Yahoo Tuning Groups Ultimate Backup tuning Approximating (or replacing) tempered intervals with harmonic ratios.

Hi everyone. I was wondering if you could help me answer a question.
It never made sense to me that the tempered intervals in the 12 tone
tuning system correspond to the irrational numbers 2 to power of n/12
(n = 1,2,...12). This is because these numbers are by definition
aperiodic and therefore without a tonic (e.g. Pythagoras' proof that
sqrt 2 or flat 5th interval has no rational number corresponding to
it). I suspected that, context permitting, these intervals might
instead correspond to harmonics in the rarefied upper registers. I
developed a loose technique for finding such numbers: given 2 to power
of n/12, take 2 to power of N + n/12, N = 0,1,2,3...If the whole-part
of the number is odd then ignore the remainder and divide by 2 to
power of N. Or else we can round off to the nearest odd number. It
seemes that the higher N, the closer the approx. to the original
tempered interval (their difference becoming smaller).
E.G. For minor 3rd, N = 4 gives 16:19 or 19/16=1.1875 (much closer
than 6/5=1.2, and with the further advantage that 16 is octave
equivalent to the tonic-fundamental while 5 is not). N=9 gives
609/512=1.1894...and so on.
My question is: Is there some way of determining some realistic limit,
either in our hearing or even in the waves themselves? In other words,
perhaps we can only perceive to a few decimal places so that the
intervals are rational and periodic after all?
Excuse my ignorance cause I'm only a beginner here (and as a jazz
guitarist I obviously cannot test things for myself), but this
question seems to apply to any tuning system whatsoever. Thanks folks
and hope to hear from you.

🔗Aaron Wolf <wolftune@...>

Dear Rick,

Download yourself a free or cheap tone generator program. You really
should be able to test these things. It isn't that complex.

Yes, the limits of this are related to human psychology and
physiology. Sure, *some* of the 12ET notes are close enough to
harmonic to have essentially the same effect on listeners, but that
doesn't prove anything.

Take a jazz guitarist who plays through an amp with a chorus effect...
The chorus detunes the note slightly and creates a busy, slightly
wavering sound. That has a certain level of just a little tension for
the listener, and that's pleasing in some ways. And in that case,
there is a decent range of a few cents in which no particular precise
point has any change for the listener because it is all wavering and
detuned anyway.

There is simply a lot more to this than theoretical numbers and
tuning. If you TELL people VERBALLY that they should listen for a
particular sound and it is the main resolution, they'll relate
everything to that and try to predict when it is coming and assess the
best they can when it comes. And if they believe that it arrived,
then they'll feel a sense of resolution. This is completely
independent of whether the same sound did in fact arrive or what
happens with the whole rest of the musical context.

The question is what expectations and predictions listeners develop
unconsciously when they are not told anything. And this relates to
MANY factors, from being trained musicians or not, stylistically
biased or experienced, having heard the piece before, the recording
before, and yes it also includes things like harmonic support.

After you take into account all these related factors you will realize
that maybe there are more important questions than finding some very
complex rational number to justify 12ET intellectually. There's very
little chance that listeners experience it that way.

Best,
Aaron Wolf

--- In tuning@yahoogroups.com, "rick_ballan" <rick_ballan@...> wrote:
>
> Hi everyone. I was wondering if you could help me answer a question.
> It never made sense to me that the tempered intervals in the 12 tone
> tuning system correspond to the irrational numbers 2 to power of n/12
> (n = 1,2,...12). This is because these numbers are by definition
> aperiodic and therefore without a tonic (e.g. Pythagoras' proof that
> sqrt 2 or flat 5th interval has no rational number corresponding to
> it). I suspected that, context permitting, these intervals might
> instead correspond to harmonics in the rarefied upper registers. I
> developed a loose technique for finding such numbers: given 2 to power
> of n/12, take 2 to power of N + n/12, N = 0,1,2,3...If the whole-part
> of the number is odd then ignore the remainder and divide by 2 to
> power of N. Or else we can round off to the nearest odd number. It
> seemes that the higher N, the closer the approx. to the original
> tempered interval (their difference becoming smaller).
> E.G. For minor 3rd, N = 4 gives 16:19 or 19/16=1.1875 (much closer
> than 6/5=1.2, and with the further advantage that 16 is octave
> equivalent to the tonic-fundamental while 5 is not). N=9 gives
> 609/512=1.1894...and so on.
> My question is: Is there some way of determining some realistic limit,
> either in our hearing or even in the waves themselves? In other words,
> perhaps we can only perceive to a few decimal places so that the
> intervals are rational and periodic after all?
> Excuse my ignorance cause I'm only a beginner here (and as a jazz
> guitarist I obviously cannot test things for myself), but this
> question seems to apply to any tuning system whatsoever. Thanks folks
> and hope to hear from you.
>

🔗Carl Lumma <carl@...>

Numbers can approximate other numbers. In this case, 2^7/12
can approximate 3/2. The ear is plenty forgiving to hear it
this way, though it also detects it isn't quite right by hearing
beats.

> I suspected that, context permitting, these intervals might
> instead correspond to harmonics in the rarefied upper registers.

The intervals correspond to harmonics in the low
registers... 5/4, 3/2, etc. That's why we use 12-ET; because
it gives approximations to just intonation.

> My question is: Is there some way of determining some realistic
> limit, either in our hearing or even in the waves themselves? In
> other words, perhaps we can only perceive to a few decimal places
> so that the intervals are rational and periodic after all?

Auditory neuroscience is now probing the limits of periodicity
detection in the ear/brain. If the period of a waveform given
by a fraction n/d is n*d, then my experience tells me the ear
starts to give up the ghost with n*d ~ 70.

-Carl

🔗Michael Sheiman <djtrancendance@...>

A side question...about how much difference can you get away with left of right of a low-numbered fraction IE 3/2, 4/5...can you get before your brain sees two different notes and/or roughness?

Suppose we actually take "advantage" of the fact the ears seems to round notes up or down to the nearest fraction to make extra notes both about the limit ABOVE and BELOW the perfect fraction thus doubling the number of possible notes in the scale while still fooling the brain that they are "in tune"?

Note I'm not familiar with n*d notation...and I'm guessing n * d = x number of cents

For example, suppose the magic ratio is 1.02 and it's inverse, 0.98/ And you could take everything in a 7 note Just Intonation times those two ratios to create a 14-note scale.

Any ideas how this would/could work?

Carl Lumma <carl@...> wrote: --- In tuning@yahoogroups.com, "rick_ballan" <rick_ballan@...> wrote:
>
> Hi everyone. I was wondering if you could help me answer a question.
> It never made sense to me that the tempered intervals in the 12 tone
> tuning system correspond to the irrational numbers 2 to power of
> n/12 (n = 1,2,...12). This is because these numbers are by
> definition aperiodic and therefore without a tonic

Numbers can approximate other numbers. In this case, 2^7/12
can approximate 3/2. The ear is plenty forgiving to hear it
this way, though it also detects it isn't quite right by hearing
beats.

> I suspected that, context permitting, these intervals might
> instead correspond to harmonics in the rarefied upper registers.

The intervals correspond to harmonics in the low
registers... 5/4, 3/2, etc. That's why we use 12-ET; because
it gives approximations to just intonation.

-Carl

🔗Charles Lucy <lucy@...>

It does seem that our brains could be "centering" onto familiar intervals, yet in your posting you are assuming that those "roundings" are to (JI) small integer ratios.

I accept that beating is "created" by small integer ratios, yet to assume that "ears seem to round notes up or down" to integer ratios or even that "musical harmonics" are to be found at these same intervals is questionable.

Maybe "perfect" musical intervals and "harmonics" are elsewhere.

On 26 May 2008, at 18:29, Michael Sheiman wrote:

> A side question...about how much difference can you get away with > left of right of a low-numbered fraction IE 3/2, 4/5...can you get > before your brain sees two different notes and/or roughness?
>
> Suppose we actually take "advantage" of the fact the ears seems > to round notes up or down to the nearest fraction to make extra > notes both about the limit ABOVE and BELOW the perfect fraction thus > doubling the number of possible notes in the scale while still > fooling the brain that they are "in tune"?
>
> Note I'm not familiar with n*d notation...and I'm guessing n * d = x > number of cents
>
> For example, suppose the magic ratio is 1.02 and it's inverse, > 0.98/ And you could take everything in a 7 note Just Intonation > times those two ratios to create a 14-note scale.
>
> Any ideas how this would/could work?
>
> Carl Lumma <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "rick_ballan" <rick_ballan@...> wrote:
> >
> > Hi everyone. I was wondering if you could help me answer a question.
> > It never made sense to me that the tempered intervals in the 12 tone
> > tuning system correspond to the irrational numbers 2 to power of
> > n/12 (n = 1,2,...12). This is because these numbers are by
> > definition aperiodic and therefore without a tonic
>
> Numbers can approximate other numbers. In this case, 2^7/12
> can approximate 3/2. The ear is plenty forgiving to hear it
> this way, though it also detects it isn't quite right by hearing
> beats.
>
> > I suspected that, context permitting, these intervals might
> > instead correspond to harmonics in the rarefied upper registers.
>
> The intervals correspond to harmonics in the low
> registers... 5/4, 3/2, etc. That's why we use 12-ET; because
> it gives approximations to just intonation.
>
> > My question is: Is there some way of determining some realistic
> > limit, either in our hearing or even in the waves themselves? In
> > other words, perhaps we can only perceive to a few decimal places
> > so that the intervals are rational and periodic after all?
>
> Auditory neuroscience is now probing the limits of periodicity
> detection in the ear/brain. If the period of a waveform given
> by a fraction n/d is n*d, then my experience tells me the ear
> starts to give up the ghost with n*d ~ 70.
>
> -Carl
>
>
>
>
Charles Lucy
lucy@...

- Promoting global harmony through LucyTuning -

for information on LucyTuning go to:
http://www.lucytune.com

For LucyTuned Lullabies go to:
http://www.lullabies.co.uk

🔗Mike Battaglia <battaglia01@...>

🔗Aaron Wolf <wolftune@...>

Yeah, good points. I find as I play around with my TPX keyboard that
switching from using 7/6 m-thirds to 6/5 m-thirds can totally modify
my own expectations, and after a "shifting" period I can see one or
the other as the more expected in-tune version. And each note
in-between or above or below can itself do that as well to a
reasonable degree.

In fact, when I play around with diatonic stuff, those changes seem
small, but after about two minutes of playing around with intervals
under 30 cents, I find that 70 cent intervals sound ENORMOUS, like a
skip actually.

Leonard Meyer in "Emotion and Meaning in Music" claims that all tuning
systems have a tendency over time to equalize, to fill in any larger
steps until a sense of complete continuity exists...

-AW

--- In tuning@yahoogroups.com, "Mike Battaglia" <battaglia01@...> wrote:
>
> Aaron is right here -- the ear has to be "guided" to these intervals
before
> it hears them as separate intervals in their own right sometimes. We are
> very used to C-Eb being a 6/5 minor third because we hear it played
that way
> on every other instrument than on a piano or a guitar. I would say
that if
> you were to play an entire piece and replace all of the 6/5 minor thirds
> with 19/16 minor thirds and the 5/4 major thirds with 32/19 major
thirds,
> you would hear the 19/16 as being an out of tune 6/5 and the 32/19
as being
> an out of tune 5/4, unless you somehow managed to create the distinction
> between 19/16 and 6/5 in people's minds. It is possible to use these
> intervals that way, but as that music hasn't been so explored yet, I
think
> that the current human condition is to hear those intervals as 6/5
and 5/4
> intervals.
>
> Although I think it is noteworthy that my 12tet ears often hear
major thirds
> that are slightly sharp as sounding very "in tune" and having a
character
> all their own that is quite distinct from 5/4 -- so you might be onto
> something. It depends.
>
> On Mon, May 26, 2008 at 4:37 AM, rick_ballan <rick_ballan@...>
> wrote:
>
> > Hi everyone. I was wondering if you could help me answer a question.
> > It never made sense to me that the tempered intervals in the 12 tone
> > tuning system correspond to the irrational numbers 2 to power of n/12
> > (n = 1,2,...12). This is because these numbers are by definition
> > aperiodic and therefore without a tonic (e.g. Pythagoras' proof that
> > sqrt 2 or flat 5th interval has no rational number corresponding to
> > it). I suspected that, context permitting, these intervals might
> > instead correspond to harmonics in the rarefied upper registers. I
> > developed a loose technique for finding such numbers: given 2 to power
> > of n/12, take 2 to power of N + n/12, N = 0,1,2,3...If the whole-part
> > of the number is odd then ignore the remainder and divide by 2 to
> > power of N. Or else we can round off to the nearest odd number. It
> > seemes that the higher N, the closer the approx. to the original
> > tempered interval (their difference becoming smaller).
> > E.G. For minor 3rd, N = 4 gives 16:19 or 19/16=1.1875 (much closer
> > than 6/5=1.2, and with the further advantage that 16 is octave
> > equivalent to the tonic-fundamental while 5 is not). N=9 gives
> > 609/512=1.1894...and so on.
> > My question is: Is there some way of determining some realistic limit,
> > either in our hearing or even in the waves themselves? In other words,
> > perhaps we can only perceive to a few decimal places so that the
> > intervals are rational and periodic after all?
> > Excuse my ignorance cause I'm only a beginner here (and as a jazz
> > guitarist I obviously cannot test things for myself), but this
> > question seems to apply to any tuning system whatsoever. Thanks folks
> > and hope to hear from you.
> >
> >
> >
>

🔗Carl Lumma <carl@...>

Michael Sheiman <djtrancendance@...> wrote:

> A side question...about how much difference can you get away
> with left of right of a low-numbered fraction IE 3/2, 4/5...can
> you get before your brain sees two different notes and/or
> roughness?

As you detune a pure interval (let's say 5:4), first you get
beating. When the beating reaches a certain rate it becomes
roughness. In the case of simple ratios like 5:4, you brain's
"virtual pitch" detector will tolerate a good deal of roughness
before giving up on the interpretation. In 12-ET 5:4 is quite
rough but is still unmistakably 5:4. The approximation will
definitely break (unless you hit the listener over the head
with "priming") when we get on the other side of a harmonic
entropy maximum from 5:4. The maximum-to-maximum width around
a ratio like 5:4 is proportional to 1/sqrt(5*4). The maxima
occur near the mediants between 5:4 and its neighbors in a
farey series of suitable order. For example, the mediant of
6:5 and 5:4 is 11:9, and this is indeed very near the harmonic
entropy maximum South of 5:4. So intervals flatter than 11:9
are more likely to be heard as 6:5 approximations than 5:4
approximations.

> Suppose we actually take "advantage" of the fact the ears
> seems to round notes up or down to the nearest fraction to make
> extra notes both about the limit ABOVE and BELOW the perfect
> fraction thus doubling the number of possible notes in the scale
> while still fooling the brain that they are "in tune"?

Yeah, just imagine!

You've discovered temperament.

> Note I'm not familiar with n*d notation...and I'm guessing
> n * d = x number of cents

No. For a ratio n/d (as in Numerator and Denominator),
it is the product n * d. For example 5:4 -> 20.

> For example, suppose the magic ratio is 1.02 and it's
> inverse, 0.98/ And you could take everything in a 7 note
> Just Intonation times those two ratios to create a 14-note
> scale.

The "magic ratio" will depend on the interval being
approximated (remember the width around a ratio is proportional
to 1/sqrt(n*d)). Nevertheless, you could say you don't
want errors over 3 cents and find temperaments with lower
error than that.

-Carl

🔗Carl Lumma <carl@...>

🔗Michael Sheiman <djtrancendance@...>

---Yeah, just imagine! You've discovered temperament.

Hehehe...
Well I already knew 12TET works because of this phenomena IE it's not JI but rather" just close enough to JI to work". Though I didn't know temperament (as it seems now) means more than "toward a clear tone" but rather "within a defined amount of space from a clear tone". In that case, I'm guessing, 7-tone JI is essentially either "non-tempered" because it does not contain any estimations?

I was just wondering if someone in history had deliberately tried to make a scale out of "maximum acceptable temperament errors" on both sides of an ideal tone from JI, not just one or the other, in the quest to achieve a higher degree of tonal freedom without sounding odd to most listeners.

If I have your article right that would mean around 3 cents and/or when you hit the other side of the "entropy maximum" or area of maximum dissonance.

Carl Lumma <carl@...> wrote: Michael Sheiman <djtrancendance@...> wrote:

Yeah, just imagine!

You've discovered temperament.

> Note I'm not familiar with n*d notation...and I'm guessing
> n * d = x number of cents

No. For a ratio n/d (as in Numerator and Denominator),
it is the product n * d. For example 5:4 -> 20.

> For example, suppose the magic ratio is 1.02 and it's
> inverse, 0.98/ And you could take everything in a 7 note
> Just Intonation times those two ratios to create a 14-note
> scale.

-Carl

🔗Carl Lumma <carl@...>

Michael wrote...

> Hehehe...
> Well I already knew 12TET works because of this phenomena IE
> it's not JI but rather" just close enough to JI to work". Though
> I didn't know temperament (as it seems now) means more than
> "toward a clear tone" but rather "within a defined amount of
> space from a clear tone". In that case, I'm guessing, 7-tone JI
> is essentially either "non-tempered" because it does not contain
> any estimations?

The point of temperament is to get pitches to serve double,
triple, etc. duty in terms of JI. It's a bit confusing because
people think of "temperaments" as tunings, but what they really
are is mappings that define the double/triple/etc. duties.

I don't know what this 7-note JI is you keep mentioning, but
you can get JI to approximate things. The best-known example
is Pythagorean tuning. 12 pure 5ths will give you usable
major 3rds in the same spots that 12-ET does. But the idea
is that making the 5ths flat 2 cents improves these 3rds more
than it damages the 5ths. That's the idea behind tempered
tunings. But either way the mapping is the same.

Two pure 4ths will give you something that sounds like a 7:4,
but if we make the 4ths flat we can improve the 7:4 more than
we damage the 4ths. Or so the theory goes.

> I was just wondering if someone in history had deliberately
> tried to make a scale out of "maximum acceptable temperament
> errors" on both sides of an ideal tone from JI, not just one
> or the other, in the quest to achieve a higher degree of tonal
> freedom without sounding odd to most listeners.

Yes some people do advocate high-error temperaments. Herman
Miller is one person who's used them in inspiring ways (check
out his music!). Easley Blackwood also did some nice work
in this space with 15-ET.

> If I have your article right that would mean around 3 cents
> and/or when you hit the other side of the "entropy maximum"
> or area of maximum dissonance.

In most cases you'd hit 3 cents first. You seemed to be asking
about a constant fudge factor so I gave that as an example.
You could use something I'd call "inverse weighted error", which
would temper simple intervals more than complex ones (since their
basins of attraction, if you will, are wider). But what is
usually done is "weighted error", which tempers simple intervals
less. This is because even though their basins are wider they
are also much deeper. So a given amount of tuning error increases
the dissonance more (because the walls of basin are steeper) and
the idea is to minimize the total amount of dissonance doled out
to the listener.

Hope that makes sense,

-Carl

🔗Kraig Grady <kraiggrady@...>

harmonics exist, pluck your guitar.
It is a gestalt question also. if i show you an 89 degree angle you are going to see right angleness in it.
While often Harmonics do not land at exactly whole number ratios, they vary no more than people keeping a beat. No two beats are ever exactly whole number relations if we start looking at the most precise measurements we have, but we all hear it as.
but yes the implication that only whole numbers are the only type of acoustical phenomenon is mistaken.

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Charles Lucy wrote:
>
> It does seem that our brains could be "centering" onto familiar > intervals, yet in your posting you are assuming that those "roundings" > are to (JI) small integer ratios.
>
>
> I accept that beating is "created" by small integer ratios, yet to > assume that "ears seem to round notes up or down" to integer ratios or > even that "musical harmonics" are to be found at these same intervals > is questionable.
>
> Maybe "perfect" musical intervals and "harmonics" are elsewhere.
>
>
>
>
> On 26 May 2008, at 18:29, Michael Sheiman wrote:
>
>> A side question...about how much difference can you get away with >> left of right of a low-numbered fraction IE 3/2, 4/5...can you get >> before your brain sees two different notes and/or roughness?
>>
>> Suppose we actually take "advantage" of the fact the ears seems to >> round notes up or down to the nearest fraction to make extra notes >> both about the limit ABOVE and BELOW the perfect fraction thus >> doubling the number of possible notes in the scale while still >> fooling the brain that they are "in tune"?
>>
>> Note I'm not familiar with n*d notation...and I'm guessing n * d = x >> number of cents
>>
>> For example, suppose the magic ratio is 1.02 and it's inverse, >> 0.98/ And you could take everything in a 7 note Just Intonation >> times those two ratios to create a 14-note scale.
>>
>> Any ideas how this would/could work?
>>
>> */Carl L! umma < ;carl@... <mailto:carl@...>>/* wrote:
>>
>> --- In tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com>,
>> "rick_ballan" <rick_ballan@...> wrote:
>> >
>> > Hi everyone. I was wondering if you could help me answer a question.
>> > It never made sense to me that the tempered intervals in the 12 tone
>> > tuning system correspond to the irrational numbers 2 to power of
>> > n/12 (n = 1,2,...12). This is because these numbers are by
>> > definition aperiodic and therefore without a tonic
>>
>> Numbers can approximate other numbers. In this case, 2^7/12
>> can approximate 3/2. The ear is plenty forgiving to hear it
>> this way, though it also detects it isn't quite right by hearing
>> beats.
>>
>> > I suspected that, context permitting, these intervals might
>> > instead correspond to harmonics in the rarefied upper registers.
>>
>> The intervals correspond to harmonics in the low >> registers... 5/4, 3/2, etc. That's why we use 12-ET; because
>> it gives approximations to just intonation.
>>
>> > My question is: Is there some way of determining some realistic
>> > limit, either in our hearing or even in the waves themselves? In
>> > other words, perhaps we can only perceive to a few decimal places
>> > so that the intervals are rational and periodic after all?
>>
>> Auditory neuroscience is now probing the limits of periodicity
>> detection in the ear/brain. If the period of a waveform given
>> by a fraction n/d is n*d, then my experience tells me the ear
>> starts to give up the ghost with n*d ~ 70.
>>
>> -Carl
>>
>>
>>
>
> Charles Lucy
> lucy@... <mailto:lucy@...>
>
> - Promoting global harmony through LucyTuning -
>
> for information on LucyTuning go to:
> http://www.lucytune.com <http://www.lucytune.com>
>
> For LucyTuned Lullabies go to:
> http://www.lullabies.co.uk <http://www.lullabies.co.uk>
>
>
>
>

🔗Kraig Grady <kraiggrady@...>

a 6/5 has never sounded like a minor note to me, it is too happy. the 19/16 sounds much better and closer to what me are used to the minor third saying

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Mike Battaglia wrote:
>
> Aaron is right here -- the ear has to be "guided" to these intervals > before it hears them as separate intervals in their own right > sometimes. We are very used to C-Eb being a 6/5 minor third because we > hear it played that way on every other instrument than on a piano or a > guitar. I would say that if you were to play an entire piece and > replace all of the 6/5 minor thirds with 19/16 minor thirds and the > 5/4 major thirds with 32/19 major thirds, you would hear the 19/16 as > being an out of tune 6/5 and the 32/19 as being an out of tune 5/4, > unless you somehow managed to create the distinction between 19/16 and > 6/5 in people's minds. It is possible to use these intervals that way, > but as that music hasn't been so explored yet, I think that the > current human condition is to hear those intervals as 6/5 and 5/4 > intervals.
>
> Although I think it is noteworthy that my 12tet ears often hear major > thirds that are slightly sharp as sounding very "in tune" and having a > character all their own that is quite distinct from 5/4 -- so you > might be onto something. It depends.
>
> On Mon, May 26, 2008 at 4:37 AM, rick_ballan <rick_ballan@... > <mailto:rick_ballan@...>> wrote:
>
> Hi everyone. I was wondering if you could help me answer a question.
> It never made sense to me that the tempered intervals in the 12 tone
> tuning system correspond to the irrational numbers 2 to power of n/12
> (n = 1,2,...12). This is because these numbers are by definition
> aperiodic and therefore without a tonic (e.g. Pythagoras' proof that
> sqrt 2 or flat 5th interval has no rational number corresponding to
> it). I suspected that, context permitting, these intervals might
> instead correspond to harmonics in the rarefied upper registers. I
> developed a loose technique for finding such numbers: given 2 to power
> of n/12, take 2 to power of N + n/12, N = 0,1,2,3...If the whole-part
> of the number is odd then ignore the remainder and divide by 2 to
> power of N. Or else we can round off to the nearest odd number. It
> seemes that the higher N, the closer the approx. to the original
> tempered interval (their difference becoming smaller).
> E.G. For minor 3rd, N = 4 gives 16:19 or 19/16=1.1875 (much closer
> than 6/5=1.2, and with the further advantage that 16 is octave
> equivalent to the tonic-fundamental while 5 is not). N=9 gives
> 609/512=1.1894...and so on.
> My question is: Is there some way of determining some realistic limit,
> either in our hearing or even in the waves themselves? In other words,
> perhaps we can only perceive to a few decimal places so that the
> intervals are rational and periodic after all?
> Excuse my ignorance cause I'm only a beginner here (and as a jazz
> guitarist I obviously cannot test things for myself), but this
> question seems to apply to any tuning system whatsoever. Thanks folks
> and hope to hear from you.
>
>
>

🔗Kraig Grady <kraiggrady@...>

That gaps will be filled in is correct but not equally, More along the lines of MOS and related ways it appears. The pentatonic was not filled in with equal steps, neither was the diatonic, especially by ear. It is only "rational" approaches that lead to such processes.

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Aaron Wolf wrote:
>
>
> Leonard Meyer in "Emotion and Meaning in Music" claims that all tuning
> systems have a tendency over time to equalize, to fill in any larger
> steps until a sense of complete continuity exists...
>
> -AW
>
>

> >
>

>
> > > and hope to hear from you.
> > >
> > >
> > >
> >
>
>

🔗Aaron Wolf <wolftune@...>

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
>
> a 6/5 has never sounded like a minor note to me, it is too happy. the
> 19/16 sounds much better and closer to what me are used to the minor
> third saying
>

Me too. 19 sounds like the "dark", "sad" and such that we are used to
minor being. And 7/6 sounds like the minor bluesy sound of blues and
jazz, and 6/5 is sort happy I guess.

But I was intrigued by Leonard Meyers' claim in Emotion and Meaning In
Music that explaining the minor by mathematical ratios is erroneous.
His hypothesis (50 years ago to be certain) was that the minor feel of
minor is based more on the dynamic flexibility of the mode and the
contrasts within it and a few other melodic aspects that he thinks may
have much more prominance than the issue of harmonic tuning...
I think he's right to a degree, that these ideas can't be explained in
isolated out-of-context ratios.

-AW

>
> /^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
> _'''''''_ ^North/Western Hemisphere:
> North American Embassy of Anaphoria Island <http://anaphoria.com/>
>
> _'''''''_ ^South/Eastern Hemisphere:
> Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>
>
> ',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
>
>
>
>
> Mike Battaglia wrote:
> >
> > Aaron is right here -- the ear has to be "guided" to these intervals
> > before it hears them as separate intervals in their own right
> > sometimes. We are very used to C-Eb being a 6/5 minor third
because we
> > hear it played that way on every other instrument than on a piano
or a
> > guitar. I would say that if you were to play an entire piece and
> > replace all of the 6/5 minor thirds with 19/16 minor thirds and the
> > 5/4 major thirds with 32/19 major thirds, you would hear the 19/16 as
> > being an out of tune 6/5 and the 32/19 as being an out of tune 5/4,
> > unless you somehow managed to create the distinction between 19/16
and
> > 6/5 in people's minds. It is possible to use these intervals that
way,
> > but as that music hasn't been so explored yet, I think that the
> > current human condition is to hear those intervals as 6/5 and 5/4
> > intervals.
> >
> > Although I think it is noteworthy that my 12tet ears often hear major
> > thirds that are slightly sharp as sounding very "in tune" and
having a
> > character all their own that is quite distinct from 5/4 -- so you
> > might be onto something. It depends.
> >
> > On Mon, May 26, 2008 at 4:37 AM, rick_ballan <rick_ballan@...
> > <mailto:rick_ballan@...>> wrote:
> >
> > Hi everyone. I was wondering if you could help me answer a
question.
> > It never made sense to me that the tempered intervals in the
12 tone
> > tuning system correspond to the irrational numbers 2 to power
of n/12
> > (n = 1,2,...12). This is because these numbers are by definition
> > aperiodic and therefore without a tonic (e.g. Pythagoras'
proof that
> > sqrt 2 or flat 5th interval has no rational number
corresponding to
> > it). I suspected that, context permitting, these intervals might
> > instead correspond to harmonics in the rarefied upper registers. I
> > developed a loose technique for finding such numbers: given 2
to power
> > of n/12, take 2 to power of N + n/12, N = 0,1,2,3...If the
whole-part
> > of the number is odd then ignore the remainder and divide by 2 to
> > power of N. Or else we can round off to the nearest odd number. It
> > seemes that the higher N, the closer the approx. to the original
> > tempered interval (their difference becoming smaller).
> > E.G. For minor 3rd, N = 4 gives 16:19 or 19/16=1.1875 (much closer
> > than 6/5=1.2, and with the further advantage that 16 is octave
> > equivalent to the tonic-fundamental while 5 is not). N=9 gives
> > 609/512=1.1894...and so on.
> > My question is: Is there some way of determining some
realistic limit,
> > either in our hearing or even in the waves themselves? In
other words,
> > perhaps we can only perceive to a few decimal places so that the
> > intervals are rational and periodic after all?
> > Excuse my ignorance cause I'm only a beginner here (and as a jazz
> > guitarist I obviously cannot test things for myself), but this
> > question seems to apply to any tuning system whatsoever.
Thanks folks
> > and hope to hear from you.
> >
> >
> >
>

🔗Aaron Wolf <wolftune@...>

Kraig,

Meyer's claim isn't that the ear does this. His claim is that
cultures over time (generations maybe) tend to little by little evolve
their systems to fill in the gaps. Partly this happens simply due to
creative people looking for ways to deviate from existing norms. But
his claim (not mine - not sure I agree) is that this can end when
something equal is reached because there is no longer a sense of gap...

-AW

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
>
>
> That gaps will be filled in is correct but not equally, More along the
> lines of MOS and related ways it appears. The pentatonic was not filled
> in with equal steps, neither was the diatonic, especially by ear. It is
> only "rational" approaches that lead to such processes.
>
> /^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
> _'''''''_ ^North/Western Hemisphere:
> North American Embassy of Anaphoria Island <http://anaphoria.com/>
>
> _'''''''_ ^South/Eastern Hemisphere:
> Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>
>
> ',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
>
>
>
>
> Aaron Wolf wrote:
> >
> >
> > Leonard Meyer in "Emotion and Meaning in Music" claims that all tuning
> > systems have a tendency over time to equalize, to fill in any larger
> > steps until a sense of complete continuity exists...
> >
> > -AW
> >
> >
>
> > >
> >
>
> >
> > > > and hope to hear from you.
> > > >
> > > >
> > > >
> > >
> >
> >
>

🔗Kraig Grady <kraiggrady@...>

As i mentioned in my other post, being of 'stacked thirds' we accept as a 'variation' of the major, which 'rationally' we try to explain as an inversion. but empirically does not seem to be.

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Aaron Wolf wrote:
>
> --- In tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com>, Kraig > Grady <kraiggrady@...> wrote:
> >
> > a 6/5 has never sounded like a minor note to me, it is too happy. the
> > 19/16 sounds much better and closer to what me are used to the minor
> > third saying
> >
>
> Me too. 19 sounds like the "dark", "sad" and such that we are used to
> minor being. And 7/6 sounds like the minor bluesy sound of blues and
> jazz, and 6/5 is sort happy I guess.
>
> But I was intrigued by Leonard Meyers' claim in Emotion and Meaning In
> Music that explaining the minor by mathematical ratios is erroneous.
> His hypothesis (50 years ago to be certain) was that the minor feel of
> minor is based more on the dynamic flexibility of the mode and the
> contrasts within it and a few other melodic aspects that he thinks may
> have much more prominance than the issue of harmonic tuning...
> I think he's right to a degree, that these ideas can't be explained in
> isolated out-of-context ratios.
>
> -AW
>
> >
> > /^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
> > _'''''''_ ^North/Western Hemisphere:
> > North American Embassy of Anaphoria Island <http://anaphoria.com/ > <http://anaphoria.com/>>
> >
> > _'''''''_ ^South/Eastern Hemisphere:
> > Austronesian Outpost of Anaphoria > <http://anaphoriasouth.blogspot.com/ > <http://anaphoriasouth.blogspot.com/>>
> >
> > ',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
> >
> >
> >
> >
> > Mike Battaglia wrote:
> > >
> > > Aaron is right here -- the ear has to be "guided" to these intervals
> > > before it hears them as separate intervals in their own right
> > > sometimes. We are very used to C-Eb being a 6/5 minor third
> because we
> > > hear it played that way on every other instrument than on a piano
> or a
> > > guitar. I would say that if you were to play an entire piece and
> > > replace all of the 6/5 minor thirds with 19/16 minor thirds and the
> > > 5/4 major thirds with 32/19 major thirds, you would hear the 19/16 as
> > > being an out of tune 6/5 and the 32/19 as being an out of tune 5/4,
> > > unless you somehow managed to create the distinction between 19/16
> and
> > > 6/5 in people's minds. It is possible to use these intervals that
> way,
> > > but as that music hasn't been so explored yet, I think that the
> > > current human condition is to hear those intervals as 6/5 and 5/4
> > > intervals.
> > >
> > > Although I think it is noteworthy that my 12tet ears often hear major
> > > thirds that are slightly sharp as sounding very "in tune" and
> having a
> > > character all their own that is quite distinct from 5/4 -- so you
> > > might be onto something. It depends.
> > >
> > > On Mon, May 26, 2008 at 4:37 AM, rick_ballan <rick_ballan@...
> > > <mailto:rick_ballan@...>> wrote:
> > >
> > > Hi everyone. I was wondering if you could help me answer a
> question.
> > > It never made sense to me that the tempered intervals in the
> 12 tone
> > > tuning system correspond to the irrational numbers 2 to power
> of n/12
> > > (n = 1,2,...12). This is because these numbers are by definition
> > > aperiodic and therefore without a tonic (e.g. Pythagoras'
> proof that
> > > sqrt 2 or flat 5th interval has no rational number
> corresponding to
> > > it). I suspected that, context permitting, these intervals might
> > > instead correspond to harmonics in the rarefied upper registers. I
> > > developed a loose technique for finding such numbers: given 2
> to power
> > > of n/12, take 2 to power of N + n/12, N = 0,1,2,3...If the
> whole-part
> > > of the number is odd then ignore the remainder and divide by 2 to
> > > power of N. Or else we can round off to the nearest odd number. It
> > > seemes that the higher N, the closer the approx. to the original
> > > tempered interval (their difference becoming smaller).
> > > E.G. For minor 3rd, N = 4 gives 16:19 or 19/16=1.1875 (much closer
> > > than 6/5=1.2, and with the further advantage that 16 is octave
> > > equivalent to the tonic-fundamental while 5 is not). N=9 gives
> > > 609/512=1.1894...and so on.
> > > My question is: Is there some way of determining some
> realistic limit,
> > > either in our hearing or even in the waves themselves? In
> other words,
> > > perhaps we can only perceive to a few decimal places so that the
> > > intervals are rational and periodic after all?
> > > Excuse my ignorance cause I'm only a beginner here (and as a jazz
> > > guitarist I obviously cannot test things for myself), but this
> > > question seems to apply to any tuning system whatsoever.
> Thanks folks
> > > and hope to hear from you.
> > >
> > >
> > >
> >
>
>

🔗Charles Lucy <lucy@...>

It works for me Carl.

On 26 May 2008, at 19:56, Carl Lumma wrote:

> > Maybe "perfect" musical intervals and "harmonics" are elsewhere.
>
> >Yeah! Maybe they're found in the scale you've named after
> >yourself, and patented and trademarked! What are the odds?
>

> What's your stake Carl?
>

> Is this another case of sour grapes, N.I.H. (not invented here)
>

> Or was that the old prune's idea of a wind up?
>

BTW (In colloquial West Midlands English) What's the odds? means "What is the difference?"

It's an old usage, but you still sometimes hear it used in Worcestershire, Gloucestershire, Hereford area.

best wishes Carl;-)
Charles Lucy
lucy@...

- Promoting global harmony through LucyTuning -

for information on LucyTuning go to:
http://www.lucytune.com

For LucyTuned Lullabies go to:
http://www.lullabies.co.uk

🔗Kraig Grady <kraiggrady@...>

I like Meyers work very much BTW.
This is pretty much what Moments of Symmetry is about as well as the others that have been mentioned. Yasser was Erv's springboard as well as it was for Kornerup, although there is a debate as to who came first. Viggo Bruns Algorithm does a good job with this too in a related and rational way.
Cris Forster points out that Meyer laid out the triadic diamond in 1927. While Novaro mapped out the 7 limit ( plus filling the gaps to make a constant structure) in the same year, Meyers diagram more resembles Partch's in terms of visualization.

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Aaron Wolf wrote:
>
> Kraig,
>
> Meyer's claim isn't that the ear does this. His claim is that
> cultures over time (generations maybe) tend to little by little evolve
> their systems to fill in the gaps. Partly this happens simply due to
> creative people looking for ways to deviate from existing norms. But
> his claim (not mine - not sure I agree) is that this can end when
> something equal is reached because there is no longer a sense of gap...
>
> -AW
>
> --- In tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com>, Kraig > Grady <kraiggrady@...> wrote:
> >
> >
> > That gaps will be filled in is correct but not equally, More along the
> > lines of MOS and related ways it appears. The pentatonic was not filled
> > in with equal steps, neither was the diatonic, especially by ear. It is
> > only "rational" approaches that lead to such processes.
> >
> > /^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
> > _'''''''_ ^North/Western Hemisphere:
> > North American Embassy of Anaphoria Island <http://anaphoria.com/ > <http://anaphoria.com/>>
> >
> > _'''''''_ ^South/Eastern Hemisphere:
> > Austronesian Outpost of Anaphoria > <http://anaphoriasouth.blogspot.com/ > <http://anaphoriasouth.blogspot.com/>>
> >
> > ',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
> >
> >
> >
> >
> > Aaron Wolf wrote:
> > >
> > >
> > > Leonard Meyer in "Emotion and Meaning in Music" claims that all tuning
> > > systems have a tendency over time to equalize, to fill in any larger
> > > steps until a sense of complete continuity exists...
> > >
> > > -AW
> > >
> > >
> >
> > > >
> > >
> >
> > >
> > > > > and hope to hear from you.
> > > > >
> > > > >
> > > > >
> > > >
> > >
> > >
> >
>
>

🔗Kraig Grady <kraiggrady@...>

P.S.> I do think that the ear does do this with culture being a 'collective' ear

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

🔗Cameron Bobro <misterbobro@...>

> Carl Lumma <carl@...> wrote:

>
> The intervals correspond to harmonics in the low
> registers... 5/4, 3/2, etc. That's why we use 12-ET; because
> it gives approximations to just intonation.

12-tET works because it most cleverly and excellently does NOT give
approximations to Just intonation.

The octave is pure. 3/2 and 4/3 are damn near pure. The third and
sixths are vague and uncommited as far as Just identities, and that
is why they work- because they can be interpreted in so many ways,
either in the imagination, or literally when performed by flexibly-
pitched instruments. Same thing with seconds and sevenths- they're
quite accurate when they appear in terms of fourths and fifths or
9/8, and in other contexts, altered tones etc. they're vague enough
to do many duties.

Equating 400 cents with 5/4 is just plain boloney. Sometimes 400
cents suggests 5/4, sometimes 400 cents on paper IS 5/4 in actual
physical performance. Sometimes 400 cents is doing a fine job as
81/64, or suggesting 9/7- or anything inbetween.

> Auditory neuroscience is now probing the limits of periodicity
> detection in the ear/brain. If the period of a waveform given
> by a fraction n/d is n*d, then my experience tells me the ear
> starts to give up the ghost with n*d ~ 70.
>
> -Carl
>

11/9 sounds Just to me. Of course I don't know if that's due to
periodicity detection, coincident partials, etc. or some combination
of things. I of course go for the combination of things approach and
find Terhardt's to be the most reasonable "school" but anyway.

-Cameron Bobro

Approximating (or replacing) tempered intervals with harmonic ratios.

🔗rick_ballan <rick_ballan@...>

🔗Aaron Wolf <wolftune@...>

🔗Carl Lumma <carl@...>

🔗Michael Sheiman <djtrancendance@...>

🔗Charles Lucy <lucy@...>

🔗Mike Battaglia <battaglia01@...>

🔗Aaron Wolf <wolftune@...>

🔗Carl Lumma <carl@...>

🔗Carl Lumma <carl@...>

🔗Michael Sheiman <djtrancendance@...>

🔗Carl Lumma <carl@...>

🔗Kraig Grady <kraiggrady@...>

🔗Kraig Grady <kraiggrady@...>

🔗Kraig Grady <kraiggrady@...>

🔗Kraig Grady <kraiggrady@...>

🔗Aaron Wolf <wolftune@...>

🔗Aaron Wolf <wolftune@...>

🔗Kraig Grady <kraiggrady@...>

🔗Charles Lucy <lucy@...>

🔗Kraig Grady <kraiggrady@...>

🔗Kraig Grady <kraiggrady@...>

🔗Cameron Bobro <misterbobro@...>

🔗Carl Lumma <carl@...>

🔗Cameron Bobro <misterbobro@...>