Re: Interval vs interval

Hi Paul,

I have finally found time to continue with this thread.

-----

You wrote in 21106 :

<< but . . . do you agree with how I understand your definition
of "interval", as I characterized with the Zarlino JI scale
example? >>

I'm not sure really how you understand my definition since you use the term
"scale" while I use the term "mode" in a mathematical sense. So let us look
at your example. I presume you talk about what you wrote in 20976 :

<< I don't think that is what you want to use. If the Zarlino
JI heptatonic scale is extended over several octaves, then
its intervals modulo 2 would be:

-------------upper note--------------
C D E F G A B

1:1 9:8 5:4 4:3 3:2 5:3 15:8 C }
16:9 1:1 10:9 32:27 4:3 40:27 5:3 D }
8:5 9:5 1:1 16:15 6:5 4:3 3:2 E }
3:2 27:16 15:8 1:1 9:8 5:4 45:32 F } lower
4:3 3:2 5:3 16:9 1:1 10:9 5:4 G } note
6:5 27:20 3:2 8:5 9:5 1:1 9:8 A }
16:15 6:5 4:3 64:45 8:5 16:9 1:1 B } >>

I agree this array corresponds to the interval modulo 2 space spanned by
the Zarlino scale. Consequently this array contains in the first row the
values 1:1 9:8 5:4 4:3 3:2 5:3 15:8 representing the intervals C:C C:D C:E
C:F C:G C:A C:B.

( I would add the fourth row contains as
F:F F:G F:A F:B F:C F:D F:E the intervals
1:1 9:8 5:4 45:32 3:2 27:16 15:8
which are the following harmonics of F :
1 9 5 45 3 27 15. See forward. )

When you talk about interval you refer always, if I have well understood,
to such concrete pitch dyads like

D:D = 1:1
F:G = 9:8
G:B = 5:4
B:E = 4:3

and you distinguish the tone 9/8 with the interval C:D = 9:8 for a scale is
a concrete acoustical object made with pitches. So the tones 1 9/8 5/4 4/3
3/2 5/3 15/8 are only "designators" of pitch heights "measured" from the
tonic pitch.

I have absolutely nothing against this viewpoint, but a "mode", in my
terms, is very different. Even using the expression "musical mode" it's
essentially a mathematical object defined in the pure context of an
algebraic structure. (It's what is astonishing to find closed numerical
relations between a such complex axiomatic object and concrete pitch
intervals used by musicians.)

Let us talk, for instance, about our 1/1 uses. For you, the tone 1/1
represents the tonic while you use 1:1 with the unison intervals C:C or D:D
or E:E . . . For me, in the context of a structure having "modes", I talk
always about the neutral element of that singular mathematical structure.
As you know, the neutral element 1 or 0 is defined by the following
property : for any x in the structure we have

1 * x = x * 1 = x (with multiplication)
0 + x = x + 0 = x (with addition)

When I talk about unison I refer uniquely to that neutral element having to
exist in the structured set of intervals, and that, without any reference
to a tonic or a pitch height for I use always an internal composition law.
And there is no place for pitches in a such model since it makes no sense
to use an internal composition law (addition or multiplication) with pitches.

The pitch question may come afterwards when musical scales and "sui generis
modes" are compared. I would like to remark however that the comparison is
made between intervals : my intervals as elements in a mathematical
structure and your intervals you understand as pitch indicators.

Perhaps the better way to understand my 1 or 1/1, when scales and modes are
compared, would be to see it as representing the class of your intervals
{C:C, D:D, E:E, F:F, G:G, A:A, B:B}. That class is the neutral element in
the algebraic structure built on the classes of your concrete intervals.

I add there exist 49 concrete X:Y intervals in your matrix while it is
partitioned in 19 interval classes. The following set of rationals
represents that set of classes :

{1, 16/5, 10/9, 9/8, 32/27, 6/5, 5/4, 4/3, 27/20, 45/32,
15/8, 9/5, 16/9, 27/16, 5/3, 8/5, 3/2, 40/27, 64/45}

This set, with multiplication, is a gammier for all axioms of the gammier
structure are respected. Besides, (since a gammier is an harmoid with
supplemental axioms) there exist, as harmoid, an odd generator which is here

<1 3 5 9 15 27 45>

( Cf. the precedent remark about
F:F F:G F:A F:B F:C F:D F:E
as harmonics of F. )

With the appropriate ordering you can then surely recognize the following
array as identical to that one of your concrete pitch height intervals.
However, I emphasize, there exist neither pitch reference nor concrete
classes here : that are only classes of odd ratios modulo 2.

3 27 15 1 9 5 45

3 1 9/8 5/4 4/3 3/2 5/3 15/8
27 16/9 1 10/9 32/27 4/3 40/27 5/3
15 8/5 9/5 1 16/15 6/5 4/3 3/2
1 3/2 27/16 15/8 1 9/8 5/4 45/32
9 4/3 3/2 5/3 16/9 1 10/9 5/4
5 6/5 27/20 3/2 8/5 9/5 1 9/8
45 16/15 6/5 4/3 64/45 8/5 16/9 1

-----

Q. Is it the same stuff with only a more abstracted formulation?

A. I don't think so. A (concrete) scale spans a set of intervals
while a (virtual) structure spans a set of modes. The precedent
structured set of 19 intervals spans 84 distinct modes.

Q. What are these (sui generis) modes?

A. All ordered subsets of intervals X(i) such that any ratio

X(i+1) / X(i)

between two successive intervals in the subset (or the
limit 2) is an atom of the set.

-- I recall my definition of an "atom" : any interval k
greater than unison that cannot be splitted in two other
intervals x and y greater than unison, where
x < k and y < k --

-- In gammoids and gammiers the total amount of intervals
in any mode is always the same and atoms are synonymous
with steps --

-----

(Naturally, the properties of each mode are not equivalent particularly
about harmonicity and transposability. Historically, the reduction to a
bimodal system in the tonal language corresponds, I presume, to an emphasis
on these properties since the modes retained maximize it.)

-----

Here is the modal lattice showing the 84 modes in <1 3 5 9 15 19 27 45>
whose 19 interval set is also spanned by the Zarlino scale.

10/9====5/4===45/32 40/27===5/3===15/8
/ \ / \ \ / / \ / \
/ \/ \ \/ / \/ \
/ /\ \ /\ / /\ \
/ / \ \ / \ / / \ \
1====9/8 32/27====4/3====3/2====27/16 16/9====2
\ \ / / \ / \ \ / /
\ \/ / \/ \ \/ /
\ /\ / /\ \ /\ /
\ / \ / / \ \ / \ /
16/15===6/5===27/20 64/45===8/5====9/5

Here is the minimal substructure (the hard kernel of the structure)
retaining only 13 intervals and 16 modes (or 12 by comma reduction). The
Zarlino mode is yet there and its existence as sui generis mode depends on
the global configuration of these 13 intervals.

10/9====5/4 5/3====15/8
/ / \ / \ \
/ / \ / \ \
/ / \ / \ \
/ / \ / \ \
1=====9/8 4/3======3/2 16/9=====2
\ \ / \ / /
\ \ / \ / /
\ \ / \ / /
\ \ / \ / /
16/15===6/5 8/5=====9/5

How the structure is maintained may be seen in the matrix obtained after
erasing the contribution of 27 and 45 in the precedent matrix.

3 (27) 15 1 9 5 (45)

3 1 - 5/4 4/3 3/2 5/3 -
(27) - - - - - - -
15 8/5 - 1 16/15 6/5 4/3 -
1 3/2 - 15/8 1 9/8 5/4 -
9 4/3 - 5/3 16/9 1 10/9 -
5 6/5 - 3/2 8/5 9/5 1 -
(45) - - - - - - -

So the minimal generator of that gammier having rank 9 in the infinite
space of gammiers is <1 3 5 9 15>. The odd generator <1 5 9 15> gives also
the same 13 intervals but all the composition triplets (like 4/3 * 9/8 =
3/2) are not chordically defined on a row or a column.

--------------------

II

Quoting what I wrote :

<< The ratio 4:5:6 is also synonynous with the major quality
of a chord in relational paradigm. However the existence
itself of a such chord requires much more than 3 elements
having the ratio 4:5:6. In the relational paradigm an object
like a chord exist within a finite set S of intervals having
a partial composition law. If 3 elements <a b c> exist in S
such that a:b:c = 4:5:6 then <a b c> is a chord if and only
if the 7 intervals <1 6/5 5/4 4/3 3/2 8/5 5/3> exist in S,
otherwise <a b c> is called a discordance relatively to S. >>

You wrote in 21106 :

<< I don't understand this. Can you give a musical example?
For instance, in the key of C major, is the G major chord
a discordance? >>

Absolutely not. Any chord being a subchord of 1:5:3:15:9 or its dual (in
any mode or combination of modes) in the gammier <1 3 5 9 15> is concordant
: so not only any major and minor triads but also any maj7, min7, 6 and
min6 tetrads or maj9 and min9 pentads. This is the constitutive harmony of
that gammier.

The other chords are discordant. That is the extension harmony relatively
to that gammier. For instance, the triad G B D F, the dominant 7th chord on
G, is discordant not only for F = 21/16 (near from 4/3) don't exist
precisely in the set, but for the intervals 7/6, 7/5, 7/4 (and reverses)
don't exist in the set. The tones (my "tons") 7/6 and 7/4 are near from 6/5
and 16/9 but the triton 7/5 appears more easily as a stranger. In the
following matrix I retained the notes in the key of C major and its
relative A minor.

1 5 3 15 9

1 C E G B D (?)
5 - C - G -
3 F A C E G
15 - F - C -
9 - D F A C

We see clearly the distinction between the major tone D = 9/8 and the minor
tone D = 10/9 in the two following submatrices showing the major harmony in
the key of C major (with an extension for F = 21/16)

1 5 3 15 9 21

1 C E G B D (F)
3 F A C E G

and the minor harmony in the relative key of A minor

5 15

1 E B
5 C G
3 A E
15 F C
9 D A

Since G7 is discordant, the F in (G B D F) may not appear on a row of the
matrix. Its place indicated by (?) outside the matrix corresponds to the
harmonic 7 of G and then would have to correspond to the odd 21 in the
generator. As you know the 21 --> 5 and 15 --> 1 resolution way of that
discordance were abundandtly used in the tonal music.

The chord G7 is not discordant in itself : the discordance is not a
microtonal property but a macrotonal property (i.e. depending of context
configuration). For instance, in the gammier <3 5 7 15 21> which is the
minimal structure containing the blues mode, the chord G7 is not a
discordance for the internal intervals spanned by the dominant 7th chord is
a subset of that gammier.

-----

Hoping it's more clear.

Maybe the next step would be I would write a developed text about my
musical interval concept which uses a three levels definition :
parametrical (as used in acoustics), relational (as I use it in algebra)
and functional (as we have to use to explain why intervals are used in music).

Probably I will not have time for that before june or july.

Pierre

--- In tuning@y..., Pierre Lamothe <plamothe@a...> wrote:
>
> Hi Paul,
>
> I have finally found time to continue with this thread.
>
>
> -----
>
> You wrote in 21106 :
>
> << but . . . do you agree with how I understand your definition
> of "interval", as I characterized with the Zarlino JI scale
> example? >>
>
> I'm not sure really how you understand my definition since you use
the term
> "scale" while I use the term "mode" in a mathematical sense. So let
us look
> at your example. I presume you talk about what you wrote in 20976 :
>
> << I don't think that is what you want to use. If the Zarlino
> JI heptatonic scale is extended over several octaves, then
> its intervals modulo 2 would be:
>
> -------------upper note--------------
> C D E F G A B
>
> 1:1 9:8 5:4 4:3 3:2 5:3 15:8 C }
> 16:9 1:1 10:9 32:27 4:3 40:27 5:3 D }
> 8:5 9:5 1:1 16:15 6:5 4:3 3:2 E }
> 3:2 27:16 15:8 1:1 9:8 5:4 45:32 F } lower
> 4:3 3:2 5:3 16:9 1:1 10:9 5:4 G } note
> 6:5 27:20 3:2 8:5 9:5 1:1 9:8 A }
> 16:15 6:5 4:3 64:45 8:5 16:9 1:1 B } >>
>
> I agree this array corresponds to the interval modulo 2 space
spanned by
> the Zarlino scale. Consequently this array contains in the first row
the
> values 1:1 9:8 5:4 4:3 3:2 5:3 15:8 representing the intervals C:C
C:D C:E
> C:F C:G C:A C:B.
>
> ( I would add the fourth row contains as
> F:F F:G F:A F:B F:C F:D F:E the intervals
> 1:1 9:8 5:4 45:32 3:2 27:16 15:8
> which are the following harmonics of F :
> 1 9 5 45 3 27 15. See forward. )
>
> When you talk about interval you refer always, if I have well
understood,
> to such concrete pitch dyads like
>
> D:D = 1:1
> F:G = 9:8
> G:B = 5:4
> B:E = 4:3
>
> and you distinguish the tone 9/8 with the interval C:D = 9:8 for a
scale is
> a concrete acoustical object made with pitches. So the tones 1 9/8
5/4 4/3
> 3/2 5/3 15/8 are only "designators" of pitch heights "measured" from
the
> tonic pitch.
>
> I have absolutely nothing against this viewpoint, but a "mode", in
my
> terms, is very different. Even using the expression "musical mode"
it's
> essentially a mathematical object defined in the pure context of an
> algebraic structure.

I think that most people on this list, when they see the expression
"musical mode", will give it concrete meaning in terms of actual
sounds, and not just look at it as a mathematical object in a "pure"
context. Otherwise, it loses its interest for most of the people here.

>
> Let us talk, for instance, about our 1/1 uses. For you, the tone 1/1
> represents the tonic while you use 1:1 with the unison intervals C:C
or D:D
> or E:E . . . For me, in the context of a structure having "modes", I
talk
> always about the neutral element of that singular mathematical
structure.
> As you know, the neutral element 1 or 0 is defined by the following
> property : for any x in the structure we have
>
> 1 * x = x * 1 = x (with multiplication)
> 0 + x = x + 0 = x (with addition)
>
> When I talk about unison I refer uniquely to that neutral element
having to
> exist in the structured set of intervals, and that, without any
reference
> to a tonic or a pitch height for I use always an internal
composition law.
> And there is no place for pitches in a such model since it makes no
sense
> to use an internal composition law (addition or multiplication) with
pitches.
>
> The pitch question may come afterwards when musical scales and "sui
generis
> modes" are compared.

> Q. What are these (sui generis) modes?
>
> A. All ordered subsets of intervals X(i) such that any ratio
>
> X(i+1) / X(i)
>
> between two successive intervals in the subset (or the
> limit 2) is an atom of the set.
>
> -- I recall my definition of an "atom" : any interval k
> greater than unison that cannot be splitted in two other
> intervals x and y greater than unison, where
> x < k and y < k --

You'll have to refine this definition, because clearly any interval
can be split into two smaller intervals.

> (Naturally, the properties of each mode are not equivalent
particularly
> about harmonicity and transposability. Historically, the reduction
to a
> bimodal system in the tonal language corresponds, I presume, to an
emphasis
> on these properties since the modes retained maximize it.)

Can you please demonstrate this maximization? (As you know, I have a
different point of view on this issue, but I'm more than interested in
yours.)

> --------------------
>
> II
>
> Quoting what I wrote :
>
> << The ratio 4:5:6 is also synonynous with the major quality
> of a chord in relational paradigm. However the existence
> itself of a such chord requires much more than 3 elements
> having the ratio 4:5:6. In the relational paradigm an object
> like a chord exist within a finite set S of intervals having
> a partial composition law. If 3 elements <a b c> exist in S
> such that a:b:c = 4:5:6 then <a b c> is a chord if and only
> if the 7 intervals <1 6/5 5/4 4/3 3/2 8/5 5/3> exist in S,
> otherwise <a b c> is called a discordance relatively to S. >>
>
> You wrote in 21106 :
>
> << I don't understand this. Can you give a musical example?
> For instance, in the key of C major, is the G major chord
> a discordance? >>
>
> Absolutely not. Any chord being a subchord of 1:5:3:15:9 or its dual
(in
> any mode or combination of modes) in the gammier <1 3 5 9 15> is
concordant
> : so not only any major and minor triads but also any maj7, min7, 6
and
> min6 tetrads

A min6 tetrad is, for example, D F A B. How can that be expressed as a
subchord of 1:5:3:15:9 or its dual?

But you seem to be agreeing, then, that a 4:5:6 chord would be
concordant regardless of its position relative to the mode . . . no?

And a practical problem with your theory would be the maj6/9 chord.
>
> The chord G7 is not discordant in itself : the discordance is not a
> microtonal property but a macrotonal property (i.e. depending of
context
> configuration). For instance, in the gammier <3 5 7 15 21> which is
the
> minimal structure containing the blues mode, the chord G7 is not a
> discordance for the internal intervals spanned by the dominant 7th
chord is
> a subset of that gammier.

I'm very interested in discussing this further but it's clear that in
musical practice, the "blues mode" uses neutral thirds much more
frequently than subminor thirds as you would have it.
>
>
> -----
>
> Hoping it's more clear.
>
> Maybe the next step would be I would write a developed text about my
> musical interval concept which uses a three levels definition :
> parametrical (as used in acoustics), relational (as I use it in
algebra)
> and functional (as we have to use to explain why intervals are used
in music).
>
> Probably I will not have time for that before june or july.

I'll wait in anticipation!