Just intonation and the harmonic series

I took brief part in a conversation on the Just Intonation Network group on Facebook - which seems to be perhaps not too stable because I can't find the record of the conversation - but one thing that came up was the concept that not all just intonation need be based on the harmonic series. On the surface, this looks like a fine proposition, but I just thought about it a bit, and it looks like, indeed, just intonation actually is based on the harmonic series, or at least it's impossible to get away from a fairly direct relationship.

Consider a set of ratios which do not initially appear to be on a harmonic series. The minor triad, for example:

1/1, 6/5, and 3/2.

A rather trivial operation guarantees that I can place this on a harmonic series: multiply each numerator by every unique prime in the demoninators. In this case, that means to multiple each by 10, and that gives us:

10/1, 12/1, and 15/1.

That is clearly on a harmonic series. And because of the simplicity of this operation, it looks as if there is no possible set of integer ratios which cannot be placed on a harmonic series.

Then, of course, the issue of whether or not this is a relevant relationship in terms of how our ears actually interpret the sound. I'm not sure.

One point of clarification occurs to me:

'Unique prime' perhaps isn't the correct term. It should be every prime that is present in each denominator, exclusing primes that occur in more than one denominator. For example, if a set includes both 6/5 and 27/25, it is sufficient to multiply the numerators by 25, not 125.

--- In tuning@yahoogroups.com, "bigAndrewM" <bigandrewm@...> wrote:
>
> I took brief part in a conversation on the Just Intonation Network group on Facebook - which seems to be perhaps not too stable because I can't find the record of the conversation - but one thing that came up was the concept that not all just intonation need be based on the harmonic series. On the surface, this looks like a fine proposition, but I just thought about it a bit, and it looks like, indeed, just intonation actually is based on the harmonic series, or at least it's impossible to get away from a fairly direct relationship.
>
> Consider a set of ratios which do not initially appear to be on a harmonic series. The minor triad, for example:
>
> 1/1, 6/5, and 3/2.
>
> A rather trivial operation guarantees that I can place this on a harmonic series: multiply each numerator by every unique prime in the demoninators. In this case, that means to multiple each by 10, and that gives us:
>
> 10/1, 12/1, and 15/1.
>
> That is clearly on a harmonic series. And because of the simplicity of this operation, it looks as if there is no possible set of integer ratios which cannot be placed on a harmonic series.
>
> Then, of course, the issue of whether or not this is a relevant relationship in terms of how our ears actually interpret the sound. I'm not sure.
>

Hi Andrew,

it is true that each just chord can be represented as a part of an overtone series. However, such an interpretation can result in ridiculously large frequency ratios. For example, let's take the m7b5 chord from the undertone series:

1:1 7:6 7:5 7:4 (starting from the lowest note), or
7:6 6:5 5:4 (subsequent intervals)

This is basically a mirrored variant of the 4:5:6:7 overtone chord, so the same simple 7-limit ratios occur between the notes: 5:4, 6:5, 7:6, 3:2, 7:5 and 7:4.

However, as opposed to the 4:5:6:7 overtone chord, the most simple overtone interpretation of this m7b5 chord is very complex: 60:70:84:105. And this was a rather simple undertone chord...

--- In tuning@yahoogroups.com, "bigAndrewM" wrote:

One point of clarification occurs to me:

'Unique prime' perhaps isn't the correct term. It should be every prime that is present in each denominator, exclusing primes that occur in more than one denominator. For example, if a set includes both 6/5 and 27/25, it is sufficient to multiply the numerators by 25, not 125.

The simple and correct thing to say is "least common multiple". If you multiply everything by the least common multiple of the denominators, you get a set of coprime integers.

Keenan

Hello Andrew~
Historically Just intonation has been more based on the subharmonic series than the harmonic series. The latter is a recent discovery actually. It is string lengths and the equal division of string lengths one finds in ancient history.
--
signature file

/^_,',',',_ //^/Kraig Grady_^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere:
North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

a momentary antenna as i turn to water
this evaporates - an island once again

> Consider a set of ratios which do not initially appear to be on a harmonic series. The minor triad, for example:
>
> 1/1, 6/5, and 3/2.

They are not a part of the overtone series, but the undertone series, with the root actually being the 3/2, building downwards. You would think that the undertone series for every root would build a minor chord, but it is actually the minor IV that it creates. Perhaps this is the reason why minor IV resolves so satisfactorily to the I in classical music, because they are mathematical inversions of each other:
1/1, 5/4, 3/2 (major root, building upwards from 1/1)
1/1, 8/5, 4/3 (minor IV, building downwards from 1/1)
As you can see, the numerators and denominators are inverted.
"X" major and "X" minor (using the 6/5) are related, but through the 3/2. There is also some doubt in my mind whether 6/5 is really as good as 32/27, which is an undertone directly related to the root, and closer to the equal tempered minor third we are all used to.
--Mike Nolley

On Thu, Dec 20, 2012 at 5:41 PM, MikeN <miken277@...> wrote:
>
> > Consider a set of ratios which do not initially appear to be on a
> > harmonic series. The minor triad, for example:
> >
> > 1/1, 6/5, and 3/2.
>
> They are not a part of the overtone series, but the undertone series, with
> the root actually being the 3/2, building downwards.

They're also a part of the overtone series, in that they're 10:12:15.
Think of it as the top three notes of a major 7 chord, if you like.

> You would think that
> the undertone series for every root would build a minor chord, but it is
> actually the minor IV that it creates. Perhaps this is the reason why minor
> IV resolves so satisfactorily to the I in classical music, because they are
> mathematical inversions of each other:
> 1/1, 5/4, 3/2 (major root, building upwards from 1/1)
> 1/1, 8/5, 4/3 (minor IV, building downwards from 1/1)

You can easily test if things work in quite this way by working with
other otonal/utonal inversions, such as: 4:6:7 and its utonal inverse,
4:5:7 and its utonal inverse, 6:7:9 and its utonal inverse, 5:6:7 and
its utonal inverse, 7:9:11 and its utonal inverse and so on.

-Mike

> They're also a part of the overtone series, in that they're 10:12:15.
> Think of it as the top three notes of a major 7 chord, if you like.

Yes, that's true. You're essentially modulating up a major third by multiplying by five, which has canceled out the troublesome denominator in 6/5. But, correct me if I'm wrong, I don't think that means that you can change any set of undertones into overtones by multiplying them by one or another of the denominators - you would have to multiply by all of them to cancel out all of the denominators. It works for 1, 6/5, 3/2, as related to 1/1 because 1/1 is already the undertone of its overtone 3/2, and 6/5 is canceled by the multiplier. Interesting, isn't it, that analysis proves that the stuff of music to be fraction simplification we learned in elementary schooL?
--Mike

On Thu, Dec 20, 2012 at 6:17 PM, MikeN <miken277@...> wrote:
> But, correct me if I'm wrong, I don't think that means that you can
> change any set of undertones into overtones by multiplying them by one or
> another of the denominators - you would have to multiply by all of them to
> cancel out all of the denominators. It works for 1, 6/5, 3/2, as related to
> 1/1 because 1/1 is already the undertone of its overtone 3/2, and 6/5 is
> canceled by the multiplier.

Yes, the general algorithm is to multiply by the least common multiple
(LCM) of all the denominators.

> Interesting, isn't it, that analysis proves that
> the stuff of music to be fraction simplification we learned in elementary
> schooL?

Mike, just curious, do you hear this piece that I'm playing here to be
in a minor key, all things consider?

http://www.youtube.com/watch?v=BjoNHKx7GEI&t=0m40s

-Mike

--- In tuning@yahoogroups.com, "MikeN" <miken277@...> wrote:
Interesting, isn't it, that analysis proves that the stuff of music to be fraction simplification we learned in elementary schooL?

No one ever teaches the Euclidean algorithm in elementary school.

While the relationship to string lengths is clear, I prefer to think in just intonation in terms of frequencies, not string lengths.

--- In tuning@yahoogroups.com, kraiggrady <kraiggrady@...> wrote:
>
> Hello Andrew~
> Historically Just intonation has been more based on the subharmonic
> series than the harmonic series. The latter is a recent discovery
> actually. It is string lengths and the equal division of string lengths
> one finds in ancient history.
> --
> signature file
>
> /^_,',',',_ //^/Kraig Grady_^_,',',',_
> Mesotonal Music from:
> _'''''''_ ^North/Western Hemisphere:
> North American Embassy of Anaphoria Island <http://anaphoria.com/>
>
> _'''''''_^South/Eastern Hemisphere:
> Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>
>
> ',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
>
> a momentary antenna as i turn to water
> this evaporates - an island once again
>

. . . and, we can do a similar operation from what I initially described to describe any set from an overtone series in terms of being in an undertone series. They're two sides of the same coin.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Thu, Dec 20, 2012 at 5:41 PM, MikeN <miken277@...> wrote:
> >
> > > Consider a set of ratios which do not initially appear to be on a
> > > harmonic series. The minor triad, for example:
> > >
> > > 1/1, 6/5, and 3/2.
> >
> > They are not a part of the overtone series, but the undertone series, with
> > the root actually being the 3/2, building downwards.
>
> They're also a part of the overtone series, in that they're 10:12:15.
> Think of it as the top three notes of a major 7 chord, if you like.
>
> > You would think that
> > the undertone series for every root would build a minor chord, but it is
> > actually the minor IV that it creates. Perhaps this is the reason why minor
> > IV resolves so satisfactorily to the I in classical music, because they are
> > mathematical inversions of each other:
> > 1/1, 5/4, 3/2 (major root, building upwards from 1/1)
> > 1/1, 8/5, 4/3 (minor IV, building downwards from 1/1)
>
> You can easily test if things work in quite this way by working with
> other otonal/utonal inversions, such as: 4:6:7 and its utonal inverse,
> 4:5:7 and its utonal inverse, 6:7:9 and its utonal inverse, 5:6:7 and
> its utonal inverse, 7:9:11 and its utonal inverse and so on.
>
> -Mike
>

--- In tuning@yahoogroups.com, "bigAndrewM" <bigandrewm@...> wrote:
>
> . . . and, we can do a similar operation from what I initially described to describe any set from an overtone series in terms of being in an undertone series. They're two sides of the same coin.
>
>

However, I suspect that the ear has a slight preference for interpreting the sound in terms of the overtone series, not the undertone series.

--- In tuning@yahoogroups.com, "bigAndrewM" <bigandrewm@...> wrote:
> However, I suspect that the ear has a slight preference for interpreting the sound in terms of the overtone series, not the undertone series.

Yes, our auditory system tries to identify fundamental frequencies, and sort out redundant overtones. But if every sound was always reduced to a single unique fundamental frequency, we were not able to identify multiple voices when several people are talking at the same time.

So if the root of a chord is somehow related to the fundamental frequency of an overtone series, then I don't see a reason why one shouldn't allow for multiple (stronger or weaker weighted) roots at the same time, especially in the case of utonal chords.
For example, an A minor chord could be interpreted as two overlapping overtone series, one based on A (covers A and E) and one based on C (covers C and E).

An interesting aspect of utonal chords is that the possibly harmonically most important note (= the reference tone of the undertone series) is a common overtone shared by all notes. I wouldn't call that note a 'root' though, in order to avoid unnecessary confusion...

On Sat, Dec 22, 2012 at 12:42 AM, gedankenwelt94 <gedankenwelt94@...>
wrote:
>
> Yes, our auditory system tries to identify fundamental frequencies, and
> sort out redundant overtones. But if every sound was always reduced to a
> single unique fundamental frequency, we were not able to identify multiple
> voices when several people are talking at the same time.

This is an excellent point. And furthermore, if the sound of "major"
actually required direct activation of some sort of partial fusion of
4:5:6 into a single note, then arpeggiated major chords would sound
very different to us. Thankfully, this is not the case, so we get to
have nice things like (implied) triadic harmony in two-part
inventions.

> So if the root of a chord is somehow related to the fundamental frequency
> of an overtone series, then I don't see a reason why one shouldn't allow for
> multiple (stronger or weaker weighted) roots at the same time, especially in
> the case of utonal chords.
> For example, an A minor chord could be interpreted as two overlapping
> overtone series, one based on A (covers A and E) and one based on C (covers
> C and E).

I don't think that the root of a chord is determined by the virtual
fundamental. For instance, it's pretty easy to hear the root of 5:6:9
as "5", as though it were a minor 7 chord with the 5th omitted. Of
course, you could take the approach that the One True Minor Chord is
16:19:24 or something, if you want.

-Mike

On Sat, Dec 22, 2012 at 1:08 AM, Mike Battaglia <battaglia01@...>wrote:

> I don't think that the root of a chord is determined by the virtual
> fundamental. For instance, it's pretty easy to hear the root of 5:6:9
> as "5"
>
Are you saying that the 5 in the ratio 5:6:9 is the root? Just so I
understand.

> , as though it were a minor 7 chord with the 5th omitted.
>
Do you mean a 7 chord with a missing root?
Like the V-7 chord with mising root [vii-dim].

> Of
> course, you could take the approach that the One True Minor Chord is
> 16:19:24 or something, if you want.
>
1:12:15.
ƒg

On Sat, Dec 22, 2012 at 2:00 AM, Freeman Gilmore <freeman.gilmore@...>
wrote:
>
>> I don't think that the root of a chord is determined by the virtual
>> fundamental. For instance, it's pretty easy to hear the root of 5:6:9
>> as "5"
>
> Are you saying that the 5 in the ratio 5:6:9 is the root? Just so I
> understand.

Right.

>> , as though it were a minor 7 chord with the 5th omitted.
>
> Do you mean a 7 chord with a missing root?
> Like the V-7 chord with mising root [vii-dim].

No, I mean a minor 7 chord with a missing fifth. For instance, it's
like C-Eb-G-Bb, but without the G, so it's just C-Eb-Bb. Then the Eb
and Bb are roughly 17 cents sharp of what they'd be in 12-EDO, so that
the whole thing is tuned to 5:6:9.

>> Of
>> course, you could take the approach that the One True Minor Chord is
>> 16:19:24 or something, if you want.
>
> 1:12:15.

I assume that you mean 10:12:15 here. I highly doubt that the sound of
minor is the sound of 10:12:15, unfortunately.

-Mike

--- In tuning@yahoogroups.com, "gedankenwelt94" <gedankenwelt94@...> wrote:
>
>
> Yes, our auditory system tries to identify fundamental frequencies, and sort out redundant overtones. But if every sound was always reduced to a single unique fundamental frequency, we were not able to identify multiple voices when several people are talking at the same time.
>

Yes, it's clear that our ears do not reduce every sound to a single unique fundamental frequency. My limited understanding is more in terms of illusions, that the ears and brain add artifacts to what we hear so that individual voices in a crowd are clearer. Because the human voice resonates with the overtone series and not the undertone series, that's what the artifacts enhance.

Speaking of which, how is it that we are able to distinguish (for example) two trumpets playing a fifth apart from one? The upper trumpet, if it is tune, will be sounding exactly the same overtones as the lower trumpet. The biggest thing that's different is the relative strengths of the overtones, which leads me to suspect that the brain automatically creates a 'map' of expected overtones to help sort the information. And, arguably, this map isn't perfect, because parallel intervals are sometimes used in orchestration to 'mimic' a new instrument sound. If someone who had never heard a trumpet before were to hear a recording of two trumpets playing a perfect fifth, would they conclude that it is one instrument or two?

A lot of mainstream psychoacousticians (a new aquatic mammal group? maybe involving the fluid in our middle ear where sound waves are converted into electrical impulses!) don't even think we hear intervals as frequency ratios. Not a popular thing to say on a tuning group! I"m sort of in both camps. People like Richard Parncutt feel that intervals are perceived linearly, in semitones, and not as frequency ratios. (He has a chapter on this in his harmony book.) My theory is that intervals which, as measured in semitones, divide evenly into the octave and which have multiple 'frequency' spellings (m3, M3, and M2, which are the source of dim, aug, and whole tone, which have powerful tonal properties!)...that those intervals are perceived linearly, like he says. But that P5 is a frequency ratio interval, because it is a large interval which occurs in the harmonic series (as the intervals get smaller) before the intervals get more ambiguous in terms of which frequency ratio they represent, as well as the tonal fact of small interval dissonance from symmetrical chords (M3-M3-M3; m3-m3-m3-m3) and roughness (M2-M2...; m2-m2...) That leaves the question of how to classify P4, M6, and M7...but since it doesn't involve tuning math, there may be no interest in that here! cheers, Kelly

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Sat, Dec 22, 2012 at 2:00 AM, Freeman Gilmore <freeman.gilmore@...>
> wrote:
> >
> >> I don't think that the root of a chord is determined by the virtual
> >> fundamental. For instance, it's pretty easy to hear the root of 5:6:9
> >> as "5"
> >
> > Are you saying that the 5 in the ratio 5:6:9 is the root? Just so I
> > understand.
>
> Right.
>
> >> , as though it were a minor 7 chord with the 5th omitted.
> >
> > Do you mean a 7 chord with a missing root?
> > Like the V-7 chord with mising root [vii-dim].
>
> No, I mean a minor 7 chord with a missing fifth. For instance, it's
> like C-Eb-G-Bb, but without the G, so it's just C-Eb-Bb. Then the Eb
> and Bb are roughly 17 cents sharp of what they'd be in 12-EDO, so that
> the whole thing is tuned to 5:6:9.
>
> >> Of
> >> course, you could take the approach that the One True Minor Chord is
> >> 16:19:24 or something, if you want.
> >
> > 1:12:15.
>
> I assume that you mean 10:12:15 here. I highly doubt that the sound of
> minor is the sound of 10:12:15, unfortunately.
>
> -Mike
>

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

> if the sound of "major"
> actually required direct activation of some sort of partial
> fusion of 4:5:6 into a single note, then arpeggiated major
> chords would sound very different to us. Thankfully, this is
> not the case, so we get to have nice things like (implied)
> triadic harmony in two-part inventions.

Well, arpeggiated chords do sound very different than
homophonic chords. But the key here is "direct". Memories
involve activation of the same machinery that's used when
the corresponding stimulus is present...

> I don't think that the root of a chord is determined by the
> virtual fundamental. For instance, it's pretty easy to hear
> the root of 5:6:9 as "5",

I don't disagree, but we need to know what the virtual
fundamental of a chord would even be, don't we?

Assuming there's no hierarchical analysis going on
(and there almost certainly is), a chord's spectrum still
looks a lot different than the spectrum of its tones,
even in JI...

-Carl

On Sat, Dec 22, 2012 at 1:15 PM, bigAndrewM <bigandrewm@...> wrote:

> **
>
>
>
>
> --- In tuning@yahoogroups.com, "gedankenwelt94" <gedankenwelt94@...>
> wrote:
> >
> >
> > Yes, our auditory system tries to identify fundamental frequencies, and
> sort out redundant overtones. But if every sound was always reduced to a
> single unique fundamental frequency, we were not able to identify multiple
> voices when several people are talking at the same time.
> >
>
> Yes, it's clear that our ears do not reduce every sound to a single unique
> fundamental frequency. My limited understanding is more in terms of
> illusions, that the ears and brain add artifacts to what we hear so that
> individual voices in a crowd are clearer. Because the human voice resonates
> with the overtone series and not the undertone series, that's what the
> artifacts enhance.
>
> Speaking of which, how is it that we are able to distinguish (for example)
> two trumpets playing a fifth apart from one? The upper trumpet, if it is
> tune, will be sounding exactly the same overtones as the lower trumpet. The
> biggest thing that's different is the relative strengths of the overtones,
> which leads me to suspect that the brain automatically creates a 'map' of
> expected overtones to help sort the information. And, arguably, this map
> isn't perfect, because parallel intervals are sometimes used in
> orchestration to 'mimic' a new instrument sound. If someone who had never
> heard a trumpet before were to hear a recording of two trumpets playing a
> perfect fifth, would they conclude that it is one instrument or two?
>

If you assumed that person never heard a trumpet. And all the overtones
of both are very close to each other. You would have a composite sound,
which has a missing fundamental. But is the sound of both is sufficient
to generate a fundamental in to ear/brain then it would sound complete;
with week fundamental and some strong harmonics. It may sound odd even to
the person that has heard a trumpet? That said, if you had a peace of
music and passed through a part with a fifth like that, that happen to have
very close overtones, it would go on noticed.
ƒg

On Sat, Dec 22, 2012 at 1:15 PM, bigAndrewM <bigandrewm@...> wrote:
>
> Speaking of which, how is it that we are able to distinguish (for example)
> two trumpets playing a fifth apart from one? The upper trumpet, if it is
> tune, will be sounding exactly the same overtones as the lower trumpet.

What do you mean by this, exactly? If two trumpets are in a 3/2 ratio,
then the lower trumpet will be hitting overtones that the upper one
isn't. So for instance, if the fundamentals of the two notes are 200
Hz and 300 Hz, then the lower one will have 400 Hz as a harmonic,
which will be absent from the latter one.

> If someone who had never
> heard a trumpet before were to hear a recording of two trumpets playing a
> perfect fifth, would they conclude that it is one instrument or two?

I think it'll rely heavily on cues such as how much the two trumpets
move together, how well-correlated the onset and release trails of the
two trumpets are, etc. If they consistently move together in an exact
3/2 ratio, with the upper note playing a 3/2 above the lower note even
when such a note wouldn't even fit the scale, I think that'd aid in
facilitating a perception where the two trumpets are grouped into a
single auditory stream, which might further aid the fusion of the two
notes into a single timbre.

The obvious example of this would be in Ravel's Bolero, where 1:3:5
seems to fuse into a single organ-ish timbre - but especially so in
performances where the three instruments are very "tight," hit the
notes at the same time, where the "5" on top doesn't stick out too
much, etc.

You might look up Huron's work on relating principles of voice leading
to more general psychoacoustic notions of auditory stream segregation
for some more info on stuff like this:
http://csml.som.ohio-state.edu/Huron/Publications/huron.voice.leading.html

-Mike

On Sat, Dec 22, 2012 at 6:40 PM, Carl Lumma <carl@...> wrote:
>
> > if the sound of "major"
> > actually required direct activation of some sort of partial
> > fusion of 4:5:6 into a single note, then arpeggiated major
> > chords would sound very different to us. Thankfully, this is
> > not the case, so we get to have nice things like (implied)
> > triadic harmony in two-part inventions.
>
> Well, arpeggiated chords do sound very different than
> homophonic chords. But the key here is "direct". Memories
> involve activation of the same machinery that's used when
> the corresponding stimulus is present...

Yes, I agree with this. I think it's very possible (and probably
likely) that some sort of conditioning or learning process like this
exists. My only point was that the details of such a process are
non-trivial.

Anecdotally, I've noticed that this specific thing does happen for me,
especially in decent 11-limit tunings where there are lots of dyads
that by themselves are rather discordant and inharmonic sounding, but
which suddenly come to life in the context of a larger chord. 11/8
used to sound like crap by itself, but after playing in porcupine so
much, it's hard for me to hear that dyad and NOT imagine it as part of
a whole bunch of larger "otonal" sounding chords like 8:9:10:11:12.

I never heard any of the above until I bought my AXiS and my 22-EDO
guitar though. 9/7 and 11/8 used to just sound like crap; it's only
after playing around in porcupine and with 11-limit harmony a lot that
the whole thing has changed. It's really nothing psychoacoustic and
mysterious; it's like the equivalent of playing C-Eb-Bb in 12-EDO and
imagining a "G" or something, making it a minor seventh chord, just
with all of these new 11-limit chords. It's not really like I'm
imagining VFs directly or what have you, more that I'm just imagining
other notes as existing in "the current scale" or something, and then
remembering the sound of the chord formed by those imagined notes and
the ones I'm playing now. Hopefully that seems direct and simple
enough.

I've also had the same experience with 9/7, which now always "reminds
me" of 4:7:9; it sounds much less "harsh" that way. I would be very
surprised if it were just completely random that imagining the root of
9/7 as "4", vs any other possible note in the vicinity, makes the
whole thing sound less harsh.

So perhaps this is anecdotal evidence in favor of the thing you laid
out above. I wonder if something like that is the same basic concept
as imagining triadic harmony during a two-part invention, and if the
whole thing falls under the heading of "habitually imagined notes" or
something like that.

> > I don't think that the root of a chord is determined by the
> > virtual fundamental. For instance, it's pretty easy to hear
> > the root of 5:6:9 as "5",
>
> I don't disagree, but we need to know what the virtual
> fundamental of a chord would even be, don't we?
>
> Assuming there's no hierarchical analysis going on
> (and there almost certainly is), a chord's spectrum still
> looks a lot different than the spectrum of its tones,
> even in JI...

What do you mean by a heirarchical analysis, exactly?

I used to think that 8:12:16:19, or 16:19:24, or whatever might be
somehow related to the sound of minor, at least when the root is heard
as the lowest note in the triad. I figured that the same sort of
conditioning process I laid out above for 11/8 and 9/7 could suggest
such a result. By itself, 16:19:24 is complex, but if you're imagining
other notes in the bass, then you might get something like
4:8:12:16:19:24, which is less complex than 5:10:15:20:24:30, and if
you decide to imagine notes even further down in the bass, then it
gets even less complex.

I originally started thinking along those lines because I noted above
that 9/7 sounds much more consonant (not concordant, but consonant) if
I imagine the root as "4"; in general, I doubt that this is some
random coincidence that this particular choice of imagined note in the
bass would transform the sound so dramatically vs imagining any other
random note in the bass. It's likely that 4:7:9 has something to do
with it, whether because of some past memory of hearing 4:7:9 or
because of some more "direct" thing or whatever. I figured the same
could apply to minor, when the lowest note is heard as the root, and
8:12:16:19 or 16:19:24 chords.

However, the fact that something as far out as 225 cents in 16-EDO can
sound like minor, at least to some people, would appear to cast a lot
of doubt on that theory, or at least raise a lot of questions about
how that could work...

-Mike

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

> > we need to know what the virtual
> > fundamental of a chord would even be, don't we?
> > Assuming there's no hierarchical analysis going on
> > (and there almost certainly is), a chord's spectrum still
> > looks a lot different than the spectrum of its tones,
> > even in JI...
>
> What do you mean by a heirarchical analysis, exactly?

A process that analyses subsets of the stimulus (per
frequency band, say, or per scene-located source) and then
applies a similar analysis to the results, would be
hierarchical.

-Carl

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
>
> What do you mean by this, exactly? If two trumpets are in a 3/2 ratio,
> then the lower trumpet will be hitting overtones that the upper one
> isn't. So for instance, if the fundamentals of the two notes are 200
> Hz and 300 Hz, then the lower one will have 400 Hz as a harmonic,
> which will be absent from the latter one.
>

I was thinking more along the lines that the upper trumpet will not be
producing any overtones that are absent from those produced by the lower trumpet, but only changes the relative overtone amplitudes.

> You might look up Huron's work on relating principles of voice leading
> to more general psychoacoustic notions of auditory stream segregation
> for some more info on stuff like this:
> http://csml.som.ohio-state.edu/Huron/Publications/huron.voice.leading.html
>
> -Mike
>

Thanks!

Andrew

Yes, stated much better than I did. The two trumpets produce a composite sound with a missing fundamental.

I suppose that the answer is almost self-evident. Rock guitarists use parallel fifths quite often, as a tambral effect rather than as voice-leading.

> If you assumed that person never heard a trumpet. And all the overtones
> of both are very close to each other. You would have a composite sound,
> which has a missing fundamental. But is the sound of both is sufficient
> to generate a fundamental in to ear/brain then it would sound complete;
> with week fundamental and some strong harmonics. It may sound odd even to
> the person that has heard a trumpet? That said, if you had a peace of
> music and passed through a part with a fifth like that, that happen to have
> very close overtones, it would go on noticed.
> Æ'g
>

Note, fundamental of the second trumpet is not in the harmonic series of
the first trumpet.
free

On Sun, Dec 23, 2012 at 1:22 PM, bigAndrewM <bigandrewm@...> wrote:

> **
>
>
>
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> >
> >
> > What do you mean by this, exactly? If two trumpets are in a 3/2 ratio,
> > then the lower trumpet will be hitting overtones that the upper one
> > isn't. So for instance, if the fundamentals of the two notes are 200
> > Hz and 300 Hz, then the lower one will have 400 Hz as a harmonic,
> > which will be absent from the latter one.
> >
>
> I was thinking more along the lines that the upper trumpet will not be
> producing any overtones that are absent from those produced by the lower
> trumpet, but only changes the relative overtone amplitudes.
>
> > You might look up Huron's work on relating principles of voice leading
> > to more general psychoacoustic notions of auditory stream segregation
> > for some more info on stuff like this:
> >
> http://csml.som.ohio-state.edu/Huron/Publications/huron.voice.leading.html
> >
> > -Mike
> >
>
> Thanks!
>
> Andrew
>
>
>

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

> I suppose that the answer is almost self-evident. Rock
> guitarists use parallel fifths quite often, as a tambral
> effect rather than as voice-leading.

Pipe organs too. So maybe the relative amplitudes of the
partials is not so important (many instruments, not least
the human vocal tract, have strong resonances that warp
spectra too). But in the 'power chord' case, the sounds
are coming from a single *source* in the auditory scene.
The timing of the attacks is the same, the position in the
auditory field (though some recording techniques mess
with this)... what else?

-Carl

On Sun, Dec 23, 2012 at 2:54 AM, Mike Battaglia <battaglia01@...>wrote:

> **
>
>
> On Sat, Dec 22, 2012 at 6:40 PM, Carl Lumma <carl@...> wrote:
> >
> > > if the sound of "major"
> > > actually required direct activation of some sort of partial
> > > fusion of 4:5:6 into a single note, then arpeggiated major
> > > chords would sound very different to us. Thankfully, this is
> > > not the case, so we get to have nice things like (implied)
> > > triadic harmony in two-part inventions.
> >
> > Well, arpeggiated chords do sound very different than
> > homophonic chords. But the key here is "direct". Memories
> > involve activation of the same machinery that's used when
> > the corresponding stimulus is present...
>
> Yes, I agree with this. I think it's very possible (and probably
> likely) that some sort of conditioning or learning process like this
> exists. My only point was that the details of such a process are
> non-trivial.
>
> Anecdotally, I've noticed that this specific thing does happen for me,
> especially in decent 11-limit tunings where there are lots of dyads
> that by themselves are rather discordant and inharmonic sounding, but
> which suddenly come to life in the context of a larger chord. 11/8
> used to sound like crap by itself, but after playing in porcupine so
> much, it's hard for me to hear that dyad and NOT imagine it as part of
> a whole bunch of larger "otonal" sounding chords like 8:9:10:11:12.
>
> I never heard any of the above until I bought my AXiS and my 22-EDO
> guitar though. 9/7 and 11/8 used to just sound like crap; it's only
> after playing around in porcupine and with 11-limit harmony a lot that
> the whole thing has changed. It's really nothing psychoacoustic and
> mysterious; it's like the equivalent of playing C-Eb-Bb in 12-EDO and
> imagining a "G" or something, making it a minor seventh chord, just
> with all of these new 11-limit chords. It's not really like I'm
> imagining VFs directly or what have you, more that I'm just imagining
> other notes as existing in "the current scale" or something, and then
> remembering the sound of the chord formed by those imagined notes and
> the ones I'm playing now. Hopefully that seems direct and simple
> enough.
>
> I've also had the same experience with 9/7, which now always "reminds
> me" of 4:7:9; it sounds much less "harsh" that way. I would be very
> surprised if it were just completely random that imagining the root of
> 9/7 as "4", vs any other possible note in the vicinity, makes the
> whole thing sound less harsh.
>
> So perhaps this is anecdotal evidence in favor of the thing you laid
> out above. I wonder if something like that is the same basic concept
> as imagining triadic harmony during a two-part invention, and if the
> whole thing falls under the heading of "habitually imagined notes" or
> something like that.
>
> > > I don't think that the root of a chord is determined by the
> > > virtual fundamental. For instance, it's pretty easy to hear
> > > the root of 5:6:9 as "5",
> >
> > I don't disagree, but we need to know what the virtual
> > fundamental of a chord would even be, don't we?
> >
> > Assuming there's no hierarchical analysis going on
> > (and there almost certainly is), a chord's spectrum still
> > looks a lot different than the spectrum of its tones,
> > even in JI...
>
> What do you mean by a heirarchical analysis, exactly?
>
> I used to think that 8:12:16:19, or 16:19:24, or whatever might be
> somehow related to the sound of minor, at least when the root is heard
> as the lowest note in the triad. I figured that the same sort of
> conditioning process I laid out above for 11/8 and 9/7 could suggest
> such a result. By itself, 16:19:24 is complex, but if you're imagining
> other notes in the bass, then you might get something like
> 4:8:12:16:19:24, which is less complex than 5:10:15:20:24:30, and if
> you decide to imagine notes even further down in the bass, then it
> gets even less complex.
>
> I originally started thinking along those lines because I noted above
> that 9/7 sounds much more consonant (not concordant, but consonant) if
> I imagine the root as "4"; in general, I doubt that this is some
> random coincidence that this particular choice of imagined note in the
> bass would transform the sound so dramatically vs imagining any other
> random note in the bass. It's likely that 4:7:9 has something to do
> with it, whether because of some past memory of hearing 4:7:9 or
> because of some more "direct" thing or whatever. I figured the same
> could apply to minor, when the lowest note is heard as the root, and
> 8:12:16:19 or 16:19:24 chords.
>
> However, the fact that something as far out as 225 cents in 16-EDO can
> sound like minor, at least to some people, would appear to cast a lot
> of doubt on that theory, or at least raise a lot of questions about
> how that could work...
>
> -Mike
>

I find what is said by Carl and Mike, interesting; because what they [as
well as others of this thread] have said about the roots of chords, fits
with much of my thoughts.

I would like to add my 3 cents about chords and roots. I am somewhat new
to the group. I do not have the advantage of all that has been said on
this subject in the past; so feedback would be appreciated.

I see a chord as nothing more than a group of intervals. Trying to place
all the notes in a harmonic series of a remote, lower note [root]; is an
exercise in futility. If the notes do fit in a harmonic series of a
relatively close root then the chord is harmoniously smooth. In this case
the root *may* be generated by the ear/brain and add to the sound of the
chord. The smaller ratio intervals [i.e. 1:2, 2:3, 3,4] are the most
important for the harmonic function of the chord. Note, I use ratios with
the higher frequency to the right. For some examples of chords mentioned
in this tread:

Major chord,

4:5:6 has intervals,

2:3 [lowest first]

4:5

5:6.

First inversion,

5:6:8 has intervals,

3:4

5:6

5:8.

Second inversion,

3:4:5 has the ratios,

3:4

3:5

4:5.

Minor chord,

5:6|4:5 [for math use 10:12:15] has the ratios,

2:3

4:5

5:6,

[note, same as the major chord, but the order in the chord changes the
sound of the chord].

First inversion,

4:5|3:4 has ratios,

3:4

3:5

4:5,

[Note, same as the *second* inversion of the major chord.

Second inversion,

3:4|5:6 has ratios,

3:4

5:6

5:8,

[You guessed it].

All six of these chords were thought to be 6 different chords, tell the
idea of the root was discovered.

Dominate 7 chord [major-7 chord],

4:5:6|5:6 has ratios,

2:3

4:5

5:6 [two times]

5:9

25:36 [to week].

7-limit, dominate-7 chord [major-7 chord],

4:5:6:7 has ratios,

2:3

4:5

5:6 [two times]

4:7

5:7

6:7.

4:6:9 or 2:3|2:3 chord has ratios,

2:3 [two times]

4:9.

Major-9 chord

4:5:6:9

2:3 [two times]

4:5

5:6

4:9.

5:6:9 chord has ratios,

2:3

5:6

5:9

Mike’s minor-7 chord, 10:12:(15):18

5:6|4:(5)|(2):3 has ratios ,

2:(3)

(2):3

4:5

5:6

(5):6

5:9

To determine the root of a chord look for the strongest interval of the
chord. The strongest interval is the interval that has the lowest ratio
of two notes, which are not octave related, in the chord. 2:3 is the
strongest and the inverse 3:4 is next strongest. Most all chords have an
interval of 2:3 or 3:4. For 2:3 you have the series 1:2:3. For 3:4 you
have 1:2:3:4. Theses are the strongest series in the chord. The two
notes may prepuce the lower notes of the series in the ear/brain if they
are loud enough but this is not necessary. Note, the 1, 2, and 4 are the
same note at different octaves. The root of the chord is the note that
corresponds to the 2 or 4 of these ratios. If there are two of these
ratios in the chord use the one that comes first, lowest frequencies.

If the chord does not have one of the two above ratios,* *use the nest
strongest ratio and place it in a series. The 2 or 4 of this series will
correspond to the note that is the root.

[I have not tested this completely.]

The harmonic function of the chord may determine the root of the chord in
some cases.

Take Mike’s chord, he hears it as a as a minor chord, with implied fifth. The
chord, 5:6:9 [10:12:(15):18] has the root 10:(15) = 2:(3), with 2
corresponding to lowest not of the chord.

The same chord 5:6:9 has the ratio 6:9 = 2:3, with the 2 corresponding to
the second note of the chord. You can look at it this way 5:6:9 is the
second inversion of 6:9:10, with the root on the bottom. Thus the
harmonic function may affect the root of the chord.

Another example is V and vii-dim have the same harmonic function. [The
vii-dim chord can be though of as a V-7 with a missing root, which is
implied.]

free

>
>

On Sun, Dec 23, 2012 at 3:44 PM, Carl Lumma <carl@...> wrote:
>
> Pipe organs too. So maybe the relative amplitudes of the
> partials is not so important (many instruments, not least
> the human vocal tract, have strong resonances that warp
> spectra too). But in the 'power chord' case, the sounds
> are coming from a single *source* in the auditory scene.
> The timing of the attacks is the same, the position in the
> auditory field (though some recording techniques mess
> with this)... what else?

I remember being at an ASA convention a few years back when there was
a paper talking about this, with the main conclusion being that onset
and release cues were one of the most important things to assist in
timbral fusion/auditory stream integration. I wish I remembered the
paper but can't find it now.

There's a pretty classic study from Bergman doing some tests to see
how this exact thing works:
http://webpages.mcgill.ca/staff/Group2/abregm1/web/pdf/1978_Bregman_Pinker.pdf

-Mike

Cool paper - thanks. -C.

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

> There's a pretty classic study from Bergman doing some tests to see
> how this exact thing works:
> http://webpages.mcgill.ca/staff/Group2/abregm1/web/pdf/
> 1978_Bregman_Pinker.pdf
>
> -Mike
>