Comma and Interval

Hello ALL, I understand that all commas are intervals, but all intervals are NOT commas. My question is: When does an interval become a comma?
How are commas and intervals related/differentiated?

Thanks for your time, and regards,
Haresh.

--- In tuning@yahoogroups.com, "Haresh BAKSHI" <hareshbakshi@h...>
wrote:
> Hello ALL, I understand that all commas are intervals, but all
intervals are NOT commas. My question is: When does an interval
become a comma?

When it is small enough to count as one in the appropriate prime
limit.

> How are commas and intervals related/differentiated?

I draw the line at whether or not the corresponding temperament makes
sense. If that definition is adopted, we are arguing over whether or
not 27/25 is a comma just now.

-----Urspr�ngliche Nachricht-----
Von: Gene Ward Smith [mailto:gwsmith@svpal.org]
Gesendet: Montag, 12. Januar 2004 19:25
An: tuning@yahoogroups.com
Betreff: [tuning] Re: Comma and Interval

--- In tuning@yahoogroups.com, "Haresh BAKSHI" <hareshbakshi@h...>
wrote:
>> Hello ALL, I understand that all commas are intervals, but all
>> intervals are NOT commas. My question is: When does an interval
>> become a comma?

> When it is small enough to count as one in the appropriate prime
limit.

>> How are commas and intervals related/differentiated?

> I draw the line at whether or not the corresponding temperament makes
> sense. If that definition is adopted, we are arguing over whether or
> not 27/25 is a comma just now.

Hi,

I am sure that the definition of a comma has been published
in more than one book of science in tuning and temperaments.
But as I don't find it actually, we could try to define it by
ourselves again. For instance (only my proposal):

"A *Comma* is the difference between two different means to tune
an interval as a result of a chain of multiplications by
"natural" frequency ratios." For instance:

1. Calculating a chain of 12 fifth intervals by a frequency ratio
of 3/2, beginning with a "C" , than the 13th Note B# is 24 Cents
higher than the start C in its 7th octave.
Or: (3/2)12 is 24 Cents higher than 27
(the pythagorean comma).

2. Calculating a chain of two fifths intervals by a frequency
ratio of 3/2, beginning with a "C" and ending with a D
and on the other hand calculating a sub-fifth "F" to
this start C by 2/3 and a sub-minor-third "D" with 5/6 to this F -
and then setting both D into the same octave position.
The result of the first calculation is 9/8, (3/2 * 3/2 *1/2)
The result of the second one is 10/9, (2/3*5/6*2)
(the syntonic comma).

3. The difference between a chain of three major
thirds, tuned 5/4 against the octave, tuned 2/1
(the little diesis)

and so on.

Werner

--- In tuning@yahoogroups.com, "Haresh BAKSHI" <hareshbakshi@h...>
wrote:

> Hello ALL, I understand that all commas are intervals, but all
>intervals are NOT commas.

Not all intervals are commas. Normally, only JI intervals (and only
relatively small ones at that) can even *qualify* as commas. However,
said JI interval, once it's identified as a comma, may then be
tempered out -- in fact this is sometimes used as the definition of
comma, in which case all commas are actually the same interval: 0
cents. More typical, though, is to allow it to *not* be tempered out
as well. Then we can also have the case where some *other* comma is
being tempered out, which can cause the original comma to depart from
its JI in magnitude and even direction.

It gets worse -- in JI, commas are categorized under different names,
depending on size or other, sometimes obscure (e.g. accident of
history), considerations. These names include diesis, kleisma,
and . . . comma. Clearly a definitional nightmare.

To attempt to answer your question, . . . If we are strictly talking
about JI, I might consider any JI interval as small as, say 9:8 or
smaller as a potential comma for delimiting a periodicity block --
the larger end of the range here, though, would only be relevant for
pentatonic scales and the like, while for defining large JI scales,
only the smaller part of this range would be sensible to look at. If,
on the other hand, we are talking about tempering JI intervals out,
then I would set a maximum on induced error; so for a ratio in lowest
terms n/d, I would require

log(n/d)/log(n*d) < .006

for example.

But then again you may not have been talking about periodicity blocks
or tempering . . .

Sorry this subject is so lexicographically challenged :)

--- In tuning@yahoogroups.com, "Werner Mohrlok" <wmohrlok@h...> wrote:
>
> -----Ursprüngliche Nachricht-----
> Von: Gene Ward Smith [mailto:gwsmith@s...]
> Gesendet: Montag, 12. Januar 2004 19:25
> An: tuning@yahoogroups.com
> Betreff: [tuning] Re: Comma and Interval
>
>
> --- In tuning@yahoogroups.com, "Haresh BAKSHI" <hareshbakshi@h...>
> wrote:
> >> Hello ALL, I understand that all commas are intervals, but all
> >> intervals are NOT commas. My question is: When does an
interval
> >> become a comma?
>
> > When it is small enough to count as one in the appropriate
prime
> limit.
>
> >> How are commas and intervals related/differentiated?
>
> > I draw the line at whether or not the corresponding
temperament makes
> > sense. If that definition is adopted, we are arguing over
whether or
> > not 27/25 is a comma just now.
>
> Hi,
>
> I am sure that the definition of a comma has been published
> in more than one book of science in tuning and temperaments.
> But as I don't find it actually, we could try to define it by
> ourselves again. For instance (only my proposal):
>
> "A *Comma* is the difference between two different means to tune
> an interval as a result of a chain of multiplications by
> "natural" frequency ratios."

How do you define "an interval" here? According to 12-tone category?
If so, this definition may correspond to what some book of 'science'
in tuning and temperaments may say, but around here we're less
conservative, and some intervals like 2401:2400 and 3125:3072 qualify
as 'commas' even though they don't fit the 'conservative' definition.

If you don't restrict in advance how the intervals are going to be
categorized, then *any* small rational interval can qualify. If you
don't believe me, name one, and we'll give you an example of a
categorization of intervals that makes it 'the difference between two
different means to tune an interval as a result of a chain of
multiplications by "natural" frequency ratios.'

-----Urspr�ngliche Nachricht-----
Von: wallyesterpaulrus [mailto:paul@stretch-music.com]
Gesendet: Montag, 12. Januar 2004 20:37
An: tuning@yahoogroups.com
Betreff: [tuning] Re: Comma and Interval

--- In tuning@yahoogroups.com, "Werner Mohrlok" <wmohrlok@h...> wrote:
>
> -----Urspr�ngliche Nachricht-----
> Von: Gene Ward Smith [mailto:gwsmith@s...]
> Gesendet: Montag, 12. Januar 2004 19:25
> An: tuning@yahoogroups.com
> Betreff: [tuning] Re: Comma and Interval
>
>
> --- In tuning@yahoogroups.com, "Haresh BAKSHI" <hareshbakshi@h...>
> wrote:
> >> Hello ALL, I understand that all commas are intervals, but all
> >> intervals are NOT commas. My question is: When does an
interval
> >> become a comma?
>
> > When it is small enough to count as one in the appropriate
prime
> limit.
>
> >> How are commas and intervals related/differentiated?
>
> > I draw the line at whether or not the corresponding
temperament makes
> > sense. If that definition is adopted, we are arguing over
whether or
> > not 27/25 is a comma just now.
>
> > Hi,
> >
> > I am sure that the definition of a comma has been published
> > in more than one book of science in tuning and temperaments.
> > But as I don't find it actually, we could try to define it by
> > ourselves again. For instance (only my proposal):
> >
> > "A *Comma* is the difference between two different means to tune
> > an interval as a result of a chain of multiplications by
> > "natural" frequency ratios."

> How do you define "an interval" here? According to 12-tone category?
> If so, this definition may correspond to what some book of 'science'
> in tuning and temperaments may say, but around here we're less
> conservative, and some intervals like 2401:2400 and 3125:3072 qualify
> as 'commas' even though they don't fit the 'conservative' definition.

> If you don't restrict in advance how the intervals are going to be
> categorized, then *any* small rational interval can qualify. If you
> don't believe me, name one, and we'll give you an example of a
> categorization of intervals that makes it 'the difference between two
> different means to tune an interval as a result of a chain of
> multiplications by "natural" frequency ratios.'

I have no problem with your progressive interpretation
of a "comma".

Best,
Werner

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:
> --- In tuning@yahoogroups.com, "Haresh BAKSHI" <hareshbakshi@h...>
> wrote:
>
> > Hello ALL, I understand that all commas are intervals, but all
> >intervals are NOT commas.
>
> Not all intervals are commas. Normally, only JI intervals (and only
> relatively small ones at that) can even *qualify* as commas.
However,
> said JI interval, once it's identified as a comma, may then be
> tempered out

If so, we typically exclude *powers* of other commas from counting as
commas. So (81/80)^2 = 6561/6400 is not included as a separate comma
in this sense, because tempering it out is the same thing as
tempering 81/80 out.