Yahoo Tuning Groups Ultimate Backup tuning the "math" of the minor triad

--- In tuning@yahoogroups.com, "octatonic10" <octatonic10@...> wrote:
>
> Hello! i'm doing a research paper for school on the harmonic
> series and logarithmic perception of pitch, comparing the major
> and minor triads, and this list came up in my search.

Welcome!

> For the root position, closed, major triad (eg, c4-e4-g4), the
> major 3rd and minor 3rd actually have the same frequency ratio,
> but because of our logarithmic perception of pitch, they are of
> different size (4, as compared to 3 semitones.)

They don't have the same frequency ratio. In equal temperament,
the major third's frequency ratio is about 1.2599 and the minor
third's is about 1.1892. In just intonation they are 1.25 and
1.2 exactly.

But you might have meant that, in just intonation, each pitch
in a major triad is higher than the previous one by a constant
number of Hertz. For instance, 100 Hz, 125Hz & 150 Hz is a
major triad in just intonation. The major and minor thirds
each span 25 Hz.

This is true of any chord that is made of successive harmonics.
The major triad is 4:5:6, but the chord 7:8:9 has the same
property.

It isn't quite true in equal temperament though. The major
triad on 100 Hz would be 100 Hz, 125.9920 Hz & 149.8273 Hz.
Here the major third spans 25.9920 Hz, while the minor third
spans 23.8389 Hz.

Let me know if you need any help with the arithmetic to check
these figures.

-Carl

🔗Carl Lumma <carl@...>

🔗Marcel de Velde <m.develde@...>

Great explenation by Carl allready.But may I add that in Just Intonation /
rational intonation / harmonic series there are several possibilities for a
minor triad.
The 10:12:15 mentioned by Carl is the most used and a mirror image of the
4:5:6 major triad.
Other possibilities are 6:7:9, 1/1 7/6 3/2 with a septimal minor third that
is much lower than the regular 6/5 minor third (though I'm personally unsure
about the use of septimal intervals in common practice music).
Or 27:32:40, 1/1 32/27 40/27 with a minor third very close to the 12 tone
equal tempered minor third but wit a wolf fifth (I personally belief this
one is much used in common practice music alongside 10:12:15).
Some belief there are other possibilities aswell like 16:19:24.
All of the above minor triads have the same keys on a equal tempered piano
for instance d-f-a.
The possibilities and the use/place of these minor triads is often debated
on this list and will likely continue to be, there is no clear agreed upon
by all answer.
I don't know if all this information is of any use to you, but incase it is
here you have it :)

Marcel

2009/10/14 Carl Lumma <carl@...>

> They don't have the same frequency ratio. In equal temperament,
> the major third's frequency ratio is about 1.2599 and the minor
> third's is about 1.1892. In just intonation they are 1.25 and
> 1.2 exactly.
>
> But you might have meant that, in just intonation, each pitch
> in a major triad is higher than the previous one by a constant
> number of Hertz. For instance, 100 Hz, 125Hz & 150 Hz is a
> major triad in just intonation. The major and minor thirds
> each span 25 Hz.
>

🔗octatonic10 <octatonic10@...>

🔗Carl Lumma <carl@...>

Parncutt is a good one to read. Let us know how your
project/report turns out! -Carl

--- In tuning@yahoogroups.com, "octatonic10" <octatonic10@...> wrote:
>
> reading a little more on the subject, eg, Richard Parncutt's
> Harmony and Psychoacoutics, I think the minor triad is a good
> example of a chord which suggests we may not hear the intervals
> as frequency ratios. In any case, I"m struck by the vast
> difference in music psychology between tuning theorists and
> psychoacousticians who relate acoustics to tonality.
>
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
> >
> > Sorry, octatonic10, missed your second paragraph:
> >
> > > As you know, with a root position, closed, minor triad
> > > (eg, d4-f4-a4), the frequency of the two intervals is not
> > > the same.
> >
> > I think the way to say this might be, "With a root position
> > minor triad (e.g. 10:12:15), the frequency differences making
> > up the major and minor thirds are not the same".
> >
> > > but is their perceived size as measure by scale step altered,
> > > due to logarithimc perception, as with the minor triad?
> >
> > Yes, the logarithmic perception of pitch is always working.
> > A minor third is perceived to be the same size regardless of
> > what pitch it is rooted on (at least within the center few
> > octaves of the typical range of musical pitches). And major
> > thirds are always perceived to be larger, regardless of
> > where they are rooted.
> >
> > -Carl
> >
>

🔗Chris <chrisvaisvil@...>

May not hear the intervals as frequency ratios?

Could someone expnad on that idea / statement a little bit?

Thanks

Chris.
Sent via BlackBerry from T-Mobile

-----Original Message-----
From: "Carl Lumma" <carl@lumma.org>
Date: Fri, 16 Oct 2009 17:31:03
To: <tuning@yahoogroups.com>
Subject: [tuning] Re: the "math" of the minor triad

Parncutt is a good one to read. Let us know how your
project/report turns out! -Carl

🔗octatonic10 <octatonic10@...>

🔗Carl Lumma <carl@...>

🔗Marcel de Velde <m.develde@...>

🔗Chris Vaisvil <chrisvaisvil@...>

Not sure if this was answered and I missed it or not.

If " the minor triad is a good example of a chord which suggests we may not
hear the intervals as frequency ratios' is true - doesn't that have huge
implications for JI? Or am I just uninformed?

On Fri, Oct 16, 2009 at 1:57 PM, Chris <chrisvaisvil@...> wrote:

> May not hear the intervals as frequency ratios?
>
> Could someone expnad on that idea / statement a little bit?
>
> Thanks
>
> Chris.
>
> Sent via BlackBerry from T-Mobile
> ------------------------------
> *From: * "Carl Lumma" <carl@...>
> *Date: *Fri, 16 Oct 2009 17:31:03 -0000
> *To: *<tuning@yahoogroups.com>
> *Subject: *[tuning] Re: the "math" of the minor triad
>
>
>
> Parncutt is a good one to read. Let us know how your
> project/report turns out! -Carl
>
> --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, "octatonic10"
> <octatonic10@...> wrote:
> >
> > reading a little more on the subject, eg, Richard Parncutt's
> > Harmony and Psychoacoutics, I think the minor triad is a good
> > example of a chord which suggests we may not hear the intervals
> > as frequency ratios. In any case, I"m struck by the vast
> > difference in music psychology between tuning theorists and
> > psychoacousticians who relate acoustics to tonality.
> >
> >
> > --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, "Carl Lumma"
> <carl@> wrote:
> > >
> > > Sorry, octatonic10, missed your second paragraph:
> > >
> > > > As you know, with a root position, closed, minor triad
> > > > (eg, d4-f4-a4), the frequency of the two intervals is not
> > > > the same.
> > >
> > > I think the way to say this might be, "With a root position
> > > minor triad (e.g. 10:12:15), the frequency differences making
> > > up the major and minor thirds are not the same".
> > >
> > > > but is their perceived size as measure by scale step altered,
> > > > due to logarithimc perception, as with the minor triad?
> > >
> > > Yes, the logarithmic perception of pitch is always working.
> > > A minor third is perceived to be the same size regardless of
> > > what pitch it is rooted on (at least within the center few
> > > octaves of the typical range of musical pitches). And major
> > > thirds are always perceived to be larger, regardless of
> > > where they are rooted.
> > >
> > > -Carl
> > >
> >
>
>
>

🔗Mike Battaglia <battaglia01@...>

What exactly does it mean that we "don't hear the intervals as frequency
ratios"? We don't hear anything as frequency ratios. We hear frequency
ratios as intervals...
-Mike

On Mon, Oct 19, 2009 at 7:59 PM, Chris Vaisvil <chrisvaisvil@...>wrote:

>
>
> Not sure if this was answered and I missed it or not.
>
> If " the minor triad is a good example of a chord which suggests we may not
> hear the intervals as frequency ratios' is true - doesn't that have huge
> implications for JI? Or am I just uninformed?
>
>
> On Fri, Oct 16, 2009 at 1:57 PM, Chris <chrisvaisvil@...> wrote:
>
>> May not hear the intervals as frequency ratios?
>>
>> Could someone expnad on that idea / statement a little bit?
>>
>> Thanks
>>
>> Chris.
>>
>> Sent via BlackBerry from T-Mobile
>> ------------------------------
>> *From: *"Carl Lumma" <carl@...>
>> *Date: *Fri, 16 Oct 2009 17:31:03 -0000
>> *To: *<tuning@yahoogroups.com>
>> *Subject: *[tuning] Re: the "math" of the minor triad
>>
>>
>>
>> Parncutt is a good one to read. Let us know how your
>> project/report turns out! -Carl
>>
>> --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, "octatonic10"
>> <octatonic10@...> wrote:
>> >
>> > reading a little more on the subject, eg, Richard Parncutt's
>> > Harmony and Psychoacoutics, I think the minor triad is a good
>> > example of a chord which suggests we may not hear the intervals
>> > as frequency ratios. In any case, I"m struck by the vast
>> > difference in music psychology between tuning theorists and
>> > psychoacousticians who relate acoustics to tonality.
>> >
>> >
>> > --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, "Carl Lumma"
>> <carl@> wrote:
>> > >
>> > > Sorry, octatonic10, missed your second paragraph:
>> > >
>> > > > As you know, with a root position, closed, minor triad
>> > > > (eg, d4-f4-a4), the frequency of the two intervals is not
>> > > > the same.
>> > >
>> > > I think the way to say this might be, "With a root position
>> > > minor triad (e.g. 10:12:15), the frequency differences making
>> > > up the major and minor thirds are not the same".
>> > >
>> > > > but is their perceived size as measure by scale step altered,
>> > > > due to logarithimc perception, as with the minor triad?
>> > >
>> > > Yes, the logarithmic perception of pitch is always working.
>> > > A minor third is perceived to be the same size regardless of
>> > > what pitch it is rooted on (at least within the center few
>> > > octaves of the typical range of musical pitches). And major
>> > > thirds are always perceived to be larger, regardless of
>> > > where they are rooted.
>> > >
>> > > -Carl
>> > >
>> >
>>
>>
>
>

🔗Carl Lumma <carl@...>

🔗Chris Vaisvil <chrisvaisvil@...>

here you go Carl.

Its your reply to the original - the original has that statement.

On Fri, Oct 16, 2009 at 1:31 PM, Carl Lumma <carl@...> wrote:

>
>
> Parncutt is a good one to read. Let us know how your
> project/report turns out! -Carl
>
> --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, "octatonic10"
> <octatonic10@...> wrote:
> >
> > reading a little more on the subject, eg, Richard Parncutt's
> > Harmony and Psychoacoutics, I think the minor triad is a good
> > example of a chord which suggests we may not hear the intervals
> > as frequency ratios. In any case, I"m struck by the vast
> > difference in music psychology between tuning theorists and
> > psychoacousticians who relate acoustics to tonality.
> >
> >
> > --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, "Carl Lumma"
> <carl@> wrote:
> > >
> > > Sorry, octatonic10, missed your second paragraph:
> > >
> > > > As you know, with a root position, closed, minor triad
> > > > (eg, d4-f4-a4), the frequency of the two intervals is not
> > > > the same.
> > >
> > > I think the way to say this might be, "With a root position
> > > minor triad (e.g. 10:12:15), the frequency differences making
> > > up the major and minor thirds are not the same".
> > >
> > > > but is their perceived size as measure by scale step altered,
> > > > due to logarithimc perception, as with the minor triad?
> > >
> > > Yes, the logarithmic perception of pitch is always working.
> > > A minor third is perceived to be the same size regardless of
> > > what pitch it is rooted on (at least within the center few
> > > octaves of the typical range of musical pitches). And major
> > > thirds are always perceived to be larger, regardless of
> > > where they are rooted.
> > >
> > > -Carl
> > >
> >
>
>
>

🔗Marcel de Velde <m.develde@...>

🔗Carl Lumma <carl@...>

🔗monz <joemonz@...>

I didn't really follow this thread much, but from what
i've read here, my suspicion is that it may be a reference
to dualism -- the possibility that we can perceive ratios
(or really, proportions containing 2 or more ratios)
as either otonalities or utonalities, that is, either
harmonic or subharmonic.

For example, the typical JI minor triad with ratios
1:1 - 6:5 - 3:2 can be analyzed as a proportion either
harmonically as 10:12:15, or subharmonically as
1/6:1/5:1/4 or to write it more neatly 1/(6:5:4).
Obviously in this case the subharmonic (i.e., utonal)
analysis provides smaller numbers as therefore can be
construed as the more valid interpretation.

However, there is a _lot_ of debate among tuning theorists,
and among music theorists in general, about this duality theory.
Some subscribe to it wholeheartedly (Oettingen, Hugo Riemann,
Partch), others think it's a bunch of nonsense, and most fall
somewhere in between. I personally am close to the pole
that subscribes to it.

For those who do subscribe to dualism, it clearly points
to 1:1 - 6:5 - 3:2 as being the most consonant set of
ratios which will provide the aesthetic experience of
a "minor triad", because the utonal 1/(6:5:4) analysis
provides the smallest integer ratio terms which can
result in a "minor triad" sound.

Those who do not subscribe to dualism would probably
say that 1:1 - 7:6 - 3:2 is a much more consonant
minor-triad, because its otonal (harmonic) analysis
is 6:7:9, compared to the 10:12:15 proportion of the
previous example.

The crux of the debate centers around how strongly
the harmonic series acts as a template or archtype
into which our auditory perception of simultaneous
(or arpeggiated) sounds tries to fit those sounds.
Some people just refuse to accept that we can hear
proportions "upside down", and thus they discard dualism.

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

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> OK, I see it. I don't know exactly what Melford meant by
> that. I can speculate he's referring to the issue of there
> being several equally valid JI triads that sound like
> 'the minor triad'.
>
> -Carl
>
> --- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@> wrote:
> >
> > here you go Carl.
> >
> > Its your reply to the original - the original has that statement.
> >
> > > --- In tuning@yahoogroups.com, "octatonic10" wrote:
> > > >
> > > > reading a little more on the subject, eg, Richard Parncutt's
> > > > Harmony and Psychoacoutics, I think the minor triad is a good
> > > > example of a chord which suggests we may not hear the intervals
> > > > as frequency ratios.
>

🔗gdsecor <gdsecor@...>

--- In tuning@yahoogroups.com, "monz" <joemonz@...> wrote:
>
> I didn't really follow this thread much, but from what
> i've read here, my suspicion is that it may be a reference
> to dualism -- the possibility that we can perceive ratios
> (or really, proportions containing 2 or more ratios)
> as either otonalities or utonalities, that is, either
> harmonic or subharmonic.
>
> For example, the typical JI minor triad with ratios
> 1:1 - 6:5 - 3:2 can be analyzed as a proportion either
> harmonically as 10:12:15, or subharmonically as
> 1/6:1/5:1/4 or to write it more neatly 1/(6:5:4).
> Obviously in this case the subharmonic (i.e., utonal)
> analysis provides smaller numbers as therefore can be
> construed as the more valid interpretation.
>
> However, there is a _lot_ of debate among tuning theorists,
> and among music theorists in general, about this duality theory.
> Some subscribe to it wholeheartedly (Oettingen, Hugo Riemann,
> Partch), others think it's a bunch of nonsense, and most fall
> somewhere in between. I personally am close to the pole
> that subscribes to it.
>
> For those who do subscribe to dualism, it clearly points
> to 1:1 - 6:5 - 3:2 as being the most consonant set of
> ratios which will provide the aesthetic experience of
> a "minor triad", because the utonal 1/(6:5:4) analysis
> provides the smallest integer ratio terms which can
> result in a "minor triad" sound.
>
> Those who do not subscribe to dualism would probably
> say that 1:1 - 7:6 - 3:2 is a much more consonant
> minor-triad, because its otonal (harmonic) analysis
> is 6:7:9, compared to the 10:12:15 proportion of the
> previous example.
>
> The crux of the debate centers around how strongly
> the harmonic series acts as a template or archtype
> into which our auditory perception of simultaneous
> (or arpeggiated) sounds tries to fit those sounds.
> Some people just refuse to accept that we can hear
> proportions "upside down", and thus they discard dualism.
>
> -monz
> http://tonalsoft.com/tonescape.aspx
> Tonescape microtonal music software

Regardless of what view one subscribes to, one must take into account that there is virtually unanimous agreement that the root of a 1/(6:5:4) (or 10:12:15) triad is 1/6 (or 10), not 1/4 (or 15).

--George

🔗a_sparschuh <a_sparschuh@...>

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

> Regardless of what view one subscribes to,
> one must take into account
> that there is virtually unanimous agreement
> that the root of a 1/(6:5:4) (or 10:12:15) triad
> is 1/6 (or 10), not 1/4 (or 15) ...
>
Hi George,
...or even taken as common root 4
alike in the major chord 4:5:6
when understanding the minor 3rd as
an chromtically 25:24 downwards alterated

http://en.wikipedia.org/wiki/Picardy_third

6:5 = (5/4):(25/24)

alike explained in
http://de.wikipedia.org/wiki/Moll
by Paul Hindemith.

hence the correspoding minor-triad

[10:12:15] = 10*[1:(6/5):(3/2)] = (5/2)*[4 : 4.8 : 6]

can also be understand as the double ratio

[4 : 4/(25:24) : 6]
==
[4 : 4.8 : 6]
==
(2/5)*[10:12:15]
==
4*[1:(6/5):(3/2)]

which are all equivalent representations of the minor-chord.
Which of that do you prefer?

bye
A.S.

🔗Michael <djtrancendance@...>

Far as dualism, I am of the opinion that
1) A chord formed out of sub-harmonics will be less consonant than one formed by harmonics IE 1/(6:5:4) would be less consonant than the chord 4:5:6. I suspect this is because
a) Harmonics of real life instruments, which the ear seems to be designed to hear clearly, are in non-reverse order
b) In non-reverse order smaller intervals take place at higher frequencies, where the ear is able to tolerate closer intervals between frequencies without beating issues (of course, beating contributes to dissonance).
For example if you compare the chord 400hz:500hz:600hz to 400hz :481hz:600hz (apx. a 1/(6:5:4) chord)...the most beating ratio in the first chord (600/500) is higher while the most beating ratio in the second (481/400) is lower and, as we know from Plomp and Llevelt's dissonance curves, the ear has more tolerance for close ratios at higher frequencies.

2) It's not an either/or situation. 1/(6:5:4) may very well be more consonant than, say 6:7:9 much because the 7:6 ratio beats more than any two frequencies in the 1/(6:5:4). In that case, psy-acoustics (in terms of Helmholtz's beating theory) seems to more than cancel out the advantages of periodicity gained by using harmonics instead of sub-harmonics.

So, to try and sum it up, I believe that chords based on harmonics generally sound better UNLESS the compared chord with sub-harmonics which either beat significantly less and/or have their closest ratios at much lower frequencies.

________________________________
From: monz <joemonz@...>
To: tuning@yahoogroups.com
Sent: Tue, October 20, 2009 12:22:44 PM
Subject: [tuning] Re: the "math" of the minor triad

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

--- In tuning@yahoogroups. com, "Carl Lumma" <carl@...> wrote:
>
> OK, I see it. I don't know exactly what Melford meant by
> that. I can speculate he's referring to the issue of there
> being several equally valid JI triads that sound like
> 'the minor triad'.
>
> -Carl
>
> --- In tuning@yahoogroups. com, Chris Vaisvil <chrisvaisvil@ > wrote:
> >
> > here you go Carl.
> >
> > Its your reply to the original - the original has that statement.
> >
> > > --- In tuning@yahoogroups. com, "octatonic10" wrote:
> > > >
> > > > reading a little more on the subject, eg, Richard Parncutt's
> > > > Harmony and Psychoacoutics, I think the minor triad is a good
> > > > example of a chord which suggests we may not hear the intervals
> > > > as frequency ratios.
>

🔗Marcel de Velde <m.develde@...>

>
> Regardless of what view one subscribes to, one must take into account that
> there is virtually unanimous agreement that the root of a 1/(6:5:4) (or
> 10:12:15) triad is 1/6 (or 10), not 1/4 (or 15).

What exactly is the defenition of the root of a chord in JI sense?

Wiki says the following:
When the root is the bass note <http://en.wikipedia.org/wiki/Bass_note>, or
lowest note, of the expressed chord, the chord is in *root position*. This
may also be described as in normal
form<http://en.wikipedia.org/wiki/Normal_form>,
as opposed to inverted form <http://en.wikipedia.org/wiki/Inversion_(music)>.
When the root is not the lowest pitch played in a chord, it is
inverted<http://en.wikipedia.org/wiki/Inversion_(music)>
.

The concept of root has some basis in the physical properties of waves. When
two notes of an interval <http://en.wikipedia.org/wiki/Interval_(music)> from
the harmonic series are played at the same time, people sometimes perceive
the fundamental note of the interval. For example, if notes with frequency
ratios of 7:6 (a septimal minor
third<http://en.wikipedia.org/wiki/Septimal_minor_third>)
were played, people could perceive a note whose frequency was 1/6th of the
lower interval. The following sound file demonstrates this phenomenon,
using sine waves <http://en.wikipedia.org/wiki/Sine_waves>, pure and simple
waves for which this phenomena is most easily evident.

But I'm not sure if I can agree to a link between the phantom fundamental
and the root of a chord.
Especially in case fo the 10:12:15 minor chord which doesn't have a phantom
fundamental of 10.
So I may be a subscriber of utonal.

But about the "root" of a chord.
I think I'd like to think more in terms of tonic of a chord.
In this sense the "root" tonic of a 1/1 6/5 3/2 chord is 1/1 only if 1/1 is
the tonic.
We also find 10:12:15 chord as 1/1 5/4 5/3 where 1/1 is the tonic. I don't
think the "root" of the chord is 5/3 in this case.
We also find it at 1/1 4/3 8/5 and 9/8 3/2 9/5.
And also the wolf minor triad of 9/8 4/3 5/3, and an unnamed minor of 8/5
15/8 6/5
I think one could very well say that all of these above chords in the tonic
of 1/1 have their origin in 1/1 and therefore one could say 1/1 is the
"root" or the source of the chord.

As for how the origin of these chord could come from the tonic, I recently
realised that my old harmonic interval permutation theory could explain it
all.
Take the harmonic series and construct intervals be rearanging the order of
the harmonics (on can describe it in many ways but this one is easy to
understand)
So take for instance 1/1 2/1 3/1 harmonic series.
Then one can also make 1/1 3/2 3/1.
We could say that when limiting the harmonic series to the third harmonic
that one gets the intervals 1/1 3/2 2/1 3/1 by all possible permutations of
these harmonics.
We could say 3/1 has as it's source 1/1 and can be constructed of either a
2/1 and a 3/2, or a 3/2 and a 2/1
When limiting till the fourth harmonic you get the following harmonic
interval permutation product set:
1/1 4/3 3/2 2/1 8/3 3/1 4/1

When limiting till the fifth harmonic you get the following harmonic
interval permutation product set:
1/1 5/4 4/3 3/2 5/3 15/8 2/1 5/2 8/3 3/2 10/3 15/4 4/1 5/1

When limiting till the sixth harmonic you get the following harmonic
interval permutation product set:
1/1 6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8 2/1 9/4 12/5 5/2 8/3 3/1 16/5 10/3 18/5
15/4 4/1 9/2 24/5 5/1 6/1

Put in one octave you get 1/1 9/8 6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8 2/1

These are 100% utonal <-> tonal equal sets btw.

All of these intervals in the above scales a the simplest intervals comming
from 1/1 in their construction withing their harmonic limit.
All of these intervals in these scales can be related to 1/1 and can be said
to be comming from 1/1.
I recently realised this is a great defenition of tonality and tonic!
I now use the above 6-limit tonality scale as my basis for tuning and
determining the tonic and it's amazing how well it works and makes sense.
It fits all chords it seems, for instance 9/8 4/3 8/5 15/8 we only find in
this place, and it indeed has as it's tonic 1/1 etc etc
I also find most scales, except some maqam, gypsy scales and the whole tone
scale.
These scales may come from a modulation, or they may be harmonic 7-limit.
Harmonic 7-limit will give the following tonality scale in one octave btw:
1/1 21/20 35/32 9/8 7/6 6/5 5/4 21/16 4/3 7/5 35/24 3/2 14/9 8/5 5/3 7/4 9/5
28/15 15/8 35/18 2/1

Still experimenting a lot but this all could explain a lot.

Marcel

🔗monz <joemonz@...>

🔗Mike Battaglia <battaglia01@...>

> Regardless of what view one subscribes to, one must take into account that there is virtually unanimous agreement that the root of a 1/(6:5:4) (or 10:12:15) triad is 1/6 (or 10), not 1/4 (or 15).
>
> --George

An interesting thing I've always noticed about 10:12:15 is that I can
feel it out as having two distinct roots - the 10 and the 12. If the
chord is inverted, as in 12:15:20, the strength of the 12 becomes
considerably stronger than that of the 10. This is even more so if a
fifth is added above the 12, making it into a major 6 chord -- I think
that chord is a case study in how the brain perceives multiple
harmonic structures in one chord, while never really "fusing" the
whole thing together into a solid 1. An m7 and a maj6 chord are
perceived with very different roots, although they are harmonic
inversions of one another. All of this disregards the bottom bass note
that the listener "imagines" when trying to harmonically analyze a
chord (even in a rootless voicing), and is probably what people argue
about when they talk about how to define the "root" of a chord.

I tried this experiment one time where I played a 10:12:15 polyrhythm
- I played accents on every 4th sixteenth note, every 5th, and every
6th. While grasping the periodicity of the whole pattern (hearing the
"1") was way beyond me, I was easily drawn to the repeating pattern of
the 2:3 polyrhythm, and the 4:5 polyrhythm as well -- the 5:6 was
there, but wasn't as easy to "find". I actually heard the 4:5 first,
with the other rhythm just being perceived as noise in the background,
but once I heard the 2:3, it was hard to get away from it and focus on
anything else.

I do agree though with the fact that whatever a root is (if it can
even be rigorously defined at all), it certainly isn't the 15 of
10:12:15. The only duality I've ever sensed in the minor chord is the
conflict between the 10 and the 12 as being the root (and sometimes a
2, rarely a 1). But then again, doesn't that have to do with how we
imagine the chord anyway?

-Mike

🔗Petr Parízek <p.parizek@...>

🔗a_sparschuh <a_sparschuh@...>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> The only duality I've ever sensed in the minor chord is the
> conflict between the 10 and the 12 as being the root
> (and sometimes a 2, rarely a 1).
> But then again, doesn't that have to do with how we
> imagine the chord anyway?
>
in deed Mike,
simply compare the many views of various theorists
about voice-doublings, as listed in:

http://mto.societymusictheory.org/issues/mto.04.10.2/mto.04.10.2.aarden_hippel_2.1.2.html

that stem from the different emphases in opinions about

http://mto.societymusictheory.org/issues/mto.04.10.2/mto.04.10.2.aarden_hippel_2.1.1.html
"
2.1.1 Past Doubling Theories

[1] There are four general categories of theories that are used to justify various rules of chord-tone doubling. These are based on:

* harmonic series,
* sonority,
* key structure, and
* voice-leading.

Doubling to Fit a Harmonic Series

[2] The most popular theory of voice-doubling, one that now dominates modern thinking about doubling rules, holds that a triad is an extension of its root, and the organization of pitches within a triad serves to emphasize the root. The scaffolding for that organization is the harmonic series, and the fundamental of the harmonics is the root of the triad. Thus the most obvious PC to double is that of the root....

Approach:
Take the 5-limit restricted partials
out of the general overtone series by
firstly neglecting everything above 7-limit:

1/1 : 2 : 3 : [4:5:6] : 8 : 9 : [10:12:15] : 16 : 18 : 20 : 24 : 25...
http://www.research.att.com/~njas/sequences/A051037

then in that initially sense
all the two triads of
major as minor do both
refer to the common root base 1/1
simplified to 3-chords:

1/1...[4:5:6]...[10:12:15]...

Then construct more properly two better suited subsets
out of the 7-limit partial-series:

1/1:2:3:[4:5:6]:(7):8:9:[10:12...]:(14):[...15]:16:18:20:21...
http://www.research.att.com/~njas/sequences/A002473

That concept allows to consider and compare the similar series

1/1 ... [4:5:6:7] ... [10:12:14:15] ....

in the both modes: major and minor
when understood as real 4-chord tonality generalization
over the same common root 1/1 respectively.

My personal conclusion:
The above given 7-limit generalizations of the two modes
major-minor do allow to get rid of the tonality-confusion,
that arised when persisting within 5-limit restriction,
because it turns out to be easier
to distinct [4:5:6:7] vs. [10:12:14:15] quad-chordic modes,
than barely [4:5:6] vs. [10:12:15] tri-chordic subsets,
at least in my own ears perception.

http://dict.leo.org/forum/selectThreadByDate.php?lp=ende&date=20060324104352
"
4-note chord, 5-note chord
Kommentar If hexachord means any 6-note chord, as the Wikipedia entry for Chord (music) says, then triangle, quadrilateral, pentagon, hexagon
implies trichord, quadrichord, pentachord, hexachord.
But frankly I doubt Wikipedia - hexachord is just a series of notes by my dictionary. I'm playing safe....
Kommentar support Paul
We have "triad" for Dreiklang, but it stops there. German is more flexible here....
...not to be confused with the ancient:
http://www.medieval.org/emfaq/harmony/pyth2.html
'In fact, Pythagorean tuning is described in the medieval sources as being based on four numbers: 12:9:8:6. Jacobus of Liege (c. 1325) describes a "quadrichord" with four strings having these lengths:'

bye
A.S.

🔗monz <joemonz@...>

Hi Marcel,

This is exactly the idea behind Paul Erlich's
idea of "relative harmonic entropy". I have a
definition of it in my Encyclopedia:

http://tonalsoft.com/enc/h/harmonic-entropy.aspx

... but i give the link with the caveat that, regarding
the more detailed pages that i wrote on harmonic entropy
which can be found by a Google search, Paul expressed
some problems with my explanation and i never got around
to correcting it.

There's a whole Yahoo group devoted to discussing
harmonic entropy:

/harmonic_entropy/

... i might add that since Paul disappeared from these
groups a couple of years ago, there's been essentially
no activity on that one.

-- so you'd better look thru the archives
of the Yahoo group in addition to reading my pages.

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

--- In tuning@yahoogroups.com, Marcel de Velde <m.develde@...> wrote:
>
> I don't think the ear is precise enough to hear JI.
> But even very mistuned intervals can still be interpreted
> by the brain as JI I think, so that when you listen to
> 12tet you still hear the musical structure (which I think
> is a pure JI structure)
>
> Marcel

🔗Michael <djtrancendance@...>

🔗Marcel de Velde <m.develde@...>

🔗gdsecor <gdsecor@...>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > Regardless of what view one subscribes to, one must take into account that there is virtually unanimous agreement that the root of a 1/(6:5:4) (or 10:12:15) triad is 1/6 (or 10), not 1/4 (or 15).
> >
> > --George

I've been very pressed for time lately & didn't intend to get into a detailed discussion about this, since this is *not* a subject about which I'm strongly opinionated. I was just trying to contribute what I thought was a simple and clear-cut observation that would shed some light on the matter. If some don't agree, that's okay by me -- I have to pick & choose my battles, and this isn't one of them.

> An interesting thing I've always noticed about 10:12:15 is that I can
> feel it out as having two distinct roots - the 10 and the 12. If the
> chord is inverted, as in 12:15:20, the strength of the 12 becomes
> considerably stronger than that of the 10. This is even more so if a
> fifth is added above the 12, making it into a major 6 chord -- I think
> that chord is a case study in how the brain perceives multiple
> harmonic structures in one chord, while never really "fusing" the
> whole thing together into a solid 1. An m7 and a maj6 chord are
> perceived with very different roots, although they are harmonic
> inversions of one another.

I agree, and in response I'll contribute another, more general, observation that I made in a discussion on tuning-math about 6 years ago (concerning constant structures):
/tuning-math/messages/7664
I noted that:

<< Now we could go on to ask why this scale-member identity or functionality is so important, and this is the point at which I really had to dig deep for an answer. I believe that, at least with the examples given above, it has something to do with the role that the simplest ratios of 3 play in establishing the roots of chords. If a chord contains a *single* 2:3 or 3:4 (whether just or tempered), then I can almost guarantee that the tone represented by the 3 will *never* be heard as the root of the chord. (There are instances, e.g., 8:10:15, that a tone not in the 2:3 interval will be perceived as the root, but that's not critical to the point that I'm making.) It is this property of the simple ratios of 3 that makes it possible to *invert* many conventional triads and seventh chords *without* changing our *perception* of which note of the chord functions as the *root*. >>

My observation doesn't tell you what be the root of a chord *is*, but rather what it *isn't*.

Just my 2 cents, for what it's worth.

--George

🔗Charles Lucy <lucy@...>

The root is only significant if one is considering scales/triads etc.

The way in which I have constructed my scalecoding enables you to immediately recognise patterns (or megmodes as I call them) regardless of the root.

You can also find all scales which will contain particular intervals by doing a search. e.g. all scales which use the bIII interval (L+s).

Check it out on FileMaker from:

http://www.lucytune.com/scales/

The emotional aspects seem to derived from scales in particular, although other aspects are also involved; particularly rhythmic elements including tempo/time sig etc.

On 21 Oct 2009, at 14:31, Michael wrote:

>
> These posts about "what's the true perceived root of certain minor > triads" all beg the question: why is it so important which note in > the chord the brain finds to be the "root"?
> And that includes the sub questions
> A) Does which note is the root effect perceived consonance?
> B) Does it effect tonal color?
> C) Is there any gain (listed above or otherwise) to be had from > convincing the brain different notes are roots?
>
> Sometimes I worry...this forum reaches too far into "tuning math > for the sake of math alone" and veers away from how it relates to > emotional moods,
> tension, and other concepts musicians actually use on a regular > basis regardless of tuning (or even in drum-based music with no > obvious tuning). What's the greater goal here?
>
>
Charles Lucy
lucy@...

- Promoting global harmony through LucyTuning -

for information on LucyTuning go to:
http://www.lucytune.com

For LucyTuned Lullabies go to:
http://www.lullabies.co.uk

🔗Chris Vaisvil <chrisvaisvil@...>

"why is it so important which note in the chord the brain finds to be the
"root"?"

If you are writing tonal music it can be important. It determines what chord
progression you can write under common practice.

"A) Does which note is the root effect perceived consonance?"

Yes. Inversions sound different.

"B) Does it effect tonal color?"

Yes, any difference does.

"C) Is there any gain (listed above or otherwise) to be had from convincing
the brain different notes are roots?"

Yes. I think it would be fair to say lots of Jazz and Impressionistic music
does that - but with extended chords.
If one is talking only in the realm of 3 note chords built of consecutive
thirds (and inversion) there is less to play with.

On Wed, Oct 21, 2009 at 9:31 AM, Michael <djtrancendance@...> wrote:

>
>
> These posts about "what's the true perceived root of certain minor triads"
> all beg the question: why is it so important which note in the chord the
> brain finds to be the "root"?
> And that includes the sub questions
> A) Does which note is the root effect perceived consonance?
> B) Does it effect tonal color?
> C) Is there any gain (listed above or otherwise) to be had from convincing
> the brain different notes are roots?
>
> Sometimes I worry...this forum reaches too far into "tuning math for the
> sake of math alone" and veers away from how it relates to emotional moods,
> tension, and other concepts musicians actually use on a regular basis
> regardless of tuning (or even in drum-based music with no obvious tuning).
> What's the greater goal here?
>
>

🔗duckfeetbilly <billygard@...>

🔗Kraig Grady <kraiggrady@...>

🔗a_sparschuh <a_sparschuh@...>

🔗cameron <misterbobro@...>

Any universal claims about chord roots are silly. Take for instance the cadential six-four: even tuned in the most tonally unambiguous way possible, with the octaves of the 3d and 5th partials as the chord members, the most simple major triad does not survive all voicings with root clarity intact.

Especially voiced in a low register, the six-four, aka second inversion, with the 3/2 in the bass, making the lowest interval a fourth, is tonally ambiguous and tends strongly, depending on the situation, to sound like a 4-3 sus. Common practice 101, surely everyone here remembers this.

On the other hand you can verify for youself that wacko dissonant chords, say high-limit chords of a heavily "diminished" nature, can sometimes, depending on all the factors involved, have roots clearly where they "should" be according to the mostly, but not entirely, idiotic concept of chords being in "first position" at their most dense voicing.

So simply universal claims about what makes roots: I don't buy it.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> These posts about "what's the true perceived root of certain minor triads" all beg the question: why is it so important which note in the chord the brain finds to be the "root"?
> And that includes the sub questions
> A) Does which note is the root effect perceived consonance?
> B) Does it effect tonal color?
> C) Is there any gain (listed above or otherwise) to be had from convincing the brain different notes are roots?
>
> Sometimes I worry...this forum reaches too far into "tuning math for the sake of math alone" and veers away from how it relates to emotional moods,
> tension, and other concepts musicians actually use on a regular basis regardless of tuning (or even in drum-based music with no obvious tuning). What's the greater goal here?
>

🔗cameron <misterbobro@...>

You are presenting a false dichotomy.

--- In tuning@yahoogroups.com, "monz" <joemonz@...> wrote:
>
> I didn't really follow this thread much, but from what
> i've read here, my suspicion is that it may be a reference
> to dualism -- the possibility that we can perceive ratios
> (or really, proportions containing 2 or more ratios)
> as either otonalities or utonalities, that is, either
> harmonic or subharmonic.
>
> For example, the typical JI minor triad with ratios
> 1:1 - 6:5 - 3:2 can be analyzed as a proportion either
> harmonically as 10:12:15, or subharmonically as
> 1/6:1/5:1/4 or to write it more neatly 1/(6:5:4).
> Obviously in this case the subharmonic (i.e., utonal)
> analysis provides smaller numbers as therefore can be
> construed as the more valid interpretation.
>
> However, there is a _lot_ of debate among tuning theorists,
> and among music theorists in general, about this duality theory.
> Some subscribe to it wholeheartedly (Oettingen, Hugo Riemann,
> Partch), others think it's a bunch of nonsense, and most fall
> somewhere in between. I personally am close to the pole
> that subscribes to it.
>
> For those who do subscribe to dualism, it clearly points
> to 1:1 - 6:5 - 3:2 as being the most consonant set of
> ratios which will provide the aesthetic experience of
> a "minor triad", because the utonal 1/(6:5:4) analysis
> provides the smallest integer ratio terms which can
> result in a "minor triad" sound.
>
> Those who do not subscribe to dualism would probably
> say that 1:1 - 7:6 - 3:2 is a much more consonant
> minor-triad, because its otonal (harmonic) analysis
> is 6:7:9, compared to the 10:12:15 proportion of the
> previous example.
>
> The crux of the debate centers around how strongly
> the harmonic series acts as a template or archtype
> into which our auditory perception of simultaneous
> (or arpeggiated) sounds tries to fit those sounds.
> Some people just refuse to accept that we can hear
> proportions "upside down", and thus they discard dualism.
>
>
> -monz
> http://tonalsoft.com/tonescape.aspx
> Tonescape microtonal music software
>
>
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
> >
> > OK, I see it. I don't know exactly what Melford meant by
> > that. I can speculate he's referring to the issue of there
> > being several equally valid JI triads that sound like
> > 'the minor triad'.
> >
> > -Carl
> >
> > --- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@> wrote:
> > >
> > > here you go Carl.
> > >
> > > Its your reply to the original - the original has that statement.
> > >
> > > > --- In tuning@yahoogroups.com, "octatonic10" wrote:
> > > > >
> > > > > reading a little more on the subject, eg, Richard Parncutt's
> > > > > Harmony and Psychoacoutics, I think the minor triad is a good
> > > > > example of a chord which suggests we may not hear the intervals
> > > > > as frequency ratios.
> >
>

🔗monz <joemonz@...>

Hi Cameron,

Well, hopefully your simple statement of a fact
is not implying that i am the inventor of that
false dichotomy, because i certainly am not.

But what makes it false? Because it certainly _is_
a dichotomy. It _is_ possible to analyze any collection
of frequencies as a subset of either the harmonic
or subharmonic series.

If the notes are tuned exactly to a rational intonation
(such as JI), then the analysis should be entirely
straightforward.

If the notes are tuned to a temperament, then which
subset is chosen depends on the arbitrary choice of
how close the approximation to JI is supposed to be,
and in that case, there is even a lot of room for
the arbitrary choice between harmonic and subharmonic.

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

--- In tuning@yahoogroups.com, "cameron" <misterbobro@...> wrote:
>
> You are presenting a false dichotomy.
>
> --- In tuning@yahoogroups.com, "monz" <joemonz@> wrote:
> >
> > I didn't really follow this thread much, but from what
> > i've read here, my suspicion is that it may be a reference
> > to dualism -- the possibility that we can perceive ratios
> > (or really, proportions containing 2 or more ratios)
> > as either otonalities or utonalities, that is, either
> > harmonic or subharmonic.
>
> <snip>

🔗Marcel de Velde <m.develde@...>

Hello Cameron,

> Any universal claims about chord roots are silly. Take for instance the
> cadential six-four: even tuned in the most tonally unambiguous way possible,
> with the octaves of the 3d and 5th partials as the chord members, the most
> simple major triad does not survive all voicings with root clarity intact.
>
> Especially voiced in a low register, the six-four, aka second inversion,
> with the 3/2 in the bass, making the lowest interval a fourth, is tonally
> ambiguous and tends strongly, depending on the situation, to sound like a
> 4-3 sus. Common practice 101, surely everyone here remembers this.
>
> On the other hand you can verify for youself that wacko dissonant chords,
> say high-limit chords of a heavily "diminished" nature, can sometimes,
> depending on all the factors involved, have roots clearly where they
> "should" be according to the mostly, but not entirely, idiotic concept of
> chords being in "first position" at their most dense voicing.
>
> So simply universal claims about what makes roots: I don't buy it.
>

What if a single chord like for instance a major chord can have many roots.
What if the root of a chord doesn't even have to be in the chord itself.

I'm thinking myself the root of all chords is the tonic.
And that there is a fixed JI scale from the tonic which has it's origin in
the tonic.
Right now I think this tonality scale for common practice music is all
possible permutations of the harmonic series limited to the sixth harmonic.
This gives the following tonality scale (when reduced to one octave): 1/1
9/8 6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8 2/1
In this system for instance a dominant major 7th chord would be 4/3 3/2 15/8
9/4 and have as it's source / root 1/1.

I can't think of a single example that would couteract this line of
thinking.
If you can please give an example.

Marcel
www.develde.net

🔗Michael <djtrancendance@...>

Marcel>"Right now I think this tonality scale for common practice music is
all possible permutations of the harmonic series limited to the sixth
harmonic.
This gives the following tonality scale (when reduced to one octave): 1/1 9/8 6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8 2/1"

I am a bit confused:
A) How do 9/8 and 15/8 fit into the 6th harmonic, rather than, say, the 8th harmonic?
B) How to 6/5, 8/5, and 9/5 fit into the idea (as I understand your saying) of everything pointing to a single root note?

I can understand how, say, 4/3 and 3/2 can point to a similar root as the third (/3) and second (/2) harmonic are a 5th apart and 5ths lend themselves well to pointing at a root tone.

However, I don't see why you are using x/5 fractions instead of, say, x/9 ones: personally I'd try 11/9 instead of 6/5, 13/8 instead of 8/5, and 16/9 instead of 9/5. The idea being that x/9 and x/8 format fractions are exponentially related to the x/3 and x/2 format fractions (since 3^2 = 9 and 2^3 = 8)...reducing back to the (3/2) fifth ratio which tends the promote/point-to a single tone.

One thing I whole-heartedly agree with you on, Marcel (if I understand you well), is that one very solid way to produce a very pure sounding scale is to arrange ratios so everything (and not just notes designed to form chords that resemble parts of a straight harmonic series) points at a single root. I'm just trying to understand how your method does that...although, again, I will say, it sounds significantly better than 12TET...I just wonder if it can be improved even further. :-)

🔗Marcel de Velde <m.develde@...>

Hi Michael,

Marcel>"Right now I think this tonality scale for common practice music is
> all possible permutations of the harmonic series limited to the sixth
> harmonic.
> This gives the following tonality scale (when reduced to one octave): 1/1
> 9/8 6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8 2/1"
>
> I am a bit confused:
> A) How do 9/8 and 15/8 fit into the 6th harmonic, rather than, say, the 8th
> harmonic?
>

They are permutations of the intervals contained in the harmonic series up
to the 6th harmonic.
Simply see the harmonic series as a sequence of intervals.
Till the 6th harmonic you have the following intervals: 2/1 + 3/2 + 4/3 +
5/4 + 6/5 making 1/1 2/1 3/1 4/1 5/1 6/1
What I'm saying is that you can place these intervals making up the harmonic
series in any order you wish (permutate the harmonic series)
So you can make 3/2 + 6/5 + 5/4 + 4/3 + 2/1 for instance, giving 1/1 3/2 9/5
9/2 3/1 6/1
All possible permutations of the harmonic series till the 6th harmonic all
put together give the following scale:
1/1 6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8 2/1 9/4 12/5 5/2 8/3 3/1 16/5 10/3 18/5
15/4 2/1 9/2 24/5 5/1 6/1
Put in one octave: 1/1 9/8 6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8 2/1

B) How to 6/5, 8/5, and 9/5 fit into the idea (as I understand your saying)
> of everything pointing to a single root note?
>

They don't point to it directly as you can find these intervals between many
notes other than only 1/1.
But they are when played above the root related by intervals.
8/5 can for instance come from the tonic in 2 ways, by adding 4/3 + 6/5, or
by 6/5 + 4/3.
There are even more ways when including octave equivalence.

> I can understand how, say, 4/3 and 3/2 can point to a similar root as
> the third (/3) and second (/2) harmonic are a 5th apart and 5ths lend
> themselves well to pointing at a root tone.
>
> However, I don't see why you are using x/5 fractions instead of, say, x/9
> ones: personally I'd try 11/9 instead of 6/5, 13/8 instead of 8/5, and 16/9
> instead of 9/5. The idea being that x/9 and x/8 format fractions are
> exponentially related to the x/3 and x/2 format fractions (since 3^2 = 9 and
> 2^3 = 8)...reducing back to the (3/2) fifth ratio which tends the
> promote/point-to a single tone.
>

I chose not to see direct harmonics as simplest / shortest way to make an
interval.
But I chose to see the harmonic intervals and all their possible
permutations / reversals as simplest series.
With this way of thinking the 11th harmonic is very far away and many (76
per octave) intervals come first.

It is a sort of utonal way of thinking.
Only not grabbing whole harmonic segments and turning them upside down, but
taking harmonic intervals and turning the order upside down in every way.

Furthermore it relates every note to it's tonic.
It doesn not relate every note that's played to eachother.
So in for instance 1/1 4/3 5/3 9/2 chord the 4/3 5/3 and 9/2 are all
individually related to / comming from 1/1, but 5/3 is not linked directly
to 9/4, only related through thesame tonic.

Now this all sounds very simple and easy to apply.
But the real difficulty now has become identifying the tonic and the
modulations.
I find that normal music theory is very vague about the tonic allowing many
possibilities.
But for tonal-JI (the name I give to this system) most of the time there is
only one good choice for tonic and finding it isn't allways easy.
It gave me many difficulties with the drei equali. (see my html
transcription on the site to see the tonics I analysed)

One thing I whole-heartedly agree with you on, Marcel (if I understand
> you well), is that one very solid way to produce a very pure sounding scale
> is to arrange ratios so everything (and not just notes designed to form
> chords that resemble parts of a straight harmonic series) points at a single
> root. I'm just trying to understand how your method does that...although,
> again, I will say, it sounds significantly better than 12TET...I just wonder
> if it can be improved even further. :-)

Oh yes please do :)
I'm thinking there are many musical rules to be gotten from this way of
thinking.
Everything can be described in mathematics this way, meleodic movements,
harmonic movements, modualtions etc.
But I don't know the best way to do this and I see it as a big task.
The more people inestigate this way of JI, the better.

Marcel
www.develde.net

🔗a_sparschuh <a_sparschuh@...>

🔗cameron <misterbobro@...>

--- In tuning@yahoogroups.com, "monz" <joemonz@...> wrote:
>
> Hi Cameron,
>
> Well, hopefully your simple statement of a fact
> is not implying that i am the inventor of that
> false dichotomy, because i certainly am not.

Obviously not- this is what, 19th Century stuff, isn't it? "O" and "U" tonality thinking is already implied in Tonnetzen I would think.

>
> But what makes it false? Because it certainly _is_
> a dichotomy. It _is_ possible to analyze any collection
> of frequencies as a subset of either the harmonic
> or subharmonic series.
>
> If the notes are tuned exactly to a rational intonation
> (such as JI), then the analysis should be entirely
> straightforward.
>
> If the notes are tuned to a temperament, then which
> subset is chosen depends on the arbitrary choice of
> how close the approximation to JI is supposed to be,
> and in that case, there is even a lot of room for
> the arbitrary choice between harmonic and subharmonic.

It is possible to analyze any collection of integer-related
frequencies as a subset of either the harmonic or subharmonic
series, as you say. However, tonality clearly functions in temperaments as well, and it can function in strange situations, for example it is possible sometimes to hear a tonal center in sonorities that approach a bunch of noise. Basing universal laws on arbitrary choices doesn't seem wise though, does it?

Anyway I already pointed out the elephant in the salon here, the
cadential six-four. Even in the strictest and simplest integer
environment, it is clear evidence that roots are NOT established
solely by harmonic or subharmonic relationship; at the very least,
you'd have to lose inversional equivalence in an attempt to maintain
this dichotomy.

🔗Kraig Grady <kraiggrady@...>

🔗cameron <misterbobro@...>

🔗Daniel Forro <dan.for@...>

I would say that in tonal and modal music which uses triadic chord structure all chords in inversion (where bass note differs from root note) are less stable, and sound more dissonant, unfinished, open... They call for resolution, therefore they are mostly used as passing chords between chords in root voicing. This instability is on harmonic domain, not melodic, and it's not connected with harmonic function in functional harmony, and concerns all chords, major, minor... Even such banal progression like D7-T will sound open and unfinished if chords are for example in inversions 2 and 6 (bass notes F-E in C major). Only progression where both chords are in root position and second chord is in unison (octave) voicing (where melody note equals root note, same as in bass) is considered to be perfectly satisfying as ending.

Daniel Forro

On 27 Oct 2009, at 8:03 PM, Kraig Grady wrote:

> Many have recognized the 6/4 as acoustically stable. It is only
> melodically unstable which points out that melody power over harmony.
> Otherwise there is little ambiguous about it.
> Take the minor version we find at both beginning and end in the sec > mvt
> of Beethooven 7th. A slight suspension but no more than that. It > ends find.
> -->

🔗Michael <djtrancendance@...>

Marcel>"They are permutations of the intervals contained in the harmonic series up to the 6th harmonic"
I think I understand now. You are swapping the intervals 2/1 3/2 4/3 5/4 6/5 in different orders IE 3/2 2/1 4/3 6/5 5/4 or 2/1 6/5 3/2 5/4 4/3 and adding them to get chords which, when summed as a set, produce

1/1 6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8 2/1 9/4 12/5 5/2 8/3 3/1 16/5 10/3 18/5 15/4 2/1 9/2 24/5 5/1 6/1
which, when reduced to an octave, becomes your scale IE
1/1 9/8 6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8 2/1
*****************************************************************
I am definitely going to look further into this though...I think you are on to something here.

>"Furthermore it relates every note to it's tonic.
It does not relate every note that's played to each other.
So
in for instance 1/1 4/3 5/3 9/2 chord the 4/3 5/3 and 9/2 are all
individually related to / comming from 1/1, but 5/3 is not linked
directly to 9/4, only related through the same tonic."

So, if I have it right, there's ambiguity as to which tonal set you are coming from despite both notes being related through the same tonic as 9/4 (or did you mean 9/2?) as they are parts of different harmonic series?

I also guess that's what makes our systems different: yours includes a u-tonal base while mine is purely o-tonally based (I ignore the degree of simplification of the upper side of the fraction). I find the u-tonal based interesting because you allow yourself to express intervals in "forward and reserve".

Again, best of luck with this theory of "tonal JI"!

________________________________
From: Marcel de Velde <m.develde@...>
To: tuning@yahoogroups.com
Sent: Mon, October 26, 2009 10:50:10 PM
Subject: Re: [tuning] Re: the "math" of the minor triad

Hi Michael,

Marcel>"Right now I think this tonality scale for common practice music is
>all possible permutations of the harmonic series limited to the sixth
>harmonic.
>This gives the following tonality scale (when reduced to one octave): 1/1 9/8 6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8 2/1"
>
>I am a bit confused:
>A) How do 9/8 and 15/8 fit into the 6th harmonic, rather than, say, the 8th harmonic?
>

They are permutations of the intervals contained in the harmonic series up to the 6th harmonic.
Simply see the harmonic series as a sequence of intervals.
Till the 6th harmonic you have the following intervals: 2/1 + 3/2 + 4/3 + 5/4 + 6/5 making 1/1 2/1 3/1 4/1 5/1 6/1
What I'm saying is that you can place these intervals making up the harmonic series in any order you wish (permutate the harmonic series)
So you can make 3/2 + 6/5 + 5/4 + 4/3 + 2/1 for instance, giving 1/1 3/2 9/5 9/2 3/1 6/1
All possible permutations of the harmonic series till the 6th harmonic all put together give the following scale:
1/1 6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8 2/1 9/4 12/5 5/2 8/3 3/1 16/5 10/3 18/5 15/4 2/1 9/2 24/5 5/1 6/1
Put in one octave: 1/1 9/8 6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8 2/1

B) How to 6/5, 8/5, and 9/5 fit into the idea (as I understand your saying) of everything pointing to a single root note?
>

They don't point to it directly as you can find these intervals between many notes other than only 1/1.
But they are when played above the root related by intervals.
8/5 can for instance come from the tonic in 2 ways, by adding 4/3 + 6/5, or by 6/5 + 4/3.
There are even more ways when including octave equivalence.

> I can understand how, say, 4/3 and 3/2 can point to a similar root as the third (/3) and second (/2) harmonic are a 5th apart and 5ths lend themselves well to pointing at a root tone.
>
> However, I don't see why you are using x/5 fractions instead of, say, x/9 ones: personally I'd try 11/9 instead of 6/5, 13/8 instead of 8/5, and 16/9 instead of 9/5. The idea being that x/9 and x/8 format fractions are exponentially related to the x/3 and x/2 format fractions (since 3^2 = 9 and 2^3 = 8)...reducing back to the (3/2) fifth ratio which tends the
> promote/point- to a single tone.
>

I chose not to see direct harmonics as simplest / shortest way to make an interval.
But I chose to see the harmonic intervals and all their possible permutations / reversals as simplest series.
With this way of thinking the 11th harmonic is very far away and many (76 per octave) intervals come first.

It is a sort of utonal way of thinking.
Only not grabbing whole harmonic segments and turning them upside down, but taking harmonic intervals and turning the order upside down in every way.

Furthermore it relates every note to it's tonic.
It doesn not relate every note that's played to eachother.
So in for instance 1/1 4/3 5/3 9/2 chord the 4/3 5/3 and 9/2 are all individually related to / comming from 1/1, but 5/3 is not linked directly to 9/4, only related through thesame tonic.

Now this all sounds very simple and easy to apply.
But the real difficulty now has become identifying the tonic and the modulations.
I find that normal music theory is very vague about the tonic allowing many possibilities.
But for tonal-JI (the name I give to this system) most of the time there is only one good choice for tonic and finding it isn't allways easy.
It gave me many difficulties with the drei equali. (see my html transcription on the site to see the tonics I analysed)

Oh yes please do :)
I'm thinking there are many musical rules to be gotten from this way of thinking.
Everything can be described in mathematics this way, meleodic movements, harmonic movements, modualtions etc.
But I don't know the best way to do this and I see it as a big task.
The more people inestigate this way of JI, the better.

Marcel
www.develde. net

🔗Michael <djtrancendance@...>

a_sparschuh>"but such a crude mismatch of (11/9):(6/5) = (55/54)"
My intention was not to produce an interval that sounded exactly alike, but rather to harmonically align it with the many x/3 format (harmonic series) o-tonal fractions in Marcel's scale.
After Marcel's explanation I realize a huge difference has to do with his trying to optimize u-tonality and not (what I was trying to do) o-tonality.

a_sparschuh>"The accuracy of discrimination should exceed those sloppy comma-precision in order to be aware of own characters"

Right (if I understand you) the tonal character would be different.
But, along the same lines, the x/7 harmonic series (where x = 7 to 14) would have far different tonal character than the major 12-TET 7-tone scale and yet sound less sour.
My point is that, while you're right I'd lose the interval's character, character does not mean purity (so far as consonance), and I'm trying to deal with optimizing that kind of purity.

In fact it's a general issue I run into a lot here...the fact that I've never headed in to making a scale or trying to give alternative suggestions for an existing one with the goal of "emulating existing intervals" even though I find people saying "oh, that (tone character emulation) must be what you were going for".
While I respect those who do that sort of that (IE say, try to find 20+ types or nearly-pure 3rds, for example), to me half the fun is finding intervals that make harmonic sense, yet have a mood completely unlike traditional intervals.

-Michael

🔗Marcel de Velde <m.develde@...>

>
> I think I understand now. You are swapping the intervals 2/1 3/2 4/3 5/4
> 6/5 in different orders IE 3/2 2/1 4/3 6/5 5/4 or 2/1 6/5 3/2 5/4 4/3 and
> adding them to get chords
>

Yes.
I don't personally know if to see them as chords though.
If one plays 1/1 3/1 I think the simplest way to see the relation is as 1/1
as source and 3/1 as comming from the harmonic division of 1/1.
One can see the whole harmonic series as a single chord, or as a series of
individual pitches all comming from 1 source.
Similarly I see all the tones produced by all permutations from the harmonic
series as comming from 1 source (the tonic).
The only difference is that I say in 1/1 3/1 the way 3/1 comes from 1/1 can
not be only through 2/1 but through several other intervals.

> which, when summed as a set, produce
> 1/1 6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8 2/1 9/4 12/5 5/2 8/3 3/1 16/5 10/3
> 18/5 15/4 2/1 9/2 24/5 5/1 6/1
> which, when reduced to an octave, becomes your scale IE
>
> 1/1 9/8 6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8 2/1
> *****************************************************************
> I am definitely going to look further into this though...I think you are on
> to something here.
>

Ok great :)
Feel free to contact me offlist to discuss things, or perhaps start a
different thread as this one will go a bit offtopic otherwise.

So, if I have it right, there's ambiguity as to which tonal set you are
> coming from despite both notes being related through the same tonic as 9/4
> (or did you mean 9/2?) as they are parts of different harmonic series?
>

I did mean to write 9/4 indeed :)
But 9/2 works too.
9/8 aswell. Perhaps 9/8 would mean the real tonic is an octave lower but
right now I'm not taking it very strict octave wise, perhaps later
meaningfull things can be said about this but for now I feel this would make
things unnessecary complex for myself and others.

As for ambiguity.
I think there's ambiguity in several ways.
First way is that when one plays 1/1 5/4 3/2, this chord can be in many ways
related to the tonic.
Where the tonic is 1/1:
1/1 5/4 3/2
6/5 3/2 9/5
4/3 5/3 2/1
3/2 15/8 9/4
8/5 2/1 12/5

And when we take into account 12tet or harmonic enthropy these also become
possible:
5/4 8/5 15/8 (32/25 third)
9/5 9/4 8/3 (40/27 wolf fifth)

Thirdly (the way you ment I think) is that there's ambiguity for every chord
in which way / structure exactly it is eminating from the tonic.
And wether al tones are comming from the tonic in thesame "structure" or if
some of the tones of the chord have a different structure as source.
For instance when playing 1/1 3/2 15/8 where 1/1 is the tonic.
Is 3/2 comming directly as 3/2 from the tonic or is it by 6/5+5/4 or even
from an octave lower 1/2 and several other intervals.
And is the 15/8 comming from 1/1+3/2+5/4 and is a continuation of the 3/2 or
does it come from 1/1+5/4+15/8 or many other options.
And does this matter?
It does seem logical that it matters and I am finding hints that it matters,
but I really don't know yet.
This is one of the things that I wish to look into in the future but see it
as a big difficult task.

I also guess that's what makes our systems different: yours includes a
> u-tonal base while mine is purely o-tonally based (I ignore the degree of
> simplification of the upper side of the fraction). I find the u-tonal based
> interesting because you allow yourself to express intervals in "forward and
> reserve".
>

Yes I find thesame.
This also makes it somewhat related to this thread I hope as it sees the
minor 1/1 6/5 3/2 triad as both a mirror of 1/1 5/4 3/2 in intervals making
up the chord, and it sees this minor triad in many diffirent ways as comming
from the tonic and in many different places to the tonic.
It also sees the minor triad as 1/1 5/4 5/3 which is not comming from a
reversal of 6/5 and 5/4 intervals and is even present in harmonic 5-limit
tonal-JI (as opposed to harmonic 6-limit tonal-JI where the 1/1 6/5 3/2
tonic minor first occurs)

>
> Again, best of luck with this theory of "tonal JI"!
>

Thank you!
You good luck with it as well :)

Marcel
www.develde.net

🔗cameron <misterbobro@...>

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

> What if a single chord like for instance a major chord can have >many roots.
> What if the root of a chord doesn't even have to be in the chord >itself.

Hehehe. If you think strongly about what you are saying, you'll realize that what I was saying in an earlier discussion about how musical structure ultimately relates to the spectra, not to "JI", is neither trivial nor wrong.
>
> I'm thinking myself the root of all chords is the tonic.
> And that there is a fixed JI scale from the tonic which has it's >origin in
> the tonic.

This is A way. Great way. Not the only way. If you think about it,
you'll realize that in certain circumstances, this is practically indistinguishable from the idea that "JI" is the deep structure. However, this (thinking they are the same because they are the same in certain music with strong limitations) is an illusion, and a deadly, or at least cheesey as hell, into which to fall.

> Right now I think this tonality scale for common practice music is >all
> possible permutations of the harmonic series limited to the sixth >harmonic.

This depends on what is meant by modulation.

> This gives the following tonality scale (when reduced to one >octave): 1/1
> 9/8 6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8 2/1
> In this system for instance a dominant major 7th chord would be 4/3 >3/2 15/8
> 9/4 and have as it's source / root 1/1.
>
> I can't think of a single example that would couteract this line of
> thinking.
> If you can please give an example.

Ravel Pavane for a Dead Princess. It fits the idea of a single complex Just sonority as the structure for the entire piece, but it doesn't work with your particular tuning, try it yourself. A lot of later tonal music is better described as being built on the Just structure derived directly from the harmonics, that is, C-D-E-F#-G-A-Bb-C, or even C-D-E-F#-G-Ab-Bb-C.

Apparently you haven't studied the history of these things enough, for if you had you'd realize that you have various possibilities in which the seventh might be derived from disjunct tetrachord thinking, not from the harmonic series, or from the harmonic series, or from a specifically meantone compromise, etc.