High third: more data

I sat down with Spectrogram last night so that I could measure what pitch
I was singing against a drone. First, I found the unison, and then
measured what intervals I could get by singing sharp. The first one that
locked-in was a 7:6. This surprised me, because I'd been getting it
before through trying to sing unisons, and had no idea it was so wide.

Then, by narrowing the interval, guided by the Spectrogram readout, I
found another lock at something like 15:14 (I can't say precisely, but it
was wider than 16:15) and then 10:9. When widening to get a third, I
found the first thing I hit was a 9:7 and then the 4:3. As I have poor
(pretty much non-existent) relative pitch, I wasn't recognising these as
thirds or fourths or anything.

It was actually quite difficult to get the 5:4, but once I did it was
obvious that it was purer than the 9:7. (I think, although it locks
strongly, there's only a very small pitch range for which does so.) Then
I tried to see what intervals locked in-between. I found an interval
around 409 or 410 cents (I had upped the precision by now) kept arising,
and also the occasional 400 cent one. There was no other pattern, and
certainly no preference for 404-5 cents. Although there was quite a bit
of vibrato, so where this led from 400 to 409 cents, you could have nailed
the average pitch at 404.5 cents. 409 cents is good for 19:15.

Finding the 6:5 was even harder, partly because of the pull of the 7:6.

The experiment Jerry suggested was to sing against the root, and then add
the fifth. Guided by the readout, I found a 5:4, then shut my eyes and
added the fifth. By opening my eyes, I could then follow the trace to see
what I had been singing. Three phenomena were observed:

1) I held the note constant. One way of doing this was to concentrate on
the note I was singing, and ignore the new one coming from the keyboard:)
Otherwise, it did sound like the note I was singing rose. I attribute
this to the increased amount of high-pitched sound reaching my ears.

2) The pitch really did rise. This tended to be to around 9:7 or 19:15,
or some inconsistent point around 400 cents.

3) Upward vibrato was added, classically resulting in the pitch
fluctuating between 5:4 and 9:7.

By consciously correcting for the sharpening, I even hit some kind of
neutral third once:)

At that point, the high third theory seemed confirmed. Then, a while
later, I realised that, as I had been using the soundcard synth, setting
the tuning on my synth keyboard would have had no effect. So I had been
tuning the fifths to golden meantone. I then repeated this experiment,
and found that the pitch did alter when the fifth was added, but not in
any consistent direction. I interpret this as meaning that there is an
optimum amount of beating that I was seeking out.

But why would a flat fifth have encouraged me to sharpen the third? I'm
not sure, but it may have increased the pull towards the 7:6 with that
fifth, as well as creating a region of ambiguity between locking with the
root and fifth. Another thing is that I may have been trying too hard to
get the right results:)

My general conclusion is that the third is much harder to lock than a
fourth or fifth, let alone a unison or octave. So, it's the most likely
note to go out of tune in a triad. It also notices less if it's out of
tune in a chord, as the consonance between the root and fifth is still
there. In addition, if the fifth is even slightly out of tune, it gets
much harder again to get the third right. The maxim "better sharp than
flat" then leads to a high third. The fact I did sometimes hit 400 cents
also suggests 12-equal conditioning. I don't know if this is stronger
than normal in me because, although I'm a keyboard player of sorts, I've
been mostly using meantone for a couple of years now. This may explain
where all the 7-prime limit intervals came from.

So there you go. Completely unscientific, but I hope interesting none the
less.

Graham
http://x31eq.com/

Graham Breed reported:
>
> I sat down with Spectrogram last night so that I could measure what pitch
> I was singing against a drone.

Graham, bless your British heart for taking me up on my suggestion to
explore the world of "ear tuning." Needless to say, I have read your
adventure with keen interest. So, if I may, I'll offer a few comments.

> First, I found the unison, and then
> measured what intervals I could get by singing sharp. The first one that
> locked-in was a 7:6. This surprised me, because I'd been getting it
> before through trying to sing unisons, and had no idea it was so wide.

If you are a "blues" fan, it shouldn't surprise you. I think this "subminor
third" (thanks, Tuning List) is the one blues singers hear when they sing
the "blue third." I am a bit surprise that you had been "getting it" by
trying to sing unisons. I don't quite understand that.
>
> Then, by narrowing the interval, guided by the Spectrogram readout, I
> found another lock at something like 15:14 (I can't say precisely, but it
> was wider than 16:15) and then 10:9.

Didn't get to 8:9? That's pretty easy to hear.

> When widening to get a third, I
> found the first thing I hit was a 9:7 and then the 4:3. As I have poor
> (pretty much non-existent) relative pitch, I wasn't recognising these as
> thirds or fourths or anything.

How did you get past the 4:5? That's the most easy to hear in that vicinity.
Interesting. Perhaps you didn't go slowly enough.
>
> It was actually quite difficult to get the 5:4, but once I did it was
> obvious that it was purer than the 9:7. (I think, although it locks
> strongly, there's only a very small pitch range for which does so.)

Oh, good. Yes, I agree that the 4:5 locks very securely to a sounding root.

> Then
> I tried to see what intervals locked in-between. I found an interval
> around 409 or 410 cents (I had upped the precision by now)

Meaning you had become more exacting in your judgements?

> kept arising,
> and also the occasional 400 cent one. There was no other pattern, and
> certainly no preference for 404-5 cents. Although there was quite a bit
> of vibrato,

Uuuuugh! Not good for this kind of thing.

> so where this led from 400 to 409 cents, you could have nailed
> the average pitch at 404.5 cents. 409 cents is good for 19:15.

Interesting, needless to say.
>
> Finding the 6:5 was even harder, partly because of the pull of the 7:6.

You ARE a blues kind of guy.
>
> The experiment Jerry suggested was to sing against the root, and then add
> the fifth. Guided by the readout, I found a 5:4, then shut my eyes and
> added the fifth. By opening my eyes, I could then follow the trace to see
> what I had been singing. Three phenomena were observed:
>
> 1) I held the note constant. One way of doing this was to concentrate on
> the note I was singing, and ignore the new one coming from the keyboard:)
> Otherwise, it did sound like the note I was singing rose. I attribute
> this to the increased amount of high-pitched sound reaching my ears.

High pitched sound? Can you be specific?
>
> 2) The pitch really did rise. This tended to be to around 9:7 or 19:15,
> or some inconsistent point around 400 cents.

Welcome to the twilight zone.
>
> 3) Upward vibrato was added, classically resulting in the pitch
> fluctuating between 5:4 and 9:7.

That's a pretty wide area. Vibrato is definitely not good for this kind of
research. My experience in this regard is with vibratoless vocal tone (to
the extent that is possible).
>
> By consciously correcting for the sharpening, I even hit some kind of
> neutral third once:)

Hold it right there! "Neutral" third? What's that?
>
> At that point, the high third theory seemed confirmed. Then, a while
> later, I realised that, as I had been using the soundcard synth, setting
> the tuning on my synth keyboard would have had no effect. So I had been
> tuning the fifths to golden meantone. I then repeated this experiment,
> and found that the pitch did alter when the fifth was added, but not in
> any consistent direction. I interpret this as meaning that there is an
> optimum amount of beating that I was seeking out.

I'm pretty sure that the "high third" I hear locks (appears to lock) in the
same way that any interval locks--namely, because beating is minimal.
>
> But why would a flat fifth have encouraged me to sharpen the third? I'm
> not sure, but it may have increased the pull towards the 7:6 with that
> fifth, as well as creating a region of ambiguity between locking with the
> root and fifth. Another thing is that I may have been trying too hard to
> get the right results:)

Always a possibility that must be considered.
>
> My general conclusion is that the third is much harder to lock than a
> fourth or fifth, let alone a unison or octave.

Makes sense.

> So, it's the most likely
> note to go out of tune in a triad.

Assuming there is a "right" tuning for the third.

> It also notices less if it's out of
> tune in a chord, as the consonance between the root and fifth is still
> there.

Apparently, that is true.

> In addition, if the fifth is even slightly out of tune, it gets
> much harder again to get the third right.

That, also, seems to be true. Although you seemed to have some "fun" using
the meantone fifth.

> The maxim "better sharp than
> flat" then leads to a high third.

From what I've read around here, that would seem to be the consensus.

> The fact I did sometimes hit 400 cents
> also suggests 12-equal conditioning.

Always to be considered.

> I don't know if this is stronger
> than normal in me because, although I'm a keyboard player of sorts, I've
> been mostly using meantone for a couple of years now. This may explain
> where all the 7-prime limit intervals came from.

Could be. Maybe you're not as "bluesy" as I thought. :-)
>
> So there you go. Completely unscientific, but I hope interesting none the
> less.

Very interesting, indeed, Graham. Hopefully, your lead will encourage others
to jump into the pool.

Jerry

In-Reply-To: <200003010635.WAA21451@harrier.prod.itd.earthlink.net>
Gerald Eskelin wrote:

> > First, I found the unison, and then
> > measured what intervals I could get by singing sharp. The first one
> > that
> > locked-in was a 7:6. This surprised me, because I'd been getting it
> > before through trying to sing unisons, and had no idea it was so wide.
>
> If you are a "blues" fan, it shouldn't surprise you. I think this
> "subminor
> third" (thanks, Tuning List) is the one blues singers hear when they
> sing
> the "blue third." I am a bit surprise that you had been "getting it" by
> trying to sing unisons. I don't quite understand that.

I'm not heavily into blues. I've got the requisite Robert Johnson CDs,
but that's about it. This interval sounded more Indian to me, probably
because that's the drone-based music I'm most familiar.

I've been trying ... for quite a while now ... to sing unisons against a
drone. Actually, I stopped practicing for a long time because I was in a
flat with thin walls, and I'm bashful about these things. But yes, I'd
eventually arrive in the general vicinity of the note I was supposed to be
singing, but sometimes lock into a note I knew to be sharp. I assumed it
was some kind of quartertone, but turns out it's a subminor third.

When I try to tune my guitar by ear, I find it's very difficult to tell if
the same note played on two different strings is tuned the same. However,
it's a lot easier when I play a tune and alternate the notes. So I've
obviously built up some idea of melody over the years. So I must have
some kind of relative pitch, but I can't put names to intervals.

> > Then, by narrowing the interval, guided by the Spectrogram readout, I
> > found another lock at something like 15:14 (I can't say precisely,
> > but it
> > was wider than 16:15) and then 10:9.
>
> Didn't get to 8:9? That's pretty easy to hear.

Yes, I'm surprised at how difficult it is to sing. As soon as I get
sharper than 10:9 I slide into a 7:6. I think I did get it last night,
although the differences are very small. I think I was singing a neutral
second as well, somewhere around 150 cents.

> > When widening to get a third, I
> > found the first thing I hit was a 9:7 and then the 4:3. As I have
> > poor
> > (pretty much non-existent) relative pitch, I wasn't recognising these
> > as
> > thirds or fourths or anything.
>
> How did you get past the 4:5? That's the most easy to hear in that
> vicinity.
> Interesting. Perhaps you didn't go slowly enough.

I found the differences in pitch to be very small, and so "a bit higher"
would take me straight from 7:6 to 9:7.

I checked the sound I was singing to last night, and it has a weak 5th
partial, which would explain my difficulty in locking the 5-limit.
Switching from "reed organ" to "accordion" seems to make more sense.

> > Then
> > I tried to see what intervals locked in-between. I found an interval
> > around 409 or 410 cents (I had upped the precision by now)
>
> Meaning you had become more exacting in your judgements?

No, I'd altered the Spectrogram settings to give finer pitch resolution.

> > 1) I held the note constant. One way of doing this was to
> > concentrate on
> > the note I was singing, and ignore the new one coming from the
> > keyboard:)
> > Otherwise, it did sound like the note I was singing rose. I attribute
> > this to the increased amount of high-pitched sound reaching my ears.
>
> High pitched sound? Can you be specific?

I mean the fifth. As that's the highest note in the chord, and new, it
makes everything sound higher.

> > 3) Upward vibrato was added, classically resulting in the pitch
> > fluctuating between 5:4 and 9:7.
>
> That's a pretty wide area. Vibrato is definitely not good for this kind
> of
> research. My experience in this regard is with vibratoless vocal tone
> (to
> the extent that is possible).

I tried my best to minimize it. This is an extreme case, although there
was always a small amount of fuzziness according to the readout.

> > By consciously correcting for the sharpening, I even hit some kind of
> > neutral third once:)
>
> Hold it right there! "Neutral" third? What's that?

Between major and minor. I'm doing a fair amount of work with neutral
third scales, and would like to be able to sing them. I think the
interval I got here was somewhere between 6:5 and 11:9, although I didn't
hold it for long enough to be that interested.

Last night, I tried singing a tune with neutral thirds either doubling on
the keyboard, or singing to chords. I think I got it, although I haven't
measured the results.

> > So, it's the most likely
> > note to go out of tune in a triad.
>
> Assuming there is a "right" tuning for the third.

As I was trying to get the cleanest sound, that should have been the
5-limit. That's certainly been my experience with keyboard tunings.
Otherwise, I think the optimum mistuning depends on the timbre.