Why does a perfect minor triad sound worse than a major one even with pure sine waves?

perfect major triad = 1 1.25 1.5
perfect minor triad = 1 1.2 1.5

Two ideas:
1)  The major triad represent the harmonic series and u-tonal relationships IE the gap between 1 and 1.5 is exactly twice the size of the gap between 1 and 1.25.  This seems directly rooted in JI and the fact the first overtone is 2 times the root tone.
2)  Related but a bit different: the pattern in the major triad that 1.25 is right in the middle and predictably splits the two sections thus obtaining a sense of symmetry between the sections.

Consider the "PHI" based triad:
PHI-based triad: 1.23607 1.38907 1.618034
    ...it certainly has no ties with the idea 1) above, but you can see that 1.23607 + 1.38907 = 1.625 (very close to the third note).  This implies that idea #2 above is coming into play.

    I think it may well be...that sort of symmetry is every bit as important in music as the sort of critical-band-roughness elimination that goes on in matching harmonic overtones in JI through the use of low-numbered fractions.  Otherwise...you'd think the above PHI chord would sound much worse than it does in reality (in the same sort of way you'd think a major and minor chord would sound alike so far as dissonance since they just have the same two intervals in reverse order...until you actually tried them).

-Michael

Hi Mike,
You're going to have a hard time convincing me that minor triads sound
worse than major triads.

-Mike

On Thu, May 14, 2009 at 5:25 PM, <djtrancendance@...> wrote:
>
>
>
> perfect major triad = 1 1.25 1.5
> perfect minor triad = 1 1.2 1.5
>
> Two ideas:
> 1)  The major triad represent the harmonic series and u-tonal relationships
> IE the gap between 1 and 1.5 is exactly twice the size of the gap between 1
> and 1.25.  This seems directly rooted in JI and the fact the first overtone
> is 2 times the root tone.
> 2)  Related but a bit different: the pattern in the major triad that 1.25 is
> right in the middle and predictably splits the two sections thus obtaining a
> sense of symmetry between the sections.
>
> Consider the "PHI" based triad:
> PHI-based triad: 1.23607 1.38907 1.618034
>     ...it certainly has no ties with the idea 1) above, but you can see that
> 1.23607 + 1.38907 = 1.625 (very close to the third note).  This implies that
> idea #2 above is coming into play.
>
>     I think it may well be...that sort of symmetry is every bit as important
> in music as the sort of critical-band-roughness elimination that goes on in
> matching harmonic overtones in JI through the use of low-numbered
> fractions.  Otherwise...you'd think the above PHI chord would sound much
> worse than it does in reality (in the same sort of way you'd think a major
> and minor chord would sound alike so far as dissonance since they just have
> the same two intervals in reverse order...until you actually tried them).
>
> -Michael
>
>

I thought that too until I remember that the major chord is considered a
more stable resolution - thus the picardy third.

Though... one might argue - its just another collection of intervals

Chris

On Thu, May 14, 2009 at 9:42 PM, Mike Battaglia <battaglia01@...>wrote:

>
>
> Hi Mike,
> You're going to have a hard time convincing me that minor triads sound
> worse than major triads.
>
> -Mike
>
>
> On Thu, May 14, 2009 at 5:25 PM, <djtrancendance@...<djtrancendance%40yahoo.com>>
> wrote:
> >
> >
> >
> > perfect major triad = 1 1.25 1.5
> > perfect minor triad = 1 1.2 1.5
> >
> > Two ideas:
> > 1) The major triad represent the harmonic series and u-tonal
> relationships
> > IE the gap between 1 and 1.5 is exactly twice the size of the gap between
> 1
> > and 1.25. This seems directly rooted in JI and the fact the first
> overtone
> > is 2 times the root tone.
> > 2) Related but a bit different: the pattern in the major triad that 1.25
> is
> > right in the middle and predictably splits the two sections thus
> obtaining a
> > sense of symmetry between the sections.
> >
> > Consider the "PHI" based triad:
> > PHI-based triad: 1.23607 1.38907 1.618034
> > ...it certainly has no ties with the idea 1) above, but you can see
> that
> > 1.23607 + 1.38907 = 1.625 (very close to the third note). This implies
> that
> > idea #2 above is coming into play.
> >
> > I think it may well be...that sort of symmetry is every bit as
> important
> > in music as the sort of critical-band-roughness elimination that goes on
> in
> > matching harmonic overtones in JI through the use of low-numbered
> > fractions. Otherwise...you'd think the above PHI chord would sound much
> > worse than it does in reality (in the same sort of way you'd think a
> major
> > and minor chord would sound alike so far as dissonance since they just
> have
> > the same two intervals in reverse order...until you actually tried them).
> >
> > -Michael
> >
> >
>
>

--- In tuning@yahoogroups.com, djtrancendance@... wrote:
>
> perfect major triad = 1 1.25 1.5
> perfect minor triad = 1 1.2 1.5
>
> Two ideas:
> 1)  The major triad represent the harmonic series and u-tonal
> relationships IE the gap between 1 and 1.5 is exactly twice the
> size of the gap between 1 and 1.25.

Huh? How does the major triad illustrate utonal relationships?

>This seems directly rooted in JI

Whatever is going on with the 4:5:6 triad, it's not unlikely
to have to do with JI. (rolls eyes)

> Consider the "PHI" based triad:
> PHI-based triad: 1.23607 1.38907 1.618034
>     ...it certainly has no ties with the idea 1) above, but you
> can see that 1.23607 + 1.38907 = 1.625 (very close to the third
> note).  This implies that idea #2 above is coming into play.

? It seemed like your idea #2 was referring to the fact
that in the 4:5:6 triad, the major and minor 3rds each cover
the same distance in hz. This phi chord doesn't have this
property.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, djtrancendance@ wrote:
> >
> > perfect major triad = 1 1.25 1.5
> > perfect minor triad = 1 1.2 1.5
> >
> > Two ideas:
> > 1)  The major triad represent the harmonic series and u-tonal
> > relationships IE the gap between 1 and 1.5 is exactly twice the
> > size of the gap between 1 and 1.25.
>
> Huh? How does the major triad illustrate utonal relationships?
>
> >This seems directly rooted in JI
>
> Whatever is going on with the 4:5:6 triad, it's not unlikely
> to have to do with JI. (rolls eyes)
>
> > Consider the "PHI" based triad:
> > PHI-based triad: 1.23607 1.38907 1.618034
> >     ...it certainly has no ties with the idea 1) above, but you
> > can see that 1.23607 + 1.38907 = 1.625 (very close to the third
> > note).  This implies that idea #2 above is coming into play.
>
> ? It seemed like your idea #2 was referring to the fact
> that in the 4:5:6 triad, the major and minor 3rds each cover
> the same distance in hz. This phi chord doesn't have this
> property.
>
> -Carl
>

I don't know where the 13/8 is coming from here- in the phi chord, the frequencies given add up (combination tone) to phi*phi (with modulo2, also the logarithmic difference between phi and its ocatve inversion of course). If he means just adding the frequency ratios, it's not 1.625, but 1.62514....

Me>"This seems directly rooted in JI"

Carl>"Whatever is going on with the 4:5:6 triad, it's not unlikely

to have to do with JI. (rolls eyes)"
  Don't be a bastard.  I was simply saying "it's blatantly obvious this is consistent with very pure low limit JI" with a different tone of voice in an attempt to avoid sounding pompous or like I have all the world's answers.  Guess I should be less polite in that sense, eh?

>"It seemed like your idea #2 was referring to the fact
that in the 4:5:6 triad, the major and minor 3rds each cover
the same distance in hz. This phi chord doesn't have this
property."

It's not, read my statement again:
> "but you can see that root + 0.23607 + 0.38907 = 1.625 (very close to the third note)."
   So the point is that even though the note gaps in hz are not the same as they are in the major chord, they are still related by a the very simple mathematical relationship mentioned above.   

Hmm...I also noticed that (1.625-1.38907)/(1.38907-1.23607) almost exactly equals 3/2. Meanwhile (1.5-1.2)/(1.2-1) also equals 3/2.

So, Carl, it seems you're right in some ways that the PHI chord has more in common with the minor chord than the major one (and the major has the (1.5-1.25)/(1.25-1) = 1/1 ratio of hz gaps: the lowest limit ratio possible AKA unison).

   Maybe this 3/2 proportionality also explains why the linear golden-section split PHI scale seems so much more consonant than the others: it has low-numbered integer relationships >not< between the actual notes but between to HZ gaps between the notes.

-Michael

--- In tuning@yahoogroups.com, djtrancendance@... wrote:
>
> Me>"This seems directly rooted in JI"
>
>"Whatever is going on with the 4:5:6 triad, it's not unlikely
> to have to do with JI. (rolls eyes)"
>
>   Don't be a bastard.

Congratulations! You're on moderated status!

> Hmm...I also noticed that (1.625-1.38907)/(1.38907-1.23607)
> almost exactly equals 3/2. Meanwhile (1.5-1.2)/(1.2-1) also
> equals 3/2.

I can't make hide nor tail of these expressions.

-Carl

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> Hi Mike,
> You're going to have a hard time convincing me that minor triads sound
> worse than major triads.
>
> -Mike
>
> Have to agree with Mickey B on this one. What about minor triad as 16:19:24? Never did much like the 6th harmonic in the denominator of 7/6 for such a fundamental tonal interval.

-Rick
>
> On Thu, May 14, 2009 at 5:25 PM, <djtrancendance@...> wrote:
> >
> >
> >
> > perfect major triad = 1 1.25 1.5
> > perfect minor triad = 1 1.2 1.5
> >
> > Two ideas:
> > 1) The major triad represent the harmonic series and u-tonal relationships
> > IE the gap between 1 and 1.5 is exactly twice the size of the gap between 1
> > and 1.25. This seems directly rooted in JI and the fact the first overtone
> > is 2 times the root tone.
> > 2) Related but a bit different: the pattern in the major triad that 1.25 is
> > right in the middle and predictably splits the two sections thus obtaining a
> > sense of symmetry between the sections.
> >
> > Consider the "PHI" based triad:
> > PHI-based triad: 1.23607 1.38907 1.618034
> > ...it certainly has no ties with the idea 1) above, but you can see that
> > 1.23607 + 1.38907 = 1.625 (very close to the third note). This implies that
> > idea #2 above is coming into play.
> >
> > I think it may well be...that sort of symmetry is every bit as important
> > in music as the sort of critical-band-roughness elimination that goes on in
> > matching harmonic overtones in JI through the use of low-numbered
> > fractions. Otherwise...you'd think the above PHI chord would sound much
> > worse than it does in reality (in the same sort of way you'd think a major
> > and minor chord would sound alike so far as dissonance since they just have
> > the same two intervals in reverse order...until you actually tried them).
> >
> > -Michael
> >
> >
>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> Hi Mike,
> You're going to have a hard time convincing me that minor
> triads sound worse than major triads.
>
> -Mike

I dunno about sound worse, but they're certainly less consonant.
There's a very high level of agreement on that. -Carl

It must be a "deja vu", because I feel like this topic has been treated "ad
nauseam" in this and other forums, with all kind of acoustical arguments.
Let us remember that humanity for centuries, based on Pythagorean phylosophy
in the Middle Ages all the way through Helmholtz's Victorian times, has
considered the major triad MORE consonant.
Some modern musicians are free to disagree.
But we have always to remember that, except for very recent compositions,
the corpus of Western music is all strongly based on this fact.

Claudio

Hi there dear moderator,

Sorry for this.
Did everybody got the rubbish below?

My Outlook shows only the few lines I wrote, so the problem does not look to
originate in my PC.
There must be a malfunction somewhere in our yahoo group I guess.

Kind regards

Claudio

_____

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf Of
Claudio Di Veroli
Sent: 15 May 2009 20:40
To: tuning@yahoogroups.com
Subject: RE: [tuning] Re: Why does a perfect minor triad sound worse than a
major one even with pure s

It must be a "deja vu", because I feel like this topic has been treated "ad
nauseam" in this and other forums, with all kind of acoustical arguments.
Let us remember that humanity for centuries, based on Pythagorean phylosophy
in the Middle Ages all the way through Helmholtz's Victorian times, has
considered the major triad MORE consonant.
Some modern musicians are free to disagree.
But we have always to remember that, except for very recent compositions,
the corpus of Westernl-ff-m id=8897752-m24769 b.mx.mail.yahoo.com '250
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--- In tuning@yahoogroups.com, "Claudio Di Veroli" <dvc@...> wrote:
>
> Hi there dear moderator,
>
> Sorry for this.
> Did everybody got the rubbish below?
>
> My Outlook shows only the few lines I wrote, so the problem does
> not look to originate in my PC.
> There must be a malfunction somewhere in our yahoo group I guess.
>
> Kind regards
>
> Claudio

It's unclear where it originated, but there is a lot of gunk
in the source of your e-mail, which was apparently sent as HTML.

Depending on a user's Yahoo settings, this gunk may or may not
appear (it doesn't for me, probably because I set Yahoo to show
me plain text only).

Sending HTML e-mail to the list is not recommended, but usually
works OK. But my advice is to set Outlook to send plain text
only.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> >
> > Hi Mike,
> > You're going to have a hard time convincing me that minor
> > triads sound worse than major triads.
> >
> > -Mike
>
> I dunno about sound worse, but they're certainly less consonant.
> There's a very high level of agreement on that. -Carl
>

I agree that people must be very high to be agreeing on that. :-P

5/4 and 6/5 are about the same as far as consonance for me. And I think that I find 5/3 more consonant and "final" than either of them.

Assuming you mean 5-Limit triads...

This has obviously been huge fodder for tuning theorists
grappling over centuries, which makes for interesting reading.

A common answer is 'minor is upside down'.

My answer: it's a version of the upside down answer.
Look at the difference tones and the fundamental from the
bottom up.

Major 4:5:6 has a clear fundamental = DTs at 1 so it
sounds perfect.

Minor 10:12:15 is harmonically the upper 3 pitches of a
Major 7th chord, so it sounds like it's hanging in space
without a root (8 is missing).

So, getting rid of the 5-Limit...

Minor 16:19:24 keeps the fundamental and has DTs at 3
and 2, but 19 being higher in the series is not as simple
and correct sounding. Still it sounds better to some ears
than 10:12:15 because it sits on its fundamental rather
than hanging in space or implying the 'wrong' root.

Since 19 is 3 cents close to 12ET m3, this explains why
minor triads sound relatively good on a piano, while
perfect sounding 5-limit major triads do not exist there.

Cheers,
Aaron
=====

--- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...> wrote:
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
> >
> > --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> > >
> > > Hi Mike,
> > > You're going to have a hard time convincing me that minor
> > > triads sound worse than major triads.
> > >
> > > -Mike
> >
> > I dunno about sound worse, but they're certainly less consonant.
> > There's a very high level of agreement on that. -Carl
> >
>
> I agree that people must be very high to be agreeing on that. :-P
>
> 5/4 and 6/5 are about the same as far as consonance for me. And I think that I find 5/3 more consonant and "final" than either of them.
>

--- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...> wrote:

> 5/4 and 6/5 are about the same as far as consonance for me. And
> I think that I find 5/3 more consonant and "final" than either
> of them.

6:5 is slightly less consonant than 5:4 for me, but 5:3 clearly
more consonant than either.

But this was about triads. And the behavior of triads is not
fully captured by the behavior of their constituent intervals.

-Carl

I'm not so sure myself about the 16:19:24 , 1/1 19/16 3/2 minor triad.
I've tried to use it but in the end it didn't make sense to me.
It's also very close to 54:64:81 , 1/1 32/27 3/2 the pythagorean minor triad
which makes much more sense.
I especially like the 32/27 in there like in the V7 of 1/1 5/4 3/2 16/9.
But it's also in 27:32:40 , 1/1 32/27 40/27.

The only minor triads I'm sure of are used in common practice music a lot
are these 2:
10:12:15 , 1/1 6/5 3/2
and 27:32:40 , 1/1 32/27 40/27 (this is normally the ii chord like D F A
where I - C E G and V - G B D are 4:5:6)

For these 2 minor triads it's easy to see why they are less consonant than
4:5:6 major chord.
With normal sounds the overtones simply do not overlapp as nicely with the
minor chords as with the 4:5:6 major chord.

Marcel

Aaron>"A common answer is 'minor is upside down'."
   I agree with you, that alone is oversimplifying it...

>"Minor 10:12:15 is harmonically the upper 3 pitches of a

Major 7th chord, so it sounds like it's hanging in space

without a root (8 is missing)."
   Ah, so the mind sees it as an incomplete major chord missing the most important tone (the root)...this makes a whole lot more sense than most of the explanations I have heard and ties back to the fact the intervals are in the same order that they are in the harmonic series in the major but not the minor chord.

   I guess I'm going to have to give this one up because, in this case, you've convinced me that preference for the major triad is a mere side-effect of the fact the major triad includes the root tone it "points" to and is thus more "securely tonal" vs. minor which points to a missing tone.

-Michael

--- On Fri, 5/15/09, Aaron Andrew Hunt <aaronhunt@...> wrote:

From: Aaron Andrew Hunt <aaronhunt@...>
Subject: [tuning] Re: Why does a perfect minor triad sound worse than a major one even with pure s
To: tuning@yahoogroups.com
Date: Friday, May 15, 2009, 4:47 PM

Assuming you mean 5-Limit triads...

This has obviously been huge fodder for tuning theorists

grappling over centuries, which makes for interesting reading.

A common answer is 'minor is upside down'.

My answer: it's a version of the upside down answer.

Look at the difference tones and the fundamental from the

bottom up.

Major 4:5:6 has a clear fundamental = DTs at 1 so it

sounds perfect.

Minor 10:12:15 is harmonically the upper 3 pitches of a

Major 7th chord, so it sounds like it's hanging in space

without a root (8 is missing).

So, getting rid of the 5-Limit...

Minor 16:19:24 keeps the fundamental and has DTs at 3

and 2, but 19 being higher in the series is not as simple

and correct sounding. Still it sounds better to some ears

than 10:12:15 because it sits on its fundamental rather

than hanging in space or implying the 'wrong' root.

Since 19 is 3 cents close to 12ET m3, this explains why

minor triads sound relatively good on a piano, while

perfect sounding 5-limit major triads do not exist there.

Cheers,

Aaron

=====

--- In tuning@yahoogroups. com, "Cameron Bobro" <misterbobro@ ...> wrote:

>

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

> >

> > --- In tuning@yahoogroups. com, Mike Battaglia <battaglia01@ > wrote:

> > >

> > > Hi Mike,

> > > You're going to have a hard time convincing me that minor

> > > triads sound worse than major triads.

> > >

> > > -Mike

> >

> > I dunno about sound worse, but they're certainly less consonant.

> > There's a very high level of agreement on that. -Carl

> >

>

> I agree that people must be very high to be agreeing on that. :-P

>

> 5/4 and 6/5 are about the same as far as consonance for me. And I think that I find 5/3 more consonant and "final" than either of them.

>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@> wrote:
>
> > 5/4 and 6/5 are about the same as far as consonance for me. And
> > I think that I find 5/3 more consonant and "final" than either
> > of them.
>
> 6:5 is slightly less consonant than 5:4 for me, but 5:3 clearly
> more consonant than either.

Somehow I overlooked the "triad" bit. Well with triads you have to specify the so-called "inversion" (as dubious as that concept is).

>
> And the behavior of triads is not
> fully captured by the behavior of their constituent intervals.

That is certainly the case, maybe even a big understatement.

--- In tuning@yahoogroups.com, Marcel de Velde <m.develde@...> wrote:
>
> I'm not so sure myself about the 16:19:24 , 1/1 19/16 3/2 minor triad.
> I've tried to use it but in the end it didn't make sense to me.
> It's also very close to 54:64:81 , 1/1 32/27 3/2 the pythagorean minor triad
> which makes much more sense.
> I especially like the 32/27 in there like in the V7 of 1/1 5/4 3/2 16/9.
> But it's also in 27:32:40 , 1/1 32/27 40/27.
>
> The only minor triads I'm sure of are used in common practice music a lot
> are these 2:
> 10:12:15 , 1/1 6/5 3/2
> and 27:32:40 , 1/1 32/27 40/27 (this is normally the ii chord like D F A
> where I - C E G and V - G B D are 4:5:6)
>
> For these 2 minor triads it's easy to see why they are less consonant than
> 4:5:6 major chord.
> With normal sounds the overtones simply do not overlapp as nicely with the
> minor chords as with the 4:5:6 major chord.
>
> Marcel
>
Hi Marcel,

As someone said earlier, the first reason is that the denominator 5 in 6/5 and 6 in 7/6 don't give the correct tonic. Like 3/2 and 5/4 we need 2^(N = 0,1,2,..) in the denom for it to be 8ve equivalent to the tonic 1. Also notice that 3 and 5 in the numerators are prime and so don't factor to other keys (9/8 for example has the correct tonic 8 but 9 = 3x3, the third harmonic of the third harmonic). Thus, 19/16 meets both these requirements.

The second reason is that 19/16 is much closer to the tempered 2^(1/3) than 6/5, and since minor must nowadays enter with an equal footing with major, I figured that a new approach was required. That's all, not rocket science.

Cheers

Rick

In my list of just chord tunings I point out many chords that are simply upper notes of a chord with a missing fundamental. The septimal diminished triad is in the same boat as the minor triad, and is in fact an octave closer to the fundamental than the minor.

A 10:12:15 is still harmonically consonant enough that the ear can work with it even in such uses as a minor tonic resolution from a dominant major chord. That the minor triad is made to function in the same context as a major one could be why the ear is encouraged to hear the bottom note of this chord as the root. It is in effect a floating, or "false" root, which may be what gives the minor more "pathos" than the major.

That it is the simplest triad in the subharmonic series is an explanation that doesn't account for why the ear doesn't hear the fifth of the minor as the root, considering that it is the true fundamental.

rootless in Seattle.
Billy

--- In tuning@yahoogroups.com, Michael Sheiman <djtrancendance@...> wrote:
>
> Aaron>"A common answer is 'minor is upside down'."
>    I agree with you, that alone is oversimplifying it...
>
> >"Minor 10:12:15 is harmonically the upper 3 pitches of a
>
> Major 7th chord, so it sounds like it's hanging in space
>
> without a root (8 is missing)."
>    Ah, so the mind sees it as an incomplete major chord missing the most important tone (the root)...this makes a whole lot more sense than most of the explanations I have heard and ties back to the fact the intervals are in the same order that they are in the harmonic series in the major but not the minor chord.
>
>    I guess I'm going to have to give this one up because, in this case, you've convinced me that preference for the major triad is a mere side-effect of the fact the major triad includes the root tone it "points" to and is thus more "securely tonal" vs. minor which points to a missing tone.
>
> -Michael

Hi Rick,

As someone said earlier, the first reason is that the denominator 5 in 6/5
> and 6 in 7/6 don't give the correct tonic. Like 3/2 and 5/4 we need 2^(N =
> 0,1,2,..) in the denom for it to be 8ve equivalent to the tonic 1. Also
> notice that 3 and 5 in the numerators are prime and so don't factor to other
> keys (9/8 for example has the correct tonic 8 but 9 = 3x3, the third
> harmonic of the third harmonic). Thus, 19/16 meets both these requirements.
>
> The second reason is that 19/16 is much closer to the tempered 2^(1/3) than
> 6/5, and since minor must nowadays enter with an equal footing with major, I
> figured that a new approach was required. That's all, not rocket science.
>

Yes I know about the denominator X/2.
But I don't think this is required for the minor chord.
Why not 7/4, 11/8 13/8 17/8 etc.
There's a minor chord in the V9th chord 1/1 5/4 3/2 16/9 20/9 of 1/1 32/27
40/27
There's another minor chord in 1/1 5/4 3/2 15/8 of 1/1 6/5 3/2, and in 1/1
5/4 5/3 of 1/1 6/5 3/2
Where does the 19th come in? I've tried to use it but in the end didn't make
sense because of many reasons.

As for the root fundamental of the minor chord I think it is based on the
1/1 3/2 with the 6/5 playing a secondary role.
in the case of 1/1 32/27 40/27 it could be 32/27 making 1/1 5/4 27/16

--- In tuning@yahoogroups.com, Marcel de Velde <m.develde@...> wrote:
>
> Hi Rick,
>
> As someone said earlier, the first reason is that the denominator 5 in 6/5
> > and 6 in 7/6 don't give the correct tonic. Like 3/2 and 5/4 we need 2^(N =
> > 0,1,2,..) in the denom for it to be 8ve equivalent to the tonic 1. Also
> > notice that 3 and 5 in the numerators are prime and so don't factor to other
> > keys (9/8 for example has the correct tonic 8 but 9 = 3x3, the third
> > harmonic of the third harmonic). Thus, 19/16 meets both these requirements.
> >
> > The second reason is that 19/16 is much closer to the tempered 2^(1/3) than
> > 6/5, and since minor must nowadays enter with an equal footing with major, I
> > figured that a new approach was required. That's all, not rocket science.
> >
>
> Yes I know about the denominator X/2.
> But I don't think this is required for the minor chord.
> Why not 7/4, 11/8 13/8 17/8 etc.
> There's a minor chord in the V9th chord 1/1 5/4 3/2 16/9 20/9 of 1/1 32/27
> 40/27
> There's another minor chord in 1/1 5/4 3/2 15/8 of 1/1 6/5 3/2, and in 1/1
> 5/4 5/3 of 1/1 6/5 3/2
> Where does the 19th come in? I've tried to use it but in the end didn't make
> sense because of many reasons.
>
> As for the root fundamental of the minor chord I think it is based on the
> 1/1 3/2 with the 6/5 playing a secondary role.
> in the case of 1/1 32/27 40/27 it could be 32/27 making 1/1 5/4 27/16
>
Hi Marcel,

Rather than assuming that 12 ET gives an unsuccessful attempt of attaining small-limit JI values, what I am arguing is that it might successfully be giving JI values in the upper registers. This idea gains ground when we consider that truly irrational numbers such as in 12 ET require an infinite number of digits after the decimal point, which neither man nor computer could obtain. Further, (as someone recently said) the idea that the minor sounds 'sad' because it doesn't have a tonic sounds a bit poetic to me; tonality requires the tonic to be 8ve to 1 (10:12:15 is the minor triad from the major third degree). Therefore, instead of comparing 19/16 against standard JI and trying to see where it fits, try starting from scratch using this interval. eg 5/4 x 19/16 = 95/64 = 1.484375 or 81/64 x 19/16 = 1539/1024 = 1,502929688 (ironically, giving this fifth as even closer to 3/2). The advantage of major triad 64:80:95 and minor 64:76:95 is that they preserve the tonal integrity of both maj and min thirds (i.e. because multiplication is commutative, n x m = m x n). Of course, all other tonal chords such as 7th's, 9th's etc...can be obtained by successive application.

One more thing. In jazz arranging it is standard to consider that thirds override fifths, the latter more about relating between keys rather than within chords. For instance, C:E would be chosen over C:G every time. Better still is to include the guide tones 3rd's and 7th's, the latter being the 3rd from the fifth.

Regards

Rick

Hi Rick,

Rather than assuming that 12 ET gives an unsuccessful attempt of attaining
> small-limit JI values, what I am arguing is that it might successfully be
> giving JI values in the upper registers. This idea gains ground when we
> consider that truly irrational numbers such as in 12 ET require an infinite
> number of digits after the decimal point, which neither man nor computer
> could obtain.
>

Yes I agree it's a good line of thought.
I've tried it myself several times.
I even put several pieces in exteded JI using 19/16 as the minor third, the
drei equali piece (posted on this list), the whole lasso piece (unposted)
and a piece by bach (also unposted).

It seemed to work really well at first but l started disliking the sound of
it, it sounded tempered to me and gave problems.
for instance I've found that often the 19/16 in the minor third will become
the "fifth to the seventh" in the V7 chord, making a V7 of 1/1 5/4 3/2
57/32.
I've tried to accept this as correct as well, but in the end it doesn't work
out, the V7 in major mode should be 1/1 5/4 3/2 16/9.

> Further, (as someone recently said) the idea that the minor sounds 'sad'
> because it doesn't have a tonic sounds a bit poetic to me; tonality requires
> the tonic to be 8ve to 1 (10:12:15 is the minor triad from the major third
> degree). Therefore, instead of comparing 19/16 against standard JI and
> trying to see where it fits, try starting from scratch using this interval.
> eg 5/4 x 19/16 = 95/64 = 1.484375 or 81/64 x 19/16 = 1539/1024 = 1,502929688
> (ironically, giving this fifth as even closer to 3/2).
>

Try a minor third of 32/27.
32/27 x 81/64 = exactly 3/2

I've found for myself that to use the minor third in music it should be
either 32/27 or 6/5, both leading in different ways back to 1/1 5/4 3/2.

Yes 1/1 6/5 3/2 doesn't have a tonic X/2^ but you assume it should have so,
I agree it would make sense but in the end 1/1 6/5 3/2 makes more sense to
me than the idea the minor chord should have a tonic X/2^. Many chords don't
have such a tonic.

The question for me are now different questions.
I beleive another minor chord can be 1/1 32/27 40/27 (= also 1/1 5/4 27/16)
Is 1/1 32/27 3/2 also a much used chord and when do you get it?
Is 1215/1024 also a minor third that can occur. (perhaps here is your tonic
third? it occurs between for instance 16/9 and 135/64)

> The advantage of major triad 64:80:95 and minor 64:76:95 is that they
> preserve the tonal integrity of both maj and min thirds (i.e. because
> multiplication is commutative, n x m = m x n). Of course, all other tonal
> chords such as 7th's, 9th's etc...can be obtained by successive application.
>

I like 1/1 5/4 3/2 16/9 20/9 as a ninth chord. (Perhaps also 1/1 5/4 3/2 9/5
9/4 but not sure about this one yet.)
You can continue like this 1/1 5/4 3/2 16/9 20/9 4/3 5/3 2/1
On white keys: 3/2 15/8 9/4 8/3 10/3 4/1 5/1 6/1
All you have to do is to accept the minor ii chord is often 1/1 32/27 40/27
and everything works out like magic.

> One more thing. In jazz arranging it is standard to consider that thirds
> override fifths, the latter more about relating between keys rather than
> within chords. For instance, C:E would be chosen over C:G every time. Better
> still is to include the guide tones 3rd's and 7th's, the latter being the
> 3rd from the fifth.
>

Well that would perhaps make more sense if you were to accept that fourth
and fifths are not allways 4/3 and 3/2?

Marcel

>"55:60:63:68: 72:76:89
>is also known as "Michael's phi scale".
   Not quite.
   As I keep telling you the 72/55 AKA 1.309 does not exist in my scale (I know it exists in Temes' scale but his and mine are quite alike but not exactly the same).
   However, the tone 84/55 (apx. 1.528 AKA 89/55 - 60/55) does.

  So the correct version is
55:60:63:68:76:84:89
:-)

-Michael

--- In tuning@yahoogroups.com, Marcel de Velde <m.develde@...> wrote:
>
> Hi Rick,
>
> Rather than assuming that 12 ET gives an unsuccessful attempt of attaining
> > small-limit JI values, what I am arguing is that it might successfully be
> > giving JI values in the upper registers. This idea gains ground when we
> > consider that truly irrational numbers such as in 12 ET require an infinite
> > number of digits after the decimal point, which neither man nor computer
> > could obtain.
> >
>
> Yes I agree it's a good line of thought.
> I've tried it myself several times.
> I even put several pieces in exteded JI using 19/16 as the minor third, the
> drei equali piece (posted on this list), the whole lasso piece (unposted)
> and a piece by bach (also unposted).
>
> It seemed to work really well at first but l started disliking the sound of it, it sounded tempered to me and gave problems. for instance I've found that often the 19/16 in the minor third will become the "fifth to the seventh" in the V7 chord, making a V7 of 1/1 5/4 3/2 57/32. I've tried to accept this as correct as well, but in the end it doesn't work out, the V7 in major mode should be 1/1 5/4 3/2 16/9.
>
Hello Marcel,

Yes it will tend to sound tempered which is precisely why it shouldn't give you those types of problems. Not sure what you mean by the "fifth to the seventh". In the key of C for eg, the 7th degree of G7 is an F note and its 5th is C (which doesn't belong in G7). Or do you mean that it gives C7 rather than C maj 7?? But either way it would seem that something is amiss in your application. From C maj tonic, the chord should be a maj 7th, successive maj3 min3 maj3. Using small JI the maj 7 should be 15/8. This is because it is semi-symmetric (the maj 3 from 5th degree). Similarly, a minor 7th using 6/5 should be 1/1 6/5 3/2 9/5. Also, given C7 as 1/1 5/4 3/2 57/32, you're still using 6/5 between 5/4 and 3/2 which might cause problems down the track. But 57/32 is a good flattened 7th anyway and is in fact also closer to tempered than 7/4.

"Yes 1/1 6/5 3/2 doesn't have a tonic X/2^ but you assume it should have so, I agree it would make sense but in the end 1/1 6/5 3/2 makes more sense to me than the idea the minor chord should have a tonic X/2^. Many chords don't have such a tonic." This is not really an assumption on my part, but is a direct result of 8ve equivalence and proofs relating to the GCD frequency which becomes "1" for all harmonics. Nevertheless it is true that many intervals when heard in isolation don't have such a tonic but sound like they do. However, the reason for this might be because, as Carl has pointed out, there is an uncertainty around each interval which is why we hear for instance 6/5, 7/6, 19/16 etc...all as "minor thirds". The problem arises when we start to build larger chords and change key where those uncertainties accumulate and are no longer negligible. In this regard, you should find that 19/16 stays pretty stable if applied properly and consistently. After all, WHY should 6/5 make more sense? It is something close to a minor third from the major third degree, not from the tonic 1,2,4 etc...

Rick

Further, (as someone recently said) the idea that the minor sounds 'sad'
> > because it doesn't have a tonic sounds a bit poetic to me; tonality requires
> > the tonic to be 8ve to 1 (10:12:15 is the minor triad from the major third
> > degree). Therefore, instead of comparing 19/16 against standard JI and
> > trying to see where it fits, try starting from scratch using this interval.
> > eg 5/4 x 19/16 = 95/64 = 1.484375 or 81/64 x 19/16 = 1539/1024 = 1,502929688
> > (ironically, giving this fifth as even closer to 3/2).
> >
>
> Try a minor third of 32/27.
> 32/27 x 81/64 = exactly 3/2
>
> I've found for myself that to use the minor third in music it should be
> either 32/27 or 6/5, both leading in different ways back to 1/1 5/4 3/2.
>
> Yes 1/1 6/5 3/2 doesn't have a tonic X/2^ but you assume it should have so,
> I agree it would make sense but in the end 1/1 6/5 3/2 makes more sense to
> me than the idea the minor chord should have a tonic X/2^. Many chords don't
> have such a tonic.
>
> The question for me are now different questions.
> I beleive another minor chord can be 1/1 32/27 40/27 (= also 1/1 5/4 27/16)
> Is 1/1 32/27 3/2 also a much used chord and when do you get it?
> Is 1215/1024 also a minor third that can occur. (perhaps here is your tonic
> third? it occurs between for instance 16/9 and 135/64)
>
>
>
> > The advantage of major triad 64:80:95 and minor 64:76:95 is that they
> > preserve the tonal integrity of both maj and min thirds (i.e. because
> > multiplication is commutative, n x m = m x n). Of course, all other tonal
> > chords such as 7th's, 9th's etc...can be obtained by successive application.
> >
>
> I like 1/1 5/4 3/2 16/9 20/9 as a ninth chord. (Perhaps also 1/1 5/4 3/2 9/5
> 9/4 but not sure about this one yet.)
> You can continue like this 1/1 5/4 3/2 16/9 20/9 4/3 5/3 2/1
> On white keys: 3/2 15/8 9/4 8/3 10/3 4/1 5/1 6/1
> All you have to do is to accept the minor ii chord is often 1/1 32/27 40/27
> and everything works out like magic.
>
>
> > One more thing. In jazz arranging it is standard to consider that thirds
> > override fifths, the latter more about relating between keys rather than
> > within chords. For instance, C:E would be chosen over C:G every time. Better
> > still is to include the guide tones 3rd's and 7th's, the latter being the
> > 3rd from the fifth.
> >
>
> Well that would perhaps make more sense if you were to accept that fourth
> and fifths are not allways 4/3 and 3/2?
>
> Marcel
>

>
> This is not really an assumption on my part, but is a direct result of 8ve
> equivalence and proofs relating to the GCD frequency which becomes "1" for
> all harmonics. Nevertheless it is true that many intervals when heard in
> isolation don't have such a tonic but sound like they do. However, the
> reason for this might be because, as Carl has pointed out, there is an
> uncertainty around each interval which is why we hear for instance 6/5, 7/6,
> 19/16 etc...all as "minor thirds". The problem arises when we start to build
> larger chords and change key where those uncertainties accumulate and are no
> longer negligible. In this regard, you should find that 19/16 stays pretty
> stable if applied properly and consistently. After all, WHY should 6/5 make
> more sense? It is something close to a minor third from the major third
> degree, not from the tonic 1,2,4 etc...

Are you saying that when you play 2 notes together, they are allways X/2^
(if this is what 8ve means) in relation to one of the notes?
So 1/1 5/3 is never played? (as it's either 1/1 5/3 or 1/1 6/5 when
inverted)
Or do you mean this only for chords larger than 2 notes? If so what's the
difference between 2 and 3 notes?
This makes no sense to me at all in so many ways I won't list them or it'll
become a very large email :)
On which kind of proofs is this based?

And I get the impression you mean there's only 1 minor third? (which is
19/16)
What about 1/1 32/27? it is X/2^ aswell.
And what about the 6/5 between 5/4 and 3/2 in 1/1 5/4 3/2? It can never be
played as 5/4 3/2 15/8? or 5/3 2/1 5/2?

On Fri, May 15, 2009 at 1:49 PM, Carl Lumma <carl@...> wrote:
>
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>>
>> Hi Mike,
>> You're going to have a hard time convincing me that minor
>> triads sound worse than major triads.
>>
>> -Mike
>
> I dunno about sound worse, but they're certainly less consonant.
> There's a very high level of agreement on that. -Carl

Of course. I object to this concept where harmonic consonance
translates to an overall "sounding good." In fact, I object to the
concept that there are somehow perceptual "rules", presumably set in
stone, that determine what people are going to like. Maybe there are
perceptual guidelines that you can come up with that elucidate upon
what people have come to like, and there are perceptual mechanisms
that do exist that are commonly manipulated in modern music -- the
periodicity mechanism with harmony and the figure-ground
organizational scheme with polyrhythms being some examples. Even with
these things in place, a statement like "major chords sound 'better'
than minor chords" or "a stretched major scale played in unison sounds
'better' than a harmonic series" is bold indeed.

-Mike

--- In tuning@yahoogroups.com, Marcel de Velde <m.develde@...> wrote:
>
"Are you saying that when you play 2 notes together, they are allways X/2^ (if this is what 8ve means) in relation to one of the notes?"

Marcel, what I'm saying is that in strict JI, as opposed to ET, one needs to make a distinction between tonic and root note. For example, there is a difference between an E minor in the key of E minor and an E minor in the key of C major. In the first case, E is both tonic of the key and the root note of the chord, while in the second, E is the root note but C is the tonic. If C corresponds to 1 in the harmonic series, then 2,4,8,...2^(N = 0,1,2,3...) are also all C notes, 3,6,12,24...are all G notes (perfect fifths)and 5,10,20,40 are all E notes (major thirds or 8ve equivalents). Therefore, 5:6 or 6/5 has 5 or E as root note but not tonic which must be a C here, which means a denominator 2^(N = 0,1,2,3...). (Incidentally, you said that 32/27 also has 2^N. But 32 is in the numerator).

So, knowing that in practice minor keys enter on an equal footing with major and must have their own tonic, the question becomes; what is a good minor third ratio where C = 1,2,4...is the tonic (denominator)? The first candidate is 19/16. We have a nice prime number 19 in the numerator which wont factor out, 2^4 in the denominator therefore giving the correct tonic, and an interval which far better approximates a tempered minor third than the traditional 6/5. The next two possibilities are 37/32 or 39/32. But these are either sharper or flatter than the 19 and therefore not as good. Next are 75 or 77 over 64. Again not as good. In fact the next improvement doesn't come until 609/512, so in the scheme of things 19/16 seems like the simplest option.

One more thing. You mentioned that you didn't like this interval because it sounds tempered. But it IS a whole numbered ratio and is not the tempered interval 2^(1/4). The fact that it does not fit into the nicely packaged 5 or 7-limit JI's may just mean that these theories are not sufficiently general to account for minor keys correctly.

-Rick

> > This is not really an assumption on my part, but is a direct result of 8ve
> > equivalence and proofs relating to the GCD frequency which becomes "1" for
> > all harmonics. Nevertheless it is true that many intervals when heard in
> > isolation don't have such a tonic but sound like they do. However, the
> > reason for this might be because, as Carl has pointed out, there is an
> > uncertainty around each interval which is why we hear for instance 6/5, 7/6,
> > 19/16 etc...all as "minor thirds". The problem arises when we start to build
> > larger chords and change key where those uncertainties accumulate and are no
> > longer negligible. In this regard, you should find that 19/16 stays pretty
> > stable if applied properly and consistently. After all, WHY should 6/5 make
> > more sense? It is something close to a minor third from the major third
> > degree, not from the tonic 1,2,4 etc...
>
>
> Are you saying that when you play 2 notes together, they are allways X/2^
> (if this is what 8ve means) in relation to one of the notes?
> So 1/1 5/3 is never played? (as it's either 1/1 5/3 or 1/1 6/5 when
> inverted)
> Or do you mean this only for chords larger than 2 notes? If so what's the
> difference between 2 and 3 notes?
> This makes no sense to me at all in so many ways I won't list them or it'll
> become a very large email :)
> On which kind of proofs is this based?
>
> And I get the impression you mean there's only 1 minor third? (which is
> 19/16)
> What about 1/1 32/27? it is X/2^ aswell.
> And what about the 6/5 between 5/4 and 3/2 in 1/1 5/4 3/2? It can never be
> played as 5/4 3/2 15/8? or 5/3 2/1 5/2?
>

Hi Rick,

Marcel, what I'm saying is that in strict JI, as opposed to ET, one needs to
> make a distinction between tonic and root note. For example, there is a
> difference between an E minor in the key of E minor and an E minor in the
> key of C major. In the first case, E is both tonic of the key and the root
> note of the chord, while in the second, E is the root note but C is the
> tonic. If C corresponds to 1 in the harmonic series, then 2,4,8,...2^(N =
> 0,1,2,3...) are also all C notes, 3,6,12,24...are all G notes (perfect
> fifths)and 5,10,20,40 are all E notes (major thirds or 8ve equivalents).
> Therefore, 5:6 or 6/5 has 5 or E as root note but not tonic which must be a
> C here, which means a denominator 2^(N = 0,1,2,3...). (Incidentally, you
> said that 32/27 also has 2^N. But 32 is in the numerator).
>

Ah ok I understand a little better what you ment now.
But what you're saying is that in the key of C major only the tonic major
chord of C E G 1/1 5/4 3/2 confirms to this rule.
And when you play in C minor you expect this same rule to apply making C Eb
G 1/1 19/16 3/2.
But why then in C major key are there not all X/2^ intervals? Why is there
4/3 and 5/3?
And I find skipping a lot of prime harmonics to go to 19 all of a sudden
very unsatisfactory.
I think that intervals made up of low prime numbers are more consonant even
when the numbers are bigger. And they certainately play nicer when you
combine more intervals into chords that can divide the larger intervals,
this never happens with a large prime.

Btw i ment that 32/27 is also X/2^ when the chord is inverted. (1/1 32/27
27/20 = 1/1 5/4 27/16, 1/1 32/27 3/2 = 1/1 81/64 27/16)
Perhaps things like 1/1 675/512 405/256 make sense too (1/1 6/5 1024/675) or
1/1 21/16 63/40 (1/1 6/5 32/21)
(expecially like the sound of this when playing 135/128 675/512 405/256 15/8
- 1/1 5/4 3/2 2/1 etc)

So, knowing that in practice minor keys enter on an equal footing with major
> and must have their own tonic, the question becomes; what is a good minor
> third ratio where C = 1,2,4...is the tonic (denominator)? The first
> candidate is 19/16. We have a nice prime number 19 in the numerator which
> wont factor out, 2^4 in the denominator therefore giving the correct tonic,
> and an interval which far better approximates a tempered minor third than
> the traditional 6/5. The next two possibilities are 37/32 or 39/32. But
> these are either sharper or flatter than the 19 and therefore not as good.
> Next are 75 or 77 over 64. Again not as good. In fact the next improvement
> doesn't come until 609/512, so in the scheme of things 19/16 seems like the
> simplest option.
>
> One more thing. You mentioned that you didn't like this interval because it
> sounds tempered. But it IS a whole numbered ratio and is not the tempered
> interval 2^(1/4). The fact that it does not fit into the nicely packaged 5
> or 7-limit JI's may just mean that these theories are not sufficiently
> general to account for minor keys correctly.
>

The way I was reasoning was that the intervals for the minor third should
make sense when you change chords and do not expect a comma shift.
for instance can in 1/1 19/16 3/2 the 1/1 19/16 become the minor third in a
major chord? this would give a major chord something like 1/1 24/19 3/2 if
you use a pure fifth.
Or can in this 1/1 19/16 3/2 the minor third become the minor third that
makes the 7th in a major chord, which would then become 1/1 5/4 3/2 57/32?
Also, does this 19/16 tell you anything about music theory and things like
that? For me not, it confuses a lot.
In the end I'm not sure what the minor third should be in all cases, but
19/16 is currently way down in my list of options.

I understand we need some basic points of view to start with in JI.Yours
seems to be that the tonic chord has to be X/2^ wether it is major or minor.
And then you pick the lowest number (regardless of prime or not) that comes
close to the tempered minor third.
I simply don't know if your rule of tonic chord needing to be X/2^ is
correct, and even if it is correct if the pick of 19/16 is correct.
I find it raises more questions that it answers and I don't like its sound
and I find other starting point of view that serve as my bases of starting
with JI to weigh stronger and also conflict with 19/16.

Marcel

Arg!You've gotten me doubting again, thanks :)
Indeed something like 1/1 2/1 3/1 19/4 sounds best of all minor triads.

> And since my most important starting point for JI is that music is perfect,
that almost forces me to accept 19 limit minor chord.
I can't see 5 limit or 7 limit producing such a perfect final minor chord.

I've failed twice before when trying to use 19/16, once a couple of years
ago and once very recently.
I'll try again.
My many questions were things like how does the fundamental bass move? can
it move with any interval?
Do comma shifts accur and if so when. How is timing in music, which note
belongs to which fundamental bass (previous one, next one)
To accept high limit prime intervals requires deeeep knowledge of the
internal (unknown) musical workings of a piece it seems to me to be able to
tune it correctly.

Marcel

--- In tuning@yahoogroups.com, Marcel de Velde <m.develde@...> wrote:
>
Bonjour Marcel,

"But why then in C major key are there not all X/2^ intervals? Why is there 4/3 and 5/3?". Because F is now the tonic, C the fifth (3) and A the major third (5). Technically speaking both intervals represent a key change to the key of F. If we heard each one by itself it would sound like the key of F. But since we've been playing in the key of C then our ear still hears C as tonic. In other words, it is due to context. Generally, given x/2^N, then we can always invert the interval 2^(N+1)/x placing the original tonic x in the role of an upper harmonic to fundamental 2^0 = 1.

As a thought experiment, try thinking of JI intervals only from the tonic and ignoring others. Eg, major triad 4:5:6 has pure maj 3 and perfect 5, while min 3 as 6/5 isn't an interval from the tonic and is ignored. Similarly, minor triad 16:19:24 has pure min 3 and perfect 5, but maj 3 as 24/19* doesn't exist for the moment. (After all, tuning consecutive intervals is more of an ET problem. The second interval of a major triad IS the same as a usual minor third in ET etc...but not in JI).
*24/19 = 1.263157895 is in fact close to a major third. But I wouldn't use it. So why use 6/5 just because it has single digits, despite the fact that it has the wrong tonic? Some JI purists "won't go too far, but cannot explain why they go SO FAR". What use is using JI intervals if it is not to preserve precise tonality during the time that tonal chords are played? They fall far short when modulation is needed.

Next, if you do want to preserve the purity of both major and minor third then you can think of sacrificing the fifth as 3/2. This gives the fifth as 5/4 x 19/16 = 95/64, which is still ok since it is odd/2^6 (and 95/64 = 1.484375 approx 1.5). We then have maj triad 64:80:95 and minor 64:76:95. Or we can repeat the process with maj 3 as 81/64. These intervals will be better for things like modulation and extended 7th, 9th chords etc...that is, places where ET comes into its own. They are particularly better for symmetric chords like diminished eg: (19/16)^2 = 1.41015625, much closer to b5th as sqrt2 = 1.41421356 than (6/5)^2 = 1.44. Similarly,(19/16)^4 = 1.9885...is closer to the 8ve 2 than (6/5)^4 = 2.0736 (the differences being of the order of 0.01 and 0.07 respectively). The list goes on. All of these important issues some historical JI'ists tend to conveniently ignore.

Regards

-Rick

Hi Rick,
>
> Marcel, what I'm saying is that in strict JI, as opposed to ET, one needs to
> > make a distinction between tonic and root note. For example, there is a
> > difference between an E minor in the key of E minor and an E minor in the
> > key of C major. In the first case, E is both tonic of the key and the root
> > note of the chord, while in the second, E is the root note but C is the
> > tonic. If C corresponds to 1 in the harmonic series, then 2,4,8,...2^(N =
> > 0,1,2,3...) are also all C notes, 3,6,12,24...are all G notes (perfect
> > fifths)and 5,10,20,40 are all E notes (major thirds or 8ve equivalents).
> > Therefore, 5:6 or 6/5 has 5 or E as root note but not tonic which must be a
> > C here, which means a denominator 2^(N = 0,1,2,3...). (Incidentally, you
> > said that 32/27 also has 2^N. But 32 is in the numerator).
> >
>
> Ah ok I understand a little better what you ment now.
> But what you're saying is that in the key of C major only the tonic major
> chord of C E G 1/1 5/4 3/2 confirms to this rule.
> And when you play in C minor you expect this same rule to apply making C Eb
> G 1/1 19/16 3/2.
> But why then in C major key are there not all X/2^ intervals? Why is there
> 4/3 and 5/3?
> And I find skipping a lot of prime harmonics to go to 19 all of a sudden
> very unsatisfactory.
> I think that intervals made up of low prime numbers are more consonant even
> when the numbers are bigger. And they certainately play nicer when you
> combine more intervals into chords that can divide the larger intervals,
> this never happens with a large prime.
>
> Btw i ment that 32/27 is also X/2^ when the chord is inverted. (1/1 32/27
> 27/20 = 1/1 5/4 27/16, 1/1 32/27 3/2 = 1/1 81/64 27/16)
> Perhaps things like 1/1 675/512 405/256 make sense too (1/1 6/5 1024/675) or
> 1/1 21/16 63/40 (1/1 6/5 32/21)
> (expecially like the sound of this when playing 135/128 675/512 405/256 15/8
> - 1/1 5/4 3/2 2/1 etc)
>
> So, knowing that in practice minor keys enter on an equal footing with major
> > and must have their own tonic, the question becomes; what is a good minor
> > third ratio where C = 1,2,4...is the tonic (denominator)? The first
> > candidate is 19/16. We have a nice prime number 19 in the numerator which
> > wont factor out, 2^4 in the denominator therefore giving the correct tonic,
> > and an interval which far better approximates a tempered minor third than
> > the traditional 6/5. The next two possibilities are 37/32 or 39/32. But
> > these are either sharper or flatter than the 19 and therefore not as good.
> > Next are 75 or 77 over 64. Again not as good. In fact the next improvement
> > doesn't come until 609/512, so in the scheme of things 19/16 seems like the
> > simplest option.
> >
> > One more thing. You mentioned that you didn't like this interval because it
> > sounds tempered. But it IS a whole numbered ratio and is not the tempered
> > interval 2^(1/4). The fact that it does not fit into the nicely packaged 5
> > or 7-limit JI's may just mean that these theories are not sufficiently
> > general to account for minor keys correctly.
> >
>
> The way I was reasoning was that the intervals for the minor third should
> make sense when you change chords and do not expect a comma shift.
> for instance can in 1/1 19/16 3/2 the 1/1 19/16 become the minor third in a
> major chord? this would give a major chord something like 1/1 24/19 3/2 if
> you use a pure fifth.
> Or can in this 1/1 19/16 3/2 the minor third become the minor third that
> makes the 7th in a major chord, which would then become 1/1 5/4 3/2 57/32?
> Also, does this 19/16 tell you anything about music theory and things like
> that? For me not, it confuses a lot.
> In the end I'm not sure what the minor third should be in all cases, but
> 19/16 is currently way down in my list of options.
>

--- In tuning@yahoogroups.com, Marcel de Velde <m.develde@...> wrote:
>
> I understand we need some basic points of view to start with in JI.Yours
> seems to be that the tonic chord has to be X/2^ wether it is major or minor.
> And then you pick the lowest number (regardless of prime or not) that comes
> close to the tempered minor third.
> I simply don't know if your rule of tonic chord needing to be X/2^ is
> correct, and even if it is correct if the pick of 19/16 is correct.
> I find it raises more questions that it answers and I don't like its sound
> and I find other starting point of view that serve as my bases of starting
> with JI to weigh stronger and also conflict with 19/16.
>
> Marcel
>
Of course it's correct Marcel. It is not "my" rule but "the" basis of musical tonality. As I've said many times here before, given two waves with frequency ratio a/b, where a and b are whole, then the resulting wave addition will give a frequency corresponding to the GCD. For eg, given frequencies 6 and 9, then 9/3 = 3/2, a = 3, b = 2, therefore 9 will be the 3rd harmonic of tonic 3 and 6 will be the 2nd harmonic. In other words, the resultant wave will be 3Hz. And while we can't hear this resultant frequency directly, it does make itself felt as musical tonality. We "know" when we hit waves that are 8ve equivalent to this GCD, that they are the tonic. The fact that most music theorists and scientists have systematically overlooked this simple fact for centuries says more about their lack of true understanding of the inner workings of musical harmony than anything truly "scientific". After all, is not centuries of practising with an orchestra or instrument also counted as a legitimate "scientific experiment"?

Nor did I "pick" the number 19 at random. Without being egotistical, like all jazz musicians, I have to identify and respond to intervals in a nanosecond, and on a regular basis. So my ears are pretty sharp. I also know that in 12ET minor and major are harmonically equal. It seems obvious now, but it took me ages to work out this interval with just a calculator and guitar. I have since tested it in my microtuning program in many tunings and it sounds just fine.

For starters, try cutting back to just I, IV and V. For I, take 1:19/16:5/4:3/2 as say C:Eb:E:G. Then apply the same from IV and V, 4/3:19/12:5/3:2 and 3/2:57/32:15/8:9/4, respectively. You will find this is a perfectly valid JI tuning, one among many including this interval, and with the advantage of giving minor chords the priority they are due. (observe for eg that it gives 57/32, traditionally a good flat 7).

Rick

--- In tuning@yahoogroups.com, Marcel de Velde <m.develde@...> wrote:
>
I forgot to mention Marcel that in this tuning (last post), your other minor intervals are already there but now in the correct context as sub-keys. From C as tonic, D:F is 4/3 x 8/9 = 32/27 and E:G is 3/2 x 4/5 = 6/5. Also, instead of using baby chords such as triads (which musicians rarely play any more), try major 7, min 7, 7 in this tuning for C and F roots, and 7 for G.

C maj 7th, 1:5/4:3/2:15/8,
F maj 7th, 1:5/4:4/3:5/3, (where maj 7 interval here is the same in both cases, 4/3 x 4/5 = 2 x 8/15 = 16/15),
C7th 1:5/4:3/2:57/32, F7th 1:19/16:4/3:5/3 (again, 4/3 x 16/19 = 32/57, the same interval in both chords), and
G7th as 9/8:4/3:3/2:15/8 (again including 32/27 but where 9/8 is now the seventh).

So you see there are many other things to consider than just playing triads alone and out of context, which will lead you into trouble in the long run.

-Rick

I understand we need some basic points of view to start with in JI.Yours
> seems to be that the tonic chord has to be X/2^ wether it is major or minor.
> And then you pick the lowest number (regardless of prime or not) that comes
> close to the tempered minor third.
> I simply don't know if your rule of tonic chord needing to be X/2^ is
> correct, and even if it is correct if the pick of 19/16 is correct.
> I find it raises more questions that it answers and I don't like its sound
> and I find other starting point of view that serve as my bases of starting
> with JI to weigh stronger and also conflict with 19/16.
>
> Marcel
>

--- In tuning@yahoogroups.com, "rick_ballan" <rick_ballan@...> wrote:
> G7th as 9/8:4/3:3/2:15/8 (again including 32/27 but where 9/8 is now the seventh).

Hi Rick, is there a mistake there? If not, can you explain, as I am not getting it!
Steve M.

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

>
> C maj 7th, 1:5/4:3/2:(15/8)=(7/4)*(15/14) {~+119Cents off from 7/4}
Hi Rick,
Leonhard Euler would consider that fourfold-chord in 7-limit terms as:
(1/1):(5/4):(3/2):(7/4)*(15/14) {~+119Cents off from 7/4}
hence
in his point of view barely an crude approximation of
4:5:6:(7=(15/8)*(14/15))
Quote:
http://sonic-arts.org/monzo/euler/euler-en.htm
"... the ratios of our four sounds are now expressed by these numbers 4, 5, 6, 7 whose perception undoubtedly is confounded."

Analogous also he argues for Rameau's
Subdomiant(sixth-adjustee)
http://www.newworldencyclopedia.org/entry/Chord#Sixth_chords
"French augmented sixth: Ab, C, D, F#"

>Fmaj7th, 1/1 : 5/4 : 4/3 : 5/3=(7/4)*(20/21) {~-84Cents off from 7/4}

and probably for the 19-limit variant:

>C7th 1 : 5/4 : 3/2 : 57/32=(7/4)*(57/56) {~+31Cents off from 7/4}

Euler continues :
"Perhaps this is the foundation of the rule on the preparation and the resolution of dissonances, to inform the listener that it is the same sound, though it can serve as two different ones, so that they he may not imagine that one has introduced a completely foreign sound." [OO, 3a, I, 515]
Here is an original interpretation. One knows indeed that it is usually supposed that preparation and resolution are only used to accustom the ear , beforehand and "subsequently" as one might say, to the dissonant sound by making it hear in consonances. And Euler
concludes:
"It is commonly claimed that one makes use in music only of proportions made up of these three prime numbers 2, 3 and 5 and the great Leibniz 24 has already remarked that in music we have still not learned to count beyond 5; which is also incontestably true in the instruments tuned according to principles of the harmony. But, if my conjecture has validity, one can say that in the compositions one already counts up to 7 and that the ear thereby is already accustomed to it. 25 It is a new kind of music, one that has begun to show use and which was unkown to the ancients. In this kind the chord 4, 5, 6, 7 is the most complete harmony, since it contains numbers 2, 3, 5 and 7; but it is more complicated than the common triad which contains only numbers 2, 3 and 5. If it is a perfection in composition, perhaps one will try to carry the instruments to the same degree." [OO, 3a, I, 515]

References:

Download an facsimile of Euler's original publication from the site:
http://www.math.dartmouth.edu/~euler/pages/E033.html
Especialliy see for the above mentioned 7-limit consonances:
http://math.dartmouth.edu/~euler/docs/originals/E314.pdf
(written in French).

You can buy an 1739 original "First Edition":
http://search.abaa.org/dbp2/book336685744.html
actual "Price: USD 2,500.00 $"
or simply study the less expensive:
http://www.rev.net/~aloe/music/7limit.html
http://en.wikipedia.org/wiki/Septimal_comma
induced intervals.

bye
A.S.

>"then the resulting wave addition will give a frequency corresponding to
the GCD. For eg, given frequencies 6 and 9, then 9/3 = 3/2, a = 3, b =
2, therefore 9 will be the 3rd harmonic of tonic 3 and 6 will be the
2nd harmonic."

Just a general comment about GCD's and matching the harmonic series.  While this is a very mathematically sound idea...if you try actually playing an x/16 harmonic series with an instrument I'm pretty sure you will find it sounds rather mechanical-sounding with virtually all beating being exactly at the same rate.  It presents a similar problem to keyboardist playing every key at a fixed "mathematically perfect" velocity.
    So while I agree x/c (where c is a constant...the most widely used example being the x/16 harmonic series fragment) is a perfectly aligned harmonic series and good "limit"...it's a limit made to be approached but sounds less than perfect if you hit it dead-on (and thus up to about 15 cents of tempering can actually be desirable to preserve a soft/flowing/non-mechanic sense of beating on a 1-2 out of 7 notes in a scale and up to 7 cents tempering on the rest, for example).  Not to say 12TET, for example, is good...but that one of its advantages over a straight harmonic series, for example, is that it creates this relaxed/slow beating. 

-Michael

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:
>
> --- In tuning@yahoogroups.com, "rick_ballan" <rick_ballan@> wrote:
> > G7th as 9/8:4/3:3/2:15/8 (again including 32/27 but where 9/8 is now the seventh).
>
> Hi Rick, is there a mistake there? If not, can you explain, as I am not getting it!
> Steve M.
>
Steve it's just D:F:G:B, 3rd inversion G7 chord. I'm just grouping them in one 8ve.

--- In tuning@yahoogroups.com, Michael Sheiman <djtrancendance@...> wrote:
>
> >"then the resulting wave addition will give a frequency corresponding to
> the GCD. For eg, given frequencies 6 and 9, then 9/3 = 3/2, a = 3, b =
> 2, therefore 9 will be the 3rd harmonic of tonic 3 and 6 will be the
> 2nd harmonic."
>
> Just a general comment about GCD's and matching the harmonic series. While this is a very mathematically sound idea...if you try actually playing an x/16 harmonic series with an instrument I'm pretty sure you will find it sounds rather mechanical-sounding with virtually all beating being exactly at the same rate. It presents a similar problem to keyboardist playing every key at a fixed "mathematically perfect" velocity.
> So while I agree x/c (where c is a constant...the most widely used example being the x/16 harmonic series fragment) is a perfectly aligned harmonic series and good "limit"...it's a limit made to be approached but sounds less than perfect if you hit it dead-on (and thus up to about 15 cents of tempering can actually be desirable to preserve a soft/flowing/non-mechanic sense of beating on a 1-2 out of 7 notes in a scale and up to 7 cents tempering on the rest, for example). Not to say 12TET, for example, is good...but that one of its advantages over a straight harmonic series, for example, is that it creates this relaxed/slow beating.
>
> -Michael
>
Yeah I agree that the pure harmonic series applied mechanically can get rather boring, although there are theoretically an infinite number of harmonics to choose from (and some of these can sound pretty weird). This is why I'm interested in the idea of "almost" periodic functions and the "leverage" around each interval as in Harmonic entropy. (How far can we go before a major third sounds out of tune rather than becoming a minor third etc...? i.e. how near is "almost"?). Still, we can't just spiral off into infinity without some reference point, and fanatical avante garde stuff can become equally boring in its unpredictability because we don't "care" for one note over another. (I remember reading Wagner who said that tonal keys are like families, modulation like adolescence, and the climax is a form of maturity where the prodigal son returns home but with a new understanding). Having composed a symphony myself (just for myself), I realised that you waste nothing. This part you rejected here can come in there with a transposition etc...

OTOH, the maths of GCD is much more elegant than just applying the counting numbers. This is because if we form a GCD frequency out of whole numbers a/b with b not equal to 1 and take the harmonic series of both sine frequencies then we get a harmonic series of GCD's. Each wave has the same shape (tone) as the original while the frequencies increase harmonically i.e. (a + 2a + 3a + ...) + (b + 2b + 3b + ...) = (a + b) + 2(a + b) + 3(a + b) + ...(although this is an abbreviation for adding sine waves). Then this frequency can be added with another, then another, and so on ad musicum.

For 12 TET beating, an example that immediately springs to mind is the wonderful closed voicings of the jazz pianist Bill Evans.

-Rick

Then it's second inversion, isn't it? As far as I know there's basic root form, 1st, 2nd and 3rd inversion for seventh chord...

Daniel Forro

On 23 May 2009, at 11:24 AM, rick_ballan wrote:

>
>
> --- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:
>
> Steve it's just D:F:G:B, 3rd inversion G7 chord. I'm just grouping > them in one 8ve.
>

--- In tuning@yahoogroups.com, Marcel de Velde <m.develde@...> wrote:
>
> Arg!You've gotten me doubting again, thanks :)
> Indeed something like 1/1 2/1 3/1 19/4 sounds best of all minor triads.
>
> > And since my most important starting point for JI is that music is perfect,
> that almost forces me to accept 19 limit minor chord.
> I can't see 5 limit or 7 limit producing such a perfect final minor chord.
>
> I've failed twice before when trying to use 19/16, once a couple of years
> ago and once very recently.
> I'll try again.
> My many questions were things like how does the fundamental bass move? can
> it move with any interval?
> Do comma shifts accur and if so when. How is timing in music, which note
> belongs to which fundamental bass (previous one, next one)
> To accept high limit prime intervals requires deeeep knowledge of the
> internal (unknown) musical workings of a piece it seems to me to be able to
> tune it correctly.
>
> Marcel
>
You're welcome.

Just thinking out aloud, many of the traditional approaches to JI were based on tuning instruments that had a restricted range and fixed set of notes, that is, from a purely practical point of view. Thus, idea's like comma shifts etc...needed to come into play because intervals had to share a double function, one as notes in chords and another as notes between keys. And once the instruments were tuned, the results were "set in stone". Of course, 12 EDO is a good solution to this as each note has 12 equal functions (though sometimes this distinction is now overlooked).

However, in the computer world, we have been freed from some of these limitations. When I said that 6/5 does not have the correct tonic, I was speaking strictly in the first sense of intervals within chords, not between keys. Nowadays we can retune for each individual chord during the time duration of that chord. And for tonal chords, THIS is where JI will sound great. But for modulation intervals and symmetric chords like diminished, I'm 99% sure that ET's will sound best here. Therefore, since comma's don't exist in ET's, and are not an issue for tonal chords regarded from the tonic, then problem solved!

-Rick

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:
>
> --- In tuning@yahoogroups.com, "rick_ballan" <rick_ballan@> wrote:
> > G7th as 9/8:4/3:3/2:15/8 (again including 32/27 but where 9/8 is now the seventh).
>
> Hi Rick, is there a mistake there? If not, can you explain, as I am not getting it!
> Steve M.
>
Ah I see, sorry for being unclear. I meant 9/8 as the interval between the 4th and 5th, F and G in the key of C, (3/2)/(4/3) = 9/8, not the 9/8 from C which is the D note.

--- In tuning@yahoogroups.com, "Andreas Sparschuh" <a_sparschuh@...> wrote:
>
> --- In tuning@yahoogroups.com, "rick_ballan" <rick_ballan@> wrote:
>
> >
> > C maj 7th, 1:5/4:3/2:(15/8)=(7/4)*(15/14) {~+119Cents off from 7/4}
> Hi Rick,
> Leonhard Euler would consider that fourfold-chord in 7-limit terms as:
> (1/1):(5/4):(3/2):(7/4)*(15/14) {~+119Cents off from 7/4}
> hence
> in his point of view barely an crude approximation of
> 4:5:6:(7=(15/8)*(14/15))
> Quote:
> http://sonic-arts.org/monzo/euler/euler-en.htm
> "... the ratios of our four sounds are now expressed by these numbers 4, 5, 6, 7 whose perception undoubtedly is confounded."
>
> Analogous also he argues for Rameau's
> Subdomiant(sixth-adjustee)
> http://www.newworldencyclopedia.org/entry/Chord#Sixth_chords
> "French augmented sixth: Ab, C, D, F#"
>
> >Fmaj7th, 1/1 : 5/4 : 4/3 : 5/3=(7/4)*(20/21) {~-84Cents off from 7/4}
>
> and probably for the 19-limit variant:
>
> >C7th 1 : 5/4 : 3/2 : 57/32=(7/4)*(57/56) {~+31Cents off from 7/4}
>
> Euler continues :
> "Perhaps this is the foundation of the rule on the preparation and the resolution of dissonances, to inform the listener that it is the same sound, though it can serve as two different ones, so that they he may not imagine that one has introduced a completely foreign sound." [OO, 3a, I, 515]
> Here is an original interpretation. One knows indeed that it is usually supposed that preparation and resolution are only used to accustom the ear , beforehand and "subsequently" as one might say, to the dissonant sound by making it hear in consonances. And Euler
> concludes:
> "It is commonly claimed that one makes use in music only of proportions made up of these three prime numbers 2, 3 and 5 and the great Leibniz 24 has already remarked that in music we have still not learned to count beyond 5; which is also incontestably true in the instruments tuned according to principles of the harmony. But, if my conjecture has validity, one can say that in the compositions one already counts up to 7 and that the ear thereby is already accustomed to it. 25 It is a new kind of music, one that has begun to show use and which was unkown to the ancients. In this kind the chord 4, 5, 6, 7 is the most complete harmony, since it contains numbers 2, 3, 5 and 7; but it is more complicated than the common triad which contains only numbers 2, 3 and 5. If it is a perfection in composition, perhaps one will try to carry the instruments to the same degree." [OO, 3a, I, 515]
>
> References:
>
> Download an facsimile of Euler's original publication from the site:
> http://www.math.dartmouth.edu/~euler/pages/E033.html
> Especialliy see for the above mentioned 7-limit consonances:
> http://math.dartmouth.edu/~euler/docs/originals/E314.pdf
> (written in French).
>
> You can buy an 1739 original "First Edition":
> http://search.abaa.org/dbp2/book336685744.html
> actual "Price: USD 2,500.00 $"
> or simply study the less expensive:
> http://www.rev.net/~aloe/music/7limit.html
> http://en.wikipedia.org/wiki/Septimal_comma
> induced intervals.
>
> bye
> A.S.
>
Thanks A.S.,

Allot of reading there. The Euler article looks interesting. I'll read it more thoroughly when I've got the time.

From what I can gather, you're saying that Euler also thought extended 4 note chords were being used by his contemporaries but that 5-limit theory hadn't caught up yet? I presume that 15/14 is a comma in his system (couldn't see it in the link) and that he has applied it from 7/4 to obtain 15/8? My reasoning was a bit more simple because maj 7 chords are standard in jazz harmony and can be seen either as two 5ths a maj 3 apart or a min 3 surrounded by two majors.

Btw, was it Euler who created the 2^(n/12) system or something similar? I vaguely recall reading that he had a system based on the log of 2.

Cheers

Rick

--- In tuning@yahoogroups.com, Daniel Forro <dan.for@...> wrote:
>
> Then it's second inversion, isn't it? As far as I know there's basic
> root form, 1st, 2nd and 3rd inversion for seventh chord...
>
> Daniel Forro
>
>Sorry, you're right Daniel, second inversion.

> On 23 May 2009, at 11:24 AM, rick_ballan wrote:
>
> >
> >
> > --- In tuning@yahoogroups.com, "martinsj013" <martinsj@> wrote:
> >
> > Steve it's just D:F:G:B, 3rd inversion G7 chord. I'm just grouping
> > them in one 8ve.
> >
>

--- In tuning@yahoogroups.com, "rick_ballan" <rick_ballan@...> wrote:
>...I meant 9/8 as the interval between the 4th and 5th, F and G in the key of C, (3/2)/(4/3) = 9/8, not the 9/8 from C which is the D note...

Thanks, Rick, yes that was why I was confused. Steve M.

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

> From what I can gather,
> you're saying that Euler also thought extended 4 note chords were
> being used by his contemporaries but that 5-limit theory hadn't
> caught up yet?
Hi Rick,
as far as I do understand Euler,
it was in his view about time to
change from 5-limit to 7-limit intervals
in order to handle 4-fold-chords alike:

G:b:D:f = 4:5:6:7

more properly than in old 5-limit

G:b:D:F = 4:5:6:(64/9=7.111111111...)

>I presume that 15/14 is a comma in his system
>(couldn't see it in the link)
Not at all,
but Euler considered 64/63 as his crucial comma.
http://en.wikipedia.org/wiki/Septimal_comma
alike already Archytas before him:
http://www.ex-tempore.org/ARCHYTAS/ARCHYTAS.html

> and that he has applied it from 7/4 to obtain 15/8?
He clearly discerned inbetween both of them:

(15/8):(7/4) = 15/14 ~119.4Cents

> My reasoning was a bit more simple because maj 7 chords
> are standard in jazz harmony
in
http://en.wikipedia.org/wiki/Jazz_harmony
terms Euler replaces within the:

"C-7, Cm7 C Eb G Bb minor 7th chord"

the highest pitch Bb=16:9 by Bb\=7:4.

http://en.wikipedia.org/wiki/Harmonic_series_(music)
"Red notes are sharp. Blue notes are flat."
http://www.brefeld.homepage.t-online.de/konsonanz.html
Quote:
"Die Zweiklänge mit dem Frequenzverhältnis 7/4, 7/5 und 7/6 werden auch als "Blue Notes" bezeichnet. "
tr:
'The intervals with pitch-ratios 7/4, 7/5 and 7/6
are also labeled as so called "blue-notes".'
>
> Btw, was it Euler who created the 2^(n/12)
> system or something similar?
No, 12-EDO
http://en.wikipedia.org/wiki/Equal_temperament
"The first person known to have attempted a numerical specification for 12-TET is probably Zhu Zaiyu (朱載堉) a prince of the Ming court, who published a theory of the temperament in 1584. It is possible that this idea was spread to Europe by way of trade, which intensified just at the moment when Zhu Zaiyu published his calculations. Within fifty-two years of Zhu's publication, the same ideas had been published by Marin Mersenne and Simon Stevin."
http://en.wikipedia.org/wiki/Zhu_Zaiyu
http://en.wikipedia.org/wiki/Simon_Stevin
"Music theory
Stevin was the first author in the West (1585, simultaneously with, and independently of, Zhu Zaiyu in China) to give a mathematically accurate specification for equal temperament."
http://www.xs4all.nl/~huygensf/doc/singe.html

>I vaguely recall reading that he had a system based on the log of 2.
That's correct.

bye
A.S.

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

Oh I see A.S, 15/14 is the difference interval b/w the two 7ths in Euler's system. He wanted to bring in the 7th harmonic quite literally as 7. Interesting about Zhu Zaiyu, Mersenne and Stevin. That's the same Mersenne who calculated Mersenne's law.

Thanks

Rick

> > From what I can gather,
> > you're saying that Euler also thought extended 4 note chords were
> > being used by his contemporaries but that 5-limit theory hadn't
> > caught up yet?
> Hi Rick,
> as far as I do understand Euler,
> it was in his view about time to
> change from 5-limit to 7-limit intervals
> in order to handle 4-fold-chords alike:
>
> G:b:D:f = 4:5:6:7
>
> more properly than in old 5-limit
>
> G:b:D:F = 4:5:6:(64/9=7.111111111...)
>
> >I presume that 15/14 is a comma in his system
> >(couldn't see it in the link)
> Not at all,
> but Euler considered 64/63 as his crucial comma.
> http://en.wikipedia.org/wiki/Septimal_comma
> alike already Archytas before him:
> http://www.ex-tempore.org/ARCHYTAS/ARCHYTAS.html
>
> > and that he has applied it from 7/4 to obtain 15/8?
> He clearly discerned inbetween both of them:
>
> (15/8):(7/4) = 15/14 ~119.4Cents
>
>
> > My reasoning was a bit more simple because maj 7 chords
> > are standard in jazz harmony
> in
> http://en.wikipedia.org/wiki/Jazz_harmony
> terms Euler replaces within the:
>
> "C-7, Cm7 C Eb G Bb minor 7th chord"
>
> the highest pitch Bb=16:9 by Bb\=7:4.
>
> http://en.wikipedia.org/wiki/Harmonic_series_(music)
> "Red notes are sharp. Blue notes are flat."
> http://www.brefeld.homepage.t-online.de/konsonanz.html
> Quote:
> "Die Zweiklänge mit dem Frequenzverhältnis 7/4, 7/5 und 7/6 werden auch als "Blue Notes" bezeichnet. "
> tr:
> 'The intervals with pitch-ratios 7/4, 7/5 and 7/6
> are also labeled as so called "blue-notes".'
> >
> > Btw, was it Euler who created the 2^(n/12)
> > system or something similar?
> No, 12-EDO
> http://en.wikipedia.org/wiki/Equal_temperament
> "The first person known to have attempted a numerical specification for 12-TET is probably Zhu Zaiyu (æœ±è¼‰å ‰) a prince of the Ming court, who published a theory of the temperament in 1584. It is possible that this idea was spread to Europe by way of trade, which intensified just at the moment when Zhu Zaiyu published his calculations. Within fifty-two years of Zhu's publication, the same ideas had been published by Marin Mersenne and Simon Stevin."
> http://en.wikipedia.org/wiki/Zhu_Zaiyu
> http://en.wikipedia.org/wiki/Simon_Stevin
> "Music theory
> Stevin was the first author in the West (1585, simultaneously with, and independently of, Zhu Zaiyu in China) to give a mathematically accurate specification for equal temperament."
> http://www.xs4all.nl/~huygensf/doc/singe.html
>
> >I vaguely recall reading that he had a system based on the log of 2.
> That's correct.
>
> bye
> A.S.
>

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

> Oh I see A.S, 15/14 is the difference interval b/w
> the two 7ths in > Euler's system.
That's correct Rick,
he had even some more "7ths" than that both variants:

starting from the 'natural' 7th:
7:4 ~967C
_64/63_ Septimal-Comma ~27C
16:9 ~996C
_81/80_ Syntonic-Comma ~21.5C
9/5 ~1018C
_25/24_ Minor-Chroma semitone ~71C
15/8 ~1088C
_81/80_ Syntonic-Comma ~21.5C
and finally the Pythagorean '7th'
243/128 = 3^3/2^7 ~1110C

Appearently
> He wanted to bring in the 7th harmonic quite literally as 7.
...alike for instance:
http://trumpeter.athabascau.ca/index.php/trumpet/article/view/497/839
"The seventh partial of the series, following g', does not appear on the piano. It is somewhere between b' flat and b' natural. It may not appear in our tempered scale, but it is often heard in folk music of various peoples, and especially in the blues, as the famous blue note, neither minor nor major, associated with indefinite emotion. Further on in the series also appear untempered notes, and these too have been utilized in some kinds of music."...

...as in the early 20th century by
http://en.wikipedia.org/wiki/Bebop
http://goliath.ecnext.com/coms2/gi_0199-6314461/Bebop-a-case-in-point.html
"
"Using the higher intervals of a chord as a melody line" can therefore only mean that Parker wanted to follow the higher partials of various chords in his melodic constructions,...
Bebop musicians' rediscovery of higher partials as resources for harmonic construction was rooted in the blues and ultimately in African tonal systems. The consequences were remarkable. Blues tonality, equitonal principles, remote partials--all of this converged in a successful process of liberation that reinstated a more African approach to tonality...
4. The upward extension of triads in bebop follows two very different principles, both inherited from African practices. One could be called "the central African model" of piling thirds on top of each other; the other principle is the selective use of upper harmonics, an auditory experience in African-American traditions transmitted through the blues. Because higher partials cannot be played exactly on a piano, the pianist chooses optimal approximations. Upper partials are what Charlie Parker must have heard internally when he tried to work out something new over "Cherokee." Thus even within the tuning of the European tonal system, C (9) in bebop and earlier blues-based jazz can be taken as a column of harmonics incorporating partials 4, 5, 6, 7, and 9; and its cognate, [C.sup.+11], the augmented eleventh chord, is but a representation of the harmonic series up to partial 11. Because that partial stands at 551 cents, its pitch value cannot be played accurately on a keyboard instrument; soft is used to represent it. That note fulfills its task much better than f, because f would simply corroborate the diatonic scale....
However, because partials-derived pitch patterns can only be objectified approximately on a piano or guitar, the analysis of such chords boils down to this question: Which note represents which partial? In the key of C over the tonic chord, partial 7 can be rendered as b, although it is 31 cents lower than the tempered value; partial 9 at 204 cents is virtually identical with a d, while partial 11 at 551 cents is rendered as f# on a piano, and partial 13 (at 840 cents) as a[flat]. Remarkably, when jazz pianists began to lay out the partials-based cluster chords, they also followed the order of partials in the natural harmonic series, from bass to treble, thereby revealing the underlying (often unconscious) auditory basis....
Thus, some of the cluster chords in bebop are simply the nearest possible representations of selected partials from the natural harmonic series, extended high up. The famous flatted fifth in bebop represents partial 11; to make that clear, it is often struck together with its fundamental, for example, f# with c, or g with c[sharp] to be resolved into c. The accurate intonation of the bebop flatted fifth would be 551 cents; the approximation played on a piano is at 600 cents. The [C.sup.+11] chord in its common layout on a piano by bebop musicians plainly reveals its partials-derived structure...

http://books.google.de/books?id=K7AFBrvT9ukC&printsec=frontcover&dq=Africa+and+the+Blues#PPA118,M1
http://books.google.de/books?id=sASKbVkXw30C&pg=PA268&dq=pygmy-polyphony#PPA279,M1
http://infohost.nmt.edu/~jstarret/pygmies.html
"According to my measurements, the basic intervals used seem to be 7/6, or 267 cents (small minor third), 8/7, or 231 cents (large second), and 9/8, or 204 cents (whole tone).

If intervals between degrees are determined by differential notes which can change according to the musical context, the polyphony can produce intervals very close from each other, but with a different function : 7/6 + 8/7 (498 cents, the perfect fourth) is only a few cents apart from 8/7 + 8/7 (462 cents) or 7/6 + 9/8 (471 cents); 7/6+7/6+8/7+8/7+9/8 (1200 cents, the octave), from 7/6+8/7+8/7+8/7+9/8 (1164 cents), etc. This could explain 8/7 (462 cents) or 7/6 + 9/8 (471 cents); 7/6+7/6+8/7+8/7+9/8 (1200 cents, the octave), from 7/6+8/7+8/7+8/7+9/8 (1164 cents), etc. This could explain the often quoted presence of "major sevenths" in Pygmy polyphonies, which have no place in a anhemitonic pentatonic scale. In my opinion, this also raises the question whether it is pertinent to use the term "pentatonic" for a music in which the role of the octave isn't necessarily the same as in ours...

Other topic:
> That's the same Mersenne who calculated Mersenne's law.
http://scienceworld.wolfram.com/physics/MersennesLaws.html
http://en.wikipedia.org/wiki/Marin_Mersenne
http://www-history.mcs.st-andrews.ac.uk/Biographies/Mersenne.html

bye
A.S.

--- In tuning@yahoogroups.com, "Andreas Sparschuh" <a_sparschuh@...> wrote:
>
Hi A.S,

lets not forget the flattened major 6 as 13th harmonic giving 4:5:6:7:11:13. The problem with these, however, is that they don't explain minor that well. And when it comes to a 7th chord, the aug 11th is called a flat-5th because it is at the expense of the fifth (aug or #11 requires maj 7, which is a chord that includes the perfect fifth). The so-called blue notes are the 7th, the b5, and min3 played against a major chord (7(b5#9) type harmony), not necessarily in the chord. Honestly, I think that our extended chords have more to do with stacking alternate maj and min thirds in ET rather than anything to do with the harmonic series up to 11 or 13. So the problem comes back to finding a decent minor third ratio with the correct tonic i.e. 19/16. But thanks for your thoughts.

-Rick

> --- In tuning@yahoogroups.com, "rick_ballan" <rick_ballan@> wrote:
>
> > Oh I see A.S, 15/14 is the difference interval b/w
> > the two 7ths in > Euler's system.
> That's correct Rick,
> he had even some more "7ths" than that both variants:
>
> starting from the 'natural' 7th:
> 7:4 ~967C
> _64/63_ Septimal-Comma ~27C
> 16:9 ~996C
> _81/80_ Syntonic-Comma ~21.5C
> 9/5 ~1018C
> _25/24_ Minor-Chroma semitone ~71C
> 15/8 ~1088C
> _81/80_ Syntonic-Comma ~21.5C
> and finally the Pythagorean '7th'
> 243/128 = 3^3/2^7 ~1110C
>
> Appearently
> > He wanted to bring in the 7th harmonic quite literally as 7.
> ...alike for instance:
> http://trumpeter.athabascau.ca/index.php/trumpet/article/view/497/839
> "The seventh partial of the series, following g', does not appear on the piano. It is somewhere between b' flat and b' natural. It may not appear in our tempered scale, but it is often heard in folk music of various peoples, and especially in the blues, as the famous blue note, neither minor nor major, associated with indefinite emotion. Further on in the series also appear untempered notes, and these too have been utilized in some kinds of music."...
>
> ...as in the early 20th century by
> http://en.wikipedia.org/wiki/Bebop
> http://goliath.ecnext.com/coms2/gi_0199-6314461/Bebop-a-case-in-point.html
> "
> "Using the higher intervals of a chord as a melody line" can therefore only mean that Parker wanted to follow the higher partials of various chords in his melodic constructions,...
> Bebop musicians' rediscovery of higher partials as resources for harmonic construction was rooted in the blues and ultimately in African tonal systems. The consequences were remarkable. Blues tonality, equitonal principles, remote partials--all of this converged in a successful process of liberation that reinstated a more African approach to tonality...
> 4. The upward extension of triads in bebop follows two very different principles, both inherited from African practices. One could be called "the central African model" of piling thirds on top of each other; the other principle is the selective use of upper harmonics, an auditory experience in African-American traditions transmitted through the blues. Because higher partials cannot be played exactly on a piano, the pianist chooses optimal approximations. Upper partials are what Charlie Parker must have heard internally when he tried to work out something new over "Cherokee." Thus even within the tuning of the European tonal system, C (9) in bebop and earlier blues-based jazz can be taken as a column of harmonics incorporating partials 4, 5, 6, 7, and 9; and its cognate, [C.sup.+11], the augmented eleventh chord, is but a representation of the harmonic series up to partial 11. Because that partial stands at 551 cents, its pitch value cannot be played accurately on a keyboard instrument; soft is used to represent it. That note fulfills its task much better than f, because f would simply corroborate the diatonic scale....
> However, because partials-derived pitch patterns can only be objectified approximately on a piano or guitar, the analysis of such chords boils down to this question: Which note represents which partial? In the key of C over the tonic chord, partial 7 can be rendered as b, although it is 31 cents lower than the tempered value; partial 9 at 204 cents is virtually identical with a d, while partial 11 at 551 cents is rendered as f# on a piano, and partial 13 (at 840 cents) as a[flat]. Remarkably, when jazz pianists began to lay out the partials-based cluster chords, they also followed the order of partials in the natural harmonic series, from bass to treble, thereby revealing the underlying (often unconscious) auditory basis....
> Thus, some of the cluster chords in bebop are simply the nearest possible representations of selected partials from the natural harmonic series, extended high up. The famous flatted fifth in bebop represents partial 11; to make that clear, it is often struck together with its fundamental, for example, f# with c, or g with c[sharp] to be resolved into c. The accurate intonation of the bebop flatted fifth would be 551 cents; the approximation played on a piano is at 600 cents. The [C.sup.+11] chord in its common layout on a piano by bebop musicians plainly reveals its partials-derived structure...
>
> http://books.google.de/books?id=K7AFBrvT9ukC&printsec=frontcover&dq=Africa+and+the+Blues#PPA118,M1
> http://books.google.de/books?id=sASKbVkXw30C&pg=PA268&dq=pygmy-polyphony#PPA279,M1
> http://infohost.nmt.edu/~jstarret/pygmies.html
> "According to my measurements, the basic intervals used seem to be 7/6, or 267 cents (small minor third), 8/7, or 231 cents (large second), and 9/8, or 204 cents (whole tone).
>
> If intervals between degrees are determined by differential notes which can change according to the musical context, the polyphony can produce intervals very close from each other, but with a different function : 7/6 + 8/7 (498 cents, the perfect fourth) is only a few cents apart from 8/7 + 8/7 (462 cents) or 7/6 + 9/8 (471 cents); 7/6+7/6+8/7+8/7+9/8 (1200 cents, the octave), from 7/6+8/7+8/7+8/7+9/8 (1164 cents), etc. This could explain 8/7 (462 cents) or 7/6 + 9/8 (471 cents); 7/6+7/6+8/7+8/7+9/8 (1200 cents, the octave), from 7/6+8/7+8/7+8/7+9/8 (1164 cents), etc. This could explain the often quoted presence of "major sevenths" in Pygmy polyphonies, which have no place in a anhemitonic pentatonic scale. In my opinion, this also raises the question whether it is pertinent to use the term "pentatonic" for a music in which the role of the octave isn't necessarily the same as in ours...
>
> Other topic:
> > That's the same Mersenne who calculated Mersenne's law.
> http://scienceworld.wolfram.com/physics/MersennesLaws.html
> http://en.wikipedia.org/wiki/Marin_Mersenne
> http://www-history.mcs.st-andrews.ac.uk/Biographies/Mersenne.html
>
> bye
> A.S.
>

--- In tuning@yahoogroups.com, "rick_ballan" <rick_ballan@...> wrote:
>
> --- In tuning@yahoogroups.com, "Andreas Sparschuh" <a_sparschuh@> wrote:
> So the problem comes back
> to finding a decent minor third ratio
> with the correct tonic i.e. 19/16.
>
Ok Rick,
already Erasthostenes introduced 19-limit interval-ratios in his
http://en.wikipedia.org/wiki/Tetrachord
"Eratosthenes chromatic tetrachord 20:19, 19:18, 6:5"
and Boethius considered just that 19/16 the context of:
http://eamusic.dartmouth.edu/~larry/published_articles/divisions_of_the_tetrachord/chapter2.pdf
Quote
p.7
"Boethius's tuning for the tetrachord...

chromatic: 256/243 • 81/76 • 19/16
... p.8
... 19/16 is virtually the same size
as the Pythagorean minor third, 32/27."

Consider the quotient among them:

513/512 = (19/16):(32/27)

that is labeled as
http://www.xs4all.nl/~huygensf/doc/intervals.html
"513/512 undevicesimal comma, Boethius' comma"

and is examined in
http://sonic-arts.org/monzo/marchet/marchet.htm
"The difference between 2187/2048 and 81/76 is

2187:2048 = 2^-11 3^7
¸ 81:76 = - 2^-2 3^4 19^-1
___ ___ ___
2^-9 3^3 19^1

or 513:512 [= 3.378 cents].
Thus Marchetto's 81/76 "Diatonic Semitone"
of 3 "dieses" is indeed very close to the
"apotome" to which he equated it.
Surprisingly, however, it is even closer to the 5-limit larger semitone, which a few centuries later also became known as the "diatonic". The progression 1/1 : 16/15 : 9/8

1/1 : 16/15 : 9/8 ....

...As indicated by the arrow,
19^1 is only approximately 0.04 Semitone
513/512 = 3.378 cents
higher than 3^-3, the standard Pythagorean "minor 3rd".
This small interval may be called the nondecimal schisma,
and will be found useful in a thorough analysis of Marchetto's purported tuning, as well as several ancient Greek tunings using 19-Limit ratios.
"
Erasthostenes' analogous 19-limit approximation of 5-limit
can be found in:
http://tonalsoft.com/monzo/marchetto/marchetto.aspx
"The difference between 16/15 and 81/76 is...
...the tiny interval 1216:1215 [= 1.42 cents]."
that is labeled as:
http://www.xs4all.nl/~huygensf/doc/intervals.html
"1216/1215 Eratosthenes' comma"

See for further details about that arguments in:
/tuning/topicId_26618.html#26618
and
/tuning/topicId_66654.html#66663
"
C#> 19/18 == (256/243)*(513/512)
b > 152/81 = (15/8)*(1216/1215)
"
or reverse express that both issues for that 2 cases as

1.) Pythagorean 3-limit approximation near to 19

2^9/3^3 = 512/27 = 18+(26/27) = ~18,962963... = 19-1/27

2.) and respectively as Syntonic 5-limit approx:

5*3^5/2^6 = 1215/64 = 18+(63/64) = ~18,984375... = 19-1/64

Conclusion:
That two arithmetical almost-coincidences
19-1/27 (in 3-lim-approx.) and 19-1/64 (in 5-lim-approx.)
make the 19th partial-tone so unique within the harmonic series.

kind regards
A.S.

On 27 May 2009, at 1:02 PM, rick_ballan wrote:
> lets not forget the flattened major 6 as 13th harmonic giving
> 4:5:6:7:11:13. The problem with these, however, is that they don't
> explain minor that well. And when it comes to a 7th chord, the aug
> 11th is called a flat-5th ...
>

They are different beasts and to mix them is wrong. We have chords
like 7/5-, it has nothing to do with augmented 11th.

> ... because it is at the expense of the fifth (aug or #11 requires
> maj 7,
>

Why? Not necessarily. Normal full 11+ chord has perfect fifth, and
minor 7th, like C-E-G-Bb-D-F#. 13, 11, 9, 7, 5 are independent and
follow only rules of alternate chromatic scale and common sense
(which will not combine in one chord 13- and 5+, or 11+ and 5-).

11th and 13th chords were build originally in advanced diatonic
classical harmony only on dominant chord (Chopin, Schumann, Liszt,Wagner...), without chromatic shifts (but with omitted third to avoid
dissonance). Much later (Debussy, Skriabin...) they were chromatized,
altered, especially when used in subdominant function (for example 11
+/5+) and exploited on all chromatic steps in the key.

> which is a chord that includes the perfect fifth). The so-called
> blue notes are the 7th, the b5,
>

But in blues scale can't be diminished 5th, only augmented 4th. Only that way the rule of alternate chromatic scale is kept. So blues
scale can be for example C-(D)-Eb-F-F#-G-(A)-Bb-(B), where notes in
parenthesis could be omitted. This scale can be played over all basic
chords used in blues pattern, or transposed according to chord
changes. This depends on performer's attitude.

> and min3 played against a major chord (7(b5#9) type harmony), not
> necessarily in the chord. Honestly, I think that our extended
> chords have more to do with stacking alternate maj and min thirds
> in ET
>

Then such major chord would be derived from Lydian (C-E-G-B-D-F#-A)
and it will be C13/11+/7maj which is wrong as classical harmony derives it from Mixolydian, as 13th chord was originally used only in
dominant function (check Liszt's Love dream No. 3 for example). So
for C major it is G-B-D-F-A-C-E (G13 - normal diatonics without
altered notes). Besides in the end there will be two small thirds (F#-
A, A-C).

And keeping your rule of alternating major and minor thirds we get
for minor key C-Eb-G-Bb-D-F-A, derived from Dorian, which is again
far from reality, as 13th chord for minor key is build again on
dominant and for C minor key it will be G-B-D-F-Ab-C-Eb (G13-/9-),
based on harmonic C minor - you see we get irregular pattern of major
and minor thirds. Check Chopin's Nocturno C minor op. 48/1.

> rather than anything to do with the harmonic series up to 11 or 13.
>

11+ could be influenced by 11th harmonics, because normal 11th chord
based on major dominant chord has perfect 11th, then third is omitted
to avoid dissonance. Augmented 11th together with minor 7th could
imitate harmonic series, which is often case in the music of Debussy,Ravel, Skriabin, Stravinskij, Bartók... also Gershwin, pop and jazz
music since 20ies.

And augmented 11th together with augmented 5th is essential chord
beloved by impressionists for its power to destroy tonality, as it's
made from whole-tone scale (C-E-G#-Bb-D-F#).

Nothing to say about chords like 13/11+/9+/9- (C-E-G-Bb-Db-D#-F#-A),
13/11+/9+ (C-E-G-Bb-D#-F#-A), 13/11+/9- (C-E-G-Bb-Db-F#-A), 11+/9+
(C-E-G-Bb-D#-F#), 11+/9- (C-E-G-Bb-Db-F#) all of them opening door to
the huge world of diminished scale and bitonality explored by many
New music and jazz composers (just to name Skriabin and Messiaen).

Daniel Forró

Of course I have meant "altered chromatic scale"... Which is for C
major key:

C-Db-D-D#-E-F-F#-G-Ab-A-A#-Bb-B

For C minor:

C-Db-D-Eb-Fb-F-F#-G-Ab-A-A#-Bb-B

Daniel Forró

On 27 May 2009, at 10:17 PM, Daniel Forro wrote:

> follow only rules of alternate chromatic scale and common sense
>

> that way the rule of alternate chromatic scale is kept. So blues
>

--- In tuning@yahoogroups.com, Daniel Forro <dan.for@...> wrote:
Hi Daniel,
Yet Monteverdi used
http://en.wikipedia.org/wiki/Seventh_chord
s http://www.associatedcontent.com/article/938795/claudio_monteverdi_and_his_secconda.html
"He also expanded harmonic usage to include dominant seventh-chords and other dissonant chords without any harmonic preparation. During the early part of the seventeenth century, he developed a new compositional style he called the seconda prattica or second manner."

Hence already for Monteverdi the so called
http://en.wikipedia.org/wiki/Diatonic_functionality
theory yiels nonsense, due to the lack of
"any harmonic preparation".

Then later the
http://en.wikipedia.org/wiki/Ninth_chord
was used in Bach's time
http://books.google.de/books?id=xJxVRyiqZNIC&pg=PA205&lpg=PA205&dq=mozart+ninth-chord&source=bl&ots=hGoYcy3u1o&sig=km8zMH-60mRw-FZwiS3VpW7279Q&hl=de&ei=i2EdSrWiNoLD_Qbl2oWIDQ&sa=X&oi=book_result&ct=result&resnum=6
as later did Mozart and Beethoven too,
sometimes without careing whether 'functionality'
should be imputed or even not?

That inapplicable
http://en.wikipedia.org/wiki/Functional_harmony
theory works even worser for
> 11th
http://en.wikipedia.org/wiki/Eleventh
http://en.wikipedia.org/wiki/Elektra_chord
> and 13th
discern here among
http://en.wikipedia.org/wiki/Thirteenth
and
http://en.wikipedia.org/wiki/Dominant_thirteenth_chord

>chords were build originally in advanced diatonic
There is no such "advanced diatonic",
but only more or less generally built multi-chords from
http://en.wikipedia.org/wiki/Upper_structure_triad
s
> classical harmony only on dominant chord
>(Chopin, Schumann, Liszt, Wagner...),
But Wagner insited for his
http://en.wikipedia.org/wiki/Tristan_chord
that it has no clear root,
hence also by-no-means an well-defined 'dominant' function.
http://www.chameleongroup.org.uk/research/The_Tristan_Chord_in_Context.pdf

> without chromatic shifts (but with omitted third to avoid
> dissonance). Much later (Debussy's

> and Skriabin...)'s
http://en.wikipedia.org/wiki/Prometheus_chord
> they were chromatized,
> altered, especially when used in subdominant function
Not at all,
the Prometheus contains none intrinsic
http://en.wikipedia.org/wiki/Tonality
per-se.

> (for example 11 +/5+)
> and exploited on all chromatic steps in the key.
But is not listed among the
http://en.wikipedia.org/wiki/Chromaticism#Chromatic_chord
s
entries.
"Any chord that is not chromatic is a diatonic chord."
http://en.wikipedia.org/wiki/Diatonic_chord
but that obsolete concept depends on the historical context.
>
> > (which is a chord that includes the perfect fifth).
> The so-called blue notes are the 7th, the b5,
> >
> But in blues scale can't be diminished 5th, only augmented 4th.
Here please be aware of
http://en.wikipedia.org/wiki/Enharmonic
spelling variants for the same pitch by different note-names
due to the 12-EDO confusion that intermeshes
sharp and flat accidentialy faulty in an wishy-washy misnomer:
'Compareing apples and oranges'

> Only
> that way the rule of alternate chromatic scale is kept. So blues
> scale can be for example C-(D)-Eb-F-F#-G-(A)-Bb-(B), where notes in
> parenthesis could
should better
>be omitted.
in order keep the restriction of one single note per
http://en.wikipedia.org/wiki/Critical_band
-width
for avoiding disturbing interferences within the band-windows.

> This scale can be played over all basic
> chords used in blues pattern, or transposed according to chord
> changes. This depends on performer's attitude.
That's up to you
alike already observed in the simpler case of the easy:
http://en.wikipedia.org/wiki/Half-diminished_seventh_chord
"The half-diminished chord has three functions in contemporary harmony: predominant function, diminished, and dominant function."
>
> > and min3 played against a major chord (7(b5#9) type harmony), not
> > necessarily in the chord. Honestly, I think that our extended
> > chords have more to do with stacking alternate maj and min thirds
> > in ET

In my view,
The primitive 12-ET graduation of barely a dozen pitch-casses
is simply to much coarse for judgeing properly about
the original intensions of the above composers.
Trying to explain all the above phenomenons only by stacking of 3rds
is an undue oversimplification: Nothing doing, sorry I'm afraid.
>
> Then such major chord would be derived from Lydian (C-E-G-B-D-F#-A)
> and it will be C13/11+/7maj which is wrong as classical harmony
> derives it from Mixolydian,...

http://en.wikipedia.org/wiki/Mixolydian_mode
"ncidentally, the order of Mixolydian tones and semitones is identical to the Dominant 7th scale. In other words, the C Mixolydian mode and the C Dominant 7th scale are identical."

> ...as 13th chord was originally
> used only in dominant function (check Liszt's Love dream No. 3 for
> example).

There in List's case
the harmonic function
depends on fine-tuning of the components among the
principial constituents of that chord,
if at all there is any detectable
'harmonic-function'
present however deduced?

> So
> for C major it is G-B-D-F-A-C-E (G13 - normal diatonics without
> altered notes).
> Besides in the end there will be two small thirds
> (F#- A, A-C).
That both 3rds are only of the same seize when thrown into the
"melting-pot" in conform 12-ET of all to much raw 100Cent steps.
http://en.wikipedia.org/wiki/Procrustes
bed
"A Procrustean bed is an arbitrary standard
to which exact conformity is forced."
to the expense of microtonality.
>
> And keeping your rule of alternating major and minor thirds we get
> for minor key C-Eb-G-Bb-D-F-A, derived from Dorian,

http://en.wikipedia.org/wiki/Dorian_mode
"The Dorian mode is equivalent to the natural minor scale (or the Aeolian mode) but with the sixth degree raised a semi-tone. "

> which is again
> far from reality,
> as 13th chord for minor key...
Fully agreed.

That chord
>...is build again on
> dominant and for C minor key
> it will be G-B-D-F-Ab-C-Eb (G13-/9-),
> based on harmonic C minor
>- you see we get irregular pattern of major
> and minor thirds...
...if one restricts the own horizon
when allowing only the dual choice exclusively inbetween
minor-300- and major-400Cents "3rds" out of 12-EDO.

>Check Chopin's Nocturno C minor op. 48/1.
That composition is contains quite a lot of ambiguous enharmonics,
that no clear unique functional analysis can be achieved properly.
>
> > rather than anything
> > to do with the harmonic series up to 11 or 13.
probably because Chopin intended here to produce an obvious
http://en.wikipedia.org/wiki/Consonance_and_dissonance#Dissonance
by enriching the chords by additional inharmonics more coloured.
>
> 11+ could be influenced by 11th harmonics,
> because normal 11th chord
> based on major dominant chord has perfect 11th,
> then third (is)...
has better
> ...to be omitted to avoid ...
all to much
>... dissonance.
> Augmented 11th together with minor 7th could
> imitate harmonic series,
> which is often case in the music of Debussy,
> Ravel, Skriabin, Stravinskij, Bart�k... also Gershwin, pop and jazz
> music since 20ies.
The usage of the 11th partial appeared earlier among the folks:
http://www.people.iup.edu/rahkonen/ilwm/Switzerland.bib.htm
"The instrument restricts players to the overtone series of the 11th partial (F in a C scale) characteristically sharp, is commonly called alphorn fa. Until 1900 the alphorn was played as a solo instrument to pacify the cattle and send signals. "
alike in the early 19th-century popular-songs
http://www.informaworld.com/index/795292761.pdf
"...articular, the 11th partial or overtone, often used in Swiss music and famously known as the "Alphorn-fa" ("fa" being the fourth pitch of a scale), ..."
Listen the CD-records:
http://www.artofthemix.org/findamix/Getcontents.asp?strMixId=118436
"Three of the tones usually played on the alphorn do not occur in the tempered tone system. The 7th natural harmonic is a slightly high B; the 11th sounds distinctly higher than F, but too deep for F sharp, and the 13th is somewhat higher than A flat. The easiest of these three "wrong" tones to recognize is the 11th natural harmonic. It is known as alphorn fa."
http://www.jean-luc-darbellay.ch/frameset.php?lang=en&cat=discography&id=16
http://en.wikipedia.org/wiki/Alpine_Symphony
/tuning/topicId_63984.html#64367

> And augmented 11th together with augmented 5th is essential chord
> beloved by impressionists for its power to destroy tonality,
> as it's made from whole-tone scale (C-E-G#-Bb-D-F#).
not to mention
http://en.wikipedia.org/wiki/Twelve-tone_technique
>
> Nothing to say about chords like 13/11+/9+/9- (C-E-G-Bb-Db-D#-F#-A),
> 13/11+/9+ (C-E-G-Bb-D#-F#-A), 13/11+/9- (C-E-G-Bb-Db-F#-A), 11+/9+
> (C-E-G-Bb-D#-F#), 11+/9- (C-E-G-Bb-Db-F#) all of them opening door
> to
> the huge world of diminished scale and bitonality
or in more generally terms:
http://en.wikipedia.org/wiki/Polytonality
"Bitonality is the use of only two different keys at the same time.
....
"Pre-twentieth-century instances of polytonality, such as Biber's "Battaglia" (1673) and Mozart's Ein musikalischer Spass (1787),..
http://www.mozartforum.com/Lore/article.php?id=314
....tend to use the technique for programmatic or comic effect."
> explored by many
> New music and jazz composers (just to name Skriabin and Messiaen).
>
"The earliest uses of polytonality in non-programmatic contexts are found in the twentieth century, particularly in the work of Bartók (Fourteen Bagatelles, op. 6 [1908]), Ives (Variations on "America"), Stravinsky (Petrushka [1911]), and Debussy (Preludes, Book 2 [1913]). Ives claimed that he learned the technique of polytonality from his father, who taught him to sing popular songs in one key while harmonizing them in another...."
or even
http://en.wikipedia.org/wiki/Bimodality
"It is more general than bitonality
since the "scales" involved need not be traditional scales;
if diatonic collections are involved,
their pitch centers need not be
the familiar major and minor-scale tonics."
...
and by that digression we got completely off topic
from the original subject
that sounds even less consonant
than an 6/5 minor-3rd.

Kind Regards
A.S.

--- In tuning@yahoogroups.com, "Andreas Sparschuh" <a_sparschuh@...> wrote:
<<< "The seventh partial of the series, following g', does not appear on the piano. It is somewhere between b' flat and b' natural. It may not appear in our tempered scale, but it is often heard in folk music of various peoples, and especially in the blues, as the famous blue note, neither minor nor major, associated with indefinite emotion. Further on in the series also appear untempered notes, and these too have been utilized in some kinds of music."... >>>

I don't know if I missed something, but the 7th harmonic above a fundamental C (if that's what this is talking about) is below B-flat. That's why when a quartet sings a dominant 7th chord, the singer of the 7th is encouraged to flatten his note until the chord locks. The major third also should be flatted but less so.

<<< However, because partials-derived pitch patterns can only be objectified approximately on a piano or guitar, the analysis of such chords boils down to this question: Which note represents which partial? In the key of C over the tonic chord, partial 7 can be rendered as b, although it is 31 cents lower than the tempered value; >>>

Once again you wouldn't want to use a b to approximate a harmonic that would actually sound a hair flatter than b-flat. B would represent the 15th harmonic well.

<<< ...and partial 13 (at 840 cents) as a[flat]. >>>

I guess A-flat would be most accurate for the 13th harmonic when actual cents are calculated. But for it to sound right in ET, you really need to use an A-natural to maintain the minor-third relationship with the 11th harmonic. And a typical last chord in swing music traditionally uses the A (e.g. CEGBbDF#A).

Billy

Hello William.

I don't know if I missed something, but the 7th harmonic above a fundamental
> C (if that's what this is talking about) is below B-flat. That's why when a
> quartet sings a dominant 7th chord, the singer of the 7th is encouraged to
> flatten his note until the chord locks. The major third also should be
> flatted but less so.

Is this really how it is tought in music schools?
And they really try to sing the harmonic 7th for the dominant 7th chord in
major mode?
If so I'm very surprised by this as my ear doesn't accept the harmonic 7th
here.
G 3/2 B 15/8 D 9/4 F 21/8
Instead G 3/2 B 15/8 D 9/4 F 8/3 sounds perfect to me and makes perfect
sense in major mode of 1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1

Also the 13th chord based on the dominant 7th seems to me to be G 3/2 B
15/8 D 9/4 F 8/3 A 10/3 C 4/1 E 5/1

Marcel

--- In tuning@yahoogroups.com, Daniel Forro <dan.for@...> wrote:
>
>
> On 27 May 2009, at 1:02 PM, rick_ballan wrote:
> > lets not forget the flattened major 6 as 13th harmonic giving
> > 4:5:6:7:11:13. The problem with these, however, is that they don't
> > explain minor that well. And when it comes to a 7th chord, the aug
> > 11th is called a flat-5th ...
> >
>
> They are different beasts and to mix them is wrong. We have chords
> like 7/5-, it has nothing to do with augmented 11th.
>
> > ... because it is at the expense of the fifth (aug or #11 requires
> > maj 7,
> >
>
> Why? Not necessarily. Normal full 11+ chord has perfect fifth, and
> minor 7th, like C-E-G-Bb-D-F#. 13, 11, 9, 7, 5 are independent and
> follow only rules of alternate chromatic scale and common sense
> (which will not combine in one chord 13- and 5+, or 11+ and 5-).
>
> 11th and 13th chords were build originally in advanced diatonic
> classical harmony only on dominant chord (Chopin, Schumann, Liszt,
> Wagner...), without chromatic shifts (but with omitted third to avoid
> dissonance). Much later (Debussy, Skriabin...) they were chromatized,
> altered, especially when used in subdominant function (for example 11
> +/5+) and exploited on all chromatic steps in the key.
>
> > which is a chord that includes the perfect fifth). The so-called
> > blue notes are the 7th, the b5,
> >
>
> But in blues scale can't be diminished 5th, only augmented 4th. Only
> that way the rule of alternate chromatic scale is kept. So blues
> scale can be for example C-(D)-Eb-F-F#-G-(A)-Bb-(B), where notes in
> parenthesis could be omitted. This scale can be played over all basic
> chords used in blues pattern, or transposed according to chord
> changes. This depends on performer's attitude.
>
> > and min3 played against a major chord (7(b5#9) type harmony), not
> > necessarily in the chord. Honestly, I think that our extended
> > chords have more to do with stacking alternate maj and min thirds
> > in ET
> >
>
> Then such major chord would be derived from Lydian (C-E-G-B-D-F#-A)
> and it will be C13/11+/7maj which is wrong as classical harmony
> derives it from Mixolydian, as 13th chord was originally used only in
> dominant function (check Liszt's Love dream No. 3 for example). So
> for C major it is G-B-D-F-A-C-E (G13 - normal diatonics without
> altered notes). Besides in the end there will be two small thirds (F#-
> A, A-C).
>
> And keeping your rule of alternating major and minor thirds we get
> for minor key C-Eb-G-Bb-D-F-A, derived from Dorian, which is again
> far from reality, as 13th chord for minor key is build again on
> dominant and for C minor key it will be G-B-D-F-Ab-C-Eb (G13-/9-),
> based on harmonic C minor - you see we get irregular pattern of major
> and minor thirds. Check Chopin's Nocturno C minor op. 48/1.
>
> > rather than anything to do with the harmonic series up to 11 or 13.
> >
>
> 11+ could be influenced by 11th harmonics, because normal 11th chord
> based on major dominant chord has perfect 11th, then third is omitted
> to avoid dissonance. Augmented 11th together with minor 7th could
> imitate harmonic series, which is often case in the music of Debussy,
> Ravel, Skriabin, Stravinskij, Bartók... also Gershwin, pop and jazz
> music since 20ies.
>
> And augmented 11th together with augmented 5th is essential chord
> beloved by impressionists for its power to destroy tonality, as it's
> made from whole-tone scale (C-E-G#-Bb-D-F#).
>
> Nothing to say about chords like 13/11+/9+/9- (C-E-G-Bb-Db-D#-F#-A),
> 13/11+/9+ (C-E-G-Bb-D#-F#-A), 13/11+/9- (C-E-G-Bb-Db-F#-A), 11+/9+
> (C-E-G-Bb-D#-F#), 11+/9- (C-E-G-Bb-Db-F#) all of them opening door to
> the huge world of diminished scale and bitonality explored by many
> New music and jazz composers (just to name Skriabin and Messiaen).
>
> Daniel Forró
>
Hi Daniel,

Almost everything you say here is true, but you're taking me way out of context and wrongly concluding that I don't know my harmony or history. In the original statement of this thread (I think to Marcel) I was talking about minor third as 19/16 being a far more viable option for minor tonality in all extant 12 tone harmony than 6/5 or 32/27 etc which all have the wrong tonic. I was pointing out that while 6/5 might sound good in simple triads considered in isolation, that 19/16 explains much better extended four note or symmetric chords than successive applications of 6/5 etc which will accumulate wrong tonics over time. With A.S it became more about trying to explain these extended 4-note chords of modern jazz harmony in terms of the harmonic series and traditional precedences to be found in Euler. I then pointed out that the 7th in dom7 chords might be better approx by 3/2 x 19/16 than the 7th harmonic (and then as an aside that the same could be said of 13 chords and the 13th harmonic as 13/8).

For the record, C E G B D F# A is called maj7(#11) because it is the #11, not the b5 which comes in on dominant chords, like you said. Of course this Lydian tonality often replaces the tonic Ionian during the time it is played for added colour.

For altered chords (on the dominant) we can have flat, sharp, or normal fifths and ninths, but we'd never play altered fifths or ninths with the natural fifths or ninths because it sounds horrible. Over A7alt, the scale we'd usually play is a Bb jazz melodic minor where the maj 6 and 7 are played both up and down, essentially a Bb min maj 7 chord with A root. Of course we can play on this chord as if it were a tonic. As you said, "And augmented 11th together with augmented 5th is essential chord beloved by impressionists for its power to destroy tonality, as it's made from whole-tone scale (C-E-G#-Bb-D-F#)". And as you see, there is no fifth present. There is also no E in Bb min maj 7 (i.e. A7alt).

Another common dom chord is 13th, so called because we don't want it to clash with the b7. This goes with a jazz dim scale, semitone-tone-semi starting from the root note of the chord. We can add all the altered 5ths and 9ths except +5 which clashes. This agrees with what you said that "Nothing to say about chords like 13/11+/9+/9-" and so on. Again, notice that there's no +5 with the 13.

However, this altered harmony is different than blues harmony where I IV and V are all 7th chords and we CAN play the blue notes b3 and b5 over the maj3 and perfect 5. It can be called b5 now because of cliche'd blues phrases such as 5 b5 4 b3 3. But since the scale C Eb F Gb G Bb contains both 4th and 5th, then the naming is up for grabs. This is often a point of confusion. Besides, blues players themselves couldn't care less what its called, which also says something. So when you say "Normal full 11+ chord has perfect fifth, and
> minor 7th, like C-E-G-Bb-D-F#", you too are confusing the two worlds. Notwithstanding serialism etc..., for C root we choose F# with B OR Gb with Bb. Why? Because just as the diminished chord can go up or down min 3rds and remain the same, or the aug by maj 3rds, so too the 7(b5) can go up or down b7. This is because they are the same chord: G7(b5) = G:B:Db:F = Db7(b5) = Db:F:G:B. Adding a D natural or Ab confuses this harmonic function with a flat 9 which is diminished based (i.e. G7(b9) has Ab dim harmonicity, but Db7(b9) doesn't lead to C tonic like Db7(b5)).

Finally, as you know C Eb G Bb D F A is called minor 11, min69, depending upon extension. Of course this is Dorian. But my original point was that this chord stands on equal ground as the maj7(#11). And if we assume that there is a rational JI counterpart, then the adoption of 19/16 in EVERY SINGLE ONE of these contexts will give far better approximations to the tempered intervals than 6/5, 32/27. Eg: 3/2 x 19/16 = 57/32 for b7. Then 57/32 x 5/4 = 285/128, 285/256 gives a new whole-tone etc...Or since the fifth is often the first to go in tonality, 5/4 x 19/16 = 95/64 gives a reasonable fifth with the correct tonic, as does 81/64 x 19/16 = 1539/1024 = 1.5029...(which is actually closer to 3/2). For bitonal/atonal symmetries, (19/16)^2 = 1.4101, a better JI sqrt2 than (6/5)^2 = 36/25 = 1.44. The list goes on.

So you see that I'm not trying to find new harmonies here but trying to explain the old ones correctly in terms of upper harmonics. I'm saying that 12 tone harmony has created new approaches to JI and is not necessarily just a bad compromise to trad JI. And I'll bet that these (my) intervals also sound closer to the whole-tone music of Debussy, the 12 tone of Shoernberg, Berg, and Webern, or the diminished scale and bitonality explored by Scriabin (spelt with a C not K btw) and Messiaen, than low-limit JI.

Cheers

-Rick

--- In tuning@yahoogroups.com, "Andreas Sparschuh" <a_sparschuh@...> wrote:
>
> --- In tuning@yahoogroups.com, "rick_ballan" <rick_ballan@> wrote:
> >
> > --- In tuning@yahoogroups.com, "Andreas Sparschuh" <a_sparschuh@> wrote:
> > So the problem comes back
> > to finding a decent minor third ratio
> > with the correct tonic i.e. 19/16.
> >
> Ok Rick,
> already Erasthostenes introduced 19-limit interval-ratios in his
> http://en.wikipedia.org/wiki/Tetrachord
> "Eratosthenes chromatic tetrachord 20:19, 19:18, 6:5"
> and Boethius considered just that 19/16 the context of:
> http://eamusic.dartmouth.edu/~larry/published_articles/divisions_of_the_tetrachord/chapter2.pdf
> Quote
> p.7
> "Boethius's tuning for the tetrachord...
>
> chromatic: 256/243 • 81/76 • 19/16
> ... p.8
> ... 19/16 is virtually the same size
> as the Pythagorean minor third, 32/27."
>
> Consider the quotient among them:
>
> 513/512 = (19/16):(32/27)
>
> that is labeled as
> http://www.xs4all.nl/~huygensf/doc/intervals.html
> "513/512 undevicesimal comma, Boethius' comma"
>
> and is examined in
> http://sonic-arts.org/monzo/marchet/marchet.htm
> "The difference between 2187/2048 and 81/76 is
>
> 2187:2048 = 2^-11 3^7
> ¸ 81:76 = - 2^-2 3^4 19^-1
> ___ ___ ___
> 2^-9 3^3 19^1
>
> or 513:512 [= 3.378 cents].
> Thus Marchetto's 81/76 "Diatonic Semitone"
> of 3 "dieses" is indeed very close to the
> "apotome" to which he equated it.
> Surprisingly, however, it is even closer to the 5-limit larger semitone, which a few centuries later also became known as the "diatonic". The progression 1/1 : 16/15 : 9/8
>
> 1/1 : 16/15 : 9/8 ....
>
> ...As indicated by the arrow,
> 19^1 is only approximately 0.04 Semitone
> 513/512 = 3.378 cents
> higher than 3^-3, the standard Pythagorean "minor 3rd".
> This small interval may be called the nondecimal schisma,
> and will be found useful in a thorough analysis of Marchetto's purported tuning, as well as several ancient Greek tunings using 19-Limit ratios.
> "
> Erasthostenes' analogous 19-limit approximation of 5-limit
> can be found in:
> http://tonalsoft.com/monzo/marchetto/marchetto.aspx
> "The difference between 16/15 and 81/76 is...
> ...the tiny interval 1216:1215 [= 1.42 cents]."
> that is labeled as:
> http://www.xs4all.nl/~huygensf/doc/intervals.html
> "1216/1215 Eratosthenes' comma"
>
> See for further details about that arguments in:
> /tuning/topicId_26618.html#26618
> and
> /tuning/topicId_66654.html#66663
> "
> C#> 19/18 == (256/243)*(513/512)
> b > 152/81 = (15/8)*(1216/1215)
> "
> or reverse express that both issues for that 2 cases as
>
> 1.) Pythagorean 3-limit approximation near to 19
>
> 2^9/3^3 = 512/27 = 18+(26/27) = ~18,962963... = 19-1/27
>
> 2.) and respectively as Syntonic 5-limit approx:
>
> 5*3^5/2^6 = 1215/64 = 18+(63/64) = ~18,984375... = 19-1/64
>
> Conclusion:
> That two arithmetical almost-coincidences
> 19-1/27 (in 3-lim-approx.) and 19-1/64 (in 5-lim-approx.)
> make the 19th partial-tone so unique within the harmonic series.
>
> kind regards
> A.S.

>Thanks Andreas,

You always give me all this great historical information, particularly Eratosthenes' 19-limit which I'd never heard of before (was he a post-Pythagorean?). The Boethius pdf is just blank and the intervals didn't come out either so I didn't quite catch that.

It's surprising that in all the list of intervals with their interesting historical names that 19/16 only gains the inglorious title "19th harmonic", this despite the fact that it is audibly almost the same as a Pythagorean minor third 32/27 (though I was aware of this interval I didn't know it was called that). This to me is made all the more surprising when we consider that it is the first minor third to have prime/2^N, which seems so basic considering 2/1, 3/2, 5/4, and 9/8. I wonder why that is? Is it because it needed ET to draw our attention to it, but since it wasn't recognised historically it didn't find a place in the pantheon of JI intervals? Poor lonely 19/16. At any rate I've nick-named it the JIET minor third because it is a fairly good rational approx. to the tempered minor.

Regards

Rick

Hi Rick,
I see that you wish to see the dominant 7th as 1/1 5/4 3/2 57/32 instead of
1/1 5/4 3/2 16/9
Why is this? The minor third here is not from the tonic and shouldn't
require your 8ve right?
It seems to me it is very clear what it should be. The function of the
dominant, subdominant and tonic chords.
It's for C major C 1/1 E 5/4 G 3/2 tonic major chord
G 3/2 B 15/8 D 9/4 dominant major chord
F 4/3 A 5/3 C 2/1 subdominant major chord
Clear relations of a 3/2 fifth between all these chords
Dominant major chord 4/3 3/2 15/8 9/4 or from G 1/1 5/4 3/2 16/9.
It seems so wrong to me to have a comma shift here and use 57/32 as the
minor 7th, or 171/128 for F
This is all the very basic major mode of 1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1 in
which the dominant and subdominant chords above make perfect sense.

Also as far as the 19/16 for minor chord from the tonic.
Couldn't it be that the 1/1 3/2 in 1/1 32/27 3/2 makes the tonic, and that
the minor third is more irrelivant?
And if not, 1215/1024 is 8ve too and could make sense too. It is incredibly
close to 19/16 btw (much closer than 32/27).

Marcel

--- In tuning@yahoogroups.com, Marcel de Velde <m.develde@...> wrote:
>
William wrote:
> > the 7th harmonic above a fundamental is below B-flat.
> > That's why when a quartet sings a dominant 7th chord,
> > the singer of the 7th is encouraged to
> > flatten his note until the chord locks.
>
Marcel asked about that:
> Is this really how it is tought in music schools?
Jutta Stüber reccomends just that in hers book:
http://www.orpheus-verlag.de/html/stuber_72.html
because she argues:
Quote:
"...Ein Mozart, Beethoven, Brahms komponierte keine temperierten Terzen, Quinten, Septimen, sondern reine, und der Hörer erwartet sie.
Das Ohr hört im Sinne der kleinen ganzzahligen Verhältnisse der reinen Stimmung. Alle Intervalle unseres Konzertrepertoires lassen sich auf wenige Grundintervalle zurückführen:
auf die Oktave 1:2,
die Quinte 2:3,
die große Terz 4:5
und die kleine Septime 4:7.
Alle anderen Intervalle
sind aus diesen vier
Grundintervallen gebildet."

My tr:

'Mozart Beethoven and Brahms never composed tempered
3rds, 5ths nor 7ths, but demaned pure intervals,
and the listerner expects them.
The ear percieves in the sense of small integral JI ratios.
All intervals in our's concert-repertoiare can be deduced
to barely a few basic-intervals:
the octave 1:2
the qunite 2:3
the tierce 4:5
and the minor-septime 4:7.
All other invervals are
are built from that four
basic-intervals.'

That's what i'm teach for my conservatory students.

> And they really try to sing the harmonic 7th
> for the dominant 7th chord in major mode?
http://www.bodensee-musikversand.de/product_info.php?products_id=143640
"Stüber, Jutta
Das Chorbuch zum Singen in reiner Stimmung Teil 1
In vielen Chorwettbewerben zeigt sich, daß ein stimmlich gut ausgestatteter, intonationsbewußt geleiteter Chor auf den zweiten oder dritten Platz verwiesen wird, weil er die Stimmtonhöhe nicht halten konnte und abgesunken war. Der Grund liegt in unserer Tonbenennung und -notierung, die zwischen Terzton (ein a als Terz von f), Quintton (ein a als Quinte von d) und Septton (ein a- als Sept von h) nicht unterscheidet. Jenes unterstrichene a liegt um ein "Komma", um den vierten Teil eines Halbtons, tiefer als das a der Quintenkette; und jenes a- ist bereits einen Viertelton (zwei Kommata) tiefer zu nehmen..."

My tr:
The choir-book to sing in JI, Part 1

In many choral competitions shows that a vocally well-equipped,
choir got deranked on the second or third place,
because the correct pitch could not be hold and was dropped down.
The reason for that arises from ours nomenclauture of
labeling pitches, that does not differ inbetween:
tierce-tone (a as third of f),
quint-tone (a as fifth of d)
and septim-tone (a- as Sept of b)
without differing each others among them.
Here the '-' means to flatten down
by a a "comma" 63:64 an fourth part of a semitone,
lowered than a within the 5ths-chain,
and that a- turns out already
a quarter tone (two commas) deeper downwards...

> If so I'm very surprised by this
> as my ear doesn't accept the harmonic 7th
> here.

Historically
already many centuries ago:

http://en.wikipedia.org/wiki/Archytas
"Archytas (Greek: Ἀρχύτας; 428â€"347 BC)"
introduced and specified that
and
http://en.wikipedia.org/wiki/Leonhard_Euler
"Leonhard Paul Euler (15 April 1707 â€" 18 September 1783)"

discussed it with his coevals about
the acceptance of such septimal 7-limit JI-intervals.
Appearently that controversy still remains
about the so called "blue-notes" 7/4, 7/6 and 7/5
and persist furthermore on more or less
as actual as over the last two millenias.

Conclusion:
Including 7-limit or even not is a matter of taste.

Kind Regards
A.S.

--- In tuning@yahoogroups.com, "William Gard" <billygard@...> wrote:
>> (Andreas wrote) In the key of C over the tonic chord, partial 7 can be rendered as b, although it is 31 cents lower than the tempered value
> (Billy wrote) Once again you wouldn't want to use a b to approximate a harmonic that would actually sound a hair flatter than b-flat. B would represent the 15th harmonic well.

I hazard a guess that Andreas was using German notation (intentionally or not) B=B flat, H=B natural.

No such excuse for the A flat though :-)

--- In tuning@yahoogroups.com, "rick_ballan" <rick_ballan@...> wrote:
> ... I'll bet that these (my) intervals also sound closer to the whole-tone music of Debussy, the 12 tone of Shoernberg, Berg, and Webern, or the diminished scale and bitonality explored by Scriabin (spelt with a C not K btw) and Messiaen, than low-limit JI.

I would urge caution when "correcting" the spelling of Russian names, since they have been transliterated from the Cyrillic alphabet and so there are often reasonable alternatives. Yes, Scriabin is usual in English, but I have seen Skriabin, Skryabin, Skryabine, etc.

BTW I have a dictionary of music that includes an entry for a composer called Chaykovskiy - I thought this was quite logical, but it didn't catch on!

And, ironically, you mis-spelled Schoenberg :-)

Steve M.

On 28 May 2009, at 2:54 PM, rick_ballan wrote:
> Hi Daniel,
>
> Almost everything you say here is true, but you're taking me way
> out of context and wrongly concluding that I don't know my harmony
> or history.
>
Hi, Rick,

I didn't say anything like this.
> In the original statement of this thread (I think to Marcel) I was
> talking about minor third as 19/16 being a far more viable option
> for minor tonality in all extant 12 tone harmony than 6/5 or 32/27
> etc which all have the wrong tonic. I was pointing out that while
> 6/5 might sound good in simple triads considered in isolation, that
> 19/16 explains much better extended four note or symmetric chords
> than successive applications of 6/5 etc which will accumulate wrong
> tonics over time. With A.S it became more about trying to explain
> these extended 4-note chords of modern jazz harmony in terms of the > harmonic series and traditional precedences to be found in Euler. I
> then pointed out that the 7th in dom7 chords might be better approx
> by 3/2 x 19/16 than the 7th harmonic (and then as an aside that the
> same could be said of 13 chords and the 13th harmonic as 13/8).
>
> For the record, C E G B D F# A is called maj7(#11) because it is
> the #11, not the b5 which comes in on dominant chords, like you said.
>
I don't understand quite well what you wanted to say here... BTW I
can imagine major chord C11/9/7maj/5-, for example. More interesting
would be C11/9-/7maj/5-, as there will be inverted symmetry between C-
E-B (47 in number of halftones) and Gb-Db-F (74)...
> Of course this Lydian tonality often replaces the tonic Ionian
> during the time it is played for added colour.
>
It depends on the context, but I don't see any reason why it should
be so. But everything is possible if it's intentional.
>
> For altered chords (on the dominant) we can have flat, sharp, or
> normal fifths and ninths,
>
If you mean diminished, augmented or perfect fifth, diminished,
augmented and major ninth, then terminology is OK :-) Not always
there's flat or sharp on those altered notes. That's the reason why I
always use neutral "-" and "+" for such alterations - it's universal
system independent on keys.
> but we'd never play altered fifths or ninths with the natural
> fifths or ninths because it sounds horrible.
>
Yes, it can't be combined. But we can have double altered chords,
like C5+/5- (and all extensions), or C9-/9+, and with 7maj also 13-/13
+ is possible. Very rarely used...

> Over A7alt, the scale we'd usually play is a Bb jazz melodic minor
> where the maj 6 and 7 are played both up and down, essentially a Bb
> min maj 7 chord with A root. Of course we can play on this chord as
> if it were a tonic.
>
I personally don't like those fixed simplistic assignments "certain
chord/certain scale", as scale used should depend more on the musical
context, not much on the chord structure itself. There's always more
possibilities, the most extreme is bitonality.

> As you said, "And augmented 11th together with augmented 5th is
> essential chord beloved by impressionists for its power to destroy
> tonality, as it's made from whole-tone scale (C-E-G#-Bb-D-F#)". And
> as you see, there is no fifth present.
>

I see here one augmented fifth in root chord :-)
> Another common dom chord is 13th, so called because we don't want
> it to clash with the b7.
>
I don't follow you here...

> However, this altered harmony is different than blues harmony where
> I IV and V are all 7th chords and we CAN play the blue notes b3 and
> b5 over the maj3 and perfect 5. It can be called b5 now because of
> cliche'd blues phrases such as 5 b5 4 b3 3.
>
It can't be called diminished 5th even when used melodically. Rules
valid for altered chromatic scale allows only rising the fourth, not
lowering 5th.

> But since the scale C Eb F Gb G Bb contains both 4th and 5th, then
> the naming is up for grabs.
>

Proper spelling is C-Eb-F-F#-G-Bb.

> This is often a point of confusion. Besides, blues players
> themselves couldn't care less what its called, which also says
> something. So when you say "Normal full 11+ chord has perfect
> fifth, and
> > minor 7th, like C-E-G-Bb-D-F#", you too are confusing the two
> worlds.
>
There's only one world. I'm living proof as a musician. And harmonic
rules are valid even in jazz music. It can't be changed by improper
use from the side of uneducated jazz amateurs.

> Notwithstanding serialism etc..., for C root we choose F# with B OR
> Gb with Bb.
>
I don't understand what you mean, if not just more comfortable
reading of the score. But if harmonic rules ask it, we must combine
flats and sharps.

> Why? Because just as the diminished chord can go up or down min
> 3rds and remain the same, or the aug by maj 3rds, so too the 7(b5)
> can go up or down b7.
>
Minor 7th? That must be wrong. You mean tritone probably, then yes
(when we ignore necessary enharmonic changes).

> This is because they are the same chord: G7(b5) = G:B:Db:F = Db7
> (b5) = Db:F:G:B.
>

Wrong enharmonic spelling: it must be Db-F-Abb-Cb, otherwise it's not
derived from the seventh chord. When you write Db-F-G-B it's just
second inversion of G-B-Db-F.

> Adding a D natural or Ab confuses this harmonic function with a
> flat 9 which is diminished based (i.e. G7(b9) has Ab dim
> harmonicity, but Db7(b9) doesn't lead to C tonic like Db7(b5)).
>

Don't understand. Such chords like 9-/5- are in use, of course, why
not? It gives Phrygian or Gipsy major scale character.
>
>
> Finally, as you know C Eb G Bb D F A is called minor 11, min69,
>
11 and 9/6 chords are totally different.

> depending upon extension. Of course this is Dorian. But my original
> point was that this chord stands on equal ground as the maj7(#11).
> And if we assume that there is a rational JI counterpart, then the
> adoption of 19/16 in EVERY SINGLE ONE of these contexts will give
> far better approximations to the tempered intervals than 6/5,
> 32/27. Eg: 3/2 x 19/16 = 57/32 for b7. Then 57/32 x 5/4 = 285/128,
> 285/256 gives a new whole-tone etc...Or since the fifth is often
> the first to go in tonality, 5/4 x 19/16 = 95/64 gives a reasonable
> fifth with the correct tonic, as does 81/64 x 19/16 = 1539/1024 =
> 1.5029...(which is actually closer to 3/2). For bitonal/atonal
> symmetries, (19/16)^2 = 1.4101, a better JI sqrt2 than (6/5)^2 =
> 36/25 = 1.44. The list goes on.
>
>
High math to me. In this purely theoretical disputation I don't
consider microtones or tempering. Just standard 12 ET.

> So you see that I'm not trying to find new harmonies here but
> trying to explain the old ones correctly in terms of upper harmonics.
>

Not everything in classical thirdal harmony can be explained from
upper harmonics, but there's high probability it was derived from it.

> I'm saying that 12 tone harmony has created new approaches to JI
> and is not necessarily just a bad compromise to trad JI. And I'll
> bet that these (my) intervals also sound closer to the whole-tone
> music of Debussy, the 12 tone of Shoernberg, Berg, and Webern, or
> the diminished scale and bitonality explored by Scriabin (spelt
> with a C not K btw) and Messiaen, than low-limit JI.
>
> Cheers
>
> -Rick
>

Skriabin was Russian composer (besides other languages I'm fluent in
Russian), so his name can be transliterated from azbuka in more ways
to the other languages. Skriabin is the nearest one, there's no
reason to use "c" for Russian "k". Azbuka has also special character
for "ia" (or "ya"), that's another reason for different
transliterations. In French they use Scriabine to avoid wrong
pronunciation. And Japanese write it in katakana as Sukurya-bin
(where dash means long pronunciation).

BTW, dear friend, before giving me language lessons, try please learn
how Schönberg is spelled.

Daniel Forró

With all the respect, Andreas, your answers often full of wikipedia
links are somehow funny to me. Looks like promotion campaign for
them. Or have you written some of those articles? Hopefully not, very
often there are mistakes, strange fomulations and conclusions in
music explanations. I don't like Wiki at all and use it rarely. It's
not enough reliable source of exact information despite all the
attempts. Maybe after another 100 years...

Yes, all manneristic periods in European music are interesting, I studied them deeply because of my personal interest in anything non-
conform, unusual and bizarre. That period about 1600, especially
seconda prattica and chromatic madrigal, especially Gesualdo, is
worth of study, I have learned a lot from it and could continue in
that direction (like to combine tonality with 12tone music). But they
had predecessors, like Solage, 200 years before. As for bizarre
chords and their using.

On 28 May 2009, at 4:52 AM, Andreas Sparschuh wrote:
> -- In tuning@yahoogroups.com, Daniel Forro <dan.for@...> wrote:
> Hi Daniel,
> Yet Monteverdi used
> http://en.wikipedia.org/wiki/Seventh_chord
> s http://www.associatedcontent.com/article/938795/
> claudio_monteverdi_and_his_secconda.html
> "He also expanded harmonic usage to include dominant seventh-chords
> and other dissonant chords without any harmonic preparation. During
> the early part of the seventeenth century, he developed a new
> compositional style he called the seconda prattica or second manner."
>
> Hence already for Monteverdi the so called
> http://en.wikipedia.org/wiki/Diatonic_functionality
> theory yiels nonsense, due to the lack of
> "any harmonic preparation".
>
> Then later the
> http://en.wikipedia.org/wiki/Ninth_chord
> was used in Bach's time
> http://books.google.de/books?
> id=xJxVRyiqZNIC&pg=PA205&lpg=PA205&dq=mozart+ninth-
> chord&source=bl&ots=hGoYcy3u1o&sig=km8zMH-60mRw-
> FZwiS3VpW7279Q&hl=de&ei=i2EdSrWiNoLD_Qbl2oWIDQ&sa=X&oi=book_result&ct=
> result&resnum=6
> as later did Mozart and Beethoven too,
> sometimes without careing whether 'functionality'
> should be imputed or even not?
>
> That inapplicable
> http://en.wikipedia.org/wiki/Functional_harmony
> theory works even worser for
> > 11th
> http://en.wikipedia.org/wiki/Eleventh
> http://en.wikipedia.org/wiki/Elektra_chord
> > and 13th
> discern here among
> http://en.wikipedia.org/wiki/Thirteenth
> and
> http://en.wikipedia.org/wiki/Dominant_thirteenth_chord
>
> >chords were build originally in advanced diatonic
> There is no such "advanced diatonic",
>

I have meant altered chromatic scale with all those artificially
added leading chromatic notes, and widened tonality.

>
> but only more or less generally built multi-chords from
> http://en.wikipedia.org/wiki/Upper_structure_triad
> s
> > classical harmony only on dominant chord
> >(Chopin, Schumann, Liszt, Wagner...),
>
Sorry, not only on dominant chord!
>
> But Wagner insited for his
> http://en.wikipedia.org/wiki/Tristan_chord
> that it has no clear root,
> hence also by-no-means an well-defined 'dominant' function.
> http://www.chameleongroup.org.uk/research/
> The_Tristan_Chord_in_Context.pdf
>
> > without chromatic shifts (but with omitted third to avoid
> > dissonance). Much later (Debussy's
>
> > and Skriabin...)'s
> http://en.wikipedia.org/wiki/Prometheus_chord
> > they were chromatized,
> > altered, especially when used in subdominant function
> Not at all,
> the Prometheus contains none intrinsic
> http://en.wikipedia.org/wiki/Tonality
> per-se.
>
>
I didn't mentioned Prometheus chord, at first its structure is
quartal, at second works where Skriabin used it have less to do with
that more traditional Chopin-Wagner-Liszt-Debussy line of widened
tonality.
> > (for example 11 +/5+)
> > and exploited on all chromatic steps in the key.
> But is not listed among the
> http://en.wikipedia.org/wiki/Chromaticism#Chromatic_chord
> s
> entries.
>
OK, time to include it :-) Music has much more possibilities then
Mickeypedia can imagine.
>
> "Any chord that is not chromatic is a diatonic chord."
> http://en.wikipedia.org/wiki/Diatonic_chord
> but that obsolete concept depends on the historical context.
> >
> > > (which is a chord that includes the perfect fifth).
> > The so-called blue notes are the 7th, the b5,
> > >
> > But in blues scale can't be diminished 5th, only augmented 4th.
> Here please be aware of
> http://en.wikipedia.org/wiki/Enharmonic
> spelling variants for the same pitch by different note-names
> due to the 12-EDO confusion that intermeshes
> sharp and flat accidentialy faulty in an wishy-washy misnomer:
> 'Compareing apples and oranges'
>
???
> alike already observed in the simpler case of the easy:
> http://en.wikipedia.org/wiki/Half-diminished_seventh_chord
> "The half-diminished chord has three functions in contemporary
> harmony: predominant function, diminished, and dominant function."
>
Never heard about predominant and diminished function. Must be
something new :-)

> > > and min3 played against a major chord (7(b5#9) type harmony), not
> > > necessarily in the chord. Honestly, I think that our extended
> > > chords have more to do with stacking alternate maj and min thirds
> > > in ET
>
> In my view,
> The primitive 12-ET graduation of barely a dozen pitch-casses
> is simply to much coarse for judgeing properly about
> the original intensions of the above composers.
>
But they composed music in 12 ET, armed with all that classical
theory based on this system, so what? I highly doubt about some
intensions going in the direction of tempering.
>
> Trying to explain all the above phenomenons only by stacking of 3rds
> is an undue oversimplification: Nothing doing, sorry I'm afraid.
>
Yes, I think the same, it was not my idea concerning that stacking...
> > ...as 13th chord was originally
> > used only in dominant function (check Liszt's Love dream No. 3 for
> > example).
>
> There in List's case
> the harmonic function
> depends on fine-tuning of the components among the
> principial constituents of that chord,
> if at all there is any detectable
> 'harmonic-function'
> present however deduced?
>
>
??? In this work he used diatonic 13th chord on dominant.
> > So
> > for C major it is G-B-D-F-A-C-E (G13 - normal diatonics without
> > altered notes).
> > Besides in the end there will be two small thirds
> > (F#- A, A-C).
> That both 3rds are only of the same seize when thrown into the
> "melting-pot" in conform 12-ET of all to much raw 100Cent steps.
>
Of course, that discussion was in the frame of 12 ET, not necessary
to mention microtonality.
> >Check Chopin's Nocturno C minor op. 48/1.
> That composition is contains quite a lot of ambiguous enharmonics,
> that no clear unique functional analysis can be achieved properly.
>
I don't agree at all. I don't see there any single enharmonics. It's
one of the most easily analyzable Chopin's work.
>
> >
> > > rather than anything
> > > to do with the harmonic series up to 11 or 13.
> probably because Chopin intended here to produce an obvious
> http://en.wikipedia.org/wiki/Consonance_and_dissonance#Dissonance
> by enriching the chords by additional inharmonics more coloured.
>
It has nothing to do with inharmonics, he used common thirdal harmony
here up to 13th chord. That's all.

> The usage of the 11th partial appeared earlier among the folks:
> http://www.people.iup.edu/rahkonen/ilwm/Switzerland.bib.htm
> "The instrument restricts players to the overtone series of the > 11th partial (F in a C scale) characteristically sharp, is commonly
> called alphorn fa. Until 1900 the alphorn was played as a solo
> instrument to pacify the cattle and send signals. "
> alike in the early 19th-century popular-songs
> http://www.informaworld.com/index/795292761.pdf
> "...articular, the 11th partial or overtone, often used in Swiss
> music and famously known as the "Alphorn-fa" ("fa" being the fourth
> pitch of a scale), ..."
> Listen the CD-records:
> http://www.artofthemix.org/findamix/Getcontents.asp?strMixId=118436
> "Three of the tones usually played on the alphorn do not occur in
> the tempered tone system. The 7th natural harmonic is a slightly
> high B; the 11th sounds distinctly higher than F, but too deep for
> F sharp, and the 13th is somewhat higher than A flat. The easiest
> of these three "wrong" tones to recognize is the 11th natural> harmonic. It is known as alphorn fa."
> http://www.jean-luc-darbellay.ch/frameset.php?
> lang=en&cat=discography&id=16
> http://en.wikipedia.org/wiki/Alpine_Symphony
> /tuning/topicId_63984.html#64367
>
>
Yes, of course, in folk music, because of using natural horn,
flutes... Maybe that's the reason why classical composers influenced by folklore and ethnic music of all kind started to use such chords
and melodic tones (but only imitating them in the frame of 12 ET).
Another interesting coincidence: they were mainly Slavonic composers
(Russian, Czech, Polish), or those living under influence of
Carpathian shepherd's music culture - Romanian (Enescu) and Hungarian
(Bartók, Kodály)... Alphorn and Debussy are exceptions (but he was
very much influenced by Russian music, especially Mussorgski).

> > And augmented 11th together with augmented 5th is essential chord
> > beloved by impressionists for its power to destroy tonality,
> > as it's made from whole-tone scale (C-E-G#-Bb-D-F#).
> not to mention
> http://en.wikipedia.org/wiki/Twelve-tone_technique
>

Not to mention, because discussion was not about destroying tonality.
>
> >
> > Nothing to say about chords like 13/11+/9+/9- (C-E-G-Bb-Db-D#-F#-A),
> > 13/11+/9+ (C-E-G-Bb-D#-F#-A), 13/11+/9- (C-E-G-Bb-Db-F#-A), 11+/9+
> > (C-E-G-Bb-D#-F#), 11+/9- (C-E-G-Bb-Db-F#) all of them opening door
> > to
> > the huge world of diminished scale and bitonality
> or in more generally terms:
> http://en.wikipedia.org/wiki/Polytonality
> "Bitonality is the use of only two different keys at the same time.
>
I would just omit "only".
> ....
> "Pre-twentieth-century instances of polytonality, such as Biber's
> "Battaglia" (1673) and Mozart's Ein musikalischer Spass (1787),..
> http://www.mozartforum.com/Lore/article.php?id=314
> ....tend to use the technique for programmatic or comic effect."
>
There are even more early bizarre examples, like Hans Neusiedler's
(1508-1563) Judentantz (1540). Extremely interesting attempt to
imitate Near East Jewish music.
> > explored by many
> > New music and jazz composers (just to name Skriabin and Messiaen).
>

Skriabin and polytonality? I wouldn't say. He didn't need it, he had
his own very logical system how to broke tonality. And Messiaen? It's
rather polymodality.
> >
> "The earliest uses of polytonality in non-programmatic contexts are
> found in the twentieth century, particularly in the work of Bartók
> (Fourteen Bagatelles, op. 6 [1908]), Ives (Variations on
> "America"), Stravinsky (Petrushka [1911]), and Debussy (Preludes,
> Book 2 [1913]). Ives claimed that he learned the technique of
> polytonality from his father, who taught him to sing popular songs
> in one key while harmonizing them in another...."
> or even
> http://en.wikipedia.org/wiki/Bimodality
> "It is more general than bitonality
> since the "scales" involved need not be traditional scales;
> if diatonic collections are involved,
> their pitch centers need not be
> the familiar major and minor-scale tonics."
> ...
> and by that digression we got completely off topic
> from the original subject
> that sounds even less consonant
> than an 6/5 minor-3rd.
>
> Kind Regards
> A.S.
>
But it was interesting trip. More interesting then eternal quarrels
here about hundredths of Cent :-)

Daniel Forró

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:
>
> --- In tuning@yahoogroups.com, "rick_ballan" <rick_ballan@> wrote:
> > ... I'll bet that these (my) intervals also sound closer to the whole-tone music of Debussy, the 12 tone of Shoernberg, Berg, and Webern, or the diminished scale and bitonality explored by Scriabin (spelt with a C not K btw) and Messiaen, than low-limit JI.
>
> I would urge caution when "correcting" the spelling of Russian names, since they have been transliterated from the Cyrillic alphabet and so there are often reasonable alternatives. Yes, Scriabin is usual in English, but I have seen Skriabin, Skryabin, Skryabine, etc.
>
> BTW I have a dictionary of music that includes an entry for a composer called Chaykovskiy - I thought this was quite logical, but it didn't catch on!
>
> And, ironically, you mis-spelled Schoenberg :-)
>
> Steve M.
>
Yes well it serves me right for being a smart arse. Point taken

Rick

--- In tuning@yahoogroups.com, Marcel de Velde <m.develde@...> wrote:
>
> Hi Rick,
> I see that you wish to see the dominant 7th as 1/1 5/4 3/2 57/32 instead of
> 1/1 5/4 3/2 16/9
> Why is this? The minor third here is not from the tonic and shouldn't
> require your 8ve right?
> It seems to me it is very clear what it should be. The function of the
> dominant, subdominant and tonic chords.
> It's for C major C 1/1 E 5/4 G 3/2 tonic major chord
> G 3/2 B 15/8 D 9/4 dominant major chord
> F 4/3 A 5/3 C 2/1 subdominant major chord
> Clear relations of a 3/2 fifth between all these chords
> Dominant major chord 4/3 3/2 15/8 9/4 or from G 1/1 5/4 3/2 16/9.
> It seems so wrong to me to have a comma shift here and use 57/32 as the
> minor 7th, or 171/128 for F
> This is all the very basic major mode of 1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1 in
> which the dominant and subdominant chords above make perfect sense.
>
> Also as far as the 19/16 for minor chord from the tonic.
> Couldn't it be that the 1/1 3/2 in 1/1 32/27 3/2 makes the tonic, and that
> the minor third is more irrelivant?
> And if not, 1215/1024 is 8ve too and could make sense too. It is incredibly
> close to 19/16 btw (much closer than 32/27).
>
> Marcel
>
Howdy Marcel,

In traditional classical major tonality where I and IV are maj triads and only V is 7th, your tuning will work fine. Inverted 4/3 and 3/2 for 7th of dominant is perfect and no further complication like introducing "19/16" etc. is needed. And for key changes to other maj keys I suppose you can just retune.

In Jazz we rarely play straight triads. Unless we are playing pop or country, we would play 7ths of some sort. The reason for this is because of the mov't of what are called "Guide Tones". The rule is that, as we cycle, 7ths become 3rds and 3rds become 7ths. Eg Cmaj7 to Fmaj7, C E G B becomes C E F A. The 7th of C, here a maj 7 B, moves to the 3rd of F, here a major 3rd A, while the 3rd of C (E) becomes the 7th of F (also E). Or we could turn the C into a dominant giving Bb to A, or F7 giving E to Eb. As an exercise on a piano, start with C E G B, move top two notes down the white keys for F maj 7, then the bottom two for B half dim, top two for E min 7, etc...till you reach C maj 7 again. You can modify chords along the way to become dom 7ths instead of min 7 etc...Observe also that we can play C major over the entire progression giving all the modes in their correct place.

Obviously your JI tuning will begin to break down over such progressions, which is where 19/16 might come in.

Given 1/1 32/27 3/2 the Pyth minor triad (switched Pyth major), the first (bottom) note seems to serve a double function as both 27 and 2. Since 2 overrides 27, then what you say is probably true i.e. we would here the bottom note as the tonic because of the presence of the perfect fifth.

Try this on a piano. C7 to F7 to Bb7 to Eb7 etc...Observe that the interval b/w 3rd and 7th (the guide tones) is a tritone and that these go down chromatically. Now cut out the fifths altogether. Eg Left hand goes C F Bb Eb... and right hand goes E-Bb, Eb-A, D-Ab, Db-G,...So you see that in this harmony at least, the fifths are not important. It is the guide tones that identify the chords. It is for reasons like this that I sensed that preserving both major and minor thirds was more important than preserving fifths (which are present as a strong harmonic anyway).

Rick

--- In tuning@yahoogroups.com, Marcel de Velde <m.develde@...> wrote:
> And they really try to sing the harmonic 7th for the dominant 7th chord in
> major mode?
> If so I'm very surprised by this as my ear doesn't accept the harmonic 7th
> here.
> G 3/2 B 15/8 D 9/4 F 21/8
> Instead G 3/2 B 15/8 D 9/4 F 8/3 sounds perfect to me and makes perfect
> sense in major mode of 1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1
>
I mainly was correcting the text that said the 7th harmonic is sharper than the ET minor 7th.

By the way, if you put the dominant 7th on the 3rd degree (EG#BD), in the JI major scale, using the 25/16 tuning for the G#, the 7th would be an even wider 9/5. You'd have to take the D down a comma to give it the same tuning as the dominant.

> Also the 13th chord based on the dominant 7th seems to me to be G 3/2 B
> 15/8 D 9/4 F 8/3 A 10/3 C 4/1 E 5/1
>
> Marcel
>

Yeah that is the true dominant 11th. But it sounds better with the third gone because the 11th is perfect.

Billy

--- In tuning@yahoogroups.com, "rick_ballan" <rick_ballan@...> wrote:
>> ... Scriabin ... Skriabin, Skryabin, Skryabine, etc.
> Yes well it serves me right for being a smart arse. Point taken

Yes well I couldn't add anything useful to the musical discussion, so ...

--- In tuning@yahoogroups.com, Daniel Forr� <dan.for@...> wrote:
>
Well observed Daniel,

> ... wikipedia links are somehow funny to me.
for me also too,
hence my carefully sceptic quotations.

> Looks like promotion campaign for them.
Never so intended.
It's time to disclaim an counterstatement:
Recommendation:
Always handle wiki-utterance-types with care,
and take it only with pinch of salt.

> Or have you written some of those articles?
None of them.

> Hopefully not, very
> often there are
unsustainable
> mistakes, strange
unreasonable
>fomulations and
wrong
> conclusions in music explanations.

> I don't like Wiki at all and use it rarely.
because it's still all to much poorly ill-concieved:
I'm fully aware that's barely half-backed.
Yet...
> It's not enough reliable source of exact information
> despite all the attempts.
Evidently.

> Maybe after another 100 years...
Or even later?
>
> Yes, all manneristic periods in European music are interesting,
Each of them possess individual specific quirks.

> I studied them deeply
> because of my personal interest in anything non-
> conform, unusual and bizarre.
with own special peculiarities.

> That period about 1600, especially
> seconda prattica and chromatic madrigal, especially Gesualdo,
here some scores:
http://imslp.org/wiki/Category:Gesualdo,_Carlo
that are
> worth of study,
> I have learned a lot from it and could continue in
> that direction (like to combine tonality with 12tone music).
Me too.

> But they had predecessors, like Solage, 200 years before.
for a score click on:
http://www.cpdl.org/wiki/index.php/Solage
http://www.cpdl.org/wiki/index.php/Fumeux_fume_par_fumee_%28Solage%29

> As for bizarre chords and their using.
or try that even old fragment of an score:
http://163.1.169.40/gsdl/collect/POxy/index/assoc/HASH351a.dir/POxy.v0015.n1786.a.01.hires.jpg

> But it was interesting trip.
I can return back that compliment

> More interesting then eternal quarrels
> here about hundredths of Cent :-)
That's at any rate nitpicking of counting beans,
due to the limits of pitch-discrimiantion discerning within the
http://en.wikipedia.org/wiki/Hearing_range
when distinguishing
http://en.wikipedia.org/wiki/Pitch_(music)
"The just noticeable difference
http://en.wikipedia.org/wiki/JND
(jnd, the threshold at which a change is perceived) depends on the tone's frequency and is about 4.3 cents (hundredths of a semitone) or about 0.36 Hz in frequency within the octave of 1,000â€"2,000 Hz but within the octave 62â€"125 Hz the jnd is much coarser with some 40 cents or about 2 Hz between perceived pitch changes."

against

http://www.pykett.org.uk/temperament_-_a_study_of_anachronism.htm
"
Appendix 2 â€" Arithmetical Precision required in Temperament Studies
The degree of precision required in numbers and arithmetic operations to do with temperament arises as follows...
A frequency tolerance of about 0.1 Hz at a frequency of about 1000 Hz implies a tuning accuracy of the order of 0.0001 or 0.01%. This is therefore also the precision required in temperament calculations which have to deliver the frequencies of the notes in a particular temperament. But because there are usually several steps in the calculation of each frequency, it is necessary that the numerical precision of the numbers used in each step is greater than that required in the final answer, otherwise the answer will not be accurate enough owing to truncation or rounding errors. Therefore at least one more significant figure is required throughout the calculations, meaning that numbers must be represented to at least a precision of 0.00001 or 0.001%. This is the same as a precision of 1 part in 100,000, or 6 significant figures,...

1200Cents * ln(1.00001)/ln(2) = ~0.017...Cents or ~1/58 Cents

Kind Regards
A.S.

--- In tuning@yahoogroups.com, Marcel de Velde <m.develde@...> wrote:
>...see the dominant 7th as 1/1 5/4 3/2 57/32 ...

with 57/32 = (7/4)*(57/56)
that's ~30.6Cents sharper above 7/4

>...instead of 1/1 5/4 3/2 16/9

with 16/9 = (7/4)*(15/14)
that's ~119.4 sharper above 7/4.

But personally i do prefer to view them both
as approximations of Euler's natural 7th quad-chord
4:5:6:7
without careing about the harmonic functionality.
That 4-chords consists simply in the harmonic-overtone-series
without the slightest deviations as in the above 2 alterations.

Kind Regards
A.S.

--- In tuning@yahoogroups.com, Daniel Forro <dan.for@...> wrote:
>
>
> On 28 May 2009, at 2:54 PM, rick_ballan wrote:

Daniel,

You said <There's only one world. I'm living proof as a musician. And harmonic rules are valid even in jazz music. It can't be changed by improper use from the side of uneducated jazz amateurs.>

I hope your not talking about me here. If so, I've obviously insulted you somewhere along the way for which I apologise. But for the record, I would say that harmonic rules are valid especially in jazz music, which has been my entire point. And since I make a living playing jazz and other stuff for money (do you call that a living?), I'm also living proof as a musician. (I also have a degree and a diploma of Jazz studies at the Conservatorium so lets leave it there).
(btw, "Scriabin" comment wasn't that serious and I can't remember how to get umlauts for "Shonberg" on the computer, so opted for "Shoernberg" but forgot the "r". Steve called me on it and I said "point taken". As for the rest,...

Hi, Rick,

I didn't say anything like this.
In the original statement of this thread (I think to Marcel) I was
talking about minor third as 19/16 being a far more viable option
for minor tonality in all extant 12 tone harmony than 6/5 or 32/27
etc which all have the wrong tonic. I was pointing out that while
6/5 might sound good in simple triads considered in isolation, that 19/16 explains much better extended four note or symmetric chords
than successive applications of 6/5 etc which will accumulate wrong
tonics over time...

For the record, C E G B D F# A is called maj7(#11) because it is
the #11, not the b5 which comes in on dominant chords, like you said.

I don't understand quite well what you wanted to say here... BTW I
can imagine major chord C11/9/7maj/5-, for example. More interesting
would be C11/9-/7maj/5-, as there will be inverted symmetry between C-E-B (47 in number of halftones) and Gb-Db-F (74)...

I was originally talking about the logic behind more standard jazz chords and why I opted for a JI tuning which preserves stacked maj-min thirds rather than fifths, and min 3 as 19/16. So the F# is both maj 3rd of D and #11 from C because it is not a Gb, G being taken. Even so, why would your first chord given here be major when it has no third? Could it be that this is still C F G B D F# i.e. the original maj chord with a sus 4 (although as a guitarist I would find it difficult to play F and F# together, so the F# would go)? Or re-voiced it could be a D13(#9), perhaps with the 7th C in the bass. As for your second chord, I'm not exactly sure what you mean since there is no E note. (Two successive semitones C B Db will always be tonally ambiguous).

Of course this Lydian tonality often replaces the tonic Ionian
during the time it is played for added colour.

It depends on the context, but I don't see any reason why it should
be so. But everything is possible if it's intentional.

Because it goes without saying that you'd play Lydian over the fourth degree chord. Naturally, intentionality is almost everything as you said. However, you and I already know the groundwork of harmony and so can recognise when something might be new and interesting, intentionally original. I just get annoyed when dilettante's use the "artistic 5th amendment" and try to cover their ignorance by screaming "I have the right to be free and modern".

For altered chords (on the dominant) we can have flat, sharp, or
normal fifths and ninths,

If you mean diminished, augmented or perfect fifth, diminished,
augmented and major ninth, then terminology is OK :-) Not always
there's flat or sharp on those altered notes. That's the reason why I
always use neutral "-" and "+" for such alterations - it's universal
system independent on keys.

When reading most jazz charts at a live gig where things go by quickly, "+" means augmented 5 and occurs either as a dominant 7 chord or a "rising 5" (eg James Bond for minor, or raining in my heart for major). To write a "+9" or "+7" for eg would read #5 with ninth or seventh.

but we'd never play altered fifths or ninths with the natural
fifths or ninths because it sounds horrible.

Yes, it can't be combined. But we can have double altered chords,
like C5+/5- (and all extensions), or C9-/9+, and with 7maj also 13-/13
+ is possible. Very rarely used...

Yes double altered's are technically just altered's. It is often left up to the player as to choice since no combo's will clash. But 7maj also 13-/13+ would usually occur in the rising/falling 5, for if 13- was played for some time the chord becomes maj from 6- degree (maj 3rds being the stronger interval), while 13+ is 7 and would sound awful with maj7, so very rare.
>
Over A7alt, the scale we'd usually play is a Bb jazz melodic minor
where the maj 6 and 7 are played both up and down, essentially a Bb
min maj 7 chord with A root. Of course we can play on this chord as
if it were a tonic.

I personally don't like those fixed simplistic assignments "certain
chord/certain scale", as scale used should depend more on the musical
context, not much on the chord structure itself. There's always more
possibilities, the most extreme is bitonality.

Well of course there are options in the choice of original chords and scales which enter the world together. And there is nothing simplistic about music. I personally don't like those fixed simplistic assignments such as "scales" because it is like studying the alphabet to learn to become a writer, which is only necessary as a first step. And bitonality only has meaning against the backdrop of tonality i.e chords, not ordered notes such as scales. It is in the choice of chords that bitonality takes off. For one simple example, using diminished chords as dominant and coming out in different keys, or even two keys at once.
>
As you said, "And augmented 11th together with augmented 5th is
essential chord beloved by impressionists for its power to destroy
tonality, as it's made from whole-tone scale (C-E-G#-Bb-D-F#)". And
as you see, there is no fifth present.
> >
>
I see here one augmented fifth in root chord :-)
Another common dom chord is 13th, so called because we don't want
it to clash with the b7.
> >
I don't follow you here...

First case, I meant there is no perfect fifth present since it's been raised. Second case, called 13th not 6th because it is a dom 7 chord with a 6th up the 8ve so as not to clash with the 7-.
>
However, this altered harmony is different than blues harmony where
I IV and V are all 7th chords and we CAN play the blue notes b3 and
b5 over the maj3 and perfect 5. It can be called b5 now because of
cliche'd blues phrases such as 5 b5 4 b3 3.
> >
It can't be called diminished 5th even when used melodically. Rules
valid for altered chromatic scale allows only rising the fourth, not
lowering 5th.

What, now you're talking about rules?? What about "I personally don't like those fixed simplistic assignments"? Who made up our minds for us that "Rules valid for altered chromatic scale" are to override every other "rule"?. It IS a flat fifth because it is preceded by the fifth. Again, this scale approach does not take ordered relations into account, the true meaning of context.
>
But since the scale C Eb F Gb G Bb contains both 4th and 5th, then
the naming is up for grabs.
> >
Proper spelling is C-Eb-F-F#-G-Bb.

Not necessarily, F-F#-G on the way up, G-Gb-F on the way down. Besides, the blue note is the flat-fifth, not the sharp fourth.
>
This is often a point of confusion. Besides, blues players
themselves couldn't care less what its called, which also says
something. So when you say "Normal full 11+ chord has perfect
fifth, and minor 7th, like C-E-G-Bb-D-F#", you too are confusing the two worlds.
> >
>
Notwithstanding serialism etc..., for C root we choose F# with B OR Gb with Bb.
> >
I don't understand what you mean, if not just more comfortable
reading of the score. But if harmonic rules ask it, we must combine
flats and sharps.

Yes that's true. But #11 and b5, though they are enharmonically equal in 12 tet, are different beasts.The reason I state below.
>
Why? Because just as the diminished chord can go up or down min
3rds and remain the same, or the aug by maj 3rds, so too the 7(b5)
can go up or down b7.
> >
Minor 7th? That must be wrong. You mean tritone probably, then yes
(when we ignore necessary enharmonic changes).

Yes sorry, tritone transposition.
>
This is because they are the same chord: G7(b5) = G:B:Db:F = Db7(b5) = Db:F:G:B.
> >
>
Wrong enharmonic spelling: it must be Db-F-Abb-Cb, otherwise it's not derived from the seventh chord. When you write Db-F-G-B it's just
second inversion of G-B-Db-F.

But I am in the key of C here, and Db7(b5) is a very standard substitution leading to a C tonic. I would never write Abb in the key of C, as you wouldn't either. The horn section would probably beat me up!
>
Adding a D natural or Ab confuses this harmonic function with a
flat 9 which is diminished based (i.e. G7(b9) has Ab dim
harmonicity, but Db7(b9) doesn't lead to C tonic like Db7(b5)).
> >
Don't understand. Such chords like 9-/5- are in use, of course, why
not? It gives Phrygian or Gipsy major scale character.

I'm saying that we wouldn't include the perfect fifths from G or Db roots in the 7(b5) chord because the b5 is an alteration to the fifth. We could play G7(b5b9) with an Ab, but there goes the Db substitution.

Finally, as you know C Eb G Bb D F A is called minor 11,

min69, 11 and 9/6 chords are totally different.

No they're not. We can almost always play an 11 with -69 without any clashes. They belong to the same group. The "vagueness" in certain jazz chord symbols is there for a reason to give leverage. There's nothing worse when people write charts trying to be "jazzy" with too many fancy chords, because jazz musicians will do this themselves automatically.
>
depending upon extension. Of course this is Dorian. But my original
point was that this chord stands on equal ground as the maj7(#11).
And if we assume that there is a rational JI counterpart, then the
adoption of 19/16 in EVERY SINGLE ONE of these contexts will give
far better approximations to the tempered intervals than 6/5,
32/27. Eg: 3/2 x 19/16 = 57/32 for b7. Then 57/32 x 5/4 = 285/128,
285/256 gives a new whole-tone etc...Or since the fifth is often
the first to go in tonality, 5/4 x 19/16 = 95/64 gives a reasonable
fifth with the correct tonic, as does 81/64 x 19/16 = 1539/1024 =
1.5029...(which is actually closer to 3/2). For bitonal/atonal
symmetries, (19/16)^2 = 1.4101, a better JI sqrt2 than (6/5)^2 =
36/25 = 1.44. The list goes on.
> >
> >
High math to me. In this purely theoretical disputation I don't
consider microtones or tempering. Just standard 12 ET.

But that's just the point. To truly understand 12 tet, and not just dismiss it as a popular bias, requires seeing objectively all of the JI harmonies, both old and new, that it subsumes. As i've said repeatedly, possibilities like 19/16 have completely been ignored historically because it has been stuck in the middle b/w an ongoing argument between JIists and ETists.
>
So you see that I'm not trying to find new harmonies here but
trying to explain the old ones correctly in terms of upper harmonics.
> >
>
Not everything in classical thirdal harmony can be explained from
upper harmonics, but there's high probability it was derived from it.
>
> > I'm saying that 12 tone harmony has created new approaches to JI
> > and is not necessarily just a bad compromise to trad JI. And I'll
> > bet that these (my) intervals also sound closer to the whole-tone
> > music of Debussy, the 12 tone of Shoernberg, Berg, and Webern, or
> > the diminished scale and bitonality explored by Scriabin (spelt
> > with a C not K btw) and Messiaen, than low-limit JI.
> >
> > Cheers
> >
> > -Rick
> >
>
> Skriabin was Russian composer (besides other languages I'm fluent in
> Russian), so his name can be transliterated from azbuka in more ways
> to the other languages. Skriabin is the nearest one, there's no
> reason to use "c" for Russian "k". Azbuka has also special character
> for "ia" (or "ya"), that's another reason for different
> transliterations. In French they use Scriabine to avoid wrong
> pronunciation. And Japanese write it in katakana as Sukurya-bin
> (where dash means long pronunciation).
>
> BTW, dear friend, before giving me language lessons, try please learn
> how Schönberg is spelled.
>
> Daniel Forró
>

On 30 May 2009, at 2:07 PM, rick_ballan wrote:

> I hope your not talking about me here. If so, I've obviously
> insulted you somewhere along the way for which I apologise.
>

Not at all, everything is OK :-)

> But for the record, I would say that harmonic rules are valid
> especially in jazz music, which has been my entire point. And since
> I make a living playing jazz and other stuff for money (do you call
> that a living?), I'm also living proof as a musician. (I also have
> a degree and a diploma of Jazz studies at the Conservatorium so
> lets leave it there).
> (btw, "Scriabin" comment wasn't that serious and I can't remember
> how to get umlauts for "Shonberg" on the computer, so opted for
> "Shoernberg" but forgot the "r".
>

I see no problem.

> I don't understand quite well what you wanted to say here... BTW I
> can imagine major chord C11/9/7maj/5-, for example. More interesting
> would be C11/9-/7maj/5-, as there will be inverted symmetry between
> C-E-B (47 in number of halftones) and Gb-Db-F (74)...
>
> I was originally talking about the logic behind more standard jazz
> chords and why I opted for a JI tuning which preserves stacked maj-
> min thirds rather than fifths, and min 3 as 19/16. So the F# is
> both maj 3rd of D and #11 from C because it is not a Gb, G being
> taken. Even so, why would your first chord given here be major when
> it has no third?
>

Of course it has third, it's not necessary to express third in the
chord, unless it's not written "Cm", "Cmi", or "C4", "C2", then it's
supposed automatically it's a chord with major third.
Concerning guitar power chords without third, I have seen chord signs
like "C5" for them.

> Could it be that this is still C F G B D F# i.e. the original maj
> chord with a sus 4 (although as a guitarist I would find it
> difficult to play F and F# together, so the F# would go)? Or re-
> voiced it could be a D13(#9), perhaps with the 7th C in the bass.
> As for your second chord, I'm not exactly sure what you mean since
> there is no E note.
>

Same as previous comment, so there's major third.

> (Two successive semitones C B Db will always be tonally ambiguous).
>

Not necessarily if root note is enough emphasized, and context is clear.

> Because it goes without saying that you'd play Lydian over the
> fourth degree chord. Naturally, intentionality is almost everything
> as you said. However, you and I already know the groundwork of
> harmony and so can recognise when something might be new and
> interesting, intentionally original. I just get annoyed when
> dilettante's use the "artistic 5th amendment" and try to cover
> their ignorance by screaming "I have the right to be free and modern".
>

Yes, there's some difference in the result between a person who knows
the old rules and breaks them intentionally, and a person who is just
chaotic.

> If you mean diminished, augmented or perfect fifth, diminished,
> augmented and major ninth, then terminology is OK :-) Not always
> there's flat or sharp on those altered notes. That's the reason why I
> always use neutral "-" and "+" for such alterations - it's universal
> system independent on keys.
>
> When reading most jazz charts at a live gig where things go by
> quickly, "+" means augmented 5 and occurs either as a dominant 7
> chord or a "rising 5" (eg James Bond for minor, or raining in my
> heart for major). To write a "+9" or "+7" for eg would read #5 with
> ninth or seventh.
>

That's the reason why I always write number first and then its shift,
not +9, but 9+.

> But I am in the key of C here, and Db7(b5) is a very standard
> substitution leading to a C tonic. I would never write Abb in the
> key of C, as you wouldn't either. The horn section would probably
> beat me up!
>

Yes, of course, that's the proof there is some difference between
pure theory and practical approach. But enharmonic changes done from
practical reason (better score readability) should be done carefully.
Too often I see in officially published jazz and pop scores very
strange things, like F#mi chord in Db major key, probably because
editor considered that proper minor subdominant third Bbb too
difficult to read or what.

> But that's just the point. To truly understand 12 tet, and not just
> dismiss it as a popular bias, requires seeing objectively all of
> the JI harmonies, both old and new, that it subsumes. As i've said
> repeatedly, possibilities like 19/16 have completely been ignored
> historically because it has been stuck in the middle b/w an ongoing
> argument between JIists and ETists.
>

But jazz doesn't use microtones, at least not in the score, it's
always 12 ET only (not to count blue notes, glissandos, glides,
smears, maybe certain tendency to just tuning of section chords in
bigbands, or vocal groups a capella... or just detuned instruments).

Daniel Forró

--- In tuning@yahoogroups.com, Daniel Forro <dan.for@...> wrote:
>
>
> On 30 May 2009, at 2:07 PM, rick_ballan wrote:
>
> > I hope your not talking about me here. If so, I've obviously
> > insulted you somewhere along the way for which I apologise.
> >
>
> Not at all, everything is OK :-)

Ah that's good.
>
> > But for the record, I would say that harmonic rules are valid
> > especially in jazz music, which has been my entire point. And since
> > I make a living playing jazz and other stuff for money (do you call
> > that a living?), I'm also living proof as a musician. (I also have
> > a degree and a diploma of Jazz studies at the Conservatorium so
> > lets leave it there).
> > (btw, "Scriabin" comment wasn't that serious and I can't remember
> > how to get umlauts for "Shonberg" on the computer, so opted for
> > "Shoernberg" but forgot the "r".
> >
>
> I see no problem.
>
> > I don't understand quite well what you wanted to say here... BTW I
> > can imagine major chord C11/9/7maj/5-, for example. More interesting
> > would be C11/9-/7maj/5-, as there will be inverted symmetry between
> > C-E-B (47 in number of halftones) and Gb-Db-F (74)...
> >
> > I was originally talking about the logic behind more standard jazz
> > chords and why I opted for a JI tuning which preserves stacked maj-
> > min thirds rather than fifths, and min 3 as 19/16. So the F# is
> > both maj 3rd of D and #11 from C because it is not a Gb, G being
> > taken. Even so, why would your first chord given here be major when
> > it has no third?
> >
>
> Of course it has third, it's not necessary to express third in the
> chord, unless it's not written "Cm", "Cmi", or "C4", "C2", then it's
> supposed automatically it's a chord with major third.
> Concerning guitar power chords without third, I have seen chord signs
> like "C5" for them.

Oh I see, C as in C major.
>
> > Could it be that this is still C F G B D F# i.e. the original maj
> > chord with a sus 4 (although as a guitarist I would find it
> > difficult to play F and F# together, so the F# would go)? Or re-
> > voiced it could be a D13(#9), perhaps with the 7th C in the bass.
> > As for your second chord, I'm not exactly sure what you mean since
> > there is no E note.
> >
>
> Same as previous comment, so there's major third.
>
> > (Two successive semitones C B Db will always be tonally ambiguous).
> >
>
> Not necessarily if root note is enough emphasized, and context is clear.
>
I was thinking about this. On the one hand, if I play a C root and try to play 2 successive semitones (3 notes) over it anywhere on the piano, I can't see a single instance where it doesn't break some rule (combining altered 5ths with perfect, maj with 7- etc...). In fact, its even possible that our rules unconsciously intend to avoid this at some level. On the other, I often play phrases of semi's at either side of notes, and who knows what will sound right with other weird and wonderful tunings.

> > Because it goes without saying that you'd play Lydian over the
> > fourth degree chord. Naturally, intentionality is almost everything
> > as you said. However, you and I already know the groundwork of
> > harmony and so can recognise when something might be new and
> > interesting, intentionally original. I just get annoyed when
> > dilettante's use the "artistic 5th amendment" and try to cover
> > their ignorance by screaming "I have the right to be free and modern".
> >
>
> Yes, there's some difference in the result between a person who knows
> the old rules and breaks them intentionally, and a person who is just
> chaotic.
>
> > If you mean diminished, augmented or perfect fifth, diminished,
> > augmented and major ninth, then terminology is OK :-) Not always
> > there's flat or sharp on those altered notes. That's the reason why I
> > always use neutral "-" and "+" for such alterations - it's universal
> > system independent on keys.
> >
> > When reading most jazz charts at a live gig where things go by
> > quickly, "+" means augmented 5 and occurs either as a dominant 7
> > chord or a "rising 5" (eg James Bond for minor, or raining in my
> > heart for major). To write a "+9" or "+7" for eg would read #5 with
> > ninth or seventh.
> >
>
> That's the reason why I always write number first and then its shift,
> not +9, but 9+.
>
> > But I am in the key of C here, and Db7(b5) is a very standard
> > substitution leading to a C tonic. I would never write Abb in the
> > key of C, as you wouldn't either. The horn section would probably
> > beat me up!
> >
>
> Yes, of course, that's the proof there is some difference between
> pure theory and practical approach. But enharmonic changes done from
> practical reason (better score readability) should be done carefully.
> Too often I see in officially published jazz and pop scores very
> strange things, like F#mi chord in Db major key, probably because
> editor considered that proper minor subdominant third Bbb too
> difficult to read or what.

Tell me about it. Most of those scores I suspect are written by old ladies who make a living accompanying high school vocalists on the piano, you know great readers, more like a human piano accordians. What's often worse is the choice of chords, you know "I've got rythm" (Bb Eb F) X 4.
>
> > But that's just the point. To truly understand 12 tet, and not just
> > dismiss it as a popular bias, requires seeing objectively all of
> > the JI harmonies, both old and new, that it subsumes. As i've said
> > repeatedly, possibilities like 19/16 have completely been ignored
> > historically because it has been stuck in the middle b/w an ongoing
> > argument between JIists and ETists.
> >
>
> But jazz doesn't use microtones, at least not in the score, it's
> always 12 ET only (not to count blue notes, glissandos, glides,
> smears, maybe certain tendency to just tuning of section chords in
> bigbands, or vocal groups a capella... or just detuned instruments).
>
> Daniel Forró
>
Yes that's true. However my original topic was that it is impossible to play actual irrational numbers (i.e. ET intervals) in the real world, since they have an infinite number of digits after the decimal point. So for eg sqrt 2, 4+ interval, is not really 1.41421356...all the way to infinity but probably a four digit approx such as 181/128 = 1.414062, an odd harmonic over the 7th 8ve. If this is true then 12 ET is a form of JI already. Since Marcel was trying to retune Beethoven in JI as an experiment, because Ludwig also composed in 12 ET I suggested 19/16 as minor third since its much closer to 12 ET than 6/5 and has a correct tonic-denominator, that's all.

-Rick