Yahoo Tuning Groups Ultimate Backup tuning About those simple ratios:

I just realized that one of the most basic properties
of ratios seems to be overlooked in these discussions.
Certainly without taking this basic property into
consideration, the ideas I've been trying to get
across probably wouldn't make sense.

A ratio tells us where there's a unison in the partials.
Simply write out the partials of each tone and see where they match.

In harmonic tones, the (numerator) partial of the 1/1 is a unsion
with the (Denominator) partial of the higher tone.

A complex ratio can have not only a near-coincidence low down, but
other near-concidences of partials, thereby sounding more consonant
than it appears. Perhaps other relationships in the lower partials,
like the concidence of a near 3/2 relationship, also contributes to
the consonance of complex ratios?

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

> There definitely is some locking in the beating (of 19/15, ed.)
>vs. 81:64 --
> I heard it in Scala when playing these triads last night.
-Carl

There certainly is a consonance in the not-so-high partials
of a 19/15 ratio: the 15th partial of the "third" is in unison with
the 19th partial of the "tonic".

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

>Your alleged theory is that intervals like 196/169 are harmonious
>because they involve small rations, meaning three-digit number
ratios.
>The trouble with that "theory" is that between 196/169 we find this:

>29/25, 181/156, 152/131, 123/106, 94/81, 159/137, 65/56, 166/143,
>101/87, 137/118, 173/149, 36/31, 187/161, 151/130, 115/99, 194/167,
>79/68, 122/105, 165/142, 43/37, 179/154, 136/117, 93/80, 143/123,
>193/166, 50/43, 157/135, 107/92, 164/141, 57/49, 178/153, 121/104,
>185/159, 64/55, 135/116, 71/61, 149/128, 78/67, 163/140, 85/73,
>177/152, 92/79, 191/164, 99/85, 106/91, 113/97, 120/103, 127/109,
>134/115, 141/121, 148/127, 155/133, 162/139, 169/145, 176/151,
>183/157, 190/163, 197/169

>The 29/25 is a real problem for your "theory". It's 256.95 cents,
>whereas 196/169 is 256.60 cents. We have been given no reason to
>think 196/169 is especially consonant sounding, and no reason to
>imagine that if it is, it has anything to do with the particular
>ratio of 196/169 rather thsn being a feature of anything of about
>256.5 cents or so. We certainly have been given no reason to think
>that if intervals of about this size have some special woo-woo
>quality, it's not attributable to 29/25 instead.

The answer is 15/13.

Is it not clear as day that simple ratios and superparticular ratios
are no mysteries? Is it not clear that what I am doing, seeking to
create a consistency in the "Uebertimbre" that is a tuning by
emphasizing certain coincidences or regular near-concidences in the
partials throughout the tuning is a very simple concept?

The consonance of a fairly complex interval such as 210/169 is
not just due to a proximity with 5/4, it's 4th partial is
near to the 5th partial of the 1/1 as in 5/4, of course. The 13th
partial of a tone at that ratio is very close to the 16th partial of
the 1/1 as well, and the 21st with the 26th respectively.

Since 196/169 has a character like 15/13, with the 13th partial
coinciding with the 15th partial of 1/1, just a tad higher, and
210/169 also has a 13th partial just a tad higher than the 16th
partial of the 1/1, am I deaf and stupid for hearing family
relations and tuning coherency in these two intervals?

The 13th partial of the second interval in a scale is a near-unison,
a touch higher, with the 15th partial of the 1/1: the 13th partial
of the third interval in a scale is a near unison, a touch higher,
than the 16th partial of the 1/1.

Pretty elegant if you ask me. And I found first by ear.

When we emphasize certain partials, we change the timbre, if only
subtly. Voila, character families. If we do this in a consistent way
throught the tuning we change the whole shebang. And so,
the "Uebertimbre".

The first two songs here:

http://www.zebox.com/bobro/music/

are created with this approach to the tuning- the tuning of the
second tune was made as strictly as I've been able to do in
accordance with the character-family and shadow-tuning approach.

How about helping me find ways to generate tunings with these kinds
of conherencies rather than pooh-pooh everything I say?

take care,

-Cameron Bobro

🔗Aaron Krister Johnson <aaron@dividebypi.com>

Hey Cameron---

Yes what you say is true, and well known in the physics of sound
circles. These things are textbook, and were discussed in a college
course I took on music perception.

What baffles me is that it is tautological, in a way, with what Carl
is saying about small numbered ratios. It seems you guys disagree only
because you're not understanding what the other is saying.

From my angle, Carl is saying '6', and you are saying 'half dozen'

Cheers.
A.

--- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...> wrote:
>
> I just realized that one of the most basic properties
> of ratios seems to be overlooked in these discussions.
> Certainly without taking this basic property into
> consideration, the ideas I've been trying to get
> across probably wouldn't make sense.
>
> A ratio tells us where there's a unison in the partials.
> Simply write out the partials of each tone and see where they match.
>
> In harmonic tones, the (numerator) partial of the 1/1 is a unsion
> with the (Denominator) partial of the higher tone.
>
> A complex ratio can have not only a near-coincidence low down, but
> other near-concidences of partials, thereby sounding more consonant
> than it appears. Perhaps other relationships in the lower partials,
> like the concidence of a near 3/2 relationship, also contributes to
> the consonance of complex ratios?
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <clumma@> wrote:
>
> > There definitely is some locking in the beating (of 19/15, ed.)
> >vs. 81:64 --
> > I heard it in Scala when playing these triads last night.
> -Carl
>
> There certainly is a consonance in the not-so-high partials
> of a 19/15 ratio: the 15th partial of the "third" is in unison with
> the 19th partial of the "tonic".
>
>
> > --- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@>
> > wrote:
>
> >Your alleged theory is that intervals like 196/169 are harmonious
> >because they involve small rations, meaning three-digit number
> ratios.
> >The trouble with that "theory" is that between 196/169 we find this:
>
> >29/25, 181/156, 152/131, 123/106, 94/81, 159/137, 65/56, 166/143,
> >101/87, 137/118, 173/149, 36/31, 187/161, 151/130, 115/99, 194/167,
> >79/68, 122/105, 165/142, 43/37, 179/154, 136/117, 93/80, 143/123,
> >193/166, 50/43, 157/135, 107/92, 164/141, 57/49, 178/153, 121/104,
> >185/159, 64/55, 135/116, 71/61, 149/128, 78/67, 163/140, 85/73,
> >177/152, 92/79, 191/164, 99/85, 106/91, 113/97, 120/103, 127/109,
> >134/115, 141/121, 148/127, 155/133, 162/139, 169/145, 176/151,
> >183/157, 190/163, 197/169
>
> >The 29/25 is a real problem for your "theory". It's 256.95 cents,
> >whereas 196/169 is 256.60 cents. We have been given no reason to
> >think 196/169 is especially consonant sounding, and no reason to
> >imagine that if it is, it has anything to do with the particular
> >ratio of 196/169 rather thsn being a feature of anything of about
> >256.5 cents or so. We certainly have been given no reason to think
> >that if intervals of about this size have some special woo-woo
> >quality, it's not attributable to 29/25 instead.
>
> The answer is 15/13.
>
> Is it not clear as day that simple ratios and superparticular ratios
> are no mysteries? Is it not clear that what I am doing, seeking to
> create a consistency in the "Uebertimbre" that is a tuning by
> emphasizing certain coincidences or regular near-concidences in the
> partials throughout the tuning is a very simple concept?
>
> The consonance of a fairly complex interval such as 210/169 is
> not just due to a proximity with 5/4, it's 4th partial is
> near to the 5th partial of the 1/1 as in 5/4, of course. The 13th
> partial of a tone at that ratio is very close to the 16th partial of
> the 1/1 as well, and the 21st with the 26th respectively.
>
> Since 196/169 has a character like 15/13, with the 13th partial
> coinciding with the 15th partial of 1/1, just a tad higher, and
> 210/169 also has a 13th partial just a tad higher than the 16th
> partial of the 1/1, am I deaf and stupid for hearing family
> relations and tuning coherency in these two intervals?
>
> The 13th partial of the second interval in a scale is a near-unison,
> a touch higher, with the 15th partial of the 1/1: the 13th partial
> of the third interval in a scale is a near unison, a touch higher,
> than the 16th partial of the 1/1.
>
> Pretty elegant if you ask me. And I found first by ear.
>
> When we emphasize certain partials, we change the timbre, if only
> subtly. Voila, character families. If we do this in a consistent way
> throught the tuning we change the whole shebang. And so,
> the "Uebertimbre".
>
> The first two songs here:
>
> http://www.zebox.com/bobro/music/
>
> are created with this approach to the tuning- the tuning of the
> second tune was made as strictly as I've been able to do in
> accordance with the character-family and shadow-tuning approach.
>
> How about helping me find ways to generate tunings with these kinds
> of conherencies rather than pooh-pooh everything I say?
>
> take care,
>
> -Cameron Bobro
>

🔗Cameron Bobro <misterbobro@yahoo.com>

--- In tuning@yahoogroups.com, "Aaron Krister Johnson" <aaron@...>
wrote:
>
>
> Hey Cameron---
>
> Yes what you say is true, and well known in the physics of sound
> circles. These things are textbook, and were discussed in a college
> course I took on music perception.

Of course, I had assumed that everyone would grasp immediately that
my character families are addressing these well-known basics, and
threw my hands up when Gene didn't reply 15/13! because it's the
simplest description of what 196/169 does in the first two dozen
partials or so, which is to have a (near) unison of it's 13th
partial with the 15th partial of the 1/1.

Where we disagree, as far as I can make out, is how important this
is compared to sheer proximity to the most simple intervals. For
example, 196/169 is about 10 cents from 7/6, yet in some ways it is
perhaps as consonant- look at the eighth partial, as well as the
13th, for example.

And so, I maintain that if there are coincidences in the first
coupla dozen partials (these are all frequency things, it doesn't
matter if the ratio is irrational), they give both recognizble
character, via a kind of timbral change, and can even make a
particular interval a little further from one of the simpler ones
more consonant than one a little closer, by virtue of one of the
lower partials happening to coincide or nearly coincide with a
partial in the other, harmony, interval.

Simple as pie really. In the tuning I use for "Mimosa", there are
numerous instances of a thirteenth partial coinciding very closely
with another partial, and I'm continually refining things to get
more of these consistencies.

-Cameron Bobro

>
> What baffles me is that it is tautological, in a way, with what
Carl
> is saying about small numbered ratios. It seems you guys disagree
only
> because you're not understanding what the other is saying.
>
> From my angle, Carl is saying '6', and you are saying 'half dozen'
>
> Cheers.
> A.
>
> --- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@>
wrote:
> >
> > I just realized that one of the most basic properties
> > of ratios seems to be overlooked in these discussions.
> > Certainly without taking this basic property into
> > consideration, the ideas I've been trying to get
> > across probably wouldn't make sense.
> >
> > A ratio tells us where there's a unison in the partials.
> > Simply write out the partials of each tone and see where they
match.
> >
> > In harmonic tones, the (numerator) partial of the 1/1 is a
unsion
> > with the (Denominator) partial of the higher tone.
> >
> > A complex ratio can have not only a near-coincidence low down,
but
> > other near-concidences of partials, thereby sounding more
consonant
> > than it appears. Perhaps other relationships in the lower
partials,
> > like the concidence of a near 3/2 relationship, also contributes
to
> > the consonance of complex ratios?
> >
> > --- In tuning@yahoogroups.com, "Carl Lumma" <clumma@> wrote:
> >
> > > There definitely is some locking in the beating (of 19/15, ed.)
> > >vs. 81:64 --
> > > I heard it in Scala when playing these triads last night.
> > -Carl
> >
> > There certainly is a consonance in the not-so-high partials
> > of a 19/15 ratio: the 15th partial of the "third" is in unison
with
> > the 19th partial of the "tonic".
> >
> >
> > > --- In tuning@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@>
> > > wrote:
> >
> > >Your alleged theory is that intervals like 196/169 are
harmonious
> > >because they involve small rations, meaning three-digit number
> > ratios.
> > >The trouble with that "theory" is that between 196/169 we find
this:
> >
> > >29/25, 181/156, 152/131, 123/106, 94/81, 159/137, 65/56,
166/143,
> > >101/87, 137/118, 173/149, 36/31, 187/161, 151/130, 115/99,
194/167,
> > >79/68, 122/105, 165/142, 43/37, 179/154, 136/117, 93/80,
143/123,
> > >193/166, 50/43, 157/135, 107/92, 164/141, 57/49, 178/153,
121/104,
> > >185/159, 64/55, 135/116, 71/61, 149/128, 78/67, 163/140, 85/73,
> > >177/152, 92/79, 191/164, 99/85, 106/91, 113/97, 120/103,
127/109,
> > >134/115, 141/121, 148/127, 155/133, 162/139, 169/145, 176/151,
> > >183/157, 190/163, 197/169
> >
> > >The 29/25 is a real problem for your "theory". It's 256.95
cents,
> > >whereas 196/169 is 256.60 cents. We have been given no reason to
> > >think 196/169 is especially consonant sounding, and no reason to
> > >imagine that if it is, it has anything to do with the particular
> > >ratio of 196/169 rather thsn being a feature of anything of
about
> > >256.5 cents or so. We certainly have been given no reason to
think
> > >that if intervals of about this size have some special woo-woo
> > >quality, it's not attributable to 29/25 instead.
> >
> > The answer is 15/13.
> >
> > Is it not clear as day that simple ratios and superparticular
ratios
> > are no mysteries? Is it not clear that what I am doing, seeking
to
> > create a consistency in the "Uebertimbre" that is a tuning by
> > emphasizing certain coincidences or regular near-concidences in
the
> > partials throughout the tuning is a very simple concept?
> >
> > The consonance of a fairly complex interval such as 210/169 is
> > not just due to a proximity with 5/4, it's 4th partial is
> > near to the 5th partial of the 1/1 as in 5/4, of course. The
13th
> > partial of a tone at that ratio is very close to the 16th
partial of
> > the 1/1 as well, and the 21st with the 26th respectively.
> >
> > Since 196/169 has a character like 15/13, with the 13th partial
> > coinciding with the 15th partial of 1/1, just a tad higher, and
> > 210/169 also has a 13th partial just a tad higher than the 16th
> > partial of the 1/1, am I deaf and stupid for hearing family
> > relations and tuning coherency in these two intervals?
> >
> > The 13th partial of the second interval in a scale is a near-
unison,
> > a touch higher, with the 15th partial of the 1/1: the 13th
partial
> > of the third interval in a scale is a near unison, a touch
higher,
> > than the 16th partial of the 1/1.
> >
> > Pretty elegant if you ask me. And I found first by ear.
> >
> > When we emphasize certain partials, we change the timbre, if
only
> > subtly. Voila, character families. If we do this in a consistent
way
> > throught the tuning we change the whole shebang. And so,
> > the "Uebertimbre".
> >
> > The first two songs here:
> >
> > http://www.zebox.com/bobro/music/
> >
> > are created with this approach to the tuning- the tuning of the
> > second tune was made as strictly as I've been able to do in
> > accordance with the character-family and shadow-tuning
approach.
> >
> > How about helping me find ways to generate tunings with these
kinds
> > of conherencies rather than pooh-pooh everything I say?
> >
> > take care,
> >
> > -Cameron Bobro
> >
>

🔗Carl Lumma <clumma@yahoo.com>

--- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...>
wrote:
> I just realized that one of the most basic properties
> of ratios seems to be overlooked in these discussions.
> Certainly without taking this basic property into
> consideration, the ideas I've been trying to get
> across probably wouldn't make sense.
>
> A ratio tells us where there's a unison in the partials.
> Simply write out the partials of each tone and see where they match.

If you're going to talk about partials and consonance,
you need to start talking about sensory dissonance calculations,
the critical band, etc. Coinciding partials do not cause
consonance. Though I did suggest recently that a large number
of coinciding spectral energy given the condition of zero
critical band interactions may account for "equivalence".

> There certainly is a consonance in the not-so-high partials
> of a 19/15 ratio: the 15th partial of the "third" is in unison with
> the 19th partial of the "tonic".

19:15 has lots of beating partials. Pointing out one
coincidence in them is relatively meaningless. Especially
since you're not making a comparison to any other intervals.

> >The 29/25 is a real problem for your "theory". It's 256.95 cents,
> >whereas 196/169 is 256.60 cents. We have been given no reason to
> >think 196/169 is especially consonant sounding, and no reason to
> >imagine that if it is, it has anything to do with the particular
> >ratio of 196/169 rather thsn being a feature of anything of about
> >256.5 cents or so. We certainly have been given no reason to think
> >that if intervals of about this size have some special woo-woo
> >quality, it's not attributable to 29/25 instead.
>
> The answer is 15/13.

The answer to what?

> Is it not clear as day that simple ratios and superparticular
> ratios are no mysteries?

They're no mystery to me... But 15/13 is neither superparticular
nor particularly simple.

> Is it not clear that what I am doing,

Nope.

> When we emphasize certain partials, we change the timbre, if only
> subtly. Voila, character families. If we do this in a consistent
> way throught the tuning we change the whole shebang. And so,
> the "Uebertimbre".

If it were up to you, the brew would be nothing but newt eye!

-Carl

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

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

> Of course, I had assumed that everyone would grasp immediately that
> my character families are addressing these well-known basics, and
> threw my hands up when Gene didn't reply 15/13!

Oh, please. There was absolutely *nothing* in what you said to
indicate you were pointing to 15/13 by means of the complex ratio
196/169, and there *still* is nothing which singles it out. You'd
mentioned 7/6, which is sharp of 196/169, but gave no slightest clue
you intended to look flat, so I looked at the interval between
197/169 and 7/6, and found many, many simpler ratios.

You want basics? Here are some basics. The semiconvergents to 196/169
go 7/6, 8/7, 15/13, 22/19, 29/25, 51/44, 80/69, 109/94, 138/119,
167/144, 196/169. Hence if 15/13 is being pointed at, it has company.
Moreover, it isn't even a convergent! The convergents (this still is
basic stuff, by the way) are 7/6, 22/19, 29/25, 167/144, 196/169. My
29/25 is a third of a cent sharp of 196/169, whereas your 15/13 is
nine cents flat of it. Since 7/6 is ten cents sharp of it, and is a
much more dominating ratio than 15/13, so I think the notion that
15/13 is very important to the sound of 196/169 is likely baloney
anyway. But even if it isn't, *you* did not point any of this out.

> because it's the
> simplest description of what 196/169 does in the first two dozen
> partials or so, which is to have a (near) unison of it's 13th
> partial with the 15th partial of the 1/1.

It is *not* the simplest description. This is just plain wrong.

> Where we disagree, as far as I can make out, is how important this
> is compared to sheer proximity to the most simple intervals. For
> example, 196/169 is about 10 cents from 7/6, yet in some ways it is
> perhaps as consonant- look at the eighth partial, as well as the
> 13th, for example.

This "look at the partials" is not a theory! If you want to propose a
theory, you need to say exactly and precisely what looking at the
partials *means*, and how it can be quantified.

> Simple as pie really.

Then why can't you do it? Your analytical comments on 196/169 make no
sense, in that they don't point to anything special about 196/169.
Why are you talking about it and not 29/25 or 22/15 (which is the
mediant between 15/13 and 7/6?)

🔗Cameron Bobro <misterbobro@yahoo.com>

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

>
> If you're going to talk about partials and consonance,
> you need to start talking about sensory dissonance calculations,
> the critical band, etc.

Feel free to talk about these things, I will continue to LISTEN to
them as I have for so long.

It just occured to me that you might not have a simple way to do
that, I'll copy and paste the Csound L/R-sines code I wrote for this
very purpose to the bottom of this message.

>Coinciding partials do not cause
> consonance.

Please take a little time to think over the old Tonverschmelzung
theory of consonance, and the argument against it.

>Though I did suggest recently that a large number
> of coinciding spectral energy given the condition of zero
> critical band interactions may account for "equivalence".

Think very carefully about what you just said. What I have been
saying is, to use this terminology, that consistently coinciding
spectral energy IN THE REGION OF SPECIFIC PARTIALS, given the
condition of "zero" critical band interactions, at least partly
accounts for a feeling of the character of an interval, and
therefore a feeling of similarity between different intervals which
share the same and related concentrations of energy in specific
places.

Character families.

I put "zero" in quotes because I know from having done this for so
long that there is a zone of subjective interpretation.

You've done additive synthesis, you know very well that loading a
specific partial with energy changes the timbre. You also know that
a "partial" is actually a zone, and that its integer description is
really a name, or the address of the center frequency, and that you
can also load energy into that zone with slightly "detuned" partials
or even filtered noise and get the same effect as simply boosting
that partial, as long as you're close enough to avoid critical band
interactions.

Take, for example, the 196/169 with which I've been unfairly teasing
Gene. I've said all along that proximity to a simple interval isn't
enough to explain the consonance or more importantly character of an
interval, and yet brought up 196/169 in the same breath as 7/6.

Isn't it obvious why?

It takes more than ONE simple interval to explain the consonant
character of 196/169!

1/1 196/169 15/13 7/6

100 115.976 115.384 116.666
200 231.952 230.769 233.333
300 347.928 346.153 350.000
400 463.905 461.538 466.666
500 579.881 576.923 583.333
600 695.857 692.307 700.000
700 811.834 807.692 816.666
800 927.810 923.076 933.333
900 1043.786 1038.461 1050.000
1000 1159.763 1153.846 1166.666
1100 1275.739 1269.230 1283.333
1200 1391.715 1384.615 1400.000
1300 1507.692 1500.000 1516.666
1400 1623.668 1615.384 1633.333
1500 1739.644 1730.769 1750.000
1600 1855.621 1846.153 1866.666
1700 1971.597 1961.538 1983.333
1800 2087.593 2076.000 2100.000
1900 2203.550 2192.307 2216.666
2000 2319.526 2307.000 2333.333
2100 2435.502 2423.000 2450.000
2200 2551.479 2538.461 2566.666
2300 2667.455 2653.000 2683.333

Spot the coincidences and near-coincidences and listen to
individually and in groups if there's any doubt about their
status in light of critical band interactions. I think in
terms of Tonverschmelzung, sweet zone, and nasty, it doesn't
matter.

196/169, and anything in the tiny zone around, it is BOTH 16/13 and
7/6 in nature. Look and listen long and carefully before replying,
please. Listen to the action at the sixth partial, and take
into consideration the story of the Tonverscmelzung theory
of consonance before replying, isn't 22/19 better, for example.

The seventh partial (the 8/7 gene in 196/169s character so to speak)
is interesting as well as the obvious 6th, 13th and 19th partials.

Keep in mind- and listen for yourself, please- that there's a
particular point of nastiness in critical band interactions. From
experience, I can say that my perception of it agrees in general
with commonly held figures, figures which you know very well.

Look and listen very carefully what happens when you try to make
196/169 "better" by moving it closer to the nearest simplest
interval, 7/6. There's a zone where it's losing the sweet spots it
shares with 16/13 and before it falls into the logic of 7/6. Pay
attention to the nasty-points of critical band interactions.

Try it, long and carefully: as I have repeatedly said, and you,
Carl, have repeatedly pooh-poohed, it is entirely possible that
there are lumps and dips of consonance and character in zones around
simple intervals. If we are dealing with more than pure sines at the
fundamental, and if we haven't deliberately reduced our perception
of musical sound to simply counting beats.

Now, is it really stupid to think of an interval in terms of the
timbre it suggests or reinforces by concentrating energy within the
critical-band-interaction-free-zone of specific partials in the
accompanying tones?

Is it nuts to use other intervals which have a similar interaction
with the 1/1 in a tuning, to seek ways to create homogeneity in a
tuning by paying attention to what happens above the fundamentals?

Let's say we have a diad: if one interval is suggesting a certain
timbre by loading specific partials in the other tone by having near-
coincidences perviced as "one" (and without phase cancellations by
the way, being near and not at), is it crazy to seek, in a third
interval that will make the triad, such coincidences, and related
coincidences, that will increase the effect?

Or as I've been saying all along, to find character families and
thereby create an overall timbre consistency in the tuning or
scale.

Gene, if you're listening, instead of throwing convergents and semi-
convergents at me- I can hit shift+alt+G in Scala and I've been
LISTENING to them all along- why don't you help me to find the
irrational interval which is "more" 196/169 than 196/169 itself is?
I'm sure it's there.

Hell, Gene, you yourself have said things that imply that you
consider more than one simple interval at a time when dealing with
complex intervals- did you not, correctly in my view, in the 81/64 +
19/15, bring up 14/11, and not just assume 5/4 as some kind of cruel
Jupiter? Is it not concievable that we can find a third which is
the "child" of 5/4 and 14/11, which shares in its partials, within
the "sweet" zones, characteristics of both? Would it not then be
desirable, rather than treating this interval as as some poor
detuned version of one of its parents, to keep in mind its character
when tuning other intervals in the tuning?

Now someone is sure to say, everyone knows this all along, to which
I reply in advance: that's wonderful, so let us hear no more
bullshit about "approximating Just intervals" by sheer proximity.
And: "eye of newt", my skinny white ass.

By the way, all this is just "a" thing, not "the thing" of course,
and there's more than one way to skin a cat. Concentrating on
similarity of character in this way, taking into consideration the
first couple dozen partials, not just the fundamental, I get tunings
that work very well melodically and harmonically, but the harmony
can be extremely sensitive to voicing as anyone who hasn't pooh-
poohed the whole thing will immediately realize.

-Cameron Bobro

;Tonverschmelzung L/R sines Cameron Bobro

sr = 44100
kr = 44100
ksmps = 1
nchnls = 2

instr 1

kcps = p5

kenv1 expseg .0001,(p3*.1)+.1, 1, p3*.2, .4, (p3*.5)-.1, .5,
p3*.2, .0001
kamp = p4*kenv1

asine poscil3 p4, kpitch, 1

outs asine*1, asine*0

endin

instr 2

kcps = p5

kenv1 expseg .0001,(p3*.1)+.1, 1, p3*.2, .4, (p3*.5)-.1, .5,
p3*.2, .0001
kamp = p4*kenv1

asine poscil3 p4, kpitch, 1

outs asine*1, asine*0

endin

</CsInstruments>

;Tonverschmelzung L/R sines Cameron Bobro

f 1 0 8388608 9 1 1 0
f 2 0 8388608 9 1 1 33

i 1 0 6 2866 700
i 2 .1 5.9 2866 703

</CsScore>

</CsoundSynthesizer>

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

🔗Carl Lumma <clumma@yahoo.com>

> > If you're going to talk about partials and consonance,
> > you need to start talking about sensory dissonance calculations,
> > the critical band, etc.
>
> Feel free to talk about these things, I will continue to LISTEN to
> them as I have for so long.

Implying I haven't?

> > Though I did suggest recently that a large number
> > of coinciding spectral energy given the condition of zero
> > critical band interactions may account for "equivalence".
>
> Think very carefully about what you just said. What I have been
> saying is, to use this terminology, that consistently coinciding
> spectral energy IN THE REGION OF SPECIFIC PARTIALS,

Which region is that?

> given the
> condition of "zero" critical band interactions, at least partly
> accounts for a feeling of the character of an interval,

Sounds vaguely like something that's true.

> You've done additive synthesis, you know very well that loading a
> specific partial with energy changes the timbre. You also know that
> a "partial" is actually a zone, and that its integer description is
> really a name, or the address of the center frequency, and that you
> can also load energy into that zone with slightly "detuned"
> partials or even filtered noise and get the same effect as simply
> boosting that partial, as long as you're close enough to avoid
> critical band interactions.

Yes.

> I've said all along that proximity to a simple interval isn't
> enough to explain the consonance or more importantly character
> of an interval, and yet brought up 196/169 in the same breath
> as 7/6.
>
> Isn't it obvious why?

No.

> It takes more than ONE simple interval to explain the consonant
> character of 196/169!
>
> 1/1 196/169 15/13 7/6
>
> 100 115.976 115.384 116.666
> 200 231.952 230.769 233.333
> 300 347.928 346.153 350.000
> 400 463.905 461.538 466.666
> 500 579.881 576.923 583.333
> 600 695.857 692.307 700.000
> 700 811.834 807.692 816.666
> 800 927.810 923.076 933.333
> 900 1043.786 1038.461 1050.000
> 1000 1159.763 1153.846 1166.666
> 1100 1275.739 1269.230 1283.333
> 1200 1391.715 1384.615 1400.000
> 1300 1507.692 1500.000 1516.666
> 1400 1623.668 1615.384 1633.333
> 1500 1739.644 1730.769 1750.000
> 1600 1855.621 1846.153 1866.666
> 1700 1971.597 1961.538 1983.333
> 1800 2087.593 2076.000 2100.000
> 1900 2203.550 2192.307 2216.666
> 2000 2319.526 2307.000 2333.333
> 2100 2435.502 2423.000 2450.000
> 2200 2551.479 2538.461 2566.666
> 2300 2667.455 2653.000 2683.333
//
> Look and listen very carefully what happens when you try to make
> 196/169 "better" by moving it closer to the nearest simplest
> interval, 7/6. There's a zone where it's losing the sweet spots it
> shares with 16/13 and before it falls into the logic of 7/6. Pay
> attention to the nasty-points of critical band interactions.
>
> Try it, long and carefully: as I have repeatedly said, and you,
> Carl, have repeatedly pooh-poohed, it is entirely possible that
> there are lumps and dips of consonance and character in zones
> around simple intervals.

I never poo-poohed this.

> Now, is it really stupid to think of an interval in terms of the
> timbre it suggests or reinforces by concentrating energy within
> the critical-band-interaction-free-zone of specific partials in
> the accompanying tones?

Have you ever attempted to calculate the critical band
interactions in an interval like 196/169? Or the harmonic
entropy? Or anything quantitative about 196/169 whatever?

-Carl

🔗Cameron Bobro <misterbobro@yahoo.com>

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

>
> And if you explained how to find that zone, you could start making
>a theory.

Gene, thanks to your response I found precisely how it goes. :-)

> The most obvious interpretation of your question is what is half-
>way
> between 15/13 and 7/6, and the answer to that is sqrt(910)/26,
>which
> is indeed an irrational interval. However, it's pretty close to
>29/25.
> The mediant, (15+7)/(13+6) = 22/19, also comes to mind.

It looks like the simplest explanation is the best, and in this case
29/25 works perfectly.

The practical method for finding these zones goes like this:

take two different offset x/harmonic sets. They can't be offset too
much and the harmonic has to be in range where you could clearly
hear it making a distinct change in timbre, on its own, in additive
synthesis, ie. <24 in my opinion.

For example: 13/12, 14/12, 15/12, 16/12 as one set, and
14/13,15/13,16/13,17/13 as the other set. Then you find the halfway
point between each pair and that's your basic tuning.

In the scale I'm posting here, this is exactly what I did for the
lower pentachord, then above 3/2, simply duplicated the intervals.
So it's a very simple implementation. In this case I went from the
x/12 down to the nearest x/13. So, the M3 is the halfway point of
15/12 (5/4) and 16/13, the flatted 4th is halfway between 16/12
(4/3) and 17/13, etc.

A quick check through the key partials and it looks like, yes,
there's near-coincidence within the sweet zone of a few Hz.
throughout the intervals. The specific near-concidences I wanted
were based on the 1/1- they move when other intervals are played
together without the 1/1, of course. But as I continue to check,
there seems to be a logic to it, and a great deal of near-
coincidences within the first two dozen partials.

Smiling.scl
!
x/12 mates with x/13
9
!
138/125
29/25
31/25
33/25
3/2
207/125
87/50
93/50
2/1

Gene, your suggestion to use 29/25 worked out perfectly- the
irrational mid-points between the two parent tunings are expressed
with these ratios with amazing precision.

Well, according to my ears this meets all the criteria I listen for.
A couple of the intervals are within a cent or two of intervals that
I found by trial and error listening trying to find intervals in the
same "family" as the original 196/169, tickled pink about that.

The "shadow tuning" of difference tones also sounds great, IMO.

thanks again,

-Cameron Bobro

🔗Andreas Sparschuh <a_sparschuh@yahoo.com>

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

> > spectral energy IN THE REGION OF SPECIFIC PARTIALS,
>
> Which region is that?
>
Playing the horn inbetween the 13th and 14th partials is really a
challenge:
http://en.wikipedia.org/wiki/Baroque_trumpet
"The natural harmonic series has several notes that are out-of-tune to
modern western perception, notably the 7th, 11th and 13th partials.
Where a composition demands these notes, the player must 'lip' them
into tune, a notoriously difficult task."

http://de.wikipedia.org/wiki/Barocktrompete
"Das Spiel in den hohen, chromatischen Lagen (ab 13. Teilton) heißt
Clarin- oder Clarinospiel und ist Kennzeichen der barocken
Trompetenkunst."
"The play in the high chromatic region (up the 13. partial)
is called 'clarino'-play, characteristic for the Baroque
tumpeting-art.

Werckmeister remarked about that higher partial series
C:g:e:~bb:... == 4:5:6:7:....
already in his "Musicalische Temperatur"(1691) p.24 Chap. XII and
on p.25 Chap. XIII
".../wenn 1. gegen 7 solte kommen. 12 und 18 sind harmonisch /
sie werden aber verdorben / und verlieren ihre Eigenschaften wenn
sie zerissen werden / als 1. gegen 11 und die 1. gegen 17.
die Zahlen 11. 13. 17. sind gantz nicht harmonisch /wenn sie also
stehen...."
tr:
.../if 1. versus 7 sounds. 12 and 18 are harmonic /
but become ill / and lost their properties when separated /
as 1. versus 11 and 1. versus 17. the numbers 11. 13. 17
are not harmonic at all / if they stand alone....

That means i.m.o:
Partials higher than the 11th can only be detected aural and
discriminated in a context of other partials near to that region.

> > It takes more than ONE simple interval to explain the consonant
> > character of 196/169!
May be that's the reason why Werckmeister choosed
the root (Wurtzel) od his "septenarius"-tuning in 196=49*4 ?

> >
> > 1/1 196/169 15/13 7/6

> entropy? Or anything quantitative about 196/169 whatever?
>

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

🔗Cameron Bobro <misterbobro@yahoo.com>

196/169=(14/13)^2 in the partial series, was once Re: About those
simple ratios:

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

>
> http://de.wikipedia.org/wiki/Barocktrompete
> "Das Spiel in den hohen, chromatischen Lagen (ab 13. Teilton) heiß´
> Clarin- oder Clarinospiel und ist Kennzeichen der barocken
> Trompetenkunst."
> "The play in the high chromatic region (up the 13. partial)
> is called 'clarino'-play, characteristic for the Baroque
> tumpeting-art.

That would be "from" the 13th partial, to be more clear in English.
And Kennzeichen would be nice as "hallmark" in this case. Not
picking, I know that translating on the fly is a pain, God only
knows what mistakes I've made in German! :-)

>
> Werckmeister remarked about that higher partial series
> C:g:e:~bb:... == 4:5:6:7:....
> already in his "Musicalische Temperatur"(1691) p.24 Chap. XII and
> on p.25 Chap. XIII
> ".../wenn 1. gegen 7 solte kommen. 12 und 18 sind harmonisch /
> sie werden aber verdorben / und verlieren ihre Eigenschaften wenn
> sie zerissen werden / als 1. gegen 11 und die 1. gegen 17.
> die Zahlen 11. 13. 17. sind gantz nicht harmonisch /wenn sie also
> stehen...."
> tr:
> .../if 1. versus 7 sounds. 12 and 18 are harmonic /
> but become ill / and lost their properties when separated /
> as 1. versus 11 and 1. versus 17. the numbers 11. 13. 17
> are not harmonic at all / if they stand alone....

This is a book I'll have to get...
>
> That means i.m.o:
> Partials higher than the 11th can only be detected aural and
> discriminated in a context of other partials near to that region.

That isn't how I'd put it (for example the 13th partial seems very
distinctive alone but changes when its neighbors change, and above
that you start to paint in ever-broader brushstrokes,etc.) but the
idea is in keeping with my experiences synthesizing sound. About a
month ago I wrote something here about doing additive synthesis and
raising specific partials alone or with their neighbors, and how the
timbre changes differently with different pairs.

This is why I chose x/12 and x/13 as the two series to "mate" by
finding the midpoints between pairs from the two series. The ideal
would be to have near-coincidences with the pure harmonic series in
partials 6,7,12,13,19, which is what happens with 196/169,29/25, and
any interval in the tiny region between them.

In that tiny region, these near-coincidences (we want to avoid both
Tonverschmelzung and critical band nastiness at each partial, so
it's a little zone) mean that we have the character of both 7/6 and
16/13, and a little of 8/7 and 22/19 (the 7th and 19th partials).
So, played with the 1/1, a tone at the interval would,
theoretically, and audibly as far as I can tell, load these specific
regions with more energy, thereby creating a subtle timbral effect.

And if throughout the tuning you could get the same and related
timbral effects, you might get what I've been humorously calling
an "Uebertimbre" but should probably be called the Gesamtklangfarbe
der Einstimmung or some such ungodly thing.

Of course you cannot square the circle: the timbre would change with
different keys, by virtue of the emphasized regions moving. This is
desirable.

Both your equal-frequency Bach tuning and the Werckmeister 196-
monochord tuning are made in the ways which I have found by trial
and error guarantee more likelihood of this kind of cohesiveness,
which I find audible but maybe not everyone does. Both constant/x
and x/constant tunings also tend to have very cohesive and good-
sounding "shadow tunings" consisting of all the possible difference
tones octave-reduced.

There are some curious things about 14/13, and 14/13*14/13, and yet
another 14/13 gives you a wonderful M3...who knows? Perhaps the 196-
monochord is a simple practical mechanism for tempering the Just
intervals not by dividing a comma, but by pushing them toward the
regions where they start to take on the characters of the next
higher primes.

take care,

-Cameron Bobro