--- jwerntz2002 <juliawerntz@attbi.com> wrote:

>>

> We're just talking about octaves (I used octaves and fifths in the paper as an

> example) because the octave would represent the most perfect harmonic structure

> according to the premise and the underlying logic of just intonation. (Actually,

> according to Partch, maybe it would be the unison?) Anyway, I never claimed that

> the JI composers I cited in the essay maximize the octave in their music. Though I

> have heard plenty of JI music that seems to be "maximizing" partials 1-7 or 1-9. I

> don't want to get into that, though.

>

> (BTW, Here's your one person: David Doty, again, since I have his website handy.

> "Just Intonation is any system of tuning in which all of the intervals can be

> represented by ratios of whole numbers, with a strongly-implied preference for the

> smallest numbers compatible with a given musical purpose." This is the first

> statement in his definition of JI from his primer. And, further down: "The

> simple-ratio intervals upon which Just Intonation is based ...are what the human

> auditory system recognizes as consonance, if it ever has the opportunity to hear

> them in a musical context." I'm just sitting at my computer right at this moment; it

> probably wouldn't be too hard to find more.)

>

>>

Thank you for your lucid remarks.

What is the URL for David Doty's website?

Also, if you play a C octave as "the most perfect harmonic structure" as you say

"of just intonation," at least in that scale, would adding an F note in that octave,

as you play it, form for you, a consonant or dissonant chord? And what do you call the F

note relative to the C octave, as described? Would you call it a harmonic chord?

Thank you for your lucid remarks.

And thanks, in advance,

Bill Arnold

billarnoldfla@yahoo.com

http://www.cwru.edu/affil/edis/scholars/arnold.htm

Independent Scholar

Independent Scholar, Modern Language Association

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--- In tuning-math@y..., Bill Arnold <billarnoldfla@y...> wrote:

> Also, if you play a C octave as "the most perfect harmonic

structure" as you say

> "of just intonation," at least in that scale, would adding an F

note in that octave,

> as you play it, form for you, a consonant or dissonant chord?

it would be consonant, but the root would now be F.

> And what do you call the F

> note relative to the C octave, as described? Would you call it a

>harmonic chord?

it could still be seen as a harmonic chord, with proportions 3:4:6

over a fundamental _F_ two octaves lower. it could also be seen as a

subharmonic chord, with proportions (1/4):(1/3):(1/2) under a common

overtone c' an octave above the higher c in the chord. since 4, 3,

and 2 are simpler numbers than 3, 4, 6, the chord C-F-c has been

considered as a "subharmonic" or "undertonal" trine, while C-G-c

would the the "harmonic" or "overtonal" trine, with the proportions

exactly reversed.

--- wallyesterpaulrus <wallyesterpaulrus@yahoo.com> wrote:

> --- In tuning-math@y..., Bill Arnold <billarnoldfla@y...> wrote:

>

> > Also, if you play a C octave as "the most perfect harmonic

> structure" as you say

> > "of just intonation," at least in that scale, would adding an F

> note in that octave,

> > as you play it, form for you, a consonant or dissonant chord?

>

> it would be consonant, but the root would now be F.

>

> > And what do you call the F

> > note relative to the C octave, as described? Would you call it a

> >harmonic chord?

>

> it could still be seen as a harmonic chord, with proportions 3:4:6

> over a fundamental _F_ two octaves lower. it could also be seen as a

> subharmonic chord, with proportions (1/4):(1/3):(1/2) under a common

> overtone c' an octave above the higher c in the chord. since 4, 3,

> and 2 are simpler numbers than 3, 4, 6, the chord C-F-c has been

> considered as a "subharmonic" or "undertonal" trine, while C-G-c

> would the the "harmonic" or "overtonal" trine, with the proportions

> exactly reversed.

>

>

Thank you for that response.

Why is it not a C chord, with two C notes in C-F-c?

Also, what if F # or Fb were substituted for the F note:

in other words, C-F#-c?

and: C-Fb-c?

Are the more harmonic chords?

Bill Arnold

billarnoldfla@yahoo.com

http://www.cwru.edu/affil/edis/scholars/arnold.htm

Independent Scholar

Independent Scholar, Modern Language Association

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--- In tuning-math@y..., Bill Arnold <billarnoldfla@y...> wrote:

> > it could still be seen as a harmonic chord, with proportions

3:4:6

> > over a fundamental _F_ two octaves lower. it could also be seen

as a

> > subharmonic chord, with proportions (1/4):(1/3):(1/2) under a

common

> > overtone c' an octave above the higher c in the chord. since 4,

3,

> > and 2 are simpler numbers than 3, 4, 6, the chord C-F-c has been

> > considered as a "subharmonic" or "undertonal" trine, while C-G-c

> > would the the "harmonic" or "overtonal" trine, with the

proportions

> > exactly reversed.

> >

> >

> Thank you for that response.

>

> Why is it not a C chord, with two C notes in C-F-c?

i'm not sure what you're asking here.

> Also, what if F # or Fb were substituted for the F note:

>

> in other words, C-F#-c?

>

> and: C-Fb-c?

it would really depend on what tuning system you're using.

>

> Are the more harmonic chords?

huh?

--- Jon Szanto <JSZANTO@ADNC.COM> wrote:

> --- In tuning@y..., Bill Arnold <billarnoldfla@y...> wrote:

> > Why is it not a C chord, with two C notes in C-F-c?

> >

> > Also, what if F # or Fb were substituted for the F note:

> >

> > in other words, C-F#-c?

> >

> > and: C-Fb-c?

> >

> > Are the more harmonic chords?

>

> Bill, the questions you are asking involve very elementary (Western) music harmony

> principles. The standard Western triad involves three notes: a root, a perfect fifth

> above it, and a note in between that forms the third of the chord. The terms root,

> third, and fifth derive from their sequence in the scale. The chord is either major or

> minor depending on whether it has a third that is, respectively, two whole steps or one

> whole step and one half step. This, again, is all relating to standard 12tet Western

> harmony. You should also note that the root, major third and perfect fifth form the

> first triad that appears in the natural harmonic overtone series.

>

> So if the root is C, the third is either E or E-flat, and the fifth is G. If you were

> to have C and F, they form a perfect fourth. If you were to have those as two

> components of a triad, it would be F on the bottom and C as the fifth, and then either

> A or A-flat as the third.

>

> You might want to check out an elementary (Western) music theory book from a library,

> as all of this, including inversions, 7th chords, etc. would be explained.

>

> Jon

>

>

Hi, Jon. Why thank you very much for your lucid explanation. It makes perfect sense to

me now, although I will have other questions as I go along in my research. Charles Lucy

has put his URL on, and I went to :

http://www.lucytune.com/colour_and_mapping/clock_face.html

I note he has two distinct clock faces: his first clock face is 12 Tone Equal Temperment,

which I guess is what you are referring to above as Western music theory? Is that

correct?

His second clock face is what he calls: Lucy Tuning Intervals or Naturals?

In what sense are the latter "Naturals" and the "Western" are not natural?

I thank you for your suggestion that I read an elementary book. However, I am not after

an elementary understanding. I look at Charles Lucy's Charts, and they are not

elementary nor do I find any of this elementary, but complex. Not that that scares

me from investigating it.

I guess what I am after now: can you explain to me if there IS such a thing as

"Natural" scales? And "Natural" chords? Or, are they, like words, the creation

of the mind? You know, as pointed out by many authors such as Guy Murchie in Music of

the Spheres, that there are "Natural" shapes in Nature: six-sided hexagonal gems like

emeralds and eight-sided pyramidal gems like diamonds. So: are their such "Natural"

shapes in scales and chords in Nature? Or, are they creatings of minds?

Thanks in advance.

Arnold

billarnoldfla@yahoo.com

http://www.cwru.edu/affil/edis/scholars/arnold.htm

Independent Scholar

Independent Scholar, Modern Language Association

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hi Bill,

> From: "Bill Arnold" <billarnoldfla@yahoo.com>

> To: <tuning@yahoogroups.com>; <tuning-math@yahoogroups.com>

> Sent: Saturday, October 19, 2002 7:41 AM

> Subject: Re: [tuning-math] [tuning] Re: Everyone Concerned

>

>

> I thank you for your suggestion that I read an

> elementary book. However, I am not after an

> elementary understanding. I look at Charles Lucy's

> Charts, and they are not elementary nor do I find

> any of this elementary, but complex. Not that that

> scares me from investigating it.

Jon is right. it's great that you're interested in more

complex relationships, but you're asking questions from

the Remedial Music Theory course. this is extremely basic

stuff that you need to know in order to investigate the

kinds of questions and speculations you have in mind.

i suggest a splendid little book which, back in the day,

taught me the things that really got me interested in music:

Otto Karolyi. 1991.

_Introducing Music_

Penguin, London

ISBN 0-14-013520-0

(this must be a more recent revised edition)

> I guess what I am after now: can you explain to me

> if there IS such a thing as "Natural" scales? And

> "Natural" chords? Or, are they, like words, the

> creation of the mind? You know, as pointed out by

> many authors such as Guy Murchie in Music of the Spheres,

> that there are "Natural" shapes in Nature: six-sided

> hexagonal gems like emeralds and eight-sided pyramidal

> gems like diamonds. So: are their such "Natural"

> shapes in scales and chords in Nature? Or, are they

> creatings of minds?

crystalline gems and tuning-theory lattice diagrams are

expressions of the same mathematical concepts. see:

/monz/lattices/lattices.htm

for my version of tonal-lattice theory.

(please note that the lattice diagrams we use around here

are usually somewhat different from the what mathematicians

call lattices. i defer to others to explain if you need it.)

seems to me that the question you're interested in is:

is mathematics the creation of the human mind, or does

it have some objective existence in the non-human world?

-monz

--- In tuning-math@y..., "monz" <monz@a...> wrote:

> (please note that the lattice diagrams we use around here

> are usually somewhat different from the what mathematicians

> call lattices. i defer to others to explain if you need it.)

this is a canard. lattices are lattices, even for mathematicians.

it's just that if you're outside the field of geometry or related

disciplines, you find other meanings for the term "lattice". no self-

respecting mathematician would deny that our lattices are in

fact "lattices".

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

> no self-respecting mathematician would deny that our lattices

> are in fact "lattices".

Really?

--- In tuning-math@y..., "Jon Szanto" <jonszanto@y...> wrote:

> --- In tuning-math@y..., "wallyesterpaulrus"

<wallyesterpaulrus@y...> wrote:

> > no self-respecting mathematician would deny that our lattices

> > are in fact "lattices".

>

> Really?

yes, there was a lot of confusion on this point a while back, when

someone thought the algebraic definition of lattices was the only

mathematical one. they missed the geometric one, which is ours. same

as in crystallography, too.

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

> --- In tuning-math@y..., "Jon Szanto" <jonszanto@y...> wrote:

> > --- In tuning-math@y..., "wallyesterpaulrus"

> <wallyesterpaulrus@y...> wrote:

> > > no self-respecting mathematician would deny that our lattices

> > > are in fact "lattices".

> >

> > Really?

>

> yes, there was a lot of confusion on this point a while back, when

> someone thought the algebraic definition of lattices was the only

> mathematical one. they missed the geometric one, which is ours. same

> as in crystallography, too.

You should remember that many self-respecting mathematicians would not call something a lattice unless it inherited a group structure from R^n.

--- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:

> --- In tuning-math@y..., "wallyesterpaulrus"

<wallyesterpaulrus@y...> wrote:

> > --- In tuning-math@y..., "Jon Szanto" <jonszanto@y...> wrote:

> > > --- In tuning-math@y..., "wallyesterpaulrus"

> > <wallyesterpaulrus@y...> wrote:

>

> > > > no self-respecting mathematician would deny that our lattices

> > > > are in fact "lattices".

> > >

> > > Really?

> >

> > yes, there was a lot of confusion on this point a while back,

when

> > someone thought the algebraic definition of lattices was the only

> > mathematical one. they missed the geometric one, which is ours.

same

> > as in crystallography, too.

>

> You should remember that many self-respecting mathematicians would

>not call something a lattice unless it inherited a group structure

>from R^n.

any examples of one that doesn't?

Gang,

I originally asked Paul

> Really?

...and I just wanted to thank you for taking it literally (and not facetiously!). I think of this in simlar terms, though not entirely identical, to the 'discussion' (and/or disagreement) over the descriptor "scale", and other similar topics that have somewhat blurred pedegrees from discipline to discipline.

Illuminated,

Jon

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

> > You should remember that many self-respecting mathematicians would

> >not call something a lattice unless it inherited a group structure

> >from R^n.

>

> any examples of one that doesn't?

The hexagonal tiling of chords in the 5-limit, for one.

--- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:

> --- In tuning-math@y..., "wallyesterpaulrus"

<wallyesterpaulrus@y...> wrote:

>

> > > You should remember that many self-respecting mathematicians

would

> > >not call something a lattice unless it inherited a group

structure

> > >from R^n.

> >

> > any examples of one that doesn't?

>

> The hexagonal tiling of chords in the 5-limit, for one.

well, it's still a lattice in the crystallographic sense, or even the

geometric sense: a regular array of points, that is, an array of

points in which every point has exactly the same relationships with

its neighbors as any other point does with *its* neighbors.