Oh another thing about 34 and the median itnervals....

Just thought I'd mention something... if you take the harmonic series
out to, say, 25, multiply it by Pi and octave reduce, you get the very
intervals I've been wingeing about for the last year. Take this set
and and interlace it with a classic JI based on the same partials,
fill in any gaps by mirroring the intervals you have against 2/1 or
using intervals seperating the ones you already have, temper out your
tinies, ie keep things as minimal and elegant as possible, and you get
a 34, smooth it out and you have 34-EDO. You have to throw out 7/4.

-Cameron Bobro

Cameron;

Just a thought about 34 edo and 5L + 2s:

If you assume the L = 6 units and s = 2 units

3L+s will give you a fifth of 705.88 cents.

So you could treat the mapping of 34 EDO as a sort of meantone-type tuning with a very sharp fifth i.e. > 702 (Phythag?.)

(at least that approach would enable you to attach notenames and scale positions to the intervals for comparison to traditional Western harmonic structures).

Charles Lucy lucy@lucytune.com

----- Promoting global harmony through LucyTuning -----

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On 11 Aug 2007, at 10:51, Cameron Bobro wrote:

> Just thought I'd mention something... if you take the harmonic series
> out to, say, 25, multiply it by Pi and octave reduce, you get the very
> intervals I've been wingeing about for the last year. Take this set
> and and interlace it with a classic JI based on the same partials,
> fill in any gaps by mirroring the intervals you have against 2/1 or
> using intervals seperating the ones you already have, temper out your
> tinies, ie keep things as minimal and elegant as possible, and you get
> a 34, smooth it out and you have 34-EDO. You have to throw out 7/4.
>
> -Cameron Bobro
>
>
>

----- Original Message ----- From: Charles Lucy
To: tuning@yahoogroups.com
Sent: Saturday, August 11, 2007 5:18 AM
Subject: Re: [tuning] Oh another thing about 34 and the median itnervals.... 5L+2s?
>
> Cameron;
>
> Just a thought about 34 edo and 5L + 2s:
>
> If you assume the L = 6 units and s = 2 units
>
> 3L+s will give you a fifth of 705.88 cents.
>
> So you could treat the mapping of 34 EDO as a sort of meantone-type tuning > with a very sharp fifth i.e. > 702 (Phythag?.)

It wouldn't be any kind of meantone if the fifth is sharper than 3/2, and the fifth of 17/34-edo is 705.882 cents.

(I've used the term "antimeantone" for fifths tempered higher, but I'm sure there's some other name used here; I can't remember what it is.)

~D.

--- In tuning@yahoogroups.com, Charles Lucy <lucy@...> wrote:
>
> Cameron;
>
> Just a thought about 34 edo and 5L + 2s:
>
> If you assume the L = 6 units and s = 2 units
>
> 3L+s will give you a fifth of 705.88 cents.
>
> So you could treat the mapping of 34 EDO as a sort of meantone-type
> tuning with a very sharp fifth i.e. > 702 (Phythag?.)
>
> (at least that approach would enable you to attach notenames and
> scale positions to the intervals for comparison to traditional
> Western harmonic structures).

Yes I think that would work just fine. But a couple of points- if 34
has a very sharp fifth compared to 3/2, does that make the Lucy
fifth very, very flat (+3.9 vs -6.5)?

Another thing is- I think the whole circle of fifths thing is
a tragic mistake, arising from getting carried away with
ancient rules of thumb
for practical tuning, aimed at getting a couple of key tones
and then having some reference points for getting intervals
actually used.

Now, this surely must be of interest to you, if to
noone else, hit "fixed font width" to view:

Int *Pi, 8ve reduced Cents Ratio Diff. in cents

25 12271846/10000000 354.423 27/22 < .13
23 1129/1000 210.0546 35/31 < .06
21 1030835/1000000 52.5761 67/65 *
19 186532/100000 1079.3078 28/15 < 1.3
17 1668971/1000000 886.7507 242/145 < .01
15 14762/10000 674.2658 31/21 < .02
13 1276272/1000000 422.323 23/18 < .05
11 1079924/1000000 133.1157 27/25 < .13
9 17671458/10000000 985.7053 23/13 < 2.5 (53/30) < .55
7 137444678/100000000 550.6213 11/8 *
5 1963495/1000000 1168.1087 53/27 <.05
3 11780972/10000000 283.7503 33/28 < .8
2 1570796/1000000 781.795 11/7 < 1.2

This is a "circular shadow" tuning, the integer series multiplied
by Pi and octave-reduced. I am using a model of sound
based not on a string, but on an ideal point emanating ideal sound
in an ideal environment; imagine, at any given instant, spheres
within spheres, and their interrelationships at any given
instant. I propose
that spherical relationships to the spheres in this model (ie,
the diameters represent the integer harmonic series, and we're
imagining another set of nestled spheres whose diameters
are the circumfrences of the spheres we already have via
the integer harmonic series) are also percieved of as
"harmonious", or being of pleasing relative proportions.

(Everything is inside-out, so to speak, but anyone who
knows enough to pick at the model will also realize
the "bigger" partials represent greater frequency,
ergo smaller spheres, not larger diameters, but anyway).

Leaving out shadows of the seventh integer harmonic and
its multiples (I dropped 7 and 21) because I'm also
dropping 7/1 and 21/1 from the original harmonic series,
for a number of reasons, one reason being that I
know that I'm going to get too many frets on my instruments
if I try to include them, here's the resulting tuning,
in rational form because I am going to interlace it with
traditional JI intervals (and some of the correlations of
this "spherical shadows" tuning with various traditional
intervals are eerily precise):

0: 1/1 0.000 unison, perfect prime
1: 27/25 133.238 large limma, BP small semitone
2: 35/31 210.104
3: 33/28 284.447 undecimal minor third
4: 27/22 354.547 neutral third, Zalzal wosta of al-Farabi
5: 23/18 424.364 vicesimotertial major third
6: 31/21 674.255
7: 11/7 782.492 undecimal augmented fifth
8: 242/145 886.745
9: 23/13 987.747
10: 28/15 1080.557 grave major seventh
11: 53/27 1167.640
12: 2/1 1200.000 octave
2/1

Go ahead and listen to this one. Interlace this with 3/2, 4/3, 5/4,
6/5, then fill out your gaps using material you've got (there's a
24/23 between 23/18 and 4/3, for example), and you'll get a 34.

Now I know that although the model is cool, the assumption
that this can be percieved may be far-fetched and can only
be "verified" subjectively. But it is a "music of the shperes",
without mysticism.

Now someone is going to say how obvious this is and everyone
knows that, to which I'll say, then how come I've been
wanking on about and making music with these very intervals,
and tunings related to Pi, for the last year here and noone
mentioned this once? :-P

-Cameron Bobro

PS, I do realize that when I said "emenating from an ideal point"
that it implies integer harmonics measured at the radii, but I did
say that I'm octave reducing.

--- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...>
wrote:
>
> --- In tuning@yahoogroups.com, Charles Lucy <lucy@> wrote:
> >
> > Cameron;
> >
> > Just a thought about 34 edo and 5L + 2s:
> >
> > If you assume the L = 6 units and s = 2 units
> >
> > 3L+s will give you a fifth of 705.88 cents.
> >
> > So you could treat the mapping of 34 EDO as a sort of meantone-
type
> > tuning with a very sharp fifth i.e. > 702 (Phythag?.)
> >
> > (at least that approach would enable you to attach notenames
and
> > scale positions to the intervals for comparison to traditional
> > Western harmonic structures).
>
> Yes I think that would work just fine. But a couple of points- if
34
> has a very sharp fifth compared to 3/2, does that make the Lucy
> fifth very, very flat (+3.9 vs -6.5)?
>
> Another thing is- I think the whole circle of fifths thing is
> a tragic mistake, arising from getting carried away with
> ancient rules of thumb
> for practical tuning, aimed at getting a couple of key tones
> and then having some reference points for getting intervals
> actually used.
>
> Now, this surely must be of interest to you, if to
> noone else, hit "fixed font width" to view:
>
> Int *Pi, 8ve reduced Cents Ratio Diff. in cents
>
> 25 12271846/10000000 354.423 27/22 < .13
> 23 1129/1000 210.0546 35/31 < .06
> 21 1030835/1000000 52.5761 67/65 *
> 19 186532/100000 1079.3078 28/15 < 1.3
> 17 1668971/1000000 886.7507 242/145 < .01
> 15 14762/10000 674.2658 31/21 < .02
> 13 1276272/1000000 422.323 23/18 < .05
> 11 1079924/1000000 133.1157 27/25 < .13
> 9 17671458/10000000 985.7053 23/13 < 2.5 (53/30) < .55
> 7 137444678/100000000 550.6213 11/8 *
> 5 1963495/1000000 1168.1087 53/27 <.05
> 3 11780972/10000000 283.7503 33/28 < .8
> 2 1570796/1000000 781.795 11/7 < 1.2
>
> This is a "circular shadow" tuning, the integer series multiplied
> by Pi and octave-reduced. I am using a model of sound
> based not on a string, but on an ideal point emanating ideal sound
> in an ideal environment; imagine, at any given instant, spheres
> within spheres, and their interrelationships at any given
> instant. I propose
> that spherical relationships to the spheres in this model (ie,
> the diameters represent the integer harmonic series, and we're
> imagining another set of nestled spheres whose diameters
> are the circumfrences of the spheres we already have via
> the integer harmonic series) are also percieved of as
> "harmonious", or being of pleasing relative proportions.
>
> (Everything is inside-out, so to speak, but anyone who
> knows enough to pick at the model will also realize
> the "bigger" partials represent greater frequency,
> ergo smaller spheres, not larger diameters, but anyway).
>
> Leaving out shadows of the seventh integer harmonic and
> its multiples (I dropped 7 and 21) because I'm also
> dropping 7/1 and 21/1 from the original harmonic series,
> for a number of reasons, one reason being that I
> know that I'm going to get too many frets on my instruments
> if I try to include them, here's the resulting tuning,
> in rational form because I am going to interlace it with
> traditional JI intervals (and some of the correlations of
> this "spherical shadows" tuning with various traditional
> intervals are eerily precise):
>
> 0: 1/1 0.000 unison, perfect prime
> 1: 27/25 133.238 large limma, BP small semitone
> 2: 35/31 210.104
> 3: 33/28 284.447 undecimal minor third
> 4: 27/22 354.547 neutral third, Zalzal wosta of al-Farabi
> 5: 23/18 424.364 vicesimotertial major third
> 6: 31/21 674.255
> 7: 11/7 782.492 undecimal augmented fifth
> 8: 242/145 886.745
> 9: 23/13 987.747
> 10: 28/15 1080.557 grave major seventh
> 11: 53/27 1167.640
> 12: 2/1 1200.000 octave
> 2/1
>
> Go ahead and listen to this one. Interlace this with 3/2, 4/3,
5/4,
> 6/5, then fill out your gaps using material you've got (there's a
> 24/23 between 23/18 and 4/3, for example), and you'll get a 34.
>
> Now I know that although the model is cool, the assumption
> that this can be percieved may be far-fetched and can only
> be "verified" subjectively. But it is a "music of the shperes",
> without mysticism.
>
> Now someone is going to say how obvious this is and everyone
> knows that, to which I'll say, then how come I've been
> wanking on about and making music with these very intervals,
> and tunings related to Pi, for the last year here and noone
> mentioned this once? :-P
>
> -Cameron Bobro
>

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

> Another thing is- I think the whole circle of fifths thing is
> a tragic mistake, arising from getting carried away with
> ancient rules of thumb
> for practical tuning, aimed at getting a couple of key tones
> and then having some reference points for getting intervals
> actually used.

Why is it a mistake? What are these alleged rules of thumb which lead
to circles of fifths?

> This is a "circular shadow" tuning, the integer series multiplied
> by Pi and octave-reduced. I am using a model of sound
> based not on a string, but on an ideal point emanating ideal sound
> in an ideal environment...

I see no model of sound here.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@>
wrote:
>
> > Another thing is- I think the whole circle of fifths thing is
> > a tragic mistake, arising from getting carried away with
> > ancient rules of thumb
> > for practical tuning, aimed at getting a couple of key tones
> > and then having some reference points for getting intervals
> > actually used.
>
> Why is it a mistake?

Read the current sruti thread for an example.

>What are these alleged rules of thumb which >lead
> to circles of fifths?

What do you mean, "alleged"? Musicians all over the world
still tune open strings to fifths and fourths, and use
pure fifths as reference points within tunings.

>
> > This is a "circular shadow" tuning, the integer series multiplied
> > by Pi and octave-reduced. I am using a model of sound
> > based not on a string, but on an ideal point emanating ideal
>sound
> > in an ideal environment...
>
> I see no model of sound here.

Huh? It's a bog standard model in acoustics, recording and
some DSP. Take a look at the classic Jecklin recording book,
there are lovely illustrations of the deviations from this
obvious model, deviations typical of common instruments
(of course the deviations are extremely important in
microphone placement).

Meanwhile you're ignoring the actual tuning I'm talking about,
LOL.

-Cameron Bobro

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

> >What are these alleged rules of thumb which >lead
> > to circles of fifths?
>
> What do you mean, "alleged"? Musicians all over the world
> still tune open strings to fifths and fourths, and use
> pure fifths as reference points within tunings.

This makes zero sense. If you tune using pure fifths and fourths, you
cannot get a circle. Of course, 53-et is close enough for government
work.

> >
> > > This is a "circular shadow" tuning, the integer series
multiplied
> > > by Pi and octave-reduced. I am using a model of sound
> > > based not on a string, but on an ideal point emanating ideal
> >sound
> > > in an ideal environment...
> >
> > I see no model of sound here.
>
> Huh? It's a bog standard model in acoustics, recording and
> some DSP.

Oh, please. None of those would touch multiplying by pi with a ten
foot barge pole.

> Meanwhile you're ignoring the actual tuning I'm talking about,
> LOL.

Multiplying by pi? Spare me.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@>
> wrote:
> > --- In tuning@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@>
> > wrote:
>
> > >What are these alleged rules of thumb which >lead
> > > to circles of fifths?
> >
> > What do you mean, "alleged"? Musicians all over the world
> > still tune open strings to fifths and fourths, and use
> > pure fifths as reference points within tunings.
>
> This makes zero sense. If you tune using pure fifths and fourths,
you
> cannot get a circle. Of course, 53-et is close enough for
>government
> work.

Who said circle?
>
> > >
> > > > This is a "circular shadow" tuning, the integer series
> multiplied
> > > > by Pi and octave-reduced. I am using a model of sound
> > > > based not on a string, but on an ideal point emanating ideal
> > >sound
> > > > in an ideal environment...
> > >
> > > I see no model of sound here.
> >
> > Huh? It's a bog standard model in acoustics, recording and
> > some DSP.
>
> Oh, please. None of those would touch multiplying by pi with a ten
> foot barge pole.

Once again, huh? I was talking about a certain ideal-sound model,
which certainly is bog-standard. What you call "multiplying by pi"
isn't standard, it's a simplified and simple way of relating one set
of spheres to another. Reverberation DSP usually reduces things to a
finite set of rays for example, it would take a hell of a lot of
calculation to deal with things via spheres. IIRC it was an interview
with one of the Lexicon engineers that really got me thinking about
this.

> > Meanwhile you're ignoring the actual tuning I'm talking about,
> > LOL.
>
> Multiplying by pi? Spare me.

Oh yeah I'm such a dolt- what do spheres and sines and all that crap
have to do with sound anyway? And a tuning that actually sounds good
and truly "xenharmonic" rather than like an out-of-tune 12-tET,
that's just downright offensive, isn't it. Hahaha!

-Cameron Bobro

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@>
> wrote:
> > --- In tuning@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@>
> > wrote:
>
> > >What are these alleged rules of thumb which >lead
> > > to circles of fifths?
> >
> > What do you mean, "alleged"? Musicians all over the world
> > still tune open strings to fifths and fourths, and use
> > pure fifths as reference points within tunings.
>
> This makes zero sense. If you tune using pure fifths and fourths,
>you
> cannot get a circle. Of course, 53-et is close enough for
>government
> work.

Didn't I say "spiral" and "chains" many times? Obviously you don't
get a CLOSED circle, and I never said you did, except in practical
real life, with 53 as you say.

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

> Didn't I say "spiral" and "chains" many times? Obviously you don't
> get a CLOSED circle, and I never said you did, except in practical
> real life, with 53 as you say.

I read "circle", which I interpret to mean circle.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@> wrote:
>
> > Didn't I say "spiral" and "chains" many times? Obviously you don't
> > get a CLOSED circle, and I never said you did, except in practical
> > real life, with 53 as you say.
>
> I read "circle", which I interpret to mean circle.
>

Sorry, I should be more precise, for example, I do say
things like such-and-such produces such an interval, when I actually
mean an interval .13 cents away, and I should always specify.

Anyway, the model of concentric and otherwise nestled spheres
(reduced to cirlces) is something I find very interesting. The
immediate appearance of the half-octave is a real curiosity, for
example.

Hey, Joe Monzo, remember how long ago you were talking about a kind
of ideal low minor third around 280-something cents, found by
experiment? I've had the same "ideal" for a long time, for me it
kept turning up around 282-284 cents, and have been tinkering with
it continuously.

If you look at this classic image

http://cda.morris.umn.edu/~mcquarrb/Precalculus/Animations/SineCosine
Anim.html

you'll find that the point marking the valley of the sine, octave-
reduced, gives you 283.75 cents. Now Gene finds all this beneath
contempt, but I find that this is precisely the low minor third I've
been looking for, and I find its similarity to the low third of 34
intriging as well. Extrapolating from this simple model gives you
all kinds of goodies as well.

I also don't think it's terribly far-fetched to imagine that
these things were explored in the ancient world, but I don't know
of any pertinent records.

Have fun and thanks for all your long and detailed recents posts,

-Cameron Bobro

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

> I also don't think it's terribly far-fetched to imagine that
> these things were explored in the ancient world

Hans Kayser's The textbook of harmonics, translated by Ariel &
Joscelyn Godwin:
http://www.sacredscience.com/archive/Kayser.htm
http://www.sacredscience.com/archive/ToneSpiral.htm

Funny pictures

Clark

--- In tuning@yahoogroups.com, "threesixesinarow" <CACCOLA@...>
wrote:
>
> --- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@>
> wrote:
>
> > I also don't think it's terribly far-fetched to imagine that
> > these things were explored in the ancient world
>
> Hans Kayser's The textbook of harmonics, translated by Ariel &
> Joscelyn Godwin:
> http://www.sacredscience.com/archive/Kayser.htm
> http://www.sacredscience.com/archive/ToneSpiral.htm
>
> Funny pictures
>
> Clark
>

Thanks for the links, the sacred science link (except for the
mistake of fiddling with degrees instead of cutting to
chase with radians) looks very good, have to examine it!

What makes everything an uphill battle for me in this place,
and a downhill battle in real life where I have to perform
all this crap in front of people, is that I'm continually
searching for a rhyme and reason to things I've long heard,
rather than trying to make a music from a scale or something
like that. For example the low and high thirds I keep harping
on were things I sang and found on a "monochord" (fretless guitar
with bottleneck) long before I had anything but the most general
ideas about tuning theory. Finding them truly exactly
and integrating them into a flexible system, that's the thing.

--- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...> wrote:
>
> ..... talking about a kind
> of ideal low minor third around 280-something cents, found by
> experiment? I've had the same "ideal" for a long time, for me it
> kept turning up around 282-284 cents, and have been tinkering with
> it continuously.

Dear Cameron,

what about the difference of the partials 5th and 17th harmonics:

(5/4) : (17/16) = (20/17) approx: [~386C - ~105C = ~281Cents]

or theoretically more precisely

1200Cents * ln(20/17) / ln2 = ~281.358304...Cents ?

http://www.xs4all.nl/~huygensf/doc/intervals.html
labels "20/17 (as) septendecimal augmented second"

Nonlinear inharmonicity effects on the basilar-membrane
for the 17th partial may explain the resulting
additional ~0.6 ... ~2.7 Cents
deviation above that ~281.4 Cents
in yours real acustically perception values
obtained by hearing experiments.

Not to be confused with the perhaps more prominet so called:
subdominant "7/6 septimal minor third"
of Archytas, Helmholtz &ct:

1200C * ln(7/6) / ln(2) = ~266.870906...Cents

that turns out to lie in difference about ~2/3 comma
120:119 = (20/17) : (7/6)
1200C * ln(120 / 119) / ln(2) = ~14.4873988...Cents
lower flat than yours preferred 20:17.

Do you discern that 120:119 discrepancy too?
A.S.

--- In tuning@yahoogroups.com, "Andreas Sparschuh" <a_sparschuh@...>
wrote:
>
> --- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@>
wrote:
> >
> > ..... talking about a kind
> > of ideal low minor third around 280-something cents, found by
> > experiment? I've had the same "ideal" for a long time, for me it
> > kept turning up around 282-284 cents, and have been tinkering
with
> > it continuously.
>
> Dear Cameron,
>
> what about the difference of the partials 5th and 17th harmonics:
>
> (5/4) : (17/16) = (20/17) approx: [~386C - ~105C = ~281Cents]
>
> or theoretically more precisely
>
> 1200Cents * ln(20/17) / ln2 = ~281.358304...Cents ?
>
> http://www.xs4all.nl/~huygensf/doc/intervals.html
> labels "20/17 (as) septendecimal augmented second"

Servus, Andreas- I've used this very interval, arriving at
it as a median of 13/11 and 7/6. Although an online perception
test that's been linked here several times states that I can
consistently hear a .75 cent difference at 500 Hz, I don't
claim to be even nearly that precise- in other words, when I first
used 20/17, and to this day, it sounds "that's it!" on its own, but
in a complete tuning and musical context there's always
something not quite right.

>
> Nonlinear inharmonicity effects on the basilar-membrane
> for the 17th partial may explain the resulting
> additional ~0.6 ... ~2.7 Cents
> deviation above that ~281.4 Cents
> in yours real acustically perception values
> obtained by hearing experiments.

That's beyond me- and I think Occam's razor would favor
my simpler explanation (which of course is only simple
if you accept Pi as a simple proportion, which I believe
it is: only the number describing Pi isn't simple).
>
>
> Not to be confused with the perhaps more prominet so called:
> subdominant "7/6 septimal minor third"
> of Archytas, Helmholtz &ct:
>
> 1200C * ln(7/6) / ln(2) = ~266.870906...Cents
>
> that turns out to lie in difference about ~2/3 comma
> 120:119 = (20/17) : (7/6)
> 1200C * ln(120 / 119) / ln(2) = ~14.4873988...Cents
> lower flat than yours preferred 20:17.
>
> Do you discern that 120:119 discrepancy too?

120/119 on it's own? I doubt it, but 7/6 is a very specific
interval, a strange combination of maximum consonance and
dissonance, and it's definitely not the one I'm hearing,
13/11 being the closest "simple" interval in character (it
also sounds just right isolated and out of context- as I said,
I don't claim to hear tiny differences in isolated intervals).

-Cameron Bobro

--- In tuning@yahoogroups.com, "threesixesinarow" <CACCOLA@...> wrote:
>
on
> http://www.sacredscience.com/archive/ToneSpiral.htm
>
> Funny pictures
>
Dears Clark & Cameron,

that appears to be again an independent rediscovery of 360-EDO:

"Drobisch Angle: 1/360 part of an octave"
http://www.xs4all.nl/~huygensf/doc/measures.html

"The Angle was proposed by Moritz Dröbisch in the 19th century as a
cycle of 360 degrees to the octave. Andrew Pikler has suggested this
name in his article "Logarithmic Frequency Systems" (1966). "

Conversions-Factors:

1. Cents2Degrees: 1200/360 = 10/3 = 3.333333333....
2. Degrees2Cents: 360/1200 = 3/10 = 0.3

but
http://de.wikipedia.org/wiki/Moritz_Wilhelm_Drobisch
http://www.leipzig-lexikon.de/PERSONEN/18020816.htm
himself preferred personally 665-EDO rather than 360-EDO:
"# Delfi unit: 1/665 part of an octave

Used in Byzantine music theory? Approximately 1/12 part of the
syntonic comma and 1/13 part of the Pythagorean comma."

Drobisch obtained 665 from considering the series:
http://www.research.att.com/~njas/sequences/?q=1+2+5+12+41+53+306+665&sort=0&fmt=0&language=english&go=Search

A.S.

--- In tuning@yahoogroups.com, "Andreas Sparschuh" <a_sparschuh@...>
wrote:
>
> --- In tuning@yahoogroups.com, "threesixesinarow" <CACCOLA@> wrote:
> >
> on
> > http://www.sacredscience.com/archive/ToneSpiral.htm
> >
> > Funny pictures
> >
> Dears Clark & Cameron,
>
> that appears to be again an independent rediscovery of 360-EDO:
>
> "Drobisch Angle: 1/360 part of an octave"
> http://www.xs4all.nl/~huygensf/doc/measures.html
>
> "The Angle was proposed by Moritz Dröbisch in the 19th century as a
> cycle of 360 degrees to the octave. Andrew Pikler has suggested
this
> name in his article "Logarithmic Frequency Systems" (1966). "
>
> Conversions-Factors:
>
> 1. Cents2Degrees: 1200/360 = 10/3 = 3.333333333....
> 2. Degrees2Cents: 360/1200 = 3/10 = 0.3
>
> but
> http://de.wikipedia.org/wiki/Moritz_Wilhelm_Drobisch
> http://www.leipzig-lexikon.de/PERSONEN/18020816.htm
> himself preferred personally 665-EDO rather than 360-EDO:
> "# Delfi unit: 1/665 part of an octave
>
> Used in Byzantine music theory? Approximately 1/12 part of the
> syntonic comma and 1/13 part of the Pythagorean comma."
>
> Drobisch obtained 665 from considering the series:
> http://www.research.att.com/~njas/sequences/?
q=1+2+5+12+41+53+306+665&sort=0&fmt=0&language=english&go=Search
>
> A.S.
>

Haha! That's very interesting, thanks. But now I must go scrape
old paint for days while organizing a little series of concerts
(non-12-tET of course!) so I'll have to get back in a day or two!

take care,

-Cameron Bobro

Hi Cameron,

--- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...> wrote:
>
> Hey, Joe Monzo, remember how long ago you were talking
> about a kind of ideal low minor third around 280-something
> cents, found by experiment?

Wow, that left a real impression on you! That *was* a
long time ago.

Cameron is talking about one particular chord in the
JI retuning of my piece _3 Plus 4_ ... you can download
an mp3 of it here:

http://sonic-arts.org/monzo/3plus4/3_plus_4_by_monz.mp3

The first time the chord appears is 20 seconds into
the tune, and at every other occurrence of this chord.

The original post i sent to which Cameron is referring
is the bottom half of this one:

/tuning/topicId_7293.html#7324

And some of my other observations about it are here:

/tuning/topicId_15732.html#15740
/tuning/topicId_15976.html#16015
/tuning/topicId_15976.html#16030
/tuning/topicId_20929.html#21713
/tuning/topicId_46912.html#47060

Cameron, the ratio i wanted was actually a bit narrower
than yours: 75/64 ratio, ~275 cents. It was found empirically
by experimenting with pitch-bend and then choosing the
nearest 5-limit ratio which seemed to my ears to have a
similar effect/affect.

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

--- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...> wrote:
>
Fürti Cameron,
> > 20/17
>
> I've used this very interval, arriving at
> it as a median of 13/11 and 7/6.
do you mean by "median" concrete the 'arithmethic-mean'?

((13/11)+(7/6))/2 = 155/132
(1 200 * ln(155 / 132)) / ln(2) = ~278.076343....Cents

or the 'geometic-mean':

(1 200 * ln(sqrt((13 / 11) * (7 / 6)))) / ln(2) = ~278.040313...Cents

> Although an online perception
> test that's been linked here several times states that I can
> consistently hear a .75 cent difference at 500 Hz,
I'm barely able to distinguish inbetween 256-512Hz
that 53 absolute pitch-classes alone by ears:
/tuning/topicId_63593.html#72755
within ~1 Hz accuracy ~0.5% relative accuracy
"513/512 undevicesimal comma, Boethius' comma"

(1 200 * ln(513 / 512)) / ln(2) = ~3.37801873...Cents

Appearently yours ears seem to possess a ~5times
more precisely resolution than my own pair:
Congratulation!

for a finer absolute resolution i need the aid of a
tuning fork in order to count the relative
resulting beats exactly up to the:
http;//www.plainsound.de/research/legendD.pdf
" 41-limit Schisma.
(32/41)·(81/64)·(81/80) = (6561/6560) b  ± 0,3 Cents"

> I don't
> claim to be even nearly that precise- in other words, when I first
> used 20/17, and to this day, it sounds "that's it!" on its own, but
> in a complete tuning and musical context there's always
> something not quite right.
In deed. Fully agreed!
That's the reason why i do prefer 53-divisions of the octave,
rather than any insufficient 12-divisions.
> >
> > Nonlinear inharmonicity effects on the basilar-membrane...

> That's beyond me- and I think Occam's razor would favor
> my simpler explanation (which of course is only simple
> if you accept Pi as a simple proportion, which I believe
> it is: only the number describing Pi isn't simple).
http://en.wikipedia.org/wiki/Image:Pi-unrolled-720.gif
> >
nonlinear deviation effects,
due to the inharmonic stiffness of the:
http://en.wikipedia.org/wiki/Basilar_membrane
http://hyperphysics.phy-astr.gsu.edu/hbase/sound/corti.html
demands compensation of that effect in pianos by:
http://www.postpiano.com/support/updates/tech/Tuning.htm
more precisely specified in
http://en.wikipedia.org/wiki/Piano_acoustics
http://upload.wikimedia.org/wikipedia/commons/a/ae/Railsback2.png
>
> >
> > Do you discern that 120:119 discrepancy too?
>
> 120/119 on it's own? I doubt it,
that's a little bit more than the deviation error of:
12ET-3rd : just-3rd
2^(1/3):(5:4)
or diesis^(1/3)=(128:125)^(1/3)=~(127:126) using
(128:125)=(128:127)(127:126)(126:125)

> but 7/6 is a very specific
> interval, a strange combination of maximum consonance and
> dissonance,
I personally do follow
http://de.wikipedia.org/wiki/Martin_Vogel
that called the 3 'blue-notes' 7:4, 7:6 and 7:5
therefore "assonances" as already Helmholtz labeled them so too.

not to be confused with the usage of that term in poetry:
http://en.wikipedia.org/wiki/Dissonance
"# Dissonance in poetry is the deliberate avoidance of assonance,"

> and it's definitely not the one I'm hearing,
> 13/11

(1 200 * ln(13 / 11)) / ln(2) = ~289.209719...Cents

> being the closest "simple" interval in character (it
> also sounds just right isolated and out of context-
that's as
http://en.wikipedia.org/wiki/Sesquitertium
term
6.6:5.5 = 13:11

also do sound pretty well lower ratio invervals of that kind:
1:1
1.5:0.5 = 3:1 duodecime
2:1
2.5:1.5 = 5:3 sixth
3:2
3.5:2.5 = 7:6
4:3
4.5:3.5 = 9:7
5:4
5.5:4.5 = 11:9
6:5
6.5:5.5 = 13:11

the harmonic spectre of
http://de.wikipedia.org/wiki/Bild:StoppedOrganPipe.svg
contains almost barely the odd partials:

...13:11(:9:7:5:3:1)

hence it can be interpreted as lacking
http://en.wikipedia.org/wiki/Missing_fundamental

> as I said,
> I don't claim to hear tiny differences in isolated intervals).
>
the "Synchronization effect, described in:
http://front.math.ucdavis.edu/0506.7094
prohibits to observe such vanishing differences
if the distance becomes to much tiny, due to the fusing of the
interval into barely an unison per-se alone.
For such very tiny intervals the property of the interval-quality
gets lost by physically diapparence of the difference
already somewhat before yielding zero-distance.
Henc it becomes impossible to gain information by counting beats,
that are no more any longer present when the tones do fuse
by synchronization.
A.S.

Andreas Sparschuh wrote:
> --- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...> wrote:
> > F�rti Cameron,
> >>> 20/17 >>> >> I've used this very interval, arriving at
>> it as a median of 13/11 and 7/6.
>> > do you mean by "median" concrete the 'arithmethic-mean'?
>
> As I've told him before, the term is "mediant" with a 't' on the end---not "median" which is incorrect, and is a term from statistics...

The mediant is (n1+n2)/(d1+d2) e.g. the mediant of 13/11 and 7/6 is (13+7)/(11+6) = 20/17

-A.

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

http://cda.morris.umn.edu/~mcquarrb/Precalculus/Animations/SineCosine
> Anim.html
>
> you'll find that the point marking the valley of the sine, octave-
> reduced, gives you 283.75 cents. Now Gene finds all this beneath
> contempt, but I find that this is precisely the low minor third I've
> been looking for, and I find its similarity to the low third of 34
> intriging as well.

Funny, I don't recall saying this. Must be some other Gene.

sorry everyone---

Gene,

Can you check your private email...I'm trying to reach you regarding midwestmicrofest.

-A.

--- In tuning@yahoogroups.com, "Andreas Sparschuh" <a_sparschuh@...>
wrote:
>
> --- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@>
wrote:
> >
> Fürti Cameron,
> > > 20/17
> >
> > I've used this very interval, arriving at
> > it as a median of 13/11 and 7/6.
> do you mean by "median" concrete the 'arithmethic-mean'?
>
> ((13/11)+(7/6))/2 = 155/132
> (1 200 * ln(155 / 132)) / ln(2) = ~278.076343....Cents
>
> or the 'geometic-mean':
>
> (1 200 * ln(sqrt((13 / 11) * (7 / 6)))) / ln(2) =
~278.040313...Cents

Sorry, mediant, as Aaron posted-

"The mediant is (n1+n2)/(d1+d2) e.g. the mediant of 13/11 and 7/6 is
(13+7)/(11+6) = 20/17"

(BTW Aaron, is the adjective also "mediant", so that we'd say
for example "mediant third"?)

The harmonic mean would be, lessee... "To find the harmonic mean of
a set of n numbers, add the reciprocals of the numbers in the set,
divide the sum by n, then take the reciprocal of the result",
which would be, in the case of 7/6 and 13/11, 182/155 at 278.00
cents, correct me if this is wrong!
>
>
> > Although an online perception
> > test that's been linked here several times states that I can
> > consistently hear a .75 cent difference at 500 Hz,
> I'm barely able to distinguish inbetween 256-512Hz
> that 53 absolute pitch-classes alone by ears:
> /tuning/topicId_63593.html#72755
> within ~1 Hz accuracy ~0.5% relative accuracy
> "513/512 undevicesimal comma, Boethius' comma"
>
> (1 200 * ln(513 / 512)) / ln(2) = ~3.37801873...Cents
>
> Appearently yours ears seem to possess a ~5times
> more precisely resolution than my own pair:
> Congratulation!

Although I tend to do pretty well on tests, I
think that they're actually fairly bogus and don't
necessarily relate to real-life performance. I suspect
that accuracy of pitch perception in actual practice
is probably about the same for most everyone here.
>
> for a finer absolute resolution i need the aid of a
> tuning fork in order to count the relative
> resulting beats exactly up to the:
> http;//www.plainsound.de/research/legendD.pdf
> " 41-limit Schisma.
> (32/41)·(81/64)·(81/80) = (6561/6560) b  ± 0,3 Cents"

I've never bothered with counting beats, at least not
conciously.

>
> > I don't
> > claim to be even nearly that precise- in other words, when I
>first
> > used 20/17, and to this day, it sounds "that's it!" on its own,
but
> > in a complete tuning and musical context there's always
> > something not quite right.
> In deed. Fully agreed!
> That's the reason why i do prefer 53-divisions of the octave,
> rather than any insufficient 12-divisions.
> > >
> > > Nonlinear inharmonicity effects on the basilar-membrane...
>
> > That's beyond me- and I think Occam's razor would favor
> > my simpler explanation (which of course is only simple
> > if you accept Pi as a simple proportion, which I believe
> > it is: only the number describing Pi isn't simple).
> http://en.wikipedia.org/wiki/Image:Pi-unrolled-720.gif
> > >
> nonlinear deviation effects,
> due to the inharmonic stiffness of the:
> http://en.wikipedia.org/wiki/Basilar_membrane
> http://hyperphysics.phy-astr.gsu.edu/hbase/sound/corti.html
> demands compensation of that effect in pianos by:
> http://www.postpiano.com/support/updates/tech/Tuning.htm
> more precisely specified in
> http://en.wikipedia.org/wiki/Piano_acoustics
> http://upload.wikimedia.org/wikipedia/commons/a/ae/Railsback2.png

But is it really true that harmonies are always literal, and
preferring deviations from integer proportions must be due to
other deviations? I simply don't believe so- I believe that there
are other "simple proportions" which we percieve, which really are
simple, physically speaking, and complex only in number. I believe
for example that the success of the 600-cent half-octave isn't
due to its dubious proximity to a "simple ratio", but to the fact
that it IS a "simple ratio" in terms of a sine wave.

>
> > but 7/6 is a very specific
> > interval, a strange combination of maximum consonance and
> > dissonance,
> I personally do follow
> http://de.wikipedia.org/wiki/Martin_Vogel
> that called the 3 'blue-notes' 7:4, 7:6 and 7:5
> therefore "assonances" as already Helmholtz labeled them so too.

I did not know that Helmholtz had used the term assonance for these
7-family intervals- it's a great term for 7/6 and 7/5, though
I think I'd call 7/4 a consonance.

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
>
> Hi Cameron,
>
>
> --- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@>
wrote:
> >
> > Hey, Joe Monzo, remember how long ago you were talking
> > about a kind of ideal low minor third around 280-something
> > cents, found by experiment?
>
>
> Wow, that left a real impression on you! That *was* a
> long time ago.

Well this approach:

"I had tried 7/6 first, and it didn't sound right at all - the
whole chord sounded badly out of tune. Then I tried 19/16
[= ~298 cents], and that was better in-tune, but not 'dark'
enough. So then I just altered the amount of pitch-bend by
eye/ear until I got the sound I wanted, and it was darn close
to 75/64, so that's how I finally retuned the '3rd'."

is a very familiar kind of process (I can analize acoustic
sounds to .1 HZ accuracy).

>
> Cameron, the ratio i wanted was actually a bit narrower
> than yours: 75/64 ratio, ~275 cents. It was found empirically
> by experimenting with pitch-bend and then choosing the
> nearest 5-limit ratio which seemed to my ears to have a
> similar effect/affect.

I've used 27/23 a lot, fits right into suparticular 34s, but
find it too dark, especially when I'm using a dark third with
a broad second in the 245 cent range.

-Cameron Bobro

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@> wrote:
>
> http://cda.morris.umn.edu/~mcquarrb/Precalculus/Animations/SineCosine
> > Anim.html
> >
> > you'll find that the point marking the valley of the sine, octave-
> > reduced, gives you 283.75 cents. Now Gene finds all this beneath
> > contempt, but I find that this is precisely the low minor third
>I've
> > been looking for, and I find its similarity to the low third of 34
> > intriging as well.
>
>
> Funny, I don't recall saying this. Must be some other Gene.
>

Sorry- I guess "give me a break" and "I see no model" would imply
that what I'm saying is worthy of, not beneath, contempt. :-D

But I guess multiplying a diameter by pi to get a circumference IS
pretty silly.
.

-Cameron Bobro

Cameron Bobro wrote:
> Sorry, mediant, as Aaron posted- >
> "The mediant is (n1+n2)/(d1+d2) e.g. the mediant of 13/11 and 7/6 is
> (13+7)/(11+6) = 20/17"
>
> (BTW Aaron, is the adjective also "mediant", so that we'd say
> for example "mediant third"?)
>
> Yup! more below....

>>> Although an online perception >>> test that's been linked here several times states that I can >>> consistently hear a .75 cent difference at 500 Hz,
>>> >> I'm barely able to distinguish inbetween 256-512Hz >> that 53 absolute pitch-classes alone by ears:
>> /tuning/topicId_63593.html#72755
>> within ~1 Hz accuracy ~0.5% relative accuracy
>> "513/512 undevicesimal comma, Boethius' comma"
>>
>> (1 200 * ln(513 / 512)) / ln(2) = ~3.37801873...Cents
>>
>> Appearently yours ears seem to possess a ~5times
>> more precisely resolution than my own pair:
>> Congratulation!
>> >
> Although I tend to do pretty well on tests, I > think that they're actually fairly bogus and don't
> necessarily relate to real-life performance. I suspect
> that accuracy of pitch perception in actual practice > is probably about the same for most everyone here. > >
Cameron does seem to possess a very sensitive ear---he could here a real distinction between 81/64 and 19/15 in a blind test I did.

-A.

Oh, Andreas and Joe, it might be of interest to note that 7/6 is of
a distinctly different family as far as difference tones, for it
produces a 4/3, while 20/17 and the "3Pi/2" third produce what are
definitely half-octaves at 597 and 612.8 cents respectively. 20/17 has
a neat property- its difference tone with 1/1 is not only almost
exactly the half-octave, its also the harmonic mean of 4/3 and 3/2
(24/17) and quite neatly lies a 6/5 away!

At a cent and 3 cents respectively away from the 34-EDO values, I'd
say 20/17 is an obvious choice for a rational well-temperament. At any
rate, anything within 281-284 cents, where I've been lurking so long,
shares these properties in real-world situations. The diminshed triad
not surprisingly sounds great and that's probably been a big factor in
my search, in retrospect.

-Cameron Bobro

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

Servus Cameron,
> But is it really true that harmonies are always literal, and
> preferring deviations from integer proportions must be due to
> other deviations?
It's still an unsolved puzzling miracle,
how human-beeings can discern well at least the low partials 1-20,
despite the many nonlinear bias contortions in psychoacoustics,
at least in the recent hearing models.

> I simply don't believe so- I believe that there
> are other "simple proportions" which we percieve, which really are
> simple, physically speaking, and complex only in number. I believe
> for example that the success of the 600-cent half-octave isn't
> due to its dubious proximity to a "simple ratio", but to the fact
> that it IS a "simple ratio" in terms of a sine wave.

Please consider:
Just from that 600Cents arises the confusional:
http://en.wikipedia.org/wiki/Tritone_paradox
http://www.cameron.edu/~lloydd/webdoc1.html
http://www.google.de/search?hl=de&q=tritone-paradox&btnG=Google-Suche&meta=
that never occurs in my ears in the neighbourhood of that, alike:

(1 200 * ln(7 / 5)) / ln(2) = ~582.512193...C
(1 200 * ln(1 024 / 729)) / ln(2) = ~588.269995...C from 2*(9/8)^(-3)

(1 200 * ln(729 / 512)) / ln(2) = ~611.730005...C from (9/8)^3
(1 200 * ln(10 / 7)) / ln(2) = ~617.487807...C

on the one hand:
if i do focus my attentional on the 3-limit concentration
than sqrt(2) oscillates inbetween
2^11/3^6 = 1024/729 and 3^6/2^9 = 729/512.

on the other hand:
if i do focus my attention on the 7&5-limit complex,
than sqrt(2) oscillates inbetween 7/5 and 10/7 respectively.

so far about the possible alternatives to interpret the
range of ~580---~620Cents.

Personal observation:
The ambigious meaning of 600C vanishes clearly at ~+-10C deviation.
In terms of 53 it becomes interpreted either as 26 or 27 steps
respectively. Therefore the tritone-paradox may suggest, why i'm
not able to yield an precise clear-cut well-indicated judgement
interpretion second to none of sqrt(2):
http://mathworld.wolfram.com/PythagorassConstant.html
That corresponds to the fact that sqrt(2) is none
http://en.wikipedia.org/wiki/Rational_number
>
>
> ....term assonance for these
> 7-family intervals- it's a great term for 7/6 and 7/5, though
> I think I'd call 7/4 a consonance.

http://www.thegearpage.net/board/showthread.php?t=246847&page=3

The Blues Scale in C is

1:1 7:6 4:3 7:5 3:2 7:4 2:1

Every other note is a blue note.
No major/minor.

1:1 unison ; prime
7:6 between maj2nd and b3 ; blue minor 3rd
4:3 fourth
7:5 slightly below dim5th ; bebop tritone 7/5
3:2 fifth
7:4 slightly below b7 ; blue 7th
2:1 octave

In
http://en.wikipedia.org/wiki/Bebop
occurs the
http://en.wikipedia.org/wiki/Flatted_fifth
"Just interval 7:5, 10:7, 45:32..."
Appearently
http://en.wikipedia.org/wiki/Charlie_Parker
seems to be the first in modern times
that confuesed his coevals by adding
to the submediant (on root 4:5) harmonic chord
an harmonic 7th partial: (4:5)*(7:4)=7:5.

But even long time before that the old greek:
http://en.wikipedia.org/wiki/Archytas
http://de.wikipedia.org/wiki/Archytas
knew already about similar techniques in 7-limit intervals:
"
enharmonisches Tetrachord: (28:27)(36:35)(5:4)
chromatisches Tetrachord: (28:27)(15:14)(6:5)
diatonisches Tetrachord: (28:27)(8:7)(9:8)
"
Without considering such 7-limit intervals in practice,
i strongly doubt about secure and precise
intonation of the blue notes 7:4, 7:6 & 7:5.
Personally i do judge the quality of a blues interpretation
primarily on:
How well is he/she able to hit and meet 7-limit pitches.
That correspondents imho with deeper harmonically insights
in melodic and chord progression.

Nevertheless there are still some incorrigble theoreticans
that try explain the above
http://en.wikipedia.org/w/index.php?title=Blues_scale&redirect=no
in that absurd way:
http://en.wikipedia.org/wiki/Blue_notes
barely by an somehow detuned 5-3 limit restriction
http://en.wikipedia.org/wiki/Hexatonic_scale

Heaven alone knows:
How long will persist any longer
that crude 5,3-limit misconception
of ignoring 7-limit intervals among todays
"blues-musicians" that are less talented as Parker the "bird".

Hopefully not another ~2.4 millenia in vain again :-(

Sincerely waiting in patience
A.S.

--- In tuning@yahoogroups.com, "Aaron K. Johnson" <aaron@...> wrote:

> Cameron does seem to possess a very sensitive ear---
> he could here a real
> distinction between 81/64 and 19/15 in a blind test I did.
>
probably compareable alike rumors about
http://en.wikipedia.org/wiki/Eratosthenes
and
http://en.wikipedia.org/wiki/Andreas_Werckmeister
that were perhaps able to discern from
http://www.xs4all.nl/~huygensf/doc/intervals.html
even
(19/15):(81/64) = " 1216/1215 Eratosthenes' comma"

(1 200 * ln(1 216 / 1 215)) / ln(2) = ~1.42429794...C

that meets exactly the sharpness of the subdomiant
and tonic 3rds: F-A and C-E in:
/tuning-math/message/13675
"
C 2173
G (6561>)6560,3280,1640,820,410,205(>204,102,51)
D 153(>152,76,38,19)
A 57
E 171
B 513(>512,..,1) W:"...one quarter comma above the unity."
F# 3
C# 9
G# 27
Eb 81
Bb 243
F 729
C 2173
"
F-A and C-E are lower sharp than the dominant 3rd G-B with 1025/1024
(1 200 * ln(1 025 / 1 024)) / ln(2) = ~1.68983327...Cents

even more difficult to discern turns out the
http://www.google.de/search?hl=de&q=41-limit+schisma+6561%2F6560&btnG=Suche&meta=
with tiny sharpness inbetween the 5th: C-G, that amounts challengeing:

(1 200 * ln(6 561 / 6 560)) / ln(2) = ~0.263887517...Cents

A.S.

--- In tuning@yahoogroups.com, "Aaron K. Johnson" <aaron@...> wrote:
>
>
> sorry everyone---
>
> Gene,
>
> Can you check your private email...I'm trying to reach you regarding
> midwestmicrofest.

Where did you send it?

--- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...> wrote:
> "The mediant is (n1+n2)/(d1+d2) e.g. the mediant of 13/11 and 7/6 is
> (13+7)/(11+6) = 20/17"

Where it's important to keep in mind that n1/d1 and n2/d2 are reduced.

> But is it really true that harmonies are always literal, and
> preferring deviations from integer proportions must be due to
> other deviations? I simply don't believe so- I believe that there
> are other "simple proportions" which we percieve, which really are
> simple, physically speaking, and complex only in number. I believe
> for example that the success of the 600-cent half-octave isn't
> due to its dubious proximity to a "simple ratio", but to the fact
> that it IS a "simple ratio" in terms of a sine wave.

What does this mean? What success? What sine wave?

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

> But I guess multiplying a diameter by pi to get a circumference IS
> pretty silly.

Obtaining musical intervals in the form q pi, where q is rational, *is*
silly, yes. It also has exactly zero to do with your statement that you
like thirds in the general region of 20/17 or 13/11.

--- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...> wrote:
>
> Oh, Andreas and Joe, it might be of interest to note that 7/6 is of
> a distinctly different family as far as difference tones, for it
> produces a 4/3, while 20/17 and the "3Pi/2" third

3/8 pi third. If you are going to do this lunacy, at least do it right.

[ Attachment content not displayed ]

Gene Ward Smith wrote:
> --- In tuning@yahoogroups.com, "Aaron K. Johnson" <aaron@...> wrote:
> >> sorry everyone---
>>
>> Gene,
>>
>> Can you check your private email...I'm trying to reach you regarding >> midwestmicrofest.
>> > your sbcglobal address---where should I have sent it?
> >
>

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

> > But is it really true that harmonies are always literal, and
> > preferring deviations from integer proportions must be due to
> > other deviations?
> It's still an unsolved puzzling miracle,
> how human-beeings can discern well at least the low partials 1-20,
> despite the many nonlinear bias contortions in psychoacoustics,
> at least in the recent hearing models.

Not to mention the distortions in the original sound.
>
> > I simply don't believe so- I believe that there
> > are other "simple proportions" which we percieve, which really
are
> > simple, physically speaking, and complex only in number. I
believe
> > for example that the success of the 600-cent half-octave isn't
> > due to its dubious proximity to a "simple ratio", but to the
fact
> > that it IS a "simple ratio" in terms of a sine wave.
>
> Please consider:
> Just from that 600Cents arises the confusional:
> http://en.wikipedia.org/wiki/Tritone_paradox
> http://www.cameron.edu/~lloydd/webdoc1.html
> http://www.google.de/search?hl=de&q=tritone-paradox&btnG=Google-
Suche&meta=
> that never occurs in my ears in the neighbourhood of that, alike:
>
> (1 200 * ln(7 / 5)) / ln(2) = ~582.512193...C
> (1 200 * ln(1 024 / 729)) / ln(2) = ~588.269995...C from 2*(9/8)^(-
3)
>
> (1 200 * ln(729 / 512)) / ln(2) = ~611.730005...C from (9/8)^3
> (1 200 * ln(10 / 7)) / ln(2) = ~617.487807...C
>
> on the one hand:
> if i do focus my attentional on the 3-limit concentration
> than sqrt(2) oscillates inbetween
> 2^11/3^6 = 1024/729 and 3^6/2^9 = 729/512.
>
> on the other hand:
> if i do focus my attention on the 7&5-limit complex,
> than sqrt(2) oscillates inbetween 7/5 and 10/7 respectively.
>
> so far about the possible alternatives to interpret the
> range of ~580---~620Cents.
>
> Personal observation:
> The ambigious meaning of 600C vanishes clearly at ~+-10C deviation.
> In terms of 53 it becomes interpreted either as 26 or 27 steps
> respectively. Therefore the tritone-paradox may suggest, why i'm
> not able to yield an precise clear-cut well-indicated judgement
> interpretion second to none of sqrt(2):
> http://mathworld.wolfram.com/PythagorassConstant.html
> That corresponds to the fact that sqrt(2) is none
> http://en.wikipedia.org/wiki/Rational_number

I think it's interesting that mediants of 1/1 and 2/1 converge on
sqrt(2), and perhaps in tonal music with pure and diminished fourth
and fifth there's a kind of drive to continue to sqrt(2). I would
also put the percieved ambiguous half-octave well within 7/5
and 10/7.
> >
> >
> > ....term assonance for these
> > 7-family intervals- it's a great term for 7/6 and 7/5, though
> > I think I'd call 7/4 a consonance.
>
> http://www.thegearpage.net/board/showthread.php?t=246847&page=3
>
> The Blues Scale in C is
>
> 1:1 7:6 4:3 7:5 3:2 7:4 2:1
>
> Every other note is a blue note.
> No major/minor.
>
> 1:1 unison ; prime
> 7:6 between maj2nd and b3 ; blue minor 3rd
> 4:3 fourth
> 7:5 slightly below dim5th ; bebop tritone 7/5
> 3:2 fifth
> 7:4 slightly below b7 ; blue 7th
> 2:1 octave
>
> In
> http://en.wikipedia.org/wiki/Bebop
> occurs the
> http://en.wikipedia.org/wiki/Flatted_fifth
> "Just interval 7:5, 10:7, 45:32..."
> Appearently
> http://en.wikipedia.org/wiki/Charlie_Parker
> seems to be the first in modern times
> that confuesed his coevals by adding
> to the submediant (on root 4:5) harmonic chord
> an harmonic 7th partial: (4:5)*(7:4)=7:5.
>
> But even long time before that the old greek:
> http://en.wikipedia.org/wiki/Archytas
> http://de.wikipedia.org/wiki/Archytas
> knew already about similar techniques in 7-limit intervals:
> "
> enharmonisches Tetrachord: (28:27)(36:35)(5:4)
> chromatisches Tetrachord: (28:27)(15:14)(6:5)
> diatonisches Tetrachord: (28:27)(8:7)(9:8)
> "
> Without considering such 7-limit intervals in practice,
> i strongly doubt about secure and precise
> intonation of the blue notes 7:4, 7:6 & 7:5.
> Personally i do judge the quality of a blues interpretation
> primarily on:
> How well is he/she able to hit and meet 7-limit pitches.
> That correspondents imho with deeper harmonically insights
> in melodic and chord progression.

I demonstrate various tunings and intervals for people who
are not very familiar with whole idea, and I find that a 7/4
always elicits the response "blues!" .
>
> Nevertheless there are still some incorrigble theoreticans
> that try explain the above
> http://en.wikipedia.org/w/index.php?title=Blues_scale&redirect=no
> in that absurd way:
> http://en.wikipedia.org/wiki/Blue_notes
> barely by an somehow detuned 5-3 limit restriction
> http://en.wikipedia.org/wiki/Hexatonic_scale

I don't get this, for an inflected pentatonic isn't a
hexatonic except sometimes in number, and the "blues
scale" presented as a hexatonic is fundamentally different
from the other two examples of hexatonic presented in
that it is tonal while the other two scales are
well-known examples of scales that can be used to
suspend, blur or otherwise make ambiguous tonality.

-Cameron Bobro

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@> wrote:
>
> > But I guess multiplying a diameter by pi to get a circumference IS
> > pretty silly.
>
> Obtaining musical intervals in the form q pi, where q is rational,
>*is*
> silly, yes. It also has exactly zero to do with your statement that
>you
> like thirds in the general region of 20/17 or 13/11.
>

A region less than 4 cents wide is not a "general" region, it's
a specific one. And it's defined by difference tones as well,
24/17 to 10/7, so 13/11 simply doesn't belong.

Perhaps I wasn't clear: 11 of the first 13 intervals generated
by this "silliness" correspond, within .02 to 2.5 cents, with
intervals I consider "of a kind" and use in my tunings. (and I
get more by simply continuing the series).

This "lunacy" is generating intervals which are mediants, or
some other mean extremely close to the mediant, of low-ratio JI.

If I'm getting some kind of mean close to the mediant of simple ratios
by using Pi, shouldn't there be a method of getting Pi from means of
ratios? If there is, then running that method backward might give me
the correct, orderly and predictable way to do all this.

This is not silly at all, it's just a thing. Even if were all pure
coincidence it's still a curiosity. I am getting a kick out of
all kinds of "coincidences" like my ear-tempered
version of 5/4 , 363/290, and my 864/575 fifth as the arithmetic
averageof the tuning (363/290 is .001 cents off).

Scoff away, it's nothing new. :-)

-Cameron Bobro

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@> wrote:
> >
> > Oh, Andreas and Joe, it might be of interest to note that 7/6 is of
> > a distinctly different family as far as difference tones, for it
> > produces a 4/3, while 20/17 and the "3Pi/2" third
>
> 3/8 pi third. If you are going to do this lunacy, at least do it
>right.
>

Wrong. It is NOT 3/8 Pi, and your assumption that it is shows that
you've forgotten or choose to ignore some very basic things. Octave
equivalence needs to be taken with a grain of salt: when dealing with
the harmonic series, 6/4 is equal to 3/2 in ratio, but not in
quantity.

For example, 5/4 is only equivalent to the 5th partial if the
fifth partial is considered in isolation. Let's say we want the
harmonic mean of the third and fifth partials, and we're going to
use the result as a scale step within the octave. If we take the
harmonic mean of 3/2 and 5/4, we get 15/11. But the harmonic mean
of the 3rd and 5th partials, octave-reduced, is NOT 15/11, it's
15/8.

Go ahead and correct me, but correct me correctly: the actual
intervals dealt with, whether octave-reduced or taken as extremes of
means or whatever, are integer(Pi/2).

You also seem to have forgotten that "silly" processes applied
to the harmonic series are nothing new, and are musically useful.
Consider the Bode frequency shifter, which adds a FIXED Hz value
to each partial. My crude shadow spectrum here is tame in
comparison, being in the end hardly different than creating a
*11/7 freqency shifter, mixing the signal back in with the original
and applying JI to the combined sprectrum.

-Cameron Bobro

--- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...>
wrote:
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@>
> wrote:
> >
> > --- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@>
wrote:
> > >
> > > Oh, Andreas and Joe, it might be of interest to note that 7/6
is of
> > > a distinctly different family as far as difference tones, for
it
> > > produces a 4/3, while 20/17 and the "3Pi/2" third

> > 3/8 pi third. If you are going to do this lunacy, at least do it
> >right.
> >
>
> Wrong. It is NOT 3/8 Pi, and your assumption that it is shows that
> you've forgotten or choose to ignore some very basic things. Octave
> equivalence needs to be taken with a grain of salt: when dealing
with
> the harmonic series, 6/4 is equal to 3/2 in ratio, but not in
> quantity.

Oh, cut the crap, please. You were talking about an interval of
283.75 cents. This interval is 3/8 pi. It is NOT 3/2 pi. If you want
to be taken seriously, please stop playing stupid games, and say what
you mean. I'd leave pi out of it anyway.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:

> Oh, cut the crap, please. You were talking about an interval of
> 283.75 cents. This interval is 3/8 pi.

I referred to it as "the 3/2 pi third", with the quotation marks,
correctly it would be "3pi/2 third". For it is derived from the
third partial of an altered harmonic series, and the whole idea
involves, as I explained earlier, applying JI to altered harmonic
series, and working with the harmonic series BEFORE octave-reducing
it.

>It is NOT 3/2 pi.

Right, it's "the 3pi/2 third", for 3pi/2 is the science-fiction
partial from which it is derived, and it is that partial, not its
octave-reduced forms, with which I'm reckoning harmonic means,
FM synthesis modulating ratios, etc.

>If you want
> to be taken seriously,

"Taken seriously"? What a ludicrous and servile concept. I
answer to the music.

>please stop playing stupid games,

Do you call everything you are too stubborn to
acknowledge a "stupid game"?

>and say what
> you mean.

That's what I do- and because what I mean is something other than
what you're used to, I will inevitably use expressions that are
"wrong" in your book.

>I'd leave pi out of it anyway.

I'd be very happy to leave Pi out of it- I love working with
integer ratios- but I cannot avoid the bizarre coincidence
of this sci-fi harmonic series with a large set of rational
intervals I already have.

-Cameron Bobro

--- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...> wrote:
Servus Cameron & Joe,
> Oh, Andreas and Joe, it might be of interest to note that 7/6 is of
> a distinctly different family as far as difference tones, for it
> produces a 4/3, while 20/17....
>
> when dealing with
> the harmonic series, 6/4 is equal to 3/2 in ratio, but not in
> quantity.
fully agreed,
also a bar consiting of 6 crotchets(quarter-notes)
sounds different as an rythm of barely 3 half beats,
not to mention 12 quavers or even the tremolo of 24 semiquavers.
Hence we can and have to distinct inbetween:
6:4 , 3:2 , 12:6 and also 24:12 respectively.

>
> For example, 5/4 is only equivalent to the 5th partial if the
> fifth partial is considered in isolation.
More strictly spoken:
5:4 lies 2 octaves below the 5th partial 5:1.

But if you don't care about octave-equivalences modulo-2^n,
then you may consider the reduction 5:4 == 5:1

> Let's say we want the
> harmonic mean of the third and fifth partials, and we're going to
> use the result as a scale step within the octave. If we take the
> harmonic mean of 3/2 and 5/4, we get 15/11.
as far as i understood Euclid's "sectio-canonis", the
http://en.wikipedia.org/wiki/Harmonic_mean

H(A,B) := (2*A*B)/(A+B)

gets applied for adding tone-proportions
given in stringlenghts, that are in yours example:

A := 2/3
B := 4/5

H(A,B) = H((2/3),((4/5)) = (16/15)/(22/15) = 11/8

Adding the reciprocal pitch-frequency ratios with definition:

a := 1/A = 3/2
b := 1/B = 5/4

sould be executed by using the corresponding:
http://en.wikipedia.org/wiki/Arithmetic_mean
of 2 frequency ratios:

A(a,b) := (a+b)/2

concrete in yours example as 3/2 and 5/4 :

A(a,b) = A((1/A),(1/B) = (a+b)/2 = ((3/2)+(5/4))/2 = 11/8 too.

that's 11th partial ic called 'Alphorn-fa':

http://www.music.princeton.edu/~ted/alphorn.html
"Very important: the seventh partial, written Bb, middle line treble
clef, is a lowered 7th -- it sounds flat [as it should]. Also, the
Alphorn FA, the 11th partial, written F#, top line treble clef, is a
raised 4th leading to the G [written]: in the key of F# it sounds
in-between B natural and C natural; it is a very distinct sound. These
notes are obviously not out of tune but part of a natural tuning which
western music has trained musicians to think is out of tune! [just
intonation junkies can come back now. -t.]"

http://findarticles.com/p/articles/mi_m2822/is_2001_Spring-Summer/ai_100808915/pg_12
"...most horns produce pitches through the twelfth overtone. Certain
pitches in the harmonic series sound "out of tune" to western ears
(see Ex. 1). In particular, the 11th partial or overtone, often used
in Swiss music and famously known as the "Alphorn-fa" ("fa" being the
fourth pitch of a scale), contributes to the uniqueness and exotic
appeal of mountain music;..."

http://www.people.iup.edu/rahkonen/ilwm/Switzerland.bib.htm
..."An alphorn is a wooden trumpet originating back to the Neolithic
period. Old style contemporary horns in the Bundner Oberland,
Graubland, and other remote areas are made from wood (occasionally
from metal pipes) and are about 2.5 meters long. In Muotatal, two
types of alphorn are found: the grada buchel, a straight, hollowed out
fir tree trunk wrapped in birch bark, with a curved bell, is from 4-10
meters long; and the buchel, a smaller horn whose coiled shape can be
played at faster tempos and higher pitches than the grada buchel.
Mouth pieces for all the alphorns are of turned boxwood, crafted in
various shapes and sizes, depending on the intended type of play.
Players once fashioned their own horns, but specialists began to
appear around 1900 with mechanical carvings and gluing. The instrument
restricts players to the overtone series of the 11th partial (F in a C
scale) characteristically sharp, is commonly called alphorn fa. Until
1900 the alphorn was played as a solo instrument to pacify the cattle
and send signals. Today it is played by amateur duets, trios, and
quartets. The alphorn is considered Switzerland's national instrument."

2 links in german:
http://delphi.zsg-rottenburg.de/ttmusik.html
"1:11 Oberton f2xx = 551 (Alphorn-Fa)"
http://th04acc0144.swisswebaward.ch/naturtone.htm

"quartertone" 24-edo approximates the 'alphorn-comma' of

33/32 := (11/8)/(4/3)
in logarithmically units:
1 200 * (ln(33 / 32) / ln(2)) = ~53.2729432...Cents

> But the harmonic mean
> of the 3rd and 5th partials, octave-reduced, is NOT 15/11, it's
> 15/8.

with
a := 3/2
b := 5/4
we obtain the product by multiplication
15/8 = (3/2)(5/4) = a*b

That's different from the 'harmonic-mean',
that would be:
H(a,b) = 2*a*b/(a+b) = 15/11
as shown above.

Don't muddle them up.

Hope
that helps to untangle the confusion inbetween the different
concepts of means.

A.S.

Cameron Bobro wrote:
>> It is NOT 3/2 pi. >> >
> Right, it's "the 3pi/2 third", for 3pi/2 is the science-fiction
> partial from which it is derived, <snip>
Huh?

3/2 pi and 3pi/2 are mathematically the same thing!

--- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...> wrote:
Servus Cameron,
> Not to mention the distortions in the original sound.
in deed
they are always -more or less- there an didsturbe perception.
but one can't get rid of that certain amount in bias.

> > The Blues-Scale in C is

> > 1:1 unison ; prime
> > 7:6 between maj2nd and b3 ; blue minor 3rd
> > 4:3 fourth
> > 7:5 slightly below dim5th ; bebop tritone 7/5
> > 3:2 fifth
> > 7:4 slightly below b7 ; blue 7th
> > 2:1 octave

> I demonstrate various tunings and intervals for people who
> are not very familiar with whole idea, and I find that a 7/4
> always elicits the response "blues!" .
/tuning/topicId_63593.html#72740
"Using 7/4 is a big decision, and I suspect that
history bears out the idea that 7/4 is generally reserved
for specific artistic purposes- the blues, the erotic,
whatever."

http://anaphoria.com/sruti.PDF
reports on p.11 (p.100 in the original document)
"The shrutis bearing septimal ratios 7/6
(280 vibrations per second sa=240)....
...7/5 (336cps)...7/4...."

The author P. Sambamoorthy remarks correctly that:
The 7=~=5 bridge 225/224 = 15^2/7/32
leads to the
http://www.xs4all.nl/~huygensf/doc/intervals.html
"225/224 septimal kleisma"

Hence 7-limit intervals got often confused with its
corresponding near double 5-limit located counterparts.
even though that difference amounts clearly percievable
(1 200 * ln(225 / 224)) / ln(2) = ~7.71152299...Cents

Conclusion:
For aware keeping apart 7-limit from 5-limit
one needs to be able in discerning
at least ~1/3 comma precisely
in accuracy of discrimination.

That's a difficult task for 12-EDO acustomed people,
that usually accept 2^(1/3) 400C generally as detuned 5/4,
despite the inherent error of ~14Cents sharp off
from the correct 5th partial ~384C.

A.S.

--- In tuning@yahoogroups.com, "Aaron K. Johnson" <aaron@...> wrote:
> Huh?
>
> 3/2 pi and 3pi/2 are mathematically the same thing!
>
first of all consider
http://gospelofreason.wordpress.com/2007/06/13/god-said-pi-3-stand-by-your-beliefs-dammit/
and accept the calculation in the book of kings that pi=3,
then it obivous to resolve from the graphics:
http://upload.wikimedia.org/wikipedia/en/b/bc/Shield-Trinity-Scutum-Fidei-English.png
http://en.wikipedia.org/wiki/Shield_of_the_Trinity
that 3=1 as the theory of
http://en.wikipedia.org/wiki/Trinity
demands:"three persons in one God"

http://en.wikipedia.org/wiki/Plato
&
http://en.wikipedia.org/wiki/Albrecht_D%C3%BCrer
http://upload.wikimedia.org/wikipedia/commons/3/3a/Melencolia_I.jpg
believed really that they solved allegdly the ancient problem of
http://en.wikipedia.org/wiki/Squaring_the_circle
by pretending the bare approximation
sqrt(2) + sqrt(3) = ~3.14626437...
as putative exact.

http://www.joyofpi.com/pilinks.html

A.S.

--- In tuning@yahoogroups.com, "Andreas Sparschuh" <a_sparschuh@...>
wrote:
>
> --- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@>
wrote:
> Servus Cameron & Joe,
> > Oh, Andreas and Joe, it might be of interest to note that 7/6 is
of
> > a distinctly different family as far as difference tones, for it
> > produces a 4/3, while 20/17....
> >
> > when dealing with
> > the harmonic series, 6/4 is equal to 3/2 in ratio, but not in
> > quantity.
> fully agreed,
> also a bar consiting of 6 crotchets(quarter-notes)
> sounds different as an rythm of barely 3 half beats,
> not to mention 12 quavers or even the tremolo of 24 semiquavers.
> Hence we can and have to distinct inbetween:
> 6:4 , 3:2 , 12:6 and also 24:12 respectively.

It's really as simple as saying that 600 Hz is not the same
as 300 Hz.
>
> >
> > For example, 5/4 is only equivalent to the 5th partial if the
> > fifth partial is considered in isolation.
> More strictly spoken:
> 5:4 lies 2 octaves below the 5th partial 5:1.
>
> But if you don't care about octave-equivalences modulo-2^n,
> then you may consider the reduction 5:4 == 5:1

But 500 Hz is not 125 Hz, that's my point.
>
> > Let's say we want the
> > harmonic mean of the third and fifth partials, and we're going to
> > use the result as a scale step within the octave. If we take the
> > harmonic mean of 3/2 and 5/4, we get 15/11.
> as far as i understood Euclid's "sectio-canonis", the
> http://en.wikipedia.org/wiki/Harmonic_mean
>
> H(A,B) := (2*A*B)/(A+B)
>
> gets applied for adding tone-proportions
> given in stringlenghts, that are in yours example:
>
> A := 2/3
> B := 4/5
>
> H(A,B) = H((2/3),((4/5)) = (16/15)/(22/15) = 11/8
>
> Adding the reciprocal pitch-frequency ratios with definition:
>
> a := 1/A = 3/2
> b := 1/B = 5/4
>
> sould be executed by using the corresponding:
> http://en.wikipedia.org/wiki/Arithmetic_mean
> of 2 frequency ratios:
>
> A(a,b) := (a+b)/2
>
> concrete in yours example as 3/2 and 5/4 :
>
> A(a,b) = A((1/A),(1/B) = (a+b)/2 = ((3/2)+(5/4))/2 = 11/8 too.
>
> that's 11th partial ic called 'Alphorn-fa':
>
> http://www.music.princeton.edu/~ted/alphorn.html
> "Very important: the seventh partial, written Bb, middle line
treble
> clef, is a lowered 7th -- it sounds flat [as it should]. Also, the
> Alphorn FA, the 11th partial, written F#, top line treble clef, is
a
> raised 4th leading to the G [written]: in the key of F# it sounds
> in-between B natural and C natural; it is a very distinct sound.
These
> notes are obviously not out of tune but part of a natural tuning
which
> western music has trained musicians to think is out of tune! [just
> intonation junkies can come back now. -t.]"
>
> http://findarticles.com/p/articles/mi_m2822/is_2001_Spring-
Summer/ai_100808915/pg_12
> "...most horns produce pitches through the twelfth overtone.
Certain
> pitches in the harmonic series sound "out of tune" to western ears
> (see Ex. 1). In particular, the 11th partial or overtone, often
used
> in Swiss music and famously known as the "Alphorn-fa" ("fa" being
the
> fourth pitch of a scale), contributes to the uniqueness and exotic
> appeal of mountain music;..."
>
> http://www.people.iup.edu/rahkonen/ilwm/Switzerland.bib.htm
> ..."An alphorn is a wooden trumpet originating back to the
Neolithic
> period. Old style contemporary horns in the Bundner Oberland,
> Graubland, and other remote areas are made from wood (occasionally
> from metal pipes) and are about 2.5 meters long. In Muotatal, two
> types of alphorn are found: the grada buchel, a straight, hollowed
out
> fir tree trunk wrapped in birch bark, with a curved bell, is from
4-10
> meters long; and the buchel, a smaller horn whose coiled shape can
be
> played at faster tempos and higher pitches than the grada buchel.
> Mouth pieces for all the alphorns are of turned boxwood, crafted in
> various shapes and sizes, depending on the intended type of play.
> Players once fashioned their own horns, but specialists began to
> appear around 1900 with mechanical carvings and gluing. The
instrument
> restricts players to the overtone series of the 11th partial (F in
a C
> scale) characteristically sharp, is commonly called alphorn fa.
Until
> 1900 the alphorn was played as a solo instrument to pacify the
cattle
> and send signals. Today it is played by amateur duets, trios, and
> quartets. The alphorn is considered Switzerland's national
instrument."
>
> 2 links in german:
> http://delphi.zsg-rottenburg.de/ttmusik.html
> "1:11 Oberton f2xx = 551 (Alphorn-Fa)"
> http://th04acc0144.swisswebaward.ch/naturtone.htm

My partner and I recorded Vinko Globokar playing the
Alpenhorn in a Baroque hall, what a sound! Unfortunately
I cannot share the recording for legal reasons, especially
unfortunate because it's by far the best sounding recording
on a technical level I've ever done, not to mention
"microtonal" as all hell, with three wildly improvising
trombones as the main instruments.

>
> > But the harmonic mean
> > of the 3rd and 5th partials, octave-reduced, is NOT 15/11, it's
> > 15/8.
>
> with
> a := 3/2
> b := 5/4
> we obtain the product by multiplication
> 15/8 = (3/2)(5/4) = a*b
>
> That's different from the 'harmonic-mean',
> that would be:
> H(a,b) = 2*a*b/(a+b) = 15/11
> as shown above.
>
> Don't muddle them up.

I'm not muddling them up, but perhaps I was not clear in
explaining. I did not say "the harmonic mean of the
octave-reduced 3d and 5th partials", I said the harmonic
mean of the 3d and 5th partials, octave-reduced, ie.
we're reducing the harmonic mean of 3/1 and 5/1, not
of 3/2 and 5/4!
a = 3/1
b = 5/1
H(a,b) = 2*a*b/(a+b) = 15/11

2*3*5= 30
/ (8)

= 30/8, ie 15/8 when octave reduced.

See what I'm saying?

>
> Hope
> that helps to untangle the confusion inbetween the different
> concepts of means.

The only confusing things are failing to differentiate
between eg 3/1 and 3/2, and the fact that different means
sometimes coincide, when denominators or numerators are equal
for example.

-Cameron Bobro

--- In tuning@yahoogroups.com, "Aaron K. Johnson" <aaron@...> wrote:
>
> Cameron Bobro wrote:
> >> It is NOT 3/2 pi.
> >>
> >
> > Right, it's "the 3pi/2 third", for 3pi/2 is the science-fiction
> > partial from which it is derived, <snip>
> Huh?
>
> 3/2 pi and 3pi/2 are mathematically the same thing!
>

Obviously- in isolation. But as I keep saying, the 3 refers to
the third partial (frequency multiplier, 1/string length). And
I'm creating "detuned" harmonic series. So an interval derived
from 3(pi/2) might be used with 3(1/2) and 3(versine of Aunt
Patsy's Bra Size), whatever.

Meanwhile Gene is trying to tell me that 1483.75 cents is 3/8pi,
and I'm insisting that it's not, it's "3(Pi/2)" in a certain octave.

-Cameron Bobro

--- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...>
wrote:
Argh, typo...
>
> I'm not muddling them up, but perhaps I was not clear in
> explaining. I did not say "the harmonic mean of the
> octave-reduced 3d and 5th partials", I said the harmonic
> mean of the 3d and 5th partials, octave-reduced, ie.
> we're reducing the harmonic mean of 3/1 and 5/1, not
> of 3/2 and 5/4!

a = 3/1
b = 5/1
H(a,b) = 2*a*b/(a+b) = 15/8

2*3*5= 30
/ (8)

= 30/8, ie 15/8 when octave reduced.

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

> Meanwhile Gene is trying to tell me that 1483.75 cents is 3/8pi,
> and I'm insisting that it's not, it's "3(Pi/2)" in a certain octave.

No, Gene is trying to tell you that the logarithm base 2^(1/1200) of
3/8 pi is 283.75035623216997536, which means the corresponding musical
interval is 283.75035623216997536 cents (not 1483.75 cents, an octave
higher.)

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@>
wrote:
>
> > Meanwhile Gene is trying to tell me that 1483.75 cents is 3/8pi,
> > and I'm insisting that it's not, it's "3(Pi/2)" in a certain
octave.
>
> No, Gene is trying to tell you that the logarithm base 2^(1/1200)
of
> 3/8 pi is 283.75035623216997536, which means the corresponding
musical
> interval is 283.75035623216997536 cents (not 1483.75 cents, an
octave
> higher.)
>

You're belaboring the obvious while entirely missing the point,
or not seeing the forest for the trees.

Both 283.75 and 1483.75 cents are simply octave versions of
the original interval at 2683.75 cents, and that interval is "3:1"
in the altered harmonic series "harmonic partial * (pi/2)". And
the reason why this matters a great deal is because, as I
have now written more than once, 3:1 and 5:1 for example are
not the same as 3:2 and 5:4! I reckon harmonic means for example
from the harmonic series itself, not just octave-reduced
intervals: the harmonic mean of 3:1 and 5:1 is not the same
as the harmonic mean of 3:2 and 5:4.

There's absolutely nothing far out or even new in what I'm
doing- Harry Partch's "subharmonic series" is also probably
more sci-fi than science. It works because it is cohesive
in and of itself and because most importantly it sounds
subjectively good when interlaced with the original harmonic
series. Not to mention that it coincides with harmonic
means of the original harmonic series, as this
particular bit of sci-fi tuning happens to do as well.

And there's nothing remotely strange or new in hearing these
intervals as "distant and soft" and somehow "harmonic",for these
kinds of intervals been described this way, in different words, for
a thousand, or thousands, of years. That's the whole point
of the means- new, different, yet somehow fitting and of pleasing
proportions.

-Cameron Bobro