Yahoo Tuning Groups Ultimate Backup tuning Re: locking in

John, Jerry, Paul,

The discussion about "locking in" on various intervals
brings up some left over issues from about a two year
series of posts we had here about "complexity measures"
of intervals, and concerns about significance of
"prime limits", "odd limits" and the overtone series.

Paul has a graph posted somewhere of an equation that
shows troughs between 1 and 2 at the expected ratios
in the expected manner, deep trough at 3/2, slightly
less deep at 4/3, slightly less at 5/4 and 8/5 (these
may not be exact, perhaps Paul will mention where and
how it was produced).

My question, to you sensitive singers over pedals, is...

...when you sing the most 'locked in' major second, is
it 9/8, 8/7 or 10/9 or is it something else?

...the most natural 'tritone' (for want of a less
weighted term), is it 7/5, 11/8, 10/7 or something
else?

The idea here is that people who believe in greater
importance for "prime limits", "odd limits" and
"overtone series significance" would probably see some
different sorting.

Bob Valentine

🔗John Link <johnlink@con2.com>

>From: Robert C Valentine <bval@iil.intel.com>
>
>My question, to you sensitive singers over pedals, is...
>
>...when you sing the most 'locked in' major second, is
>it 9/8, 8/7 or 10/9 or is it something else?
>
>...the most natural 'tritone' (for want of a less
>weighted term), is it 7/5, 11/8, 10/7 or something
>else?

I've never experimented with the intervals you mention in isolation, so I'm
not sure that the answer I'm about to give addresses your question. How the
intervals tune in the context of a chord depends upon what role the tones
play in the chord. For an example that would include all three of the major
seconds you mention, consider a dominant seventh chord with a natural
ninth. Tuned 4:5:6:7:9, we have 9/8 between the ninth and the root, 10/9
between the third and the ninth, and 8/7 between the root and the seventh
(making octave adjustments as necessary). With regard to the tritone,
whether it is 7/5 or 10/7 depends upon which note is the third and which is
the seventh. Of course I'm assuming that the interval between the root and
third is 5/4 and not the high thrid that has been discussed here recently.

John Link

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🔗Paul H. Erlich <PErlich@Acadian-Asset.com>

Robert C Valentine made mention of my harmonic entropy graphs. Those are in
connection with an as-yet not fully worked out theory -- I've e-mailed
mathematician John Conway to help me with the details. Better for now might
be to look at the many dissonance graphs in the psychoacoustical literature,
see Sethares, Kameoka & Kuriagawa, Vos, and even Helmholtz.

>My question, to you sensitive singers over pedals, is...

>...when you sing the most 'locked in' major second, is
>it 9/8, 8/7 or 10/9 or is it something else?

These are roughly equally locked in to my ears, though of a different order
of difficulty than the consonant ratios. I'd say 9/8 is special since the
virtual fundamental is 3 octaves below the "root", so it sounds most stable.

>...the most natural 'tritone' (for want of a less
>weighted term), is it 7/5, 11/8, 10/7 or something
>else?

Definitely 7/5. 10/7 requires care. 11/8 doesn't work for me as a dyad,
though if I add other notes to the drone (so as to produce, say, 8:10:11:12)
it becomes easier.

🔗Paul H. Erlich <PErlich@Acadian-Asset.com>

🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>

🔗Robert C Valentine <bval@xxx.xxxxx.xxxx>

> Robert C Valentine made mention of my harmonic entropy graphs. Those are in
> connection with an as-yet not fully worked out theory -- I've e-mailed
> mathematician John Conway to help me with the details. Better for now might
> be to look at the many dissonance graphs in the psychoacoustical literature,
> see Sethares, Kameoka & Kuriagawa, Vos, and even Helmholtz.

I should check a few of these out, but I thought of yet another algorithm
for producing an image like this (maybe its the same as others algorithms).

Make a histogram of all the unique ratios between 1 and 2 up to some limit.
Let the bins be logarithmic, for instance, somewhere in the range of 10 to
30 cents, though this could be played with.

Ratios with "lockability" should be in bins with few members. I'm not sure
if it would work exactly, since octaves and unisons have a very steep cliff
next to them if my mental image is correct.

This should also match the construction of the ear, are the hairs in the
ear logarithmically arranged in the frequency spectrum (each hair is a
bin)?

Bob Valentine

🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>

>I should check a few of these out, but I thought of yet another algorithm
>for producing an image like this (maybe its the same as others algorithms).

>Make a histogram of all the unique ratios between 1 and 2 up to some limit.
>Let the bins be logarithmic, for instance, somewhere in the range of 10 to
>30 cents, though this could be played with.

This is very similar to how harmonic entropy is computed, though with
harmonic entropy I can produce a smooth curve with any desired resolution
for any limit. It's the same idea, just "smoothed."

>Ratios with "lockability" should be in bins with few members. I'm not sure
>if it would work exactly, since octaves and unisons have a very steep cliff
>next to them if my mental image is correct.

What's wrong with that?

>This should also match the construction of the ear, are the hairs in the
>ear logarithmically arranged in the frequency spectrum (each hair is a
>bin)?

It corresponds better to the "smoothed" version -- the ear doesn't have
discrete "bins" -- many, many hairs are excited even by an absolutely pure,
unvarying tone. Plus, you're forgetting about the periodicity mechanism in
hearing: notes in the low-to-middle registers come through to the brain
through nerve impulses firing in synch with the waveforms, in addition to
the place mechanism (i.e., position of hairs on the cochlea).

🔗Robert C Valentine <bval@xxx.xxxxx.xxxx>

🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>

Robert Valentine wrote,

>Well, in the algorithm I was ponderring, the binsize would be tricky to
>tune so that octave and unison wouldn't get polluted by the sides of the
>cliff while still showing just how how wide the pocket is at 3/2 and 4/3
for
>instance. On the other hand, as the binsize goes to zero, the histogram
>turns into a continuous line and maybe the graph will be more similar to
>that you were investigating.

It's no good to let the binsize go to zero, since for small binsizes, too
many ratios will show up as "special". My harmonic entropy graphs
essentially use a continuously sliding, bell-shaped "bin" with a "width"
(standard deviation) of 1% ~= 17.3�. As you can see from the graphs, the
octave, unison, and all other intervals come out quite nicely. I'd go as low
as 0.6% ~= 10.4� for the width but any lower would probably be beyond the
capacity of normal ears. If you like, I'll go into detail into how to modify
your bin proposal in just a few easy steps so that it's the same as harmonic
entropy.

🔗Robert C Valentine <bval@xxx.xxxxx.xxxx>

Paul said :

>
> > wouldn't it be kind of natural to interpret the "high
> > third" as the product of two 8:9 steps, i.e., a Pythagorean major third?
>
> Intellectually, perhaps. Likely not "naturally" (perceptually). I'm more
> concerned about what causes, or contributes to, the "high third" experience
> as I have described it here many times--which I suspect has little or
> nothing to do with major seconds, neither intellectually nor naturally.
>

As part of this thread, people responded on how 9/8 45/32 27/16 could be
perceived more readilly than 9/8 11/8 13/8, since it tunes the triad.
Although the numbers look daunting, 27/16 is "the fifth of the fifth
of the fifth", which is sort of straightforward and the 45/32 fills
in the major triad between it and the "fifth of the fifth".

Depending on ones belief in the strength of fifths (that "up a fifth
4 times" after octave adjustment is a hearable relationship) and on
the strength of the overtone series (that the 81'st overtone still has
a reasonably good pocket to slip into), then this could seem like a
good choice for the high third.

Speaking of which, what happens to the high third in seventh chords, either
dominant or major? Does it have a high seventh that goes along with it?

> > That is the usual take on Indian music, where the just (5/4) and Pythagorean
> > (81/64) major thirds are sruti #s 7 and 8 out of a 22-sruti octave. Now it
> > is true that the Pythagorean major third wouldn't lock harmonically as part
> > of a triad, so for the third time I suggest the possibility of
> > 1/24:1/19:1/16 for your locked major triad with high third.
>
> For now the second time in a post to you (and the third if you count another
> to someone else) I am requesting a clarification of what appear to be
> fractions separated by colons.

Paul will have a better explanation of where this notation comes from
but when I see it, I just multiply by the first denominator. In this
case is yields

1/1 24/19 3/2

which is very close to the 12TET major triad. Others may like to throw out
the fractional notation altogether, in which this series is

38 48 57.

They're all the same thing (and I don't quite know when the proper time to
use slashes and colons is).

Bob Valentine

🔗Gerald Eskelin <stg3music@earthlink.net>

Bob Valentine said:

> Depending on ones belief in the strength of fifths (that "up a fifth
> 4 times" after octave adjustment is a hearable relationship) and on
> the strength of the overtone series (that the 81'st overtone still has
> a reasonably good pocket to slip into), then this could seem like a
> good choice for the high third.

Pardon my practical doubt, Bob, but the stacked fifths principle seems
"ivory tower" to me. Here's why. As one tunes "partials" over a fundamental,
the higher the numbers the less vividly intervals lock perceptually. While
the 2:3 fifth can readily be locked in by raw beginners, locking in the 8:11
interval, for example, takes considerable experience but can be mastered.

Also, while a single sounding fifth is easy to perceive, it seems to me that
imagining(?) a number of stacked fifths is unlikely. While scale step 2 can
be conceived as a stack of two fifths, it seems more practical to "hear" it
as 8:9. Even though the numbers jibe (2:3 + 2:3 = 4:9 or 8:9), my perceptual
impression appears to be a matter of tuning the 8:9 directly, and "hearing"
scale step 5 in imagination is not required for locking in the tuning.

Higher stacks would have to compete with intervals having simpler ratios, I
would think. For example, instead of hearing the A that is stacked three
fifths above C, one could more easily and immediately tune C and A as 3:5.

If I am missing something, please correct me.

Bob continues:

> Speaking of which, what happens to the high third in seventh chords, either
> dominant or major? Does it have a high seventh that goes along with it?

My observation is that either the 4:5 third or the high third can be locked
into both the triad and the dominant seventh chords. Using the high third in
combination with the 4:7 chord seventh seems to enhance the gravitational
pull toward the destination harmony. It is still a mystery, as you likely
know, as to what the real numbers (or principles) are in such a case.

Although I haven't paid much attention to the tuning of the third and
seventh in a major/major seventh chord (major triad w/ major seventh), I
suspect that the third and seventh would lean toward a 2:3 "lock," making
the 8:15 major seventh even more stable. Again, the fact that the major
seventh is aurally easier to lock when the third is present tends to support
the thesis above that lower is better.

Later, in response to this exchange:

>> Paul Erlich asked:
>>
>> > So, Gerald, to you the dominant seventh chord is 4:5:6:7, yet the major
>> > triad is not 4:5:6?
>>
>> I can "lock in" both the "low third" and the "high third" at will in both
>> the major triad and the dominant (major-minor) seventh chord. My "mystery"
>> has to do with an apparent singer preference for the high third in either or
>> both. I hope I didn't give the impression that I believe the 4:5:6 triad is
>> not the "real one." I simply would like to explore possible rationalizations
>> regarding the other one.

Bob commented:
>
> Well, IF the "high third" was 14/11, AND the minor seventh was 7/4, then
> you get an 11/8 between them... for some reason, I think I'll want some
> extra water with that.

LOL. But isn't 11/8 the "in the cracks" tritone that appears in the natural
partial series? If so, I'll have mine straight, thank you. Just how that
translates to the third/seventh of the dominant seventh chord I'm not sure,
but it seems like something worth exploring.

> Otherwise, numerical fiddling gets me into "high sevenths" to go along with
> the "high third".

I like high thirds and low sevenths where dominant function is in effect.
>
> Do you also have low sevenths (7/4) and high sevenths (9/5 or 16/9 or
> some such???)

Do you mean do I own any personally? (Sorry about that) The "low" seventh
seems to me to be "co-owned" by the dominant seventh chord and by
drone-oriented folk traditions, such as Indian (Eastern), Georgian choral
music and American blues. The "high" sevenths you mention seem at home in
non dominant functions (tonic and subdominant, for example) in Western
styles, where the tritone is normally not present.

Thanks for your thoughts, Bob.

Jerry

🔗Paul H. Erlich <PErlich@Acadian-Asset.com>

Gerald Eskelin wrote,

>Also, while a single sounding fifth is easy to perceive, it seems to me
that
>imagining(?) a number of stacked fifths is unlikely.

In the context of locking in, I agree emphatically!

>Higher stacks would have to compete with intervals having simpler ratios, I
>would think. For example, instead of hearing the A that is stacked three
>fifths above C, one could more easily and immediately tune C and A as 3:5.

Agreed again!

>I suspect that the third and seventh would lean toward a 2:3 "lock," making
>the 8:15 major seventh even more stable. Again, the fact that the major
>seventh is aurally easier to lock when the third is present tends to
support
>the thesis above that lower is better.

Yes. 8:15 would be quite difficult to lock without an intervening pitch
allowing it to be tuned as a stack of a 2:3 and a 4:5. This is like the
32:45 and 16:27 mentioned earlier -- they tunable only by means of
intervening pitches, creating stacks (2:3, 2:3, and 4:5; amd 2:3, 2:3, and
2:3, respectively). These intervening pitches must be actually sounding, not
merely imagined.

🔗Joe Monzo <monz@juno.com>

>> [Bob Valentine, TD 491.3]
>> Speaking of which, what happens to the high third in
>> seventh chords, either dominant or major? Does it have
>> a high seventh that goes along with it?

> [Jerry Eskelin, TD 492.16]
> My observation is that either the 4:5 third or the high
> third can be locked into both the triad and the dominant
> seventh chords. Using the high third in combination with
> the 4:7 chord seventh seems to enhance the gravitational
> pull toward the destination harmony. It is still a mystery,
> as you likely know, as to what the real numbers (or principles)
> are in such a case.
>
> Although I haven't paid much attention to the tuning of
> the third and seventh in a major/major seventh chord (major
> triad w/ major seventh), I suspect that the third and seventh
> would lean toward a 2:3 "lock," making the 8:15 major seventh
> even more stable. Again, the fact that the major seventh
> is aurally easier to lock when the third is present tends
> to support the thesis above that lower is better.

I have indeed found this to be the case with *lower* '3rds'
and '7ths'. Several months back, I was looking for a
chord analagous to a 'major 7th chord' that used the
9:11 'neutral 3rd', and the '7th' that I found to work
best was the 6:11 'neutral 7th', which formed a 2:3
'perfect 5th' with the 11/9. The resulting 18:22:27:33
[= 1/66:1/54:1/44:1/36] chord is a nice mellow one to
my ears.

>> [Bob Valentine, TD 491.3]
>> Well, IF the "high third" was 14/11, AND the minor seventh
>> was 7/4, then you get an 11/8 between them... for some reason,
>> I think I'll want some extra water with that.

> [Jerry Eskelin, TD 492.16]
> LOL. But isn't 11/8 the "in the cracks" tritone that appears
> in the natural partial series? If so, I'll have mine straight,
> thank you. Just how that translates to the third/seventh of
> the dominant seventh chord I'm not sure, but it seems like
> something worth exploring.

>> [Bob]
>> Otherwise, numerical fiddling gets me into "high sevenths"
>> to go along with the "high third".

> [Jerry]
> I like high thirds and low sevenths where dominant function
> is in effect.

In fact, the chord Bob suggested is just such a one.

With the ratios 1/1 - 14/11 - 3/2 - 7/4, it has the
otonal proportions 44:56:66:77 (compare 44:55:66:77
= 4:5:6:7), and can be described with smaller numbers
as an utonal proportion 1/42:1/33:1/28:1/24.

It sounds to me like a very effective 'dominant 7th' chord
in need of resolution, as opposed to the smooth and stable
4:5:6:7 'harmonic 7th chord'.

-monz

Joseph L. Monzo Philadelphia monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
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🔗Joe Monzo <monz@juno.com>

> [Daniel Wolf, TD 493.17]
> One of the most stunning things I have ever heard was a
> recording of the group including oscillators, Tony Conrad's
> violin and La Monte Young, Terry Riley and Marian Zazeela
> singing. In addition to the above, to a sustained (octaves
> ignored here) 4:6:7:9 drone, Young and Riley would take turns
> going to the 9:8 above the 7, giving a complete chord of
> 32:48:56:63:72. They would lock into this dissonance by
> listening closely to beats, making sure that the beating
> did not change speed.

I made a MIDI file of this too
http://www.ixpres.com/interval/td/monzo/young9.mid
and agree that it's a really gorgeous chord; I lust after
hearing the real thing with live instruments and voices.
With such a rich chord, it must have been a real excercise
to keep that 32:63 in tune.

If find the sonorous beauty of this chord especially
interesting, because one of the most treasured new additions
to my library is a copy of Boomsliter/Creel 1962 recently
given to me by Erv Wilson (and apparently Erv got it from
Boomsliter himself), in which they analyze Bessie Smith's
vocal in _Empty Bed Blues_ as emphasizing (in C major blues)
a 21/16 F, 75/64 Eb, and 63/32 C thru most of the song,
only going to 4/3 F, 7/6 Eb and 1/1 C at the very end.
I made a MIDI file of this too, with a very simple (since
I'm not familiar with the actual recording) 12-tET piano
accompaniment:
http://www.ixpres.com/interval/td/monzo/emptybed.mid

REFERENCE
---------
Boomsliter, Paul C. and Creel, Warren. 1962.
_Interim Report on the Project on
Organization in Auditory Perception
at the State University College, Albany, New York_.