please.

Mark

--- In tuning@yahoogroups.com, "Mohajeri Shahin" <shahinm@k...> wrote:
>
> HI
>
> AS I TOLD BEFORE THIS SYSTEM WAS FIRST STUDIED BY IBN SINA IN
TANBOUR
> BAGHDAD.
>
> THANKS
>
>
>
> ________________________________
>
> From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On
Behalf
> Of microtonalist
> Sent: Friday, October 14, 2005 4:35 PM
> To: tuning@yahoogroups.com
> Subject: [tuning] Re: equal divisions, subharmonic?
>
>
[snipt]

EXCUSE ME ...... I MADE A MISTAKE ......IN THESE 2 SYSTEMS ,WHICH THE
FIRST IS OTONAL AND THE LAST UTONAL THE NUMBER OF DEGREES ARE THE SAME
(=8) BUT THE RELATION BETWEEN THEIR NUMERARY NEXUS(N.N) OF IS SO :

N.N OF UTONAL = 2 * N.N OF OTONAL

0: 1/1 0.000 unison, perfect prime

1: 9/8 203.910 major whole tone

2: 5/4 386.314 major third

3: 11/8 551.318 undecimal semi-augmented fourth

4: 3/2 701.955 perfect fifth

5: 13/8 840.528 tridecimal neutral sixth

6: 7/4 968.826 harmonic seventh

7: 15/8 1088.269 classic major seventh

8: 2/1 1200.000 octave

|

0: 1/1 0.000 unison, perfect prime

1: 16/15 111.731 minor diatonic semitone

2: 8/7 231.174 septimal whole tone

3: 16/13 359.472 tridecimal neutral third

4: 4/3 498.045 perfect fourth

5: 16/11 648.682 undecimal semi-diminished fifth

6: 8/5 813.686 minor sixth

7: 16/9 996.090 Pythagorean minor seventh

8: 2/1 1200.000 octave

________________________________

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf
Of wallyesterpaulrus
Sent: Thursday, October 13, 2005 12:37 AM
To: tuning@yahoogroups.com
Subject: [tuning] Re: Otonal / Utonal

--- In tuning@yahoogroups.com, "Mohajeri Shahin" <shahinm@k...> wrote:

> 4-in otonal and utonal systems with same number of degrees ,
> udentity=2*odentity.

How did you come to that conclusion?

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--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > I agree with Paul 100%, about both the academic neglect
> > of "pentatonic hearing", and its relevance to the blues.
>
> Blues sounds decidedly un-pentatonic to me. I'd say a good scale for
> producing a vaguely bluesy sound is Orwell[9]; not that it's a blues
> scale, but all those 7/6s give it a sort of blue note feel. Isn't that
> what you need for blues--a few blue notes?

The "blue third" is far more commonly a neutral third rather than a
7/6. Meanwhile, alterations in the pitches don't contradict the
underlying pentatonic fabric/language, any more than chromatically
expressive common-practice music contradicts the underlying diatonicism
thereof.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
>
> > Many academics in a number of different music theory fields take
it
> > that when we hear music, we hear partly in terms of generic
diatonic
> > interval classes -- thus, a 300-cent interval will be heard as an
> > augmented second or a minor third (depending on context), which
> > correspond to 1 or 2 steps, respectively, in the 7-note diatonic
> > scale. In other words, everything is heard in terms of a presumed
> > underlying (though flexible) 7-note framework. I'd call this an
> > assumption of "diatonic hearing", and it may apply to much
Western
> > music, but even in the blues I'm pretty sure that quite a bit
> > of "pentatonic hearing" happens instead, with 5 rather than 7
generic
> > interval sizes per octave.
>
> From your language it seems the underlying assumption is some kind
of
> meantone system (since we have augmented seconds vs minor thirds,
> etc.)

My assumption was 12-equal. As far as augmented seconds actually
being tuned differently than minor thirds, you have that in plenty of
non-meantone systems, including plain old Pythagorean -- but that's
irrelevant here.

> Is this another of those Eytan Agmon deals, where people assume
> meantone without, apparently, realizing they are doing so?

It seemed to me that that claim of yours was an unfortunate result of
your fairly willful misunderstanding of his work, but I don't care to
rehash this here or now.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> > If the listener's ear is already hearing the melodic G
> > as a 7/4 above A, the melodic C as a 7/4 above D, and
> > the melodic D as a 7/4 above E, and with all the bending
> > going on, the guitarist and vocalist may indeed be
> > *playing* those notes lower in pitch so that they
> > really *are* septimal intervals.
>
> It makes sense to me, but do they?

As Monz himself pointed out, guitarists virtually always do the
opposite, and bend these notes *up* by about a quarter-tone (modulo
lots of vibrato, etc.). I'll leave my professional experience out of
this for now . . . I've heard vocalists do both the raising and the
lowering, in different modern "bluesy" contexts, but the
old "authentic" blues that I've heard has no traces of 7-limit tuning
tendencies.

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

> It seemed to me that that claim of yours was an unfortunate result of
> your fairly willful misunderstanding of his work, but I don't care to
> rehash this here or now.

I wasn't the only one to think what he did looked a hell of a lot like
a description of meantone, please recall.

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

> > > If the listener's ear is already hearing the melodic G
> > > as a 7/4 above A, the melodic C as a 7/4 above D, and
> > > the melodic D as a 7/4 above E, and with all the bending
> > > going on, the guitarist and vocalist may indeed be
> > > *playing* those notes lower in pitch so that they
> > > really *are* septimal intervals.
> >
> > It makes sense to me, but do they?
>
>
> As i said, blues musicians and vocalists bend and slide
> the pitches of the notes all over the place

But not haphazardly. The minutiae of the bending and sliding are an
integral part of the language, and learning them involves as precise
and intense a period of practice, listening, and more practice as
anything in classical music.

> ... unless
> they simply can't, as on a piano -- in which case, they
> actually often resort to playing two notes a semitone
> apart simultaneously, in order to sort of trick the ear
> into accepting that it is hearing a note that lies between
> those two.

I think it's most relevant to point out here that pianists usually
play two notes a semitone apart, *both* of which are higher than the
supposed "septimal note", and that the note that lies between the two
is the "neutral third" or other neutral scale degree that would be
achieved by *raising*, not lowering, the minor pentatonic notes.

I believe the numerary nexus is 1 for both systems if each is related
to a fixed 1/1 common to both, in some kind of Partchian diamondic
system (and if not, "numerary nexus" doesn't have much meaning here).
Why do you think otherwise?

--- In tuning@yahoogroups.com, "Mohajeri Shahin" <shahinm@k...> wrote:
>
> EXCUSE ME ...... I MADE A MISTAKE ......IN THESE 2 SYSTEMS ,WHICH
THE
> FIRST IS OTONAL AND THE LAST UTONAL THE NUMBER OF DEGREES ARE THE
SAME
> (=8) BUT THE RELATION BETWEEN THEIR NUMERARY NEXUS(N.N) OF IS SO :
>
>
>
> N.N OF UTONAL = 2 * N.N OF OTONAL
>
>
>
>
>
> 0: 1/1 0.000 unison, perfect prime
>
> 1: 9/8 203.910 major whole tone
>
> 2: 5/4 386.314 major third
>
> 3: 11/8 551.318 undecimal semi-augmented
fourth
>
> 4: 3/2 701.955 perfect fifth
>
> 5: 13/8 840.528 tridecimal neutral sixth
>
> 6: 7/4 968.826 harmonic seventh
>
> 7: 15/8 1088.269 classic major seventh
>
> 8: 2/1 1200.000 octave
>
> |
>
> 0: 1/1 0.000 unison, perfect prime
>
> 1: 16/15 111.731 minor diatonic semitone
>
> 2: 8/7 231.174 septimal whole tone
>
> 3: 16/13 359.472 tridecimal neutral third
>
> 4: 4/3 498.045 perfect fourth
>
> 5: 16/11 648.682 undecimal semi-diminished
fifth
>
> 6: 8/5 813.686 minor sixth
>
> 7: 16/9 996.090 Pythagorean minor seventh
>
> 8: 2/1 1200.000 octave
>
>
>
> ________________________________
>
> From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On
Behalf
> Of wallyesterpaulrus
> Sent: Thursday, October 13, 2005 12:37 AM
> To: tuning@yahoogroups.com
> Subject: [tuning] Re: Otonal / Utonal
>
>
>
> --- In tuning@yahoogroups.com, "Mohajeri Shahin" <shahinm@k...>
wrote:
>
> > 4-in otonal and utonal systems with same number of degrees ,
> > udentity=2*odentity.
>
> How did you come to that conclusion?
>
>
>
>
>
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Hi Paul,

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
>
> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > > > If the listener's ear is already hearing the melodic G
> > > > as a 7/4 above A, the melodic C as a 7/4 above D, and
> > > > the melodic D as a 7/4 above E, and with all the bending
> > > > going on, the guitarist and vocalist may indeed be
> > > > *playing* those notes lower in pitch so that they
> > > > really *are* septimal intervals.
> > >
> > > It makes sense to me, but do they?
> >
> >
> > As i said, blues musicians and vocalists bend and slide
> > the pitches of the notes all over the place
>
> But not haphazardly. The minutiae of the bending and sliding are an
> integral part of the language, and learning them involves as precise
> and intense a period of practice, listening, and more practice as
> anything in classical music.

Agree 100%, again.

> > ... unless
> > they simply can't, as on a piano -- in which case, they
> > actually often resort to playing two notes a semitone
> > apart simultaneously, in order to sort of trick the ear
> > into accepting that it is hearing a note that lies between
> > those two.
>
> I think it's most relevant to point out here that pianists
> usually play two notes a semitone apart, *both* of which
> are higher than the supposed "septimal note", and that the
> note that lies between the two is the "neutral third" or
> other neutral scale degree that would be achieved by
> *raising*, not lowering, the minor pentatonic notes.

Yes! Thanks for pointing that out, it's absolutely correct.

As i said, i really think the main reason so many
microtonalists think of blues as "septimal" is because
of the fact that *all* of the chords are dominant-7ths,
and the 7ths are not felt as dissonances in need of
resolution, which leads the ear into perceiving the
harmony *in a general sense* as a bunch of 4:5:6:7 ratios.

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

Hi Gene,

I guess you missed this. I asked you directly to
furnish me with a scale, because i was inspired to
start composing some Orwell Blues! Please do!

See my original reply for the quote from you:
/tuning/topicId_61293.html#61486

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> Honestly, i never tried setting up an actual tuning and
> composing a blues in it until just now. I pretty much
> just did it "on the fly" before, bending notes on either
> the guitar or synthesizer.
>
> Do you mean the 9-note orwell MOS which i show on the
> graphs at the bottom of the Encyclopedia page, with a
> generator of ~271.304 cents?
>
> http://tonalsoft.com/enc/o/orwell.aspx
>
>
> I set it up in Tonescape and started composing a blues
> in it, but i made A the tonic and it lacks anything
> close to a G.
>
> Recommend an orwell[9] for me which would be good with
> A as tonic. Cents values are fine ... if you can give
> enough info for me to set up a lattice of at least
> 2 dimensions that would be great.

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

I think one of the reason the 7/6 has been incorporated into so many peoples blues interpretation, is just what the interval 'says'.

>
>Message: 21 > Date: Mon, 17 Oct 2005 20:22:25 -0000
> From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
>Subject: Re: "pentatonic hearing" (was: Otonal / Utonal)
>
>--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> >
>>>If the listener's ear is already hearing the melodic G
>>>as a 7/4 above A, the melodic C as a 7/4 above D, and
>>>the melodic D as a 7/4 above E, and with all the bending
>>>going on, the guitarist and vocalist may indeed be >>>*playing* those notes lower in pitch so that they
>>>really *are* septimal intervals.
>>> >>>
>>It makes sense to me, but do they?
>> >>
>
>As Monz himself pointed out, guitarists virtually always do the >opposite, and bend these notes *up* by about a quarter-tone (modulo >lots of vibrato, etc.). I'll leave my professional experience out of >this for now . . . I've heard vocalists do both the raising and the >lowering, in different modern "bluesy" contexts, but the >old "authentic" blues that I've heard has no traces of 7-limit tuning >tendencies.
>
>
>
>
>
>________________________________________________________________________
>________________________________________________________________________
>
>Message: 22 > Date: Mon, 17 Oct 2005 20:27:17 -0000
> From: "Gene Ward Smith" <gwsmith@svpal.org>
>Subject: Re: Otonal / Utonal
>
>--- In tuning@yahoogroups.com, "wallyesterpaulrus"
><wallyesterpaulrus@y...> wrote:
>
> >
>>It seemed to me that that claim of yours was an unfortunate result of >>your fairly willful misunderstanding of his work, but I don't care to >>rehash this here or now.
>> >>
>
>I wasn't the only one to think what he did looked a hell of a lot like
>a description of meantone, please recall.
>
>
>
>
>
>
>________________________________________________________________________
>________________________________________________________________________
>
>Message: 23 > Date: Mon, 17 Oct 2005 20:29:20 -0000
> From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
>Subject: Re: blues (was: "pentatonic hearing" (was: Otonal / Utonal))
>
>--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> >
>>>>If the listener's ear is already hearing the melodic G
>>>>as a 7/4 above A, the melodic C as a 7/4 above D, and
>>>>the melodic D as a 7/4 above E, and with all the bending
>>>>going on, the guitarist and vocalist may indeed be >>>>*playing* those notes lower in pitch so that they
>>>>really *are* septimal intervals.
>>>> >>>>
>>>It makes sense to me, but do they?
>>> >>>
>>As i said, blues musicians and vocalists bend and slide
>>the pitches of the notes all over the place
>> >>
>
>But not haphazardly. The minutiae of the bending and sliding are an >integral part of the language, and learning them involves as precise >and intense a period of practice, listening, and more practice as >anything in classical music.
>
> >
>>... unless
>>they simply can't, as on a piano -- in which case, they
>>actually often resort to playing two notes a semitone
>>apart simultaneously, in order to sort of trick the ear >>into accepting that it is hearing a note that lies between
>>those two.
>> >>
>
>I think it's most relevant to point out here that pianists usually >play two notes a semitone apart, *both* of which are higher than the >supposed "septimal note", and that the note that lies between the two >is the "neutral third" or other neutral scale degree that would be >achieved by *raising*, not lowering, the minor pentatonic notes.
>
>
>
>
>
>________________________________________________________________________
>________________________________________________________________________
>
>Message: 24 > Date: Mon, 17 Oct 2005 20:37:07 -0000
> From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
>Subject: Re: No doubt about Lehman's Bach scale
>
>--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
> >
>>>>Wow. Are all these beat rates referring to the beating of upper >>>>partials in a not-quite-JI chord?
>>>> >>>>
>>>For example, if I play an A major triad on my piano with the
>>>A tuned to 220 Hz, the C# to 275 Hz and the E tuned to 331 Hz,
>>>then I'll get beats between the 3rd partial of the A at 660 Hz
>>>and the 2nd partial of the E at 662 Hz. These beats will be at
>>>662 - 660 = 2 Hz.
>>> >>>
>>Hi Yahya,
>>
>>I'll make a few comments. If you want to wait for Paul's reply
>>to reply, that's fine by me. I'm just thinking aloud waiting
>>for Paul's reply myself.
>>
>>I'm not sure why you're subtracting these beat rates here.
>> >>
>
>Sorry if I'm behind, but . . . subtracting beat rates? It seems Yahya >is merely calculating beat rates here (and correctly, I might add), >not subtracting any.
>
> >
>>As I think Paul was trying to say, beats are amplitude
>>modulation, which is sort of a second order sound wave.
>>They represent changes in the amplitude of a sound,
>> >>
>
>a pitch
>
> >
>>not
>>a sound
>> >>
>
>a pitch.
>
> >
>>in and of themselves. AM is used for synthesis
>>and can create new pitches, called sidebands, but they
>>aren't at the frequency of the beats.
>> >>
>
>They're at the frequency of the original sine waves in Yahya's >example, if what you're modulating is a sine wave whose frequency is >the average of the two. You've just turned the equivalence of beating >with AM around, and applied it backwards, which is of course >perfectly valid to do.
>
> >
>>Perhaps sidebands
>>could be evoked from beating in an acoustic musical
>>instrument,
>> >>
>
>I think you've gotten a bit confused. The sidebands "created" by the >AM applied to the heard frequency would be identical to the original >frequencies in the musical instrument; the AM itself would be >identical to the beating heard.
>
>
>
>
>
>________________________________________________________________________
>________________________________________________________________________
>
>Message: 25 > Date: Mon, 17 Oct 2005 20:52:34 -0000
> From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
>Subject: Re: Beats (was No doubt about Lehman's Bach scale)
>
>--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:
>
> >
>>When ANY two frequencies sound simultaneously, so do two
>>other frequencies - their sum and difference.
>> >>
>
>Yahya, this is not true in general. It's not true when the sound is >amplified or reproduced *linearly*, i.e., with no distortion. >Meanwhile, when there is a quadratic distortion of the signal, you >get not only the sum and difference, but also the 2nd partials of >each of the two frequencies. Shall I show the math?
>
> >
>>>As I think Paul was trying to say, beats are amplitude
>>>modulation, which is sort of a second order sound wave.
>>> >>>
>>Combining two pure sinusoidal waves of frequencies A and B
>>results in sinusoidal waves with components of frequencies
>>A-B and A+B. >> >>
>
>This is false, in general, as I wrote above. It's only true in the >presence of distortion. A quadratic nonlinearity is the simplest kind >of distortion, and results in not only A, B, A-B, and A+B, but also >2*A and 2*B as well. Perhaps I should review the math?
>
> >
>>If the upper and lower sidebands aren't at the >>frequencies of the combination tones, what frequencies >>are they at?
>> >>
>
>They are at the frequencies corresponding to A and B in your example.
>
>
>
>
>
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--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

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

> As i said, i really think the main reason so many
> microtonalists think of blues as "septimal" is because
> of the fact that *all* of the chords are dominant-7ths,
> and the 7ths are not felt as dissonances in need of
> resolution, which leads the ear into perceiving the
> harmony *in a general sense* as a bunch of 4:5:6:7 ratios.

I think this is correct. Septimally constructed chords and melodies
tend to sound vaguely bluesy or jazzy.

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> Hi Gene,
>
>
> I guess you missed this. I asked you directly to
> furnish me with a scale, because i was inspired to
> start composing some Orwell Blues! Please do!

I wasn't sure what you were asking, and still am not, but here is
Orwell[9]:

! or9.scl
Orwell[9] in 1728/1715 (0,-1) tuning
9
!
156.143810
271.228762
427.372572
542.457524
698.601334
813.686286
928.771238
1084.915048
1200.000000

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
> >
> > Hi Gene,
> >
> >
> > I guess you missed this. I asked you directly to
> > furnish me with a scale, because i was inspired to
> > start composing some Orwell Blues! Please do!
>
> I wasn't sure what you were asking, and still am not, but here is
> Orwell[9]:
>
> ! or9.scl
> Orwell[9] in 1728/1715 (0,-1) tuning
> 9
> !
> 156.143810
> 271.228762
> 427.372572
> 542.457524
> 698.601334
> 813.686286
> 928.771238
> 1084.915048
> 1200.000000

You guys should hear Igliashon Jones's Orwell[9] piece (in 31-equal,
only a touch different than the above). It seems to be gone from his
Soundclick page now. Igs?