How many n-limit intervals must there be in a scale/chord for it to be called

n-limit ?

Greg Schiemer wrote:

>

> How many n-limit intervals must there be in a scale/chord for it to be called

> n-limit ?

ONE! it just goes to show you how little you know by naming and classifying.

Unless one is going to investigate a particular generating pattern as in a

diamond or an Eikosany I don't see any real reason to get hung up on limits.

-- Kraig Grady

North American Embassy of Anaphoria Island

www.anaphoria.com

Greg Schiemer wrote:

>>

>> How many n-limit intervals must there be in a scale/chord for it to

be called

>> n-limit ?

All of the intervals must be n-limit for the scale or chord to be called

n-limit. Note that by the very definition of the term "limit", the

7-limit includes the 5-limit within it, and the 5-limit includes the

3-limit within it, etc. This is true for both odd and prime limits,

although I normally stick to Partch's original usage and mean odd

limits. Partch usually would refer to secondary and tertiary ratios of a

given odd limit instead of speaking of prime limits. His otonalities and

utonalities are complete chords of a given odd limit.

Kraig Grady wrote,

>ONE!

In a sense, this is correct, although the language here is tricky so it

is very important to be careful. For a scale or chord to be called

n-limit, there must be at least one interval that is outside all limits

smaller than n, and I suspect that is what Kraig meant. But there can be

no interval in the scale or chord that is outside the n-limit, if we

want to call the scale or chord 'n-limit'.

>it just goes to show you how little you know by naming and classifying.

That's a rather negative statement coming seemingly from nowhere.

>Unless one is going to investigate a particular generating pattern as

in a

>diamond or an Eikosany I don't see any real reason to get hung up on

limits.

On the contrary, there are many places the limit concept is useful, even

under its bastardized (prime) definition.

"Paul H. Erlich" wrote:

> From: "Paul H. Erlich" <PErlich@Acadian-Asset.com>

>

> Greg Schiemer wrote:

>

> >>

> >> How many n-limit intervals must there be in a scale/chord for it to

> be called

> >> n-limit ?

>

> All of the intervals must be n-limit for the scale or chord to be called

> n-limit. Note that by the very definition of the term "limit", the

> 7-limit includes the 5-limit within it, and the 5-limit includes the

> 3-limit within it, etc. This is true for both odd and prime limits,

> although I normally stick to Partch's original usage and mean odd

> limits. Partch usually would refer to secondary and tertiary ratios of a

> given odd limit instead of speaking of prime limits. His otonalities and

> utonalities are complete chords of a given odd limit.

>

> Kraig Grady wrote,

>

> >ONE!

>

> In a sense, this is correct, although the language here is tricky so it

> is very important to be careful. For a scale or chord to be called

> n-limit, there must be at least one interval that is outside all limits

> smaller than n, and I suspect that is what Kraig meant. But there can be

> no interval in the scale or chord that is outside the n-limit, if we

> want to call the scale or chord 'n-limit'.

>

> >it just goes to show you how little you know by naming and classifying.

>

> That's a rather negative statement coming seemingly from nowhere.

>

> >Unless one is going to investigate a particular generating pattern as

> in a

> >diamond or an Eikosany I don't see any real reason to get hung up on

> limits.

>

> On the contrary, there are many places the limit concept is useful, even

I am sorry you took this as a "negative" statement. I expressing my

dissatisfaction with the term. As an Aesthetic decision I use them all the

time. Without the concept Diamonds and Eikosanies would not even exist as

useful object. But frankly as an idea I have seen it do more harm than

good. It tends to erect a wall toward the future more than opening things

up. On the other hand, others will feel more "adventuresome than others

merely on the basis of them using "Higher Harmonics. Here you have the race

to the top. There is no evidence that our ears stay with limit though in

actual music. That barbershop quartets use 7s but then maybe a 17 in

diminished chords points to how the term 7 limit can interfere with our

perception of these chords. What does calling this 17 limit really tell us.

That dominants can be shown to use even higher harmonic. Especially by

string players. No string player is going to play a 5/4 in a dominant chord

but will raise it.

-- Kraig Grady

North American Embassy of Anaphoria Island

www.anaphoria.com

I'm still trying to dig myself out from stuff that's piled up from the

holidays, and only glancing at tuning-list stuff, mostly squirreling it

away for when I get a chance to deal with it, who knows when. But I

can't help but respond to this:

On Wed, 20 Jan 1999, Paul H. Erlich wrote:

> Greg Schiemer wrote:

>>> How many n-limit intervals must there be in a scale/chord for it to

>>> be called n-limit ?

>

> All of the intervals must be n-limit for the scale or chord to be called

> n-limit.

[snip]

> . . . there can be

> no interval in the scale or chord that is outside the n-limit, if we

> want to call the scale or chord 'n-limit'.

Buh? For chords I agree, but for scales I strongly disagree. Consider

the standard JI version of the major scale:

1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1

Most would consider this a 5-limit scale. But it contains intervals up

to the odd limit of 45 (the 45/32 between the 4/3 and the 15/8). Surely

no one would call this a 45-limit scale.

--pH <manynote@lib-rary.wustl.edu> http://library.wustl.edu/~manynote

O

/\ "A three-cushion player doesn't need to be married.

-\-\-- o He already has enough aggravation."

NOTE: dehyphenate node to remove spamblock. <*>

On Thu, 21 Jan 1999, Paul Hahn wrote:

> Consider

> the standard JI version of the major scale:

>

> 1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1

>

> Most would consider this a 5-limit scale. But it contains intervals up

> to the odd limit of 45 (the 45/32 between the 4/3 and the 15/8). Surely

> no one would call this a 45-limit scale.

'Course, it's only 'cos I (and Paul E.) advocate the odd-limit

interpretation over the prime-limit that this is a problem.

--pH <manynote@lib-rary.wustl.edu> http://library.wustl.edu/~manynote

O

/\ "A three-cushion player doesn't need to be married.

-\-\-- o He already has enough aggravation."

NOTE: dehyphenate node to remove spamblock. <*>

> How many n-limit intervals must there be in a scale/chord for it to be called

> n-limit ?

My impression of the term is that there needs to only be one. It's probably

safe to say that in any scale or chord with more than two pitches, if you have

one N-limit interval you pretty much have to have more than one. If for

example, you have pitches X, Y, and Z in a chord, and the interval from X to Y

is N-limit, and the interval from X to Z is not, then the interval from Y to Z

is.

Actually, that's true as long as X to Z has a lower limit than N. If it has

a higher limit, then that is the chord's limit.

A question for the JI folks.

I'm still questing for the right ratio for an interval of a ninth. My reading is bringing more questions to mind.

Question:

Does JI address intervals greater than an octave in any other way than octave equivalence?

So far it looks like to me the JI method is to collaspe an otonal or utonal series into an octave.

Thanks,

Chris

Sent via BlackBerry from T-Mobile

> A question for the JI folks.

>

> I'm still questing for the right ratio for an interval of a ninth. My reading is bringing more questions to mind.

What do you mean exactly by the "right ratio" for a ninth...? Do you

mean a major ninth, and you're unsure whether it's 9/4 or 20/9 or

something like that?

-Mike

I mean a major 9th (sorry) . And yes a ratio like that I am looking for. I don't know which would be the equivalent to a perfect 5th or octave etc. I am looking to try a tuning idea based on that interval.

Thanks

Chris

Sent via BlackBerry from T-Mobile

Hi Chris,

I mean a major 9th (sorry) . And yes a ratio like that I am looking for. I

> don't know which would be the equivalent to a perfect 5th or octave etc. I

> am looking to try a tuning idea based on that interval.

>

The 9th equivalent of the 3/2 fifth would be 9/4.

Though to play 20/9 is far less dissonant than a 40/27 fifth offcourse, and

a 20/9 would be really common.

But the "perfect" real ninth that is not an inversion of anything is 9/4.

It can be divided consonantly and without inversions in the following ways:

1/1 5/4 3/2 9/4

1/1 6/5 3/2 9/4

1/1 3/2 9/5 9/4

1/1 3/2 15/8 9/4

Marcel

Thank you Marcel (and Mike),

If I come up with something useful I'll post it!

Chris

On Sat, Apr 24, 2010 at 7:00 PM, Marcel de Velde <m.develde@gmail.com>wrote:

>

>

> Hi Chris,

>

>

> I mean a major 9th (sorry) . And yes a ratio like that I am looking for. I

>> don't know which would be the equivalent to a perfect 5th or octave etc. I

>> am looking to try a tuning idea based on that interval.

>>

>

> The 9th equivalent of the 3/2 fifth would be 9/4.

> Though to play 20/9 is far less dissonant than a 40/27 fifth offcourse, and

> a 20/9 would be really common.

> But the "perfect" real ninth that is not an inversion of anything is 9/4.

> It can be divided consonantly and without inversions in the following ways:

> 1/1 5/4 3/2 9/4

> 1/1 6/5 3/2 9/4

> 1/1 3/2 9/5 9/4

> 1/1 3/2 15/8 9/4

>

> Marcel

>

>

Yes, Chris, that's basically correct. All the fractions are based on octave equivalence--a 4/5 can be treated as an 8/5, and so on. But if you are interested in attaining pitches in a specific register (i.e., specific frequencies) then you can multiply a base frequency by fractions without octave equivalence (i.e., 9:1 for the 9th harmonic, 2/5 for a M3 and octave below that, and so forth)

Franklin

Dr. Franklin Cox

1107 Xenia Ave.

Yellow Springs, OH 45387

(937) 767-1165

franklincox@...

--- On Sat, 4/24/10, Chris <chrisvaisvil@...> wrote:

From: Chris <chrisvaisvil@...>

Subject: [tuning] JI question

To: tuning@yahoogroups.com

Date: Saturday, April 24, 2010, 10:45 PM

A question for the JI folks.

I'm still questing for the right ratio for an interval of a ninth. My reading is bringing more questions to mind.

Question:

Does JI address intervals greater than an octave in any other way than octave equivalence?

So far it looks like to me the JI method is to collaspe an otonal or utonal series into an octave.

Thanks,

Chris

Sent via BlackBerry from T-Mobile

Thanks - this does make sense. I appreciate the explanation!

Chris

On Sat, Apr 24, 2010 at 7:37 PM, Cox Franklin <franklincox@...> wrote:

>

>

> Yes, Chris, that's basically correct. All the fractions are based on

> octave equivalence--a 4/5 can be treated as an 8/5, and so on. But if you

> are interested in attaining pitches in a specific register (i.e., specific

> frequencies) then you can multiply a base frequency by fractions without

> octave equivalence (i.e., 9:1 for the 9th harmonic, 2/5 for a M3 and octave

> below that, and so forth)

>

> Franklin

>

> Dr. Franklin Cox

> 1107 Xenia Ave.

> Yellow Springs, OH 45387

> (937) 767-1165

> franklincox@yahoo.com

>

> --- On *Sat, 4/24/10, Chris <chrisvaisvil@...>* wrote:

>

>

> From: Chris <chrisvaisvil@...>

> Subject: [tuning] JI question

> To: tuning@yahoogroups.com

> Date: Saturday, April 24, 2010, 10:45 PM

>

>

>

>

> A question for the JI folks.

>

> I'm still questing for the right ratio for an interval of a ninth. My

> reading is bringing more questions to mind.

>

> Question:

>

> Does JI address intervals greater than an octave in any other way than

> octave equivalence?

>

> So far it looks like to me the JI method is to collaspe an otonal or utonal

> series into an octave.

>

> Thanks,

>

> Chris

> Sent via BlackBerry from T-Mobile

>

>

>

> Yes, Chris, that's basically correct. All the fractions are based on

> octave equivalence--a 4/5 can be treated as an 8/5, and so on. But if you

> are interested in attaining pitches in a specific register (i.e., specific

> frequencies) then you can multiply a base frequency by fractions without

> octave equivalence (i.e., 9:1 for the 9th harmonic, 2/5 for a M3 and octave

> below that, and so forth)

>

I must disagree here.

The ninth in music is NOT the 9th harmonic 9/1.

Yes it so happens that the 9th note in the diatonic system gives a 9/4, but

the 11th for instance doesn't give a 11/1 or 11/4, it gives an 8/3 or 27/10.

And the source of the 9/4 is not the ninth harmonic 9/1. It is a normal

harmonic-6-limit interval that can be 9/4 or 9/2 without inversion.

To play 9/1 would actually be an octave transposition of such a 9/4 or 9/2.

Marcel

You're saying that we tend to hear ratios of 9 as being comprised of

multiple compounded ratios of 3, rather than as octave transpositions of 9/1

directly?

-Mike

On Sat, Apr 24, 2010 at 8:26 PM, Marcel de Velde <m.develde@gmail.com>wrote:

>

>

>

> Yes, Chris, that's basically correct. All the fractions are based on

>> octave equivalence--a 4/5 can be treated as an 8/5, and so on. But if you

>> are interested in attaining pitches in a specific register (i.e., specific

>> frequencies) then you can multiply a base frequency by fractions without

>> octave equivalence (i.e., 9:1 for the 9th harmonic, 2/5 for a M3 and octave

>> below that, and so forth)

>>

>

> I must disagree here.

> The ninth in music is NOT the 9th harmonic 9/1.

> Yes it so happens that the 9th note in the diatonic system gives a 9/4, but

> the 11th for instance doesn't give a 11/1 or 11/4, it gives an 8/3 or 27/10.

> And the source of the 9/4 is not the ninth harmonic 9/1. It is a normal

> harmonic-6-limit interval that can be 9/4 or 9/2 without inversion.

> To play 9/1 would actually be an octave transposition of such a 9/4 or 9/2.

>

> Marcel

>

>

> You're saying that we tend to hear ratios of 9 as being comprised of

> multiple compounded ratios of 3, rather than as octave transpositions of 9/1

> directly?

>

> -Mike

>

Yes, the ninth in music, when it is a 9/4, and when it is consonant, then it

is heard as either of the following:

1/1 5/4 3/2 9/4

1/1 6/5 3/2 9/4

1/1 3/2 15/8 9/4

1/1 3/2 9/5 9/4

I've left out the octave, which can be put inbetween any of the intervals or

on either side.

For instance:

1/1 5/4 5/2 3/1 9/2

1/1 3/2 15/8 9/4 9/2

etc

The ninth just coincidentally can also be a "9"/4.

But the ninth only means the ninth note in the diatonic system.

The first beeing 1/1, the major second (2nd) beeing 9/8 or 10/9, the minor

second (2nd) beeing 25/24, or 16/15 or 27/25.

The minor third (3rd) beeing either 6/5 or 32/27 or 75/64

etc, there's no relation other than luck between the "9" in 9/4 and 9th note

in the diatonic system.

And furthermore there's no relation between the ninth harmonic and the 9/4

ninth in the diatonic system in common practice music.

For instance a 5/3 is when consonant either a 1/1 4/3 5/3 or 1/1 5/4 5/3 and

can be happily above the harmonic root as such.

But also consonant as an octave inversion of the 6/5 minor third (though the

harmonic root would then not be 1/1 relative to the 5/3) etc.

The ninth just happens to have some coincidences with the number 9, but this

means nothing musically.

Marcel

>

> Yes, the ninth in music, when it is a 9/4, and when it is consonant, then

> it is heard as either of the following:

>

> 1/1 5/4 3/2 9/4

> 1/1 6/5 3/2 9/4

> 1/1 3/2 15/8 9/4

>

> 1/1 3/2 9/5 9/4

>

> I've left out the octave, which can be put inbetween any of the intervals

> or on either side.

> For instance:

> 1/1 5/4 5/2 3/1 9/2

> 1/1 3/2 15/8 9/4 9/2

> etc

>

>

> The ninth just coincidentally can also be a "9"/4.

> But the ninth only means the ninth note in the diatonic system.

> The first beeing 1/1, the major second (2nd) beeing 9/8 or 10/9, the minor

> second (2nd) beeing 25/24, or 16/15 or 27/25.

> The minor third (3rd) beeing either 6/5 or 32/27 or 75/64

> etc, there's no relation other than luck between the "9" in 9/4 and 9th

> note in the diatonic system.

> And furthermore there's no relation between the ninth harmonic and the 9/4

> ninth in the diatonic system in common practice music.

> For instance a 5/3 is when consonant either a 1/1 4/3 5/3 or 1/1 5/4 5/3

> and can be happily above the harmonic root as such.

> But also consonant as an octave inversion of the 6/5 minor third (though

> the harmonic root would then not be 1/1 relative to the 5/3) etc.

> The ninth just happens to have some coincidences with the number 9, but

> this means nothing musically.

>

> Marcel

Btw, I've left out that the 9th can also be "consonant" as 20/9.

Only it is an octave inversion of the consonant 9/5 then, and the true

harmonic root of the 20/9 ninth isn't 1/1 then.

Consonant structeres underlying the 20/9 ninth are:

1/1 6/5 9/5

1/1 3/2 9/5

1/1 3/2 3/1 18/5

1/1 3/2 9/5 18/5

1/1 6/5 9/5 18/5

1/1 6/5 12/5 18/5

Octave inversion of these 9/5 or 18/5 will give the 20/9 ninth.

Marcel

> Btw, I've left out that the 9th can also be "consonant" as 20/9.

> Only it is an octave inversion of the consonant 9/5 then, and the true

> harmonic root of the 20/9 ninth isn't 1/1 then.

> Consonant structeres underlying the 20/9 ninth are:

> 1/1 6/5 9/5

>

> 1/1 3/2 9/5

> 1/1 3/2 3/1 18/5

> 1/1 3/2 9/5 18/5

> 1/1 6/5 9/5 18/5

> 1/1 6/5 12/5 18/5

> Octave inversion of these 9/5 or 18/5 will give the 20/9 ninth.

>

> Marcel

>

Just one more mail about this :)

All the structures I gave underlying both the 9/4 ninth and 20/9 are

simplifications.

So there are more possiblities where the harmonic root isn't 1/1 as in the

examples I gave.

As there are more intervals to play with and these can come before the 1/1

too.

For instance 1/1 4/3 5/3 5/2 3/1 6/1

As you can see we find the 9/4 ninth here between 4/3 and 3/1.

And the 20/9 ninth potentially in an octave inversion between 5/3 and 3/1.

Just wished to mention it to be complete.

Marcel

Marcel,

Am I missing something, or did you imagine that I wrote somewhere that the 9th harmonic is the "ninth in music"? It's important not to confuse harmonics with musical functions. The ninth in music in general comes out of a suspension figure; in C major, the 5th of the V chord (D) suspended while the bass move on to C. The standard way of representing this is indeed 9/4. However, the 9/1 in a specific set of circumstances might very well be the "ninth in music": imagine a widely spaced chord (perhaps with some octave doublings) of C2 G3 E4 D5, with the D resolving down to a C. Here the D5 is both the 9th harmonic of C2 and functioning as a 9th of the chord.

I was just giving Chris the general principle distinguishing between "octave equivalence" representation of ratios and ratios of frequencies.

Franklin

Dr. Franklin Cox

1107 Xenia Ave.

Yellow Springs, OH 45387

(937) 767-1165

franklincox@...

--- On Sun, 4/25/10, Marcel de Velde <m.develde@...> wrote:

From: Marcel de Velde <m.develde@...>

Subject: Re: [tuning] JI question

To: tuning@yahoogroups.com

Date: Sunday, April 25, 2010, 12:26 AM

Yes, Chris, that's basically correct. All the fractions are based on octave equivalence- -a 4/5 can be treated as an 8/5, and so on. But if you are interested in attaining pitches in a specific register (i.e., specific frequencies) then you can multiply a base frequency by fractions without octave equivalence (i.e., 9:1 for the 9th harmonic, 2/5 for a M3 and octave below that, and so forth)

I must disagree here.

The ninth in music is NOT the 9th harmonic 9/1.

Yes it so happens that the 9th note in the diatonic system gives a 9/4, but the 11th for instance doesn't give a 11/1 or 11/4, it gives an 8/3 or 27/10.

And the source of the 9/4 is not the ninth harmonic 9/1. It is a normal harmonic-6-limit interval that can be 9/4 or 9/2 without inversion.

To play 9/1 would actually be an octave transposition of such a 9/4 or 9/2.

Marcel

And by the way, the "ninth in music" might be a minor ninth, which would not be a 9/x ratio.

Dr. Franklin Cox

1107 Xenia Ave.

Yellow Springs, OH 45387

(937) 767-1165

franklincox@...

--- On Sun, 4/25/10, Marcel de Velde <m.develde@...> wrote:

From: Marcel de Velde <m.develde@gmail.com>

Subject: Re: [tuning] JI question

To: tuning@yahoogroups.com

Date: Sunday, April 25, 2010, 12:26 AM

Yes, Chris, that's basically correct. All the fractions are based on octave equivalence- -a 4/5 can be treated as an 8/5, and so on. But if you are interested in attaining pitches in a specific register (i.e., specific frequencies) then you can multiply a base frequency by fractions without octave equivalence (i.e., 9:1 for the 9th harmonic, 2/5 for a M3 and octave below that, and so forth)

I must disagree here.

The ninth in music is NOT the 9th harmonic 9/1.

Yes it so happens that the 9th note in the diatonic system gives a 9/4, but the 11th for instance doesn't give a 11/1 or 11/4, it gives an 8/3 or 27/10.

And the source of the 9/4 is not the ninth harmonic 9/1. It is a normal harmonic-6-limit interval that can be 9/4 or 9/2 without inversion.

To play 9/1 would actually be an octave transposition of such a 9/4 or 9/2.

Marcel

Hi Franklin,

Am I missing something, or did you imagine that I wrote somewhere that the

> 9th harmonic is the "ninth in music"? It's important not to confuse

> harmonics with musical functions. The ninth in music in general comes out of

> a suspension figure; in C major, the 5th of the V chord (D) suspended while

> the bass move on to C. The standard way of representing this is indeed

> 9/4. However, the 9/1 in a specific set of circumstances might very well be

> the "ninth in music": imagine a widely spaced chord (perhaps with some

> octave doublings) of C2 G3 E4 D5, with the D resolving down to a C. Here

> the D5 is both the 9th harmonic of C2 and functioning as a 9th of the chord.

>

> I was just giving Chris the general principle distinguishing between

> "octave equivalence" representation of ratios and ratios of frequencies.

>

Aah ok.

Yes I was minunderstanding you.

I did not think you thought the ninth in music has anything to do with the

9/4 because of the "ninth" name.

But I did think you ment the 9/4 ratio is by defenition the 9th harmonic 9/1

lowered 2 octaves, and that the "meaning" of it in music is the ninth

harmonic.

But I thought so wrongly, sorry.

Agree with everythin you say above :)

Marcel

I meant a major 9th.

In having asked the question twice I left out the qualifier the 2nd

time by accident.

The major 9th is a consonant interval to my ears - by way of the logic

of stacked 5ths, both of which I use a lot in my compositions. Also,

by following this same line of reasoning, and more importantly to my

ears, major 2nds are consonant and sus 4ths are consonant to me as

well - I treat them all as inversions of one another.

What I did with the information was create a "double gamma" tuning.

This tuning divides a major 9th into 12 steps. Given that CGD is two

fiths on top of each other. The tuning is... interesting but certainly

not earth shattering.

Thanks,

Chris

On Sat, Apr 24, 2010 at 9:23 PM, Cox Franklin <franklincox@...> wrote:

>

>

>

> And by the way, the "ninth in music" might be a minor ninth, which would not be a 9/x ratio.

>

> Dr. Franklin Cox

> 1107 Xenia Ave.

>

I am, of course, obliged to mention that the ninth in music can be

used as a stable consonance all its own as well, without being in

particular need of resolution elsewhere.

-Mike

> Marcel,

> Am I missing something, or did you imagine that I wrote somewhere that the 9th harmonic is the "ninth in music"? It's important not to confuse harmonics with musical functions. The ninth in music in general comes out of a suspension figure; in C major, the 5th of the V chord (D) suspended while the bass move on to C. The standard way of representing this is indeed 9/4. However, the 9/1 in a specific set of circumstances might very well be the "ninth in music": imagine a widely spaced chord (perhaps with some octave doublings) of C2 G3 E4 D5, with the D resolving down to a C. Here the D5 is both the 9th harmonic of C2 and functioning as a 9th of the chord.

> I was just giving Chris the general principle distinguishing between "octave equivalence" representation of ratios and ratios of frequencies.

>

> Franklin

>

>

>

> Dr. Franklin Cox

> 1107 Xenia Ave.

> Yellow Springs, OH 45387

> (937) 767-1165

> franklincox@...

>

> --- On Sun, 4/25/10, Marcel de Velde <m.develde@gmail.com> wrote:

>

> From: Marcel de Velde <m.develde@...>

> Subject: Re: [tuning] JI question

> To: tuning@yahoogroups.com

> Date: Sunday, April 25, 2010, 12:26 AM

>

>

>>

>> Yes, Chris, that's basically correct. All the fractions are based on octave equivalence- -a 4/5 can be treated as an 8/5, and so on. But if you are interested in attaining pitches in a specific register (i.e., specific frequencies) then you can multiply a base frequency by fractions without octave equivalence (i.e., 9:1 for the 9th harmonic, 2/5 for a M3 and octave below that, and so forth)

>

> I must disagree here.

> The ninth in music is NOT the 9th harmonic 9/1.

> Yes it so happens that the 9th note in the diatonic system gives a 9/4, but the 11th for instance doesn't give a 11/1 or 11/4, it gives an 8/3 or 27/10.

> And the source of the 9/4 is not the ninth harmonic 9/1. It is a normal harmonic-6-limit interval that can be 9/4 or 9/2 without inversion.

> To play 9/1 would actually be an octave transposition of such a 9/4 or 9/2.

>

> Marcel

>

>

I said that the 9th "comes out of" a suspension figure. Later on it was more freely used in appogiaturas, then later in the 19th century it began to be used in an unprepared manner. By the early 20th century some composers were regularly employing 9th chords. It's important to understand the historical derivation, though. The sort of elegant voice leading found in Classical music is based on very strict rules of dissonant treatment.

Dr. Franklin Cox

1107 Xenia Ave.

Yellow Springs, OH 45387

(937) 767-1165

franklincox@...

--- On Sun, 4/25/10, Mike Battaglia <battaglia01@gmail.com> wrote:

From: Mike Battaglia <battaglia01@...>

Subject: Re: [tuning] JI question

To: tuning@yahoogroups.com

Date: Sunday, April 25, 2010, 2:00 AM

I am, of course, obliged to mention that the ninth in music can be

used as a stable consonance all its own as well, without being in

particular need of resolution elsewhere.

-Mike

> Marcel,

> Am I missing something, or did you imagine that I wrote somewhere that the 9th harmonic is the "ninth in music"? It's important not to confuse harmonics with musical functions. The ninth in music in general comes out of a suspension figure; in C major, the 5th of the V chord (D) suspended while the bass move on to C. The standard way of representing this is indeed 9/4. However, the 9/1 in a specific set of circumstances might very well be the "ninth in music": imagine a widely spaced chord (perhaps with some octave doublings) of C2 G3 E4 D5, with the D resolving down to a C. Here the D5 is both the 9th harmonic of C2 and functioning as a 9th of the chord.

> I was just giving Chris the general principle distinguishing between "octave equivalence" representation of ratios and ratios of frequencies.

>

> Franklin

>

>

>

> Dr. Franklin Cox

> 1107 Xenia Ave.

> Yellow Springs, OH 45387

> (937) 767-1165

> franklincox@...

>

> --- On Sun, 4/25/10, Marcel de Velde <m.develde@...> wrote:

>

> From: Marcel de Velde <m.develde@...>

> Subject: Re: [tuning] JI question

> To: tuning@yahoogroups.com

> Date: Sunday, April 25, 2010, 12:26 AM

>

>

>>

>> Yes, Chris, that's basically correct. All the fractions are based on octave equivalence- -a 4/5 can be treated as an 8/5, and so on. But if you are interested in attaining pitches in a specific register (i.e., specific frequencies) then you can multiply a base frequency by fractions without octave equivalence (i.e., 9:1 for the 9th harmonic, 2/5 for a M3 and octave below that, and so forth)

>

> I must disagree here.

> The ninth in music is NOT the 9th harmonic 9/1.

> Yes it so happens that the 9th note in the diatonic system gives a 9/4, but the 11th for instance doesn't give a 11/1 or 11/4, it gives an 8/3 or 27/10.

> And the source of the 9/4 is not the ninth harmonic 9/1. It is a normal harmonic-6-limit interval that can be 9/4 or 9/2 without inversion.

> To play 9/1 would actually be an octave transposition of such a 9/4 or 9/2.

>

> Marcel

>

>

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Ok,

If anyone is interested the "double gamma" scala file and a short

pianoteq rendered improvisation in the tuning can be found here:

http://clones.soonlabel.com/public/micro/double-gamma/

I used my Korg MS2k synth so the range is a bit limited compared to 88 keys.

- And Dr Cox, I suppose then the 11 th came from a suspension of the

7th degree of V to I as you described for the 9th?

I don't think my theory class spent a lot on that aspect - no doubt it

was mentioned and I forgot. A great deal of time was spent on various

treatments of the 7th - which I interpreted as being treatments of the

tritone really. (and throw in here N6, Gr6, It6 if I remember right)

One of the main impressions I took away from the class was that

various treatments of the tritone was a driving force for innovation

and one of the engines for the music itself.

Thanks,

Chris

On Sat, Apr 24, 2010 at 10:12 PM, Cox Franklin <franklincox@...> wrote:

>

>

>

> I said that the 9th "comes out of" a suspension figure. Later on it was more freely used in appogiaturas, then later in the 19th century it began to be used in an unprepared manner. By the early 20th century some composers were regularly employing 9th chords. It's important to understand the historical derivation, though. The sort of elegant voice leading found in Classical music is based on very strict rules of dissonant treatment.

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

>

> Question:

>

> Does JI address intervals greater than an octave in any other way

> than octave equivalence?

>

> So far it looks like to me the JI method is to collaspe an otonal or

> utonal series into an octave.

Chris, I strongly recommend the JI Primer:

http://www.dbdoty.com/Words/Primer1.html#Order

To try to answer your question, octave equivalence is usually but not

necessarily assumed. Any ratio bigger than 2 is of course larger than

an octave. If I write "9:4" I mean the interval which is close to an

equal-tempered maj. 9th, and not any other interval. Similarly if I

write "9:8" we are dealing with something in the neighborhood of a

maj. 2nd.

-Carl

Yes, the 13th also was derived from a suspension figure. Even the 7th of a chord wasn't "legally" part of the chord until around Rameau's time, and even then composers treated the 7th very carefully (i.e., always resolving downward; one of the points of the German augmented 6th chord was that it is enharmonically identical to a dominant 7th, but the apparent 7th of the chord [i.e., the augmented 6th] resolved upwards rather than downwards, allowing composers to flirt with breaking the ironclad rule) for the next century or more.

Yes, treatment of the tritone and other dissonances was indeed one of the driving forces for innovation.

One of the things that troubles me about a lot of standard theory books is that they don't ground harmonic analysis in historical developments and in counterpoint. A "dominant 7th" chord was a nonentity in 1650, a new concept in 1750, and a well-worn concept in 1850.

The basics of the practical contrapuntal rules (i.e., what most composers learned by heart early on and used constantly in their compositions) lie in figured bass. I encourage my students to learn basic progressions and dissonance treatment via figured bass, because this sticks better than analysis on paper. I believe that both are necessary, but the figured bass portion usually gets neglected.

Franklin

Dr. Franklin Cox

1107 Xenia Ave.

Yellow Springs, OH 45387

(937) 767-1165

franklincox@...

--- On Sun, 4/25/10, Chris Vaisvil <chrisvaisvil@...> wrote:

From: Chris Vaisvil <chrisvaisvil@gmail.com>

Subject: Re: [tuning] JI question

To: tuning@yahoogroups.com

Date: Sunday, April 25, 2010, 2:25 AM

Ok,

If anyone is interested the "double gamma" scala file and a short

pianoteq rendered improvisation in the tuning can be found here:

http://clones. soonlabel. com/public/ micro/double- gamma/

I used my Korg MS2k synth so the range is a bit limited compared to 88 keys.

- And Dr Cox, I suppose then the 11 th came from a suspension of the

7th degree of V to I as you described for the 9th?

I don't think my theory class spent a lot on that aspect - no doubt it

was mentioned and I forgot. A great deal of time was spent on various

treatments of the 7th - which I interpreted as being treatments of the

tritone really. (and throw in here N6, Gr6, It6 if I remember right)

One of the main impressions I took away from the class was that

various treatments of the tritone was a driving force for innovation

and one of the engines for the music itself.

Thanks,

Chris

On Sat, Apr 24, 2010 at 10:12 PM, Cox Franklin <franklincox@ yahoo.com> wrote:

>

>

>

> I said that the 9th "comes out of" a suspension figure. Later on it was more freely used in appogiaturas, then later in the 19th century it began to be used in an unprepared manner. By the early 20th century some composers were regularly employing 9th chords. It's important to understand the historical derivation, though. The sort of elegant voice leading found in Classical music is based on very strict rules of dissonant treatment.

--- In tuning@yahoogroups.com, Cox Franklin <franklincox@...> wrote:

> One of the things that troubles me about a lot of standard theory books is that they don't ground harmonic analysis in historical developments and in counterpoint. Â A "dominant 7th" chord was a nonentity in 1650, a new concept in 1750, and a well-worn concept in 1850.

That doesn't seem to give much credit to Monteverdi.

Of course chords that we would consider dominant 7ths occurred before Rameau, but they tended to be treated as extensions of the triad-based system, with the extra tones (the "5" in the 6/5 chord, etc.) requiring preparation and downward resolution. A big part of figured bass training involved memorizing these rules with one's hands.

Monteverdi used unusual dissonances for his time, but they were clearly perceived as dissonances, requiring careful treatment and resolution. Actually, it was the introduction of the dissonance that was the main source of dispute, because Monteverdi used unprepared dissonances for expressive purposes. The resolutions were, proper, though. He considered this an extension of the "first practice," and was perfectly capable of going back to it to write beautiful masses in perfect sacred counterpoint.

Franklin

Dr. Franklin Cox

1107 Xenia Ave.

Yellow Springs, OH 45387

(937) 767-1165

--- On Sun, 4/25/10, genewardsmith <genewardsmith@...> wrote:

From: genewardsmith <genewardsmith@...>

Subject: [tuning] Re: JI question

To: tuning@yahoogroups.com

Date: Sunday, April 25, 2010, 3:14 AM

--- In tuning@yahoogroups. com, Cox Franklin <franklincox@ ...> wrote:

> One of the things that troubles me about a lot of standard theory books is that they don't ground harmonic analysis in historical developments and in counterpoint. Â A "dominant 7th" chord was a nonentity in 1650, a new concept in 1750, and a well-worn concept in 1850.

That doesn't seem to give much credit to Monteverdi.

Chris,

Well the major second is either 9:8 or 10:9 in JI.

Thus a "common theory correct" 9th can be either 20/9 (minor tone) or 9/4 (major tone) since

2/1 * 10/9 = 20/9

2/1 * 9/8 = 9/4

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

>

> I mean a major 9th (sorry) . And yes a ratio like that I am looking for. I don't know which would be the equivalent to a perfect 5th or octave etc. I am looking to try a tuning idea based on that interval.

>

I like the 9/4 since it is two just fifths. This is the 9th you would get by stacking alternating major (5/4) and minor (6/5) thirds. It is also the 9th of the harmonic 4:5:6:7:9 chord. There might be schools of thought that this particular tuning could be called a "perfect 9th" instead of a major one since it is two perfect 5ths stacked.

Billy

Seconda prattica is exciting and inspiring even after 400 years.

Daniel Forro

On 25 Apr 2010, at 1:03 PM, Cox Franklin wrote:

>

> Of course chords that we would consider dominant 7ths occurred > before Rameau, but

> they tended to be treated as extensions of the triad-based system, > with the extra tones (the "5" in the 6/5 chord, etc.) requiring > preparation and downward resolution. A big part of figured bass > training involved memorizing these rules with one's hands.

>

> Monteverdi used unusual dissonances for his time, but they were > clearly perceived as dissonances, requiring careful treatment and > resolution. Actually, it was the introduction of the dissonance > that was the main source of dispute, because Monteverdi used > unprepared dissonances for expressive purposes. The resolutions > were, proper, though. He considered this an extension of the > "first practice," and was perfectly capable of going back to it to > write beautiful masses in perfect sacred counterpoint.

>

> Franklin

>

>

> Dr. Franklin Cox

> 1107 Xenia Ave.

> Yellow Springs, OH 45387

> (937) 767-1165

> franklincox@...

>

We can find such chords in English music where they started to use D-T cadenza as ending pattern. But of course we can't call it that way as it was modal harmony, not yet functional harmony.

Daniel Forro

On 25 Apr 2010, at 1:03 PM, Cox Franklin wrote:

>

> Of course chords that we would consider dominant 7ths occurred > before Rameau, but

> they tended to be treated as extensions of the triad-based system, > with the extra tones (the "5" in the 6/5 chord, etc.) requiring > preparation and downward resolution. A big part of figured bass > training involved memorizing these rules with one's hands.

>

> Monteverdi used unusual dissonances for his time, but they were > clearly perceived as dissonances, requiring careful treatment and > resolution. Actually, it was the introduction of the dissonance > that was the main source of dispute, because Monteverdi used > unprepared dissonances for expressive purposes. The resolutions > were, proper, though. He considered this an extension of the > "first practice," and was perfectly capable of going back to it to > write beautiful masses in perfect sacred counterpoint.

>

> Franklin

>

>

> Dr. Franklin Cox

> 1107 Xenia Ave.

> Yellow Springs, OH 45387

> (937) 767-1165

> franklincox@...

>

Thank you Carl for the recommendation, I just ordered the book.

Chris

On Sat, Apr 24, 2010 at 10:33 PM, Carl Lumma <carl@...> wrote:

>

>

>

>

> --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, "Chris"

> <chrisvaisvil@...> wrote:

> >

> > Question:

> >

> > Does JI address intervals greater than an octave in any other way

> > than octave equivalence?

> >

> > So far it looks like to me the JI method is to collaspe an otonal or

> > utonal series into an octave.

>

> Chris, I strongly recommend the JI Primer:

>

> http://www.dbdoty.com/Words/Primer1.html#Order

>

> To try to answer your question, octave equivalence is usually but not

> necessarily assumed. Any ratio bigger than 2 is of course larger than

> an octave. If I write "9:4" I mean the interval which is close to an

> equal-tempered maj. 9th, and not any other interval. Similarly if I

> write "9:8" we are dealing with something in the neighborhood of a

> maj. 2nd.

>

> -Carl

>

>

>

>