Isoharmonics: accuracy for 18:23:28:33

Hello, there, everyone, and this is a question about the accuracy
which might be required to realize 18:23:28:33 (~0-424-765-1049 cents)
as a distinct isoharmonic chord or sonority.

Here are the deviation in cents in one tuning system I use:

(28:18) (33:18)
1:1 23:18 14:9 11:6
0 425.915 762.775 1050.488
+~1.550 -~2.141 +~1.126

This might be a question of combination tones, and I wonder if these
accuracies are fine enough to get an "isoharmonic" effect.

Most appreciatively,

Margo Schulter
mschulter@value.net

--- In tuning@y..., "M. Schulter" <MSCHULTER@V...> wrote:

> Hello, there, everyone, and this is a question about the accuracy
> which might be required to realize 18:23:28:33 (~0-424-765-1049
cents)
> as a distinct isoharmonic chord or sonority.
>
> Here are the deviation in cents in one tuning system I use:
>
> (28:18) (33:18)
> 1:1 23:18 14:9 11:6
> 0 425.915 762.775 1050.488
> +~1.550 -~2.141 +~1.126
>
> This might be a question of combination tones, and I wonder if these
> accuracies are fine enough to get an "isoharmonic" effect.
>
> Most appreciatively,
>
> Margo Schulter

one approach might be to calculate the three rates of beating between
the three first-order difference tones corresponding to the
number "5". this would require one to know the register in which you
intend to play the chord, both to determine the rates of beating and
to see if the "5" would be in the audible range at all . . .

Hello, there, Paul, and here are some frequencies for the tempered
version of 18:23:28:33 that I'm using, give or take synthesizer
factors of precision and accuracy, and based on A4=440:

473.948 Hz 606.142 Hz 736.341 Hz 869.469 Hz
B4 Eb*5 F#*5 A*5

Here an asterisk (*) shows a note raised by an artificial diesis
of about 58.68 cents. The generator for each chain is ~704.096 cents
(Wilson/Pepper Noble Fifth).

While I'm familiar with the process of calculating beats for a
two-voice interval, the calculation of beats for difference tones is
new to me. I've picked a relatively high register, and wonder if an
octave lower would help or hinder an isoharmonic effect.

Most appreciatively,

Margo Schulter
mschulter@value.net

--- In tuning@y..., "M. Schulter" <MSCHULTER@V...> wrote:
> Hello, there, Paul, and here are some frequencies for the tempered
> version of 18:23:28:33 that I'm using, give or take synthesizer
> factors of precision and accuracy, and based on A4=440:
>
> 473.948 Hz 606.142 Hz 736.341 Hz 869.469 Hz
> B4 Eb*5 F#*5 A*5
>
> Here an asterisk (*) shows a note raised by an artificial diesis
> of about 58.68 cents. The generator for each chain is ~704.096 cents
> (Wilson/Pepper Noble Fifth).
>
> While I'm familiar with the process of calculating beats for a
> two-voice interval, the calculation of beats for difference tones is
> new to me.

Margo,

For beats between difference tones, first calculate the frequencies
of the difference tones. For first-order difference tones, these
would simply be the differences between the fundamental frequencies.
Then beats would then be calculated only for the difference tones
that are approximately the same frequency. For example, between
consecutive tones in your chord, the difference tones are:

606.142 - 473.948 = 132.194 Hz
736.341 - 606.142 = 130.199
869.469 - 736.341 = 133.128

and the beat rates would be:

132.194 - 130.199 = 1.955 Hz
133.128 - 132.194 = 0.934
133.128 = 130.199 = 2.929

Then calculate:

736.341 - 473.948 = 262.393 Hz
869.469 - 606.142 = 263.327

and 263.327 - 262.393 = 0.934 Hz beat rate

> I've picked a relatively high register, and wonder if an
> octave lower would help or hinder an isoharmonic effect.

As for how successful any amount of temperament might be in
preserving the desired effect, there is no substitute for actually
trying your chord(s) in both just intonation and tempered in various
registers and judging for yourself. From my own experience with
isoharmonic chords using double-digit ratios, I think that
transposing an octave lower would decrease the consonance of the
chord, even if the beating between difference tones is cut in half.

Nice to see that you're working with some of the difference-of-5
ideas that I suggested off-list. I haven't tried any of these in
anything other than just intonation, so I don't know how successful
even the small amount of temperament you have is going to be. (I
still owe you a response to your latest, but be patient.)

--George

--- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:
> --- In tuning@y..., "M. Schulter" <MSCHULTER@V...> wrote:
> > Hello, there, Paul, and here are some frequencies for the tempered
> > version of 18:23:28:33 that I'm using, give or take synthesizer
> > factors of precision and accuracy, and based on A4=440:
> >
> > 473.948 Hz 606.142 Hz 736.341 Hz 869.469 Hz
> > B4 Eb*5 F#*5 A*5
> >
> > Here an asterisk (*) shows a note raised by an artificial diesis
> > of about 58.68 cents. The generator for each chain is ~704.096
cents
> > (Wilson/Pepper Noble Fifth).
> >
> > While I'm familiar with the process of calculating beats for a
> > two-voice interval, the calculation of beats for difference tones
is
> > new to me.
>
> Margo,
>
> For beats between difference tones, first calculate the frequencies
> of the difference tones. For first-order difference tones, these
> would simply be the differences between the fundamental
frequencies.
> Then beats would then be calculated only for the difference tones
> that are approximately the same frequency. For example, between
> consecutive tones in your chord, the difference tones are:
>
> 606.142 - 473.948 = 132.194 Hz
> 736.341 - 606.142 = 130.199
> 869.469 - 736.341 = 133.128
>
> and the beat rates would be:
>
> 132.194 - 130.199 = 1.955 Hz
> 133.128 - 132.194 = 0.934
> 133.128 = 130.199 = 2.929
>
> Then calculate:
>
> 736.341 - 473.948 = 262.393 Hz
> 869.469 - 606.142 = 263.327
>
> and 263.327 - 262.393 = 0.934 Hz beat rate

none of these beat rates portend any objectionable roughness to my
eye. but of course, in these matters, the ear always overrules the
eye!

Hello, there, George and everyone.

Please forgive me if my mention of 18:23:28:33 was at all out of
season, since it certainly depends on our offline correspondence.
However, now that you've spoken up, I'd like to confirm that I'm both
amazed and immensely enriched in my musicmaking by what you've taught
me about isoharmonic chords.

Some of this will be in our articles for Xenharmonikon 18, and this
thread is another opportunity to let people know about that issue, and
to say that your 17-tone well-temperament (17-WT) is a musical marvel
well worth reading about -- as expounded by its designer, as well as
by an enthusiastic neo-medievalist.

Anyway, I checked and noted that in our offline discussions you not
only presented the topic of isoharmonic chords with differences of 5,
but specifically mentioned 18:23:28:33.

When I play the tempered version of 18:23:28:33, by the way, it sounds
like what I call "ultra-jazz": there's a certain synergy that
intrigues me. Whether this the effect of the difference tones, I'm not
sure, but thanks for your lucid explanation of how to calculate beats
between these tones. As it happens, the least accurate adjacent
interval is 23:28, narrow by about 3.691 cents.

To conclude, your isoharmonic expertise is something else again, and I
want everyone to know this, if they haven't already realized it from
your posts. Also, that XH 18 is worth waiting for -- a friendly plug,
with many thanks to John Chalmers for all his contributions to our
community over the years, and to you as a very distinguished
contributor.

Most appreciatively,

Margo Schulter
mschulter@value.net

Hello,

Sorry for providing my homework that late (probably too late) ....

Using a brute force method (instead of difference tone heuristics used
by George) I found other values for beats. Some of those values do not
come directly from the 18:23:28:33 but from neiboughring intervals
that are simpler such as 7:9 which is 10 some ten cents above 18:23.

The "brute force method" is this: I put all the chord harmonic values
(473,95*1, 473*2... 606,142*1 606,142*2 etc.) on an excel spreadsheet,
sort them and search for those which are nearer to each other than a
given threashold, let say 35 Hz. Let say that 18 is A, 23 is B 28 is C
and 33 is D (nothing to do with the notes of scale) and notate An the
nth harmonics of A (e.g. A10 = 4739,5 Hz). Thus, for instance A14:C9
denotate the beat of the 14th harmonic of the root against the 9th
harmonic of the "fifth*5".

In the table below there are the beating frequencies up to 14 kH for a
(rather large) threshold of 40 Hz for beating frequencies. The first
column designate the harmonic pairs. Fc is the central frequency (Fx +
Fy)/2 and Fb is the beat frequency (Fx-Fy). A * in first column
denotate Hpair that belongs to the intended 18:23:28:33 chord.

Hpair Fc Fb
------------------------
A7:B9 4254,3 22,55
*A11:D6 5215,5 4,15
B10:A7 6074,5 25,78
*A14:C9 6631,2 8,24
A14:B11 6651,4 32,26
C13:D11 9569,0 6,82
B17:C14 10306,6 4,35
*A22:D12 10431,0 8,30
A23:B18 10905,7 9,70
*A28:C18 13262,4 16,48

So it may occur that some contribution to roughness come from dyads
that are foreign to the 18:23:28:33.

If the chord is lowered by an octave, all the figures above (Fc and
Fb) should be divided by two. Further, if we take the same threshold
for a lowered chord, new contribution to roughness may appear at even
lower frequencies from even simpler ratio. For instance, an octave
below, 18:28 may start to lose its "personality" and sound more as an
out of tune 2:3 as a 25Hz beat starts to be audible around 724 Hz; or
may be it will just add the ruggosity that gives the extra "jazzyness"
and delighfull ambiguity.

yours truly

François Laferrière

--- In tuning@y..., "M. Schulter" <MSCHULTER@V...> wrote:
> Hello, there, everyone, and this is a question about the accuracy
> which might be required to realize 18:23:28:33 (~0-424-765-1049
cents)
> as a distinct isoharmonic chord or sonority.
>
> Here are the deviation in cents in one tuning system I use:
>
> (28:18) (33:18)
> 1:1 23:18 14:9 11:6
> 0 425.915 762.775 1050.488
> +~1.550 -~2.141 +~1.126
>
> This might be a question of combination tones, and I wonder if these
> accuracies are fine enough to get an "isoharmonic" effect.
>
> Most appreciatively,
>
> Margo Schulter
> mschulter@v...

--- In tuning@y..., "M. Schulter" <MSCHULTER@V...> wrote:
> Hello, there, George and everyone.
>
> Please forgive me if my mention of 18:23:28:33 was at all out of
> season, since it certainly depends on our offline correspondence.
> However, now that you've spoken up, I'd like to confirm that I'm
both
> amazed and immensely enriched in my musicmaking by what you've
taught
> me about isoharmonic chords.
>
> Some of this will be in our articles for Xenharmonikon 18, and this
> thread is another opportunity to let people know about that issue,
and
> to say that your 17-tone well-temperament (17-WT) is a musical
marvel
> well worth reading about -- as expounded by its designer, as well as
> by an enthusiastic neo-medievalist.
>
> Anyway, I checked and noted that in our offline discussions you not
> only presented the topic of isoharmonic chords with differences of
5,
> but specifically mentioned 18:23:28:33.
>
> When I play the tempered version of 18:23:28:33, by the way, it
sounds
> like what I call "ultra-jazz": there's a certain synergy that
> intrigues me. Whether this the effect of the difference tones, I'm
not
> sure, but thanks for your lucid explanation of how to calculate
beats
> between these tones. As it happens, the least accurate adjacent
> interval is 23:28, narrow by about 3.691 cents.
>
> To conclude, your isoharmonic expertise is something else again,
and I
> want everyone to know this, if they haven't already realized it from
> your posts. Also, that XH 18 is worth waiting for -- a friendly
plug,
> with many thanks to John Chalmers for all his contributions to our
> community over the years, and to you as a very distinguished
> contributor.
>
> Most appreciatively,
>
> Margo Schulter

This is just to let everyone know that Margo and I participated in a
17-tone *neo-medieval revolution* last fall that delayed my showing
up here on the Tuning List until the beginning of this year.

In my 17-tone article for XH18 I speculate that, if history had taken
a different turn some centuries back, then the musical mainstream
today might be some sort of neo-medievalism -- and some on the
alternate tuning list might be advocating the 12 division of the
octave! So if you think Margo seems to be off in her own separate
musical world, you've got the general idea -- she's following the
path of an alternate history-in-the-making, for which I find that the
possibilities are quite intriguing!

(Now if I can just figure out what difference-of-5 isoharmonic chords
have to do with all of this. Our neo-medieval revolution doesn't go
beyond the 13 limit.)

--George

--- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:
> This is just to let everyone know that Margo and I participated in
a
> 17-tone *neo-medieval revolution* last fall that delayed my showing
> up here on the Tuning List until the beginning of this year.

But before anyone starts accusing her of keeping me away from here,
let it be known that she was the one who first told me (in the fall
of 2001) about the *Miracle tuning revolution* of the spring of 2001,
and it was my own decision to stay away for a season. (I can handle
only one revolution at a time -- don't want to get too dizzy.)

--George

--- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:
> --- In tuning@y..., "M. Schulter" <MSCHULTER@V...> wrote:

> In my 17-tone article for XH18 I speculate that, if history had taken
> a different turn some centuries back, then the musical mainstream
> today might be some sort of neo-medievalism -- and some on the
> alternate tuning list might be advocating the 12 division of the
> octave!

That would be too weird for many, but of course 34 and 68 would beckon.

--- In tuning@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:
> >
> > In my 17-tone article for XH18 I speculate that, if history had
taken
> > a different turn some centuries back, then the musical mainstream
> > today might be some sort of neo-medievalism -- and some on the
> > alternate tuning list might be advocating the 12 division of the
> > octave!
>
> That would be too weird for many,

That's essentially what I say in my introduction: "If we were now in
the position of evaluating 12-ET as a possible alternative to 17-ET
in the search for new tonal resources, we would probably dismiss 12-
ET just as readily [as we now dismiss 17], declaring it to be
melodically and harmonically bland and crude."

> but of course 34 and 68 would beckon.

Or 22 -- the meantone system in our alternate history.

Hope this whets your appetite!

--George