harmonic halves of the fourth

Does somebody know about a traditional music culture X in which an
instrument Y is usually - or often - tuned in such a way that it includes
the harmonic halves of the fourth (ratio 7:6 = 267 Cent PLUS ratio 8:7 = 231
Cent)?

Martin

Martin Braun wrote:

>Does somebody know about a traditional music culture X in which an
>instrument Y is usually - or often - tuned in such a way that it includes
>the harmonic halves of the fourth (ratio 7:6 = 267 Cent PLUS ratio 8:7 = 231
>Cent)?
> >
The Scala archive includes this:

! pygmie.scl
!
Pygmie scale 5
!
8/7
21/16
3/2
7/4
2/1

Graham

--- In tuning@yahoogroups.com, "Martin Braun" <nombraun@t...> wrote:
> Does somebody know about a traditional music culture X in which an
> instrument Y is usually - or often - tuned in such a way that it
includes
> the harmonic halves of the fourth (ratio 7:6 = 267 Cent PLUS ratio
8:7 = 231
> Cent)?
>
> Martin

the bagpipes are sometimes played this way, but the 7:6 is further
divided into two scale steps:

http://www-
personal.umich.edu/~emacpher/pipes/acoustics/hearscales.html

--- In tuning@yahoogroups.com, "Martin Braun" <nombraun@t...> wrote:
> Does somebody know about a traditional music culture X in which an
> instrument Y is usually - or often - tuned in such a way that it
includes
> the harmonic halves of the fourth (ratio 7:6 = 267 Cent PLUS ratio
8:7 = 231
> Cent)?
>
> Martin

I recently posted the URL of a page about pygmy scales

http://www-math.cudenver.edu/~jstarret/pygmies.html

I do not know if this really qualify as "instrument tuning" as pygmies
polyphonies I know is basically voices or voices + percussion.

It may be not what your are really looking for.

yours truly

François Laferrière

The lost chapter 21 of Boethius' music book survives the title: "How
Ptolemaeus divided the diatessaron in two parts". Ptolemaeus used
only ratios in the form x+1/x, so we can guess that the only
solutions are 16:15+5/4, 10/9+6/5, and 8/7+6/5.

Gabor

--- In tuning@yahoogroups.com, "francois_laferriere"
<francois.laferriere@o...> wrote:
> --- In tuning@yahoogroups.com, "Martin Braun" <nombraun@t...> wrote:
> > Does somebody know about a traditional music culture X in which an
> > instrument Y is usually - or often - tuned in such a way that it
> includes
> > the harmonic halves of the fourth (ratio 7:6 = 267 Cent PLUS ratio
> 8:7 = 231
> > Cent)?
> >
> > Martin

Paul:

> the bagpipes are sometimes played this way, but the 7:6 is further
> divided into two scale steps:

> http://www-personal.umich.edu/~emacpher/pipes/acoustics/hearscales.html

Thanks. This was interesting. The link to the data page is this:

http://www-personal.umich.edu/~emacpher/pipes/acoustics/chanterdata.html

We do not see a harmonic division of the fourth, though. It also seems
doubtful if the scales have a bias towards "7-limit Just Intonation". It
looks more like a "5-limit" one, with the two Gs and the high A being flat
for some reasons. In bagpipes the coexistance of the melody tones with the
"drones" may in some cases be more important than exact scale intervals.

Martin

Fran�ois:

> I recently posted the URL of a page about pygmy scales

> http://www-math.cudenver.edu/~jstarret/pygmies.html

> I do not know if this really qualify as "instrument tuning" as pygmies
> polyphonies I know is basically voices or voices + percussion.

> It may be not what your are really looking for.

Thanks. Yes, this would be an example of the harmonic division of the
fourth. But I would have to see some hard data, before I can believe that
somebody is able to sing these intervals. To tune them on a string
instrument is one thing, but to sing them seems extremely difficult. Perhaps
it's possible. If it is, it would certainly be worth documenting.

Martin

--- In tuning@yahoogroups.com, "Martin Braun" <nombraun@t...> wrote:
> Paul:
>
> > the bagpipes are sometimes played this way, but the 7:6 is further
> > divided into two scale steps:
>
> > http://www-
personal.umich.edu/~emacpher/pipes/acoustics/hearscales.html
>
> Thanks. This was interesting. The link to the data page is this:
>
> http://www-
personal.umich.edu/~emacpher/pipes/acoustics/chanterdata.html
>
> We do not see a harmonic division of the fourth, though. It also
>seems
> doubtful if the scales have a bias towards "7-limit Just
>Intonation".

the Gs usually form clear ratios of 7 with the drone.

> It
> looks more like a "5-limit" one, with the two Gs and the high A
being flat
> for some reasons. In bagpipes the coexistance of the melody tones
with the
> "drones" may in some cases be more important than exact scale
intervals.

yes, this is why the high G *is* so close to 7:4 from the drone in
general. it's a very easy interval to lock into by ear.

--- In tuning@yahoogroups.com, "Martin Braun" <nombraun@t...> wrote:

> To tune them on a string
> instrument is one thing, but to sing them seems extremely
difficult. Perhaps
> it's possible. If it is, it would certainly be worth documenting.
>
> Martin

why would you think it difficult? people learn to sing all kinds of
intervals, including our familiar minor seconds, though exposure and
training.

Paul:

> why would you think it difficult? people learn to sing all kinds of
> intervals, including our familiar minor seconds, though exposure and
> training.

If the 8/7 and 7/6 intervals are sung with as much deviations as is
customary with "our familiar minor seconds", the singers would lose the
narrow bounds of harmonicity effects with these intervals most of the time.

Then, singers of "our familiar minor seconds" are operating in a fixed
12-tone framework, where all tones are most of the time related to much
easier intervals, such as thirds, fourths and fifths.

As I said, I could imagine that it's possible to sing these intervals,
particularly in a slow tempo. But I would not take it for granted, and the
matter would need a thorough documentation.

Martin

--- In tuning@yahoogroups.com, "Martin Braun" <nombraun@t...> wrote:
> Paul:
>
> > why would you think it difficult? people learn to sing all kinds
of
> > intervals, including our familiar minor seconds, though exposure
and
> > training.
>
> If the 8/7 and 7/6 intervals are sung with as much deviations as is
> customary with "our familiar minor seconds", the singers would lose
the
> narrow bounds of harmonicity effects with these intervals most of
the time.

what harmonicity effects? we're talking *horizontal* intervals,
aren't we? if we're talking *vertical* intervals, then there's no
problem singing these intervals very well in tune, by listening to
beats and combinational tones.

and again, training makes all the difference, as most string quartets
around today (and probably for the last 200 years) typically
eschewing the "narrow bounds of harmonicity effects" possible with
thirds and sixths in favor of the significantly tempered versions of
these intervals found on the almighty piano. this becomes all too
apparent after having one's piano (contrary to over 200 years of
practice) tuned in a good meantone for a while -- the sound of scales
in parallel thirds on such an instrument makes nearly all western
performance sound "noisy" by comparison.

Paul:

>> If the 8/7 and 7/6 intervals are sung with as much deviations as is
>> customary with "our familiar minor seconds", the singers would lose
>> the narrow bounds of harmonicity effects with these intervals most of
>> the time.

> what harmonicity effects? we're talking *horizontal* intervals,
> aren't we?

Yes, of course. But we assume that also melodic ("horizontal") intervals
cause harmonicity effects in the brain, don't we? Why else should it be much
easier to sing a fourth or a fifth than a tritone? Physiologically, we
assume a "reverberation" of oscillating pitch detector neurons.

So, if a melodic 7/6 is attractive to a singer, it must be so because it
causes harmonicity effects. But to reach these effects, the singer must hit
the interval much more precisely than when hitting a melodic fourth (as seen
in your "entropy" graphs). That's why 7/6 and 8/7 are more difficult to sing
than 5/4 and 6/5.

But, as I said, it may be possible, and I would love to see some data.

Martin

--- In tuning@yahoogroups.com, "Martin Braun" <nombraun@t...> wrote:

> Paul:
>
> >> If the 8/7 and 7/6 intervals are sung with as much deviations as
is
> >> customary with "our familiar minor seconds", the singers would
lose
> >> the narrow bounds of harmonicity effects with these intervals
most of
> >> the time.
>
> > what harmonicity effects? we're talking *horizontal* intervals,
> > aren't we?
>
> Yes, of course. But we assume that also melodic ("horizontal")
intervals
> cause harmonicity effects in the brain, don't we? Why else should
it be much
> easier to sing a fourth or a fifth than a tritone?

i see no evidence that this ease carries over beyond ratios of 3 to
ratios of 5, let alone ratios of 7.

> Physiologically, we
> assume a "reverberation" of oscillating pitch detector neurons.

whatever it is, it appears far less powerful than the "harmonicity
effects" attending vertical "intervals", and doesn't extend to nearly
as complex ratios.

> So, if a melodic 7/6 is attractive to a singer, it must be so
because it
> causes harmonicity effects.

no, it would have to be experience/training.

> But to reach these effects, the singer must hit
> the interval much more precisely than when hitting a melodic fourth
(as seen
> in your "entropy" graphs).

excuse me?

> That's why 7/6 and 8/7 are more difficult to sing
> than 5/4 and 6/5.

not my much, if at all, assuming similar training. i don't hear many
people singing melodic 5/4s and 6/5s these days, except when moving
slowly against a fixed harmony from one "locked-in" pitch to another.

Paul:

>> Physiologically, we
>> assume a "reverberation" of oscillating pitch detector neurons.

> whatever it is, it appears far less powerful than the "harmonicity
> effects" attending vertical "intervals", and doesn't extend to nearly
> as complex ratios.

Less powerful than in "vertical" intervals, yes. But it must be present in
"horizontal" intervals as well. Otherwise we would not have thirds in
"horizontal" music.

>> So, if a melodic 7/6 is attractive to a singer, it must be so because it
>> causes harmonicity effects.

> no, it would have to be experience/training.

But why on earth train something, which - as you think - could not be heard
anyway? Why on earth train 231 Cent + 267 Cent, instead of anything else,
say, 245 Cent + 253 Cent?

Martin

--- In tuning@yahoogroups.com, "Martin Braun" <nombraun@t...> wrote:
> Paul:
>
> >> Physiologically, we
> >> assume a "reverberation" of oscillating pitch detector neurons.
>
> > whatever it is, it appears far less powerful than the "harmonicity
> > effects" attending vertical "intervals", and doesn't extend to
nearly
> > as complex ratios.
>
> Less powerful than in "vertical" intervals, yes. But it must be
present in
> "horizontal" intervals as well. Otherwise we would not have thirds
in
> "horizontal" music.

not true. the horizontal thirds found in world musics show no
attraction to ratios of 5. we have seconds in horizontal music and
thirds can be built up from seconds. do the seconds have a derivation
in ratios? no, and neither do the horizonatal thirds.

> >> So, if a melodic 7/6 is attractive to a singer, it must be so
because it
> >> causes harmonicity effects.
>
> > no, it would have to be experience/training.
>
> But why on earth train something, which - as you think - could not
be heard
> anyway? Why on earth train 231 Cent + 267 Cent, instead of anything
else,
> say, 245 Cent + 253 Cent?

it might be an arbitrary choice, perhaps some
musician's "philosophical" choice as some around here are fond of
saying. or, more likely, there is some *vertical* aspect in the
music, perhaps just a drone, which makes intervals like these play a
large role and therefore become familiar.

In a message dated 9/25/2003 4:58:57 PM Eastern Daylight Time,
paul@stretch-music.com writes:

> >But why on earth train something, which - as you think - could not
> be heard
> >anyway? Why on earth train 231 Cent + 267 Cent, instead of anything
> else,
> >say, 245 Cent + 253 Cent?
>
> it might be an arbitrary choice, perhaps some
> musician's "philosophical" choice as some around here are fond of
> saying. or, more likely, there is some *vertical* aspect in the
> music, perhaps just a drone, which makes intervals like these play a
> large role and therefore become familiar.
>
>

Um, why can't any musician learn any linear/melodic set of notes without any
drone basis? I don't think there needs to be any harmonic basis, or
underpinning for the choice of musical tones utilized. When Odetta rehearses her field
hollars, she sings the intervals she wants with exactitude and with
recurrence. These are not just intonation intervals, though one might find some way to
construe them as just. And they are not intellectually rendered in cents.
They are emotionally memorized by rote.

best, Johnny Reinhard

--- In tuning@yahoogroups.com, Afmmjr@a... wrote:
> In a message dated 9/25/2003 4:58:57 PM Eastern Daylight Time,
> paul@s... writes:
>
> > >But why on earth train something, which - as you think - could
not
> > be heard
> > >anyway? Why on earth train 231 Cent + 267 Cent, instead of
anything
> > else,
> > >say, 245 Cent + 253 Cent?
> >
> > it might be an arbitrary choice, perhaps some
> > musician's "philosophical" choice as some around here are fond of
> > saying. or, more likely, there is some *vertical* aspect in the
> > music, perhaps just a drone, which makes intervals like these
play a
> > large role and therefore become familiar.
> >
> >
>
> Um, why can't any musician learn any linear/melodic set of notes
without any
> drone basis? I don't think there needs to be any harmonic basis,
or
> underpinning for the choice of musical tones utilized.

i agree completely, which is why 245 Cent + 253 Cent would be a
perfectly valid choice, just as valid as 231 Cent + 267 Cent, in the
absence of any drone or vertical harmony.

> When Odetta rehearses her field
> hollars, she sings the intervals she wants with exactitude and with
> recurrence. These are not just intonation intervals,

exactly.

> They are emotionally memorized by rote.

yes, the emotional content is exactly as dictated by the interval in
the context of the culture, undiluted by any "vagueness" that would
supposedly result from deviation from JI, despite what one of the
aarons is claiming.

Paul Erlich wrote:
> --- In tuning@yahoogroups.com, "Martin Braun" <nombraun@t...> wrote:
> >>Paul:
>>
>>
>>>>Physiologically, we
>>>>assume a "reverberation" of oscillating pitch detector neurons.
>>
>>>whatever it is, it appears far less powerful than the "harmonicity
>>>effects" attending vertical "intervals", and doesn't extend to > > nearly
> >>>as complex ratios.
>>
>>Less powerful than in "vertical" intervals, yes. But it must be > > present in
> >>"horizontal" intervals as well. Otherwise we would not have thirds > > in
> >>"horizontal" music.
> > > not true. the horizontal thirds found in world musics show no > attraction to ratios of 5. we have seconds in horizontal music and > thirds can be built up from seconds. do the seconds have a derivation > in ratios? no, and neither do the horizonatal thirds.
> > >>>>So, if a melodic 7/6 is attractive to a singer, it must be so > > because it
> >>>>causes harmonicity effects.
>>
>>>no, it would have to be experience/training.
>>
>>But why on earth train something, which - as you think - could not > > be heard
> >>anyway? Why on earth train 231 Cent + 267 Cent, instead of anything > > else,
> >>say, 245 Cent + 253 Cent?
> > > it might be an arbitrary choice, perhaps some > musician's "philosophical" choice as some around here are fond of > saying. or, more likely, there is some *vertical* aspect in the > music, perhaps just a drone, which makes intervals like these play a > large role and therefore become familiar.

In other words, it's a measurement thing necessary if things should be repeatable. I just realize this is not the barbershop thread, but since I'm not going to mention comma drift anyway...

JI, the locking into small number ratios, is a direct, numberless kind of measurement; you go by your ears alone, and you can indeed sense the physics. Idiophones almost the same, minus a consistent spectrum and the much less direct physical experience and therefore probably with much effects. Plucked string instruments can produce harmonics; here there's something to count and the obvious tuning is pythagorean. A monochord collection like the piano needs to be economical with regard to number of tones/time used for retunings (and cost, and playability) and leans toward temperaments.

The physical aspect affects things like ease of production and sustain, but there is music to be made with any pitch collection. I believe that music is tuned up mainly for consistency's sake, but there are still different ways to tune that are more or less natural for a particular instrument.

klaus

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:

> the horizontal thirds found in world musics show no
> attraction to ratios of 5. we have seconds in horizontal music and
> thirds can be built up from seconds. do the seconds have a
derivation
> in ratios? no, and neither do the horizonatal thirds.

Paul, do you have evidence for this? Otherwise I won't buy it.
Without evidence to the contrary, I would assume that the "thirds
found in world musics" are in most cases subunits of fifths and are
sung under the influence of the attractors of the ratios 5/4 and 6/5.
That is, I would expect them to have a statistical bias towards these
ratios.

Martin

--- In tuning@yahoogroups.com, "Martin Braun" <nombraun@t...> wrote:
> --- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
>
> > the horizontal thirds found in world musics show no
> > attraction to ratios of 5. we have seconds in horizontal music
and
> > thirds can be built up from seconds. do the seconds have a
> derivation
> > in ratios? no, and neither do the horizonatal thirds.
>
> Paul, do you have evidence for this? Otherwise I won't buy it.
> Without evidence to the contrary, I would assume that the "thirds
> found in world musics" are in most cases subunits of fifths and are
> sung under the influence of the attractors of the ratios 5/4 and
6/5.

you would do well to sit down for hours with the "new grove
dictionary of music and musicians" and digest the ethnomusicological
articles (look up the name of just about any country). your
assumption will prove to be most incorrect.

> That is, I would expect them to have a statistical bias towards
these
> ratios.
>
> Martin

martin, glancing over just a few ethnic traditions, you will find
some which use neutral thirds, some which have no thirds at all
(scales approaching 5-equal), and just about everything in-between.
byzantine music often uses scales with intervals around 366 cents
above the tonic, persian music, 333 cents. chinese thirds tend to
assume pythagorean proportions (294 & 408 cent thirds), and this is
also roughly true of western melodic practice since about 1800 (and
probably before 1480 as well). if there is a statistical attraction
to 5/4 and 6/5 as melodic intervals, it is clearly a very weak effect
compared with the force of experience and tradition within any given
culture.

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:

if there is a statistical attraction
> to 5/4 and 6/5 as melodic intervals, it is clearly a very weak
effect
> compared with the force of experience and tradition within any
given
> culture.

This is pretty far from saying 5/4 "sounds wrong", which is what you
just said re barbershop.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
>
> if there is a statistical attraction
> > to 5/4 and 6/5 as melodic intervals, it is clearly a very weak
> effect
> > compared with the force of experience and tradition within any
> given
> > culture.
>
> This is pretty far from saying 5/4 "sounds wrong", which is what
you
> just said re barbershop.

i guess you didn't understand something. 5/4 will sound wrong if the
force of experience and tradition within a given culture is enforcing
a different standard for the major third -- for example classical (as
opposed to folk) string players today will tend to hear it as too
narrow. of course, playing around with alternate tuning systems is a
powerful experiential force itself!

Paul:

>> Without evidence to the contrary, I would assume that the "thirds
>> found in world musics" are in most cases subunits of fifths and are
>> sung under the influence of the attractors of the ratios 5/4 and 6/5.

> you would do well to sit down for hours with the "new grove
> dictionary of music and musicians" and digest the ethnomusicological
> articles (look up the name of just about any country). your
> assumption will prove to be most incorrect.

These articles are (nearly) all based on accidental observations and theory,
not on statistical analysis. If there are large data tables, such as those
of Surjodiningrat et al. (1969) on Central Javanese Gamelan tuning, people
write about them what they like without ever applying a decent statistical
analysis. [My analysis of the pelog data from last year was the first one
that was ever carried out, 33 years (!) after data publication.] So, reading
the New Grove would not decide the issue.

>> That is, I would expect them to have a statistical bias towards
>> these ratios.

> if there is a statistical attraction
> to 5/4 and 6/5 as melodic intervals, it is clearly a very weak effect
> compared with the force of experience and tradition within any given
> culture.

This is a purely empirical question. As a long as there is no evidence, I
will continue to assume that singers of a third with a mean close to the 5/4
ratio are likely to have a smaller standard deviation than singers of a
"neutral" third close to 350 Cent. Not all traditions are equally stable.
Looking at the musical world on the whole, 5/4 thirds seem to have survived
a lot more than 350 Cent thirds.

Martin

--- In tuning@yahoogroups.com, "Martin Braun" <nombraun@t...> wrote:

> Looking at the musical world on the whole, 5/4 thirds seem to have
survived
> a lot more than 350 Cent thirds.
>
> Martin

your musical world, then, is very different from mine (where blues
and middle eastern music feature strongly). i don't know of any
examples of reliable 5/4 thirds being sung today without a
harmonic/vertical reason for, or aid to, doing so (and were *were*
discussing purely the issue of melodic/horizontal intervals).

Paul:

>> Looking at the musical world on the whole, 5/4 thirds seem to have
>> survived a lot more than 350 Cent thirds.
>
> your musical world, then, is very different from mine (where blues
> and middle eastern music feature strongly). i don't know of any
> examples of reliable 5/4 thirds being sung today without a
> harmonic/vertical reason for, or aid to, doing so (and were *were*
> discussing purely the issue of melodic/horizontal intervals).

Perhaps your guess is right - that thirds of all types, in unaccompanied
solo singing, show about the same deviations around their mean. But we have
no way of knowing. We gotta wait until somebody measures this.

Anyhow, perhaps we can agree that in music cultures where string instruments
have a strong position, like in Japan, China, India, the Orient, and Europe,
there is some reason to assume that even for unaccompanied solo singers the
5/4 thirds may be more stable than other thirds.

Martin

Speaking of harmonic halves of the fourth, in any temperament with
(676/675)~1, we have (52/45)^2 ~ 4/3, which is a pretty nice 13-limit
harmonic half of a fourth.