Justification for TOPS

I was thinking about TOPS, and how it does not retain octave-
equivalence. I can think of two things that justify TOPS acoustically:

1. The chorus effect. On a pianoforte, most keys have three strings.
Piano tuners purposely tune them slightly different to produce
a "chorus effect" or block of sound. So a tone is actually
a blur of frequencies.

2. Enharmonicity on the pianoforte. (Only applies to TOP tunings
>1200 cents). For certain physical reasons, the octave is slightly
stretched on a pianoforte. This would justify TOP tunings > 1200
The octave is especially stretched more at the extremes - the very
low and the very high.

Now obviously most microtonally-tuned performances are done on
electric instruments, but the above rules could apply to synthesized
piano sounds, at least.

I thought I would start this discussion on tuning and we could go
to tuning-math if it gets mathematical.

Comments, anyone?

Paul

>I was thinking about TOPS, and how it does not retain octave-
>equivalence. I can think of two things that justify TOPS acoustically:
>
>1. The chorus effect. On a pianoforte, most keys have three strings.
>Piano tuners purposely tune them slightly different to produce
>a "chorus effect" or block of sound. So a tone is actually
>a blur of frequencies.

This is not usually the case. The ideal is tune them as close as
possible, so there is no audible chorus effect. I suspect they
ideally can phase lock... maybe Paul has something to say about
that.

>2. Enharmonicity on the pianoforte.

Usually spelled "inharmonicity". Enharmonics are like, Ab = G#.

>(Only applies to TOP tunings 1200 cents).

The reason for tempering octaves is that you're tempering everything
else, so why waste the 'room' the octaves provide to hide error?
Are octaves in some way more special than fifths? If so, you work
out a weighting to account for this. Tenney weighting says that
3:1 "tritaves" should bear log3/log2 times the error of octaves, for
example.

-Carl

--- In tuning@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> I was thinking about TOPS, and how it does not retain octave-
> equivalence. I can think of two things that justify TOPS
acoustically:
>
> 1. The chorus effect. On a pianoforte, most keys have three
strings.
> Piano tuners purposely tune them slightly different to produce
> a "chorus effect" or block of sound. So a tone is actually
> a blur of frequencies.

This is kind of a misconception. Please read

http://www.speech.kth.se/music/5_lectures/weinreic/weinreic.html

But even if you were right, I can't see how this would justify TOsP.

> 2. Enharmonicity on the pianoforte.

You mean inharmonicity, though enharmonicity is also possible on the
pianoforte :)

(Only applies to TOP tunings
> >1200 cents). For certain physical reasons, the octave is slightly
> stretched on a pianoforte. This would justify TOP tunings > 1200
> The octave is especially stretched more at the extremes - the very
> low and the very high.

Unfortunately, 12-equal TOP is *compressed*, not stretched, in the 5-
limit.

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> The reason for tempering octaves is that you're tempering everything
> else, so why waste the 'room' the octaves provide to hide error?
> Are octaves in some way more special than fifths?

Jon is obviously far more sensative to octaves mistuned than fifths,
and to a lesser degree so am I. I thought TOP meantone was great in
some of the examples, such as the infamous Haydn quartet movement
which can only be rendered using string fonts, on pain of severe pain.
Maybe I should try TOP ennealimmal and see if anyone complains. This
has octaves tuned a nerve-wracking 0.036 cents sharp.

TOP ennealimmal generators: {133.337, 49.024}
map: [<9 15 22 26|, <0 -2 -3 -2|]

It has our new, black-magic complexity rating of 39.83, which means
you should use a 36 or 45 note DE for it. I guess. Maybe I'll try,
this misty thing isn't jelling very well.

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> Tenney weighting says that
> 3:1 "tritaves" should bear log3/log2 times the error of octaves, for
> example.

Not necessarily. More accurately, it says that a just-bearable error
for the 3:1 twelfths or "tritaves" is log3/log2 times a just-bearable
error for the octaves . . . but it also can be justified in other
ways, just as 1/4-comma meantone fulfills various optimization
criteria.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> > The reason for tempering octaves is that you're tempering
everything
> > else, so why waste the 'room' the octaves provide to hide error?
> > Are octaves in some way more special than fifths?
>
> Jon is obviously far more sensative to octaves mistuned than fifths,

You're basing this on a conversation in which you two surely were
talking about different things. It's too bad you've chosen to
pigeonhole Jon in this way when he clarified his position in a couple
of ways.

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:
> --- In tuning@yahoogroups.com, "Paul G Hjelmstad"
> <paul.hjelmstad@u...> wrote:
> > I was thinking about TOPS, and how it does not retain octave-
> > equivalence. I can think of two things that justify TOPS
> acoustically:
> >
> > 1. The chorus effect. On a pianoforte, most keys have three
> strings.
> > Piano tuners purposely tune them slightly different to produce
> > a "chorus effect" or block of sound. So a tone is actually
> > a blur of frequencies.

>
> This is kind of a misconception. Please read
>
> http://www.speech.kth.se/music/5_lectures/weinreic/weinreic.html
>
> But even if you were right, I can't see how this would justify TOsP.

Okay, throw out 1.!

>
> > 2. Enharmonicity on the pianoforte.
>
> You mean inharmonicity, though enharmonicity is also possible on
the
> pianoforte :)
>
> (Only applies to TOP tunings
> > >1200 cents). For certain physical reasons, the octave is slightly
> > stretched on a pianoforte. This would justify TOP tunings > 1200
> > The octave is especially stretched more at the extremes - the
very
> > low and the very high.
>
> Unfortunately, 12-equal TOP is *compressed*, not stretched, in the
5-
> limit.

So, throw out 2.! How is 12-equal TOP in the 7-limit?

-Paul

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:

> Unfortunately, 12-equal TOP is *compressed*, not stretched, in the 5-
> limit.

Ah, but 5-limit meantone is *stretched*, as we discovered to our
horror and dismay.

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:

> You're basing this on a conversation in which you two surely were
> talking about different things. It's too bad you've chosen to
> pigeonhole Jon in this way when he clarified his position in a couple
> of ways.

It seemed to me he was saying the octaves in the Haydn example were
unacceptable to him, but it's now time for Jon to come in and explain
what he meant.

--- In tuning@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:

> So, throw out 2.! How is 12-equal TOP in the 7-limit?

Even more compressed. Worse yet in the 11-limit, in case anyone cares.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "Paul G Hjelmstad"
> <paul.hjelmstad@u...> wrote:
>
> > So, throw out 2.! How is 12-equal TOP in the 7-limit?
>
> Even more compressed. Worse yet in the 11-limit, in case anyone
cares.

Darn! Well, there's still 5-limit meantone, as you said...The truth
of the matter is that the middle octaves (around middle C) are hardly
stretched at all, its the outer octaves that are stretched the most.
I am going to find out just how much and will let you all know.

I have been told that the ear doesn't discern much of tuning
differences less that 10 cents. However, I would think an octave
tuned to 1209 cents would sound pretty bad (and have almost as bad a
beat frequency as a major third). I would like to listen to this,
can Tonalsoft detune octaves?

Paul

--- In tuning@yahoogroups.com, "Paul G Hjelmstad"

/tuning/topicId_52348.html#52363

<> I have been told that the ear doesn't discern much of tuning
> differences less that 10 cents.

***Well, around *here* there are supermen who can hear pennies,
without even looking for "spare change..."

J. Pehrson

>> Tenney weighting says that
>> 3:1 "tritaves" should bear log3/log2 times the error of octaves, for
>> example.
>
>Not necessarily. More accurately, it says that a just-bearable error
>for the 3:1 twelfths or "tritaves" is log3/log2 times a just-bearable
>error for the octaves . . .

Does this come out of the minimax aspect, or does it apply to log
weighting in general? -C.

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> Tenney weighting says that
> >> 3:1 "tritaves" should bear log3/log2 times the error of octaves, for
> >> example.
> >
> >Not necessarily. More accurately, it says that a just-bearable error
> >for the 3:1 twelfths or "tritaves" is log3/log2 times a just-bearable
> >error for the octaves . . .
>
> Does this come out of the minimax aspect, or does it apply to log
> weighting in general? -C.

It's the particular weighting for Tenney, which weighs p/q as
log2(p*q). Other ways of weighting are possible, leading to other
things analogous to TOP, but the real trick is to find something which
can plausibly claim to be as good, or almost as good, as Tenney for
evaluation of consonance and which also leads to a norm. You seem to
be interested in that and if you can think of some place to start,
tell me about it.