Yahoo Tuning Groups Ultimate Backup tuning 612-edo schisma as interval measurement unit

Hi Gene,

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "monz" <monz@> wrote:
>
> > As a theorist who has used 612-edo as a measurment unit,
> > an analogue of cents, have you ever used a distinctive name
> > for one degree of this tuning when used this way?
>
> I had a distinctive abbreviation I sometimes used:
> 5/4 was 197 sk, 3/2 was 358 sk, and so forth.

Hmm, i like that ... very compact.
I think "sk" is OK as a term.

> > If not, i suppose we could call it a "schisma" ...
> > note that i spell this with the "c", as opposed to the
> > JI skhisma with the "k" which was proposed by Ellis,
> > who AFAIK coined the term for use in tuning theory.
>
> "Schisma" is usually taken to mean 32805/32768 also.

That's exactly what Ellis meant by "skhisma". He deliberately
spelled it with a "k" instead of a "c" because he wanted
to avoid the religious connotation of the Great Schism.

> But Ellis had another definition: a fifth minus 700 cents.

That is a "grad".

> We have:
>
> cents(32805/32768) = 1.95372

skhisma

> cents(3/2) - 700 = 1.95500

grad

> 1200/612 = 1.96078

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗Mark Rankin <markrankin95511@yahoo.com>

Ladies and Gentlemen,

I have read the three recent posts concerning the
microtone known as the skhisma, the first by Monz, the
second by Gene, and the third by ThreeSixesInA- Row
Clark. I like Gene's list containing the traditional
skhisma and two similar-sized approximations. It
reminded me that there is a realm of tiny intervals
which when compared to, say, a 1/1 starting tone,
should not really be called "musical" intervals at
all.

I looked in three books to see if they could add
anything to the list or throw any light on the
subject: Herman Helmholtz's "On the Sensation of
Tone" (pages 230-238); J. Murray Barbour's "Tuning and
Temperament" (page 131-132); and Alain Danielou's
"Tableau Comparatif Des Intervalles Musicaux" (page
2).

Helmholtz's book is the oldest. I'm not sure when the
original german edition was published, but the second
edition of Ellis's english translation of "On the
Sensation of Tone" came out in 1885, and I believe
Ellis was the inventer of the Cents system. A
skhisma, equaling 1.954 cents, is mentioned on page
431 (with the skh- spelling shown here), and a
Skhismic Temperament appears on page 435. Ellis made
a distinction between rounded-off whole cents which
added-up evenly, such as 112 cents and 1088 cents
equaling 1200 cents, which he called Cyclic Cents, and
the more precise Just Cents with rounded-off decimal
remainders, such as the 50.4 cents and 1148.9 cents
from his "Expression of Just Intonation in the Cycle
of 1200" table, which do not quite add up correctly.

Barbour's book was first published in 1951. Under the
heading "Theory of Multiple Division" it presents, on
page 131, "the following excellent series of octave
divisions: 3, 5, 7, 12, 19, 31, 34, 53, 87, 118, 559,
612....". Notice that neither the word "Equal" nor
the abbreviation "EDO" appear in reference to the
divisions of the octave. The spelling "skhisma" does
not appear in Barbour's book, but "schisma" appears
seven times, first on page 64, where we're told that
the interval of the schisma is "the difference between
the syntonic and the ditonic commas". The second
mention, on page 80, describes the schisma as "about
two cents". The final five references to skhisma
aren't of much interest, and are all found two pages
beyond the page numbers listed for them in the index.

Near the top of page 131, Barbour's book offers a
reference to a division of the octave that generates a
microtone that is smaller than one 612th of an octave.
It is the 665-Tone Per Octave division which,
counting the starting tone, yields the famous "number
of man", 666, sometimes known as "the number of the
beast".

Danielou's book, a remarkable 145 page "Comparative
Table of Musical Intervals", was published in french
in India in 1958. An abbreviated version of the table
between 1/1 and 3.618 cents looks as follows:

Intervals Ratios Factors Cents

1/1 1/1 .0000

665th 5th 3^665/2^1054 .096

1025/1024 5^2*41/2^10 1.690

1024/1023 2^10/3*11*31 1.692

359th 5th 3^359/2^569 1.834

Skhisma 32805/32768 5*3^8/2^15 1.9537

513/512 3^*19/2^9 3.3781

53rd 5th 3^53/2^83 3.618

This set of miniature microtones might be called
Nanotones.

Mark Rankin

--- monz <monz@tonalsoft.com> wrote:

> Hi Gene,
>
>
> As a theorist who has used 612-edo as a measurment
> unit,
> an analogue of cents, have you ever used a
> distinctive name
> for one degree of this tuning when used this way?
>
> If not, i suppose we could call it a "schisma" ...
> note
> that i spell this with the "c", as opposed to the JI
> skhisma
> with the "k" which was proposed by Ellis, who AFAIK
> coined
> the term for use in tuning theory.
>
>
> -monz
> http://tonalsoft.com
> Tonescape microtonal music software
>
>
>
>
>

____________________________________________________________________________________
TV dinner still cooling?
Check out "Tonight's Picks" on Yahoo! TV.
http://tv.yahoo.com/

🔗Ozan Yarman <ozanyarman@ozanyarman.com>

Nanotonality... that's an interesting concept!

----- Original Message -----
From: "Mark Rankin" <markrankin95511@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: 25 Mart 2007 Pazar 4:51
Subject: Re: [tuning] 612-edo schisma as interval measurement unit

> Ladies and Gentlemen,
>
> I have read the three recent posts concerning the
> microtone known as the skhisma, the first by Monz, the
> second by Gene, and the third by ThreeSixesInA- Row
> Clark. I like Gene's list containing the traditional
> skhisma and two similar-sized approximations. It
> reminded me that there is a realm of tiny intervals
> which when compared to, say, a 1/1 starting tone,
> should not really be called "musical" intervals at
> all.
>
> I looked in three books to see if they could add
> anything to the list or throw any light on the
> subject: Herman Helmholtz's "On the Sensation of
> Tone" (pages 230-238); J. Murray Barbour's "Tuning and
> Temperament" (page 131-132); and Alain Danielou's
> "Tableau Comparatif Des Intervalles Musicaux" (page
> 2).
>
> Helmholtz's book is the oldest. I'm not sure when the
> original german edition was published, but the second
> edition of Ellis's english translation of "On the
> Sensation of Tone" came out in 1885, and I believe
> Ellis was the inventer of the Cents system. A
> skhisma, equaling 1.954 cents, is mentioned on page
> 431 (with the skh- spelling shown here), and a
> Skhismic Temperament appears on page 435. Ellis made
> a distinction between rounded-off whole cents which
> added-up evenly, such as 112 cents and 1088 cents
> equaling 1200 cents, which he called Cyclic Cents, and
> the more precise Just Cents with rounded-off decimal
> remainders, such as the 50.4 cents and 1148.9 cents
> from his "Expression of Just Intonation in the Cycle
> of 1200" table, which do not quite add up correctly.
>
>
> Barbour's book was first published in 1951. Under the
> heading "Theory of Multiple Division" it presents, on
> page 131, "the following excellent series of octave
> divisions: 3, 5, 7, 12, 19, 31, 34, 53, 87, 118, 559,
> 612....". Notice that neither the word "Equal" nor
> the abbreviation "EDO" appear in reference to the
> divisions of the octave. The spelling "skhisma" does
> not appear in Barbour's book, but "schisma" appears
> seven times, first on page 64, where we're told that
> the interval of the schisma is "the difference between
> the syntonic and the ditonic commas". The second
> mention, on page 80, describes the schisma as "about
> two cents". The final five references to skhisma
> aren't of much interest, and are all found two pages
> beyond the page numbers listed for them in the index.
>
> Near the top of page 131, Barbour's book offers a
> reference to a division of the octave that generates a
> microtone that is smaller than one 612th of an octave.
> It is the 665-Tone Per Octave division which,
> counting the starting tone, yields the famous "number
> of man", 666, sometimes known as "the number of the
> beast".
>
> Danielou's book, a remarkable 145 page "Comparative
> Table of Musical Intervals", was published in french
> in India in 1958. An abbreviated version of the table
> between 1/1 and 3.618 cents looks as follows:
>
> Intervals Ratios Factors Cents
>
> 1/1 1/1 .0000
>
> 665th 5th 3^665/2^1054 .096
>
> 1025/1024 5^2*41/2^10 1.690
>
> 1024/1023 2^10/3*11*31 1.692
>
> 359th 5th 3^359/2^569 1.834
>
> Skhisma 32805/32768 5*3^8/2^15 1.9537
>
> 513/512 3^*19/2^9 3.3781
>
> 53rd 5th 3^53/2^83 3.618
>
>
>
> This set of miniature microtones might be called
> Nanotones.
>
> Mark Rankin
>
>
>

🔗Gene Ward Smith <genewardsmith@sbcglobal.net>

🔗Mark Rankin <markrankin95511@yahoo.com>

Ladies and Gentlemen, Again,

I'm confused! The table of microtones below printed
out wrong a few minutes ago (all crammed together)
when I printed out a copy of the table for my records,
but now, when I'm printing out an apology/explanation
of the foul up, the table way down below from the
original email (not the simple list of cents) is
printed out correctly! Can anyone explain this
development?

Mark Rankin

I am replying to my own post to say that I created a
decent table starting with 1/1 and followed by 7 tiny
microtones, the largest being 3.618 cents.
Unfortunately, as I had feared, the e-mail software
crammed all the information in the horizontal rows
together. The information is all there, but it needs
to be separated horizontally, and there were some
blank spaces in parts of the table that make sorting
out the horizontal components even trickier. I wonder
if re-doing the entire table in Microsoft Word would
give back the origional table? Even the four headers
were crammed together! What *is* legible is the list
of cents:

Cents

.000

.096

1.690

1.692

1.834

1.9573

3.3781

3.618

Mark Rankin

--- Mark Rankin <markrankin95511@yahoo.com> wrote:

> Ladies and Gentlemen,
>
> I have read the three recent posts concerning the
> microtone known as the skhisma, the first by Monz,
> the
> second by Gene, and the third by ThreeSixesInA- Row
> Clark. I like Gene's list containing the
> traditional
> skhisma and two similar-sized approximations. It
> reminded me that there is a realm of tiny intervals
> which when compared to, say, a 1/1 starting tone,
> should not really be called "musical" intervals at
> all.
>
> I looked in three books to see if they could add
> anything to the list or throw any light on the
> subject: Herman Helmholtz's "On the Sensation of
> Tone" (pages 230-238); J. Murray Barbour's "Tuning
> and
> Temperament" (page 131-132); and Alain Danielou's
> "Tableau Comparatif Des Intervalles Musicaux" (page
> 2).
>
> Helmholtz's book is the oldest. I'm not sure when
> the
> original german edition was published, but the
> second
> edition of Ellis's english translation of "On the
> Sensation of Tone" came out in 1885, and I believe
> Ellis was the inventer of the Cents system. A
> skhisma, equaling 1.954 cents, is mentioned on page
> 431 (with the skh- spelling shown here), and a
> Skhismic Temperament appears on page 435. Ellis
> made
> a distinction between rounded-off whole cents which
> added-up evenly, such as 112 cents and 1088 cents
> equaling 1200 cents, which he called Cyclic Cents,
> and
> the more precise Just Cents with rounded-off decimal
> remainders, such as the 50.4 cents and 1148.9 cents
> from his "Expression of Just Intonation in the Cycle
> of 1200" table, which do not quite add up correctly.
>
>
>
> Barbour's book was first published in 1951. Under
> the
> heading "Theory of Multiple Division" it presents,
> on
> page 131, "the following excellent series of octave
> divisions: 3, 5, 7, 12, 19, 31, 34, 53, 87, 118,
> 559,
> 612....". Notice that neither the word "Equal" nor
> the abbreviation "EDO" appear in reference to the
> divisions of the octave. The spelling "skhisma"
> does
> not appear in Barbour's book, but "schisma" appears
> seven times, first on page 64, where we're told that
> the interval of the schisma is "the difference
> between
> the syntonic and the ditonic commas". The second
> mention, on page 80, describes the schisma as "about
> two cents". The final five references to schisma
> aren't of much interest, and are all found two pages
> beyond the page numbers listed for them in the
> index.
>
> Near the top of page 131, Barbour's book offers a
> reference to a division of the octave that generates
> a
> microtone that is smaller than one 612th of an
> octave.
> It is the 665-Tone Per Octave division which,
> counting the starting tone, yields the famous
> "number
> of man", 666, sometimes known as "the number of the
> beast".
>
> Danielou's book, a remarkable 145 page "Comparative
> Table of Musical Intervals", was published in french
> in India in 1958. An abbreviated version of the
> table
> between 1/1 and 3.618 cents looks as follows:
>
> Intervals Ratios Factors Cents
>
> 1/1 1/1 .0000
>
> 665th 5th 3^665/2^1054 .096
>
> 1025/1024 5^2*41/2^10 1.690
>
> 1024/1023 2^10/3*11*31 1.692
>
> 359th 5th 3^359/2^569 1.834
>
> Skhisma 32805/32768 5*3^8/2^15 1.9537
>
> 513/512 3^*19/2^9 3.3781
>
> 53rd 5th 3^53/2^83 3.618
>
>
>
> This set of miniature microtones might be called
> Nanotones.
>
> Mark Rankin
>
>
>
>
>
> --- monz <monz@tonalsoft.com> wrote:
>
> > Hi Gene,
> >
> >
> > As a theorist who has used 612-edo as a measurment
> > unit,
> > an analogue of cents, have you ever used a
> > distinctive name
> > for one degree of this tuning when used this way?
> >
> > If not, i suppose we could call it a "schisma" ...
> > note
> > that i spell this with the "c", as opposed to the
> JI
> > skhisma
> > with the "k" which was proposed by Ellis, who
> AFAIK
> > coined
> > the term for use in tuning theory.
> >
> >
> > -monz
> > http://tonalsoft.com
> > Tonescape microtonal music software
> >
> >
> >
> >
> >
>
>
>
>
>
>
____________________________________________________________________________________
> TV dinner still cooling?
> Check out "Tonight's Picks" on Yahoo! TV.
> http://tv.yahoo.com/
>

____________________________________________________________________________________
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🔗Mark Rankin <markrankin95511@yahoo.com>

Lassies and Gents,

The table has improved, but now it hasw run off the
right margin a little. I give up!

Mark Rankin

--- Mark Rankin <markrankin95511@yahoo.com> wrote:

> Ladies and Gentlemen, Again,
>
> I'm confused! The table of microtones below printed
> out wrong a few minutes ago (all crammed together)
> when I printed out a copy of the table for my
> records,
> but now, when I'm printing out an
> apology/explanation
> of the foul up, the table way down below from the
> original email (not the simple list of cents) is
> printed out correctly! Can anyone explain this
> development?
>
> Mark Rankin
>
>
> I am replying to my own post to say that I created a
> decent table starting with 1/1 and followed by 7
> tiny
> microtones, the largest being 3.618 cents.
> Unfortunately, as I had feared, the e-mail software
> crammed all the information in the horizontal rows
> together. The information is all there, but it
> needs
> to be separated horizontally, and there were some
> blank spaces in parts of the table that make sorting
> out the horizontal components even trickier. I
> wonder
> if re-doing the entire table in Microsoft Word would
> give back the origional table? Even the four
> headers
> were crammed together! What *is* legible is the
> list
> of cents:
>
> Cents
>
> .000
>
> .096
>
> 1.690
>
> 1.692
>
> 1.834
>
> 1.9573
>
> 3.3781
>
> 3.618
>
> Mark Rankin
>
>
>
> --- Mark Rankin <markrankin95511@yahoo.com> wrote:
>
> > Ladies and Gentlemen,
> >
> > I have read the three recent posts concerning the
> > microtone known as the skhisma, the first by Monz,
> > the
> > second by Gene, and the third by ThreeSixesInA-
> Row
> > Clark. I like Gene's list containing the
> > traditional
> > skhisma and two similar-sized approximations. It
> > reminded me that there is a realm of tiny
> intervals
> > which when compared to, say, a 1/1 starting tone,
> > should not really be called "musical" intervals at
> > all.
> >
> > I looked in three books to see if they could add
> > anything to the list or throw any light on the
> > subject: Herman Helmholtz's "On the Sensation of
> > Tone" (pages 230-238); J. Murray Barbour's "Tuning
> > and
> > Temperament" (page 131-132); and Alain Danielou's
> > "Tableau Comparatif Des Intervalles Musicaux"
> (page
> > 2).
> >
> > Helmholtz's book is the oldest. I'm not sure when
> > the
> > original german edition was published, but the
> > second
> > edition of Ellis's english translation of "On the
> > Sensation of Tone" came out in 1885, and I believe
> > Ellis was the inventer of the Cents system. A
> > skhisma, equaling 1.954 cents, is mentioned on
> page
> > 431 (with the skh- spelling shown here), and a
> > Skhismic Temperament appears on page 435. Ellis
> > made
> > a distinction between rounded-off whole cents
> which
> > added-up evenly, such as 112 cents and 1088 cents
> > equaling 1200 cents, which he called Cyclic Cents,
> > and
> > the more precise Just Cents with rounded-off
> decimal
> > remainders, such as the 50.4 cents and 1148.9
> cents
> > from his "Expression of Just Intonation in the
> Cycle
> > of 1200" table, which do not quite add up
> correctly.
> >
> >
> >
> > Barbour's book was first published in 1951. Under
> > the
> > heading "Theory of Multiple Division" it
> presents,
> > on
> > page 131, "the following excellent series of
> octave
> > divisions: 3, 5, 7, 12, 19, 31, 34, 53, 87, 118,
> > 559,
> > 612....". Notice that neither the word "Equal"
> nor
> > the abbreviation "EDO" appear in reference to the
> > divisions of the octave. The spelling "skhisma"
> > does
> > not appear in Barbour's book, but "schisma"
> appears
> > seven times, first on page 64, where we're told
> that
> > the interval of the schisma is "the difference
> > between
> > the syntonic and the ditonic commas". The second
> > mention, on page 80, describes the schisma as
> "about
> > two cents". The final five references to schisma
> > aren't of much interest, and are all found two
> pages
> > beyond the page numbers listed for them in the
> > index.
> >
> > Near the top of page 131, Barbour's book offers a
> > reference to a division of the octave that
> generates
> > a
> > microtone that is smaller than one 612th of an
> > octave.
> > It is the 665-Tone Per Octave division which,
> > counting the starting tone, yields the famous
> > "number
> > of man", 666, sometimes known as "the number of
> the
> > beast".
> >
> > Danielou's book, a remarkable 145 page
> "Comparative
> > Table of Musical Intervals", was published in
> french
> > in India in 1958. An abbreviated version of the
> > table
> > between 1/1 and 3.618 cents looks as follows:
> >
> > Intervals Ratios Factors
> Cents
> >
> > 1/1 1/1
> .0000
> >
> > 665th 5th 3^665/2^1054 .096
>
> >
> > 1025/1024 5^2*41/2^10 1.690
>
> >
> > 1024/1023 2^10/3*11*31 1.692
> >
> > 359th 5th 3^359/2^569 1.834
> >
> > Skhisma 32805/32768 5*3^8/2^15
> 1.9537
> >
> > 513/512 3^*19/2^9
> 3.3781
> >
> > 53rd 5th 3^53/2^83 3.618
> >
> >
> >
> > This set of miniature microtones might be called
> > Nanotones.
> >
> > Mark Rankin
> >
> >
> >
> >
> >
> > --- monz <monz@tonalsoft.com> wrote:
> >
> > > Hi Gene,
>
=== message truncated ===

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🔗monz <monz@tonalsoft.com>

Hi Mark,

--- In tuning@yahoogroups.com, Mark Rankin <markrankin95511@...> wrote:
>
> Lassies and Gents,
>
> The table has improved, but now it hasw run off the
> right margin a little. I give up!

If you're viewing the tuning list on the Yahoo web interface,
what you're seeing is a combination of Yahoo's wonderful
"features", one of which deletes "unnecessary" spaces, and
the other of which forces readers to view your post in
Times New Roman font.

On the upper right of the page, there's a link called "Option",
and if you click on it, you'll see another link called
"Use Fixed Width Font". This will change the font of your
post to Courier (or something like it), which should allow
your tables and diagrams to appear correctly. I'm not sure
if it retains the "unnecessary" spaces.

If you're not viewing on the web interface, and instead
are talking about changes in the posts which you receive
by email, i can't help you there because i don't read the
list that way.

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗monz <monz@tonalsoft.com>

Hi Mark,

--- In tuning@yahoogroups.com, Mark Rankin <markrankin95511@...> wrote:

> I looked in three books to see if they could add
> anything to the list or throw any light on the
> subject: Herman Helmholtz's "On the Sensation of
> Tone" (pages 230-238); J. Murray Barbour's "Tuning and
> Temperament" (page 131-132); and Alain Danielou's
> "Tableau Comparatif Des Intervalles Musicaux" (page
> 2).
>
> Helmholtz's book is the oldest. I'm not sure when the
> original german edition was published, but the second
> edition of Ellis's english translation of "On the
> Sensation of Tone" came out in 1885, and I believe
> Ellis was the inventer of the Cents system. A
> skhisma, equaling 1.954 cents, is mentioned on page
> 431 (with the skh- spelling shown here), and a
> Skhismic Temperament appears on page 435. Ellis made
> a distinction between rounded-off whole cents which
> added-up evenly, such as 112 cents and 1088 cents
> equaling 1200 cents, which he called Cyclic Cents, and
> the more precise Just Cents with rounded-off decimal
> remainders, such as the 50.4 cents and 1148.9 cents
> from his "Expression of Just Intonation in the Cycle
> of 1200" table, which do not quite add up correctly.

Helmholtz first published his treatise in 1863.
I have this under the appropriate year in my chronology
"A Century of New Music in Vienna":

http://tonalsoft.com/enc/v/vienna.htm

>> The 41-year-old physicist Hermann Helmholtz, teaching
>> at Heidelberg University, publishes his groundbreaking
>> study of musical acoustics, _Die Lehre von den
>> Tonempfindungen als physiologische Grundlage für die
>> Theorie der Musik_ ['treatise on sound-perception as a
>> physiological basis for the theory of music', published
>> later in English translation with "Die Lehre von den
>> Tonempfindungen" rendered as _On the Sensations of Tone_].

Note that Helmholtz used the convenient superparticular
ratio 887:886 to represent the skhisma, something which
was fairly common in those days. I mention it in the 2nd
definition on my Encyclopedia page:

http://tonalsoft.com/enc/s/schisma.aspx

The actual skhisma is 32805:32768 ratio. The difference
between this and 887:886 is really tiny: 2076088:2076087 ratio
= ~0.000833893 cent ( = ~1/1200 cent).

As you can see from my definition #1, historical priority
for the term goes all the way back to Philolaus in the
400s BC -- but he meant a far different interval,
a logarithmic 1/2 of the pythagorean comma (~11.73 cents).

I'm not sure if Ellis was aware of Philolaus's usage, but
Ellis was the first person to use the word to represent the
32805:32768 ratio, in the first edition of his English
translation of Helmholtz which appeared in 1875, and he
deliberately spelled it with a "k" to avoid the religious
connotation associated with the Great Schism. (I just
wrote that yesterday ... did you see my post?)

... and now i have to update that page to include
Gene's 612-edo "sk" as a 4th definition.

> Barbour's book was first published in 1951. Under the
> heading "Theory of Multiple Division" it presents, on
> page 131, "the following excellent series of octave
> divisions: 3, 5, 7, 12, 19, 31, 34, 53, 87, 118, 559,
> 612....". Notice that neither the word "Equal" nor
> the abbreviation "EDO" appear in reference to the
> divisions of the octave.

But EDOs are exactly what Barbour is referring to.
In fact, the major criticism of his book from the
microtonalist's perspective is that he was such a
strong advocate of 12-edo. In essence, his book is
nothing but a polemic arguing that 12-edo is the
"best" tuning ... altho it does have value for all
the data he published concerning other tunings.

> The spelling "skhisma" does not appear in Barbour's
> book, but "schisma" appears seven times, <snip>

Yes, despite Ellis's efforts, his unique spelling never
caught on and everyone just used the old Greek spelling.
I've been trying to be a counter-force to that. ;-)

> Near the top of page 131, Barbour's book offers a
> reference to a division of the octave that generates a
> microtone that is smaller than one 612th of an octave.
> It is the 665-Tone Per Octave division which,
> counting the starting tone, yields the famous "number
> of man", 666, sometimes known as "the number of the
> beast".

Marc Jones, with his sardonic sense of humor, dubbed
the characteristic small interval of that tuning the
"satanic comma" (a pun on the familiar "syntonic comma").
You can see his definition on my webpage:

http://tonalsoft.com/enc/j/marc-jones-defs.aspx

> Danielou's book, a remarkable 145 page "Comparative
> Table of Musical Intervals", was published in french
> in India in 1958. An abbreviated version of the table
> between 1/1 and 3.618 cents looks as follows:
>
> Intervals Ratios Factors Cents
>
> 1/1 1/1 .0000
>
> 665th 5th 3^665/2^1054 .096
>
> 1025/1024 5^2*41/2^10 1.690
> <snip>

You can see a lot more intervals in that range in my
"Big List of 11-limit Intervals":

http://tonalsoft.com/enc/i/interval-list.aspx

However, i haven't yet gotten around to adding descriptive
names to the intervals which have them. (I also haven't
yet finished formatting that table.)

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗monz <monz@tonalsoft.com>

🔗Carl Lumma <clumma@yahoo.com>

🔗Mark Rankin <markrankin95511@yahoo.com>

Monz,

Thank you very much for these three posts in a row.
I'll begin replying at the latest of them and work
backwards.

You're quite right about the originality of Ellis's
Appendix XX that he tacked onto his translation of
Helmholtz's masterpiece. I find Ellis's Appendix to
be a masterpiece in it's own right!

Your Big List of 11-limit Intervals is a treat. It
reminds me of John Chalmers's early lists which he
passed around to Ivor and Irv.

The Marc Jones-Orphon material was completely new to
me - what a cornucopia of mixing scales and
temperaments! I noticed Orphon's clever choice of D
as the starting tone because of it's being centered in
the scale's nomenclature: A B C *D* E F G.
My 88-year old friend Ernest McClain has based all of
his musical mathematics on the key of D for three or
more decades for exactly the same reason. This is the
first time I've seen anyone else do it - I applaud the
Orphon folks for it, they have some deep thinkers!
I'd like to meet them someday. Are based anywhere in
particular, or are they an internet phenomenon?

Finally, I'd like to thank you for explaining Yahoo's
"wonderful features". I'm running a four year old
version of XP Home Edition, but unfortunately I don't
get any mention of any "Option" when I click on Links
in the upper right corner of my screen, nor when I
click on the >> symbol right below Links.

Mark Rankin

--- monz <monz@tonalsoft.com> wrote:

> --- In tuning@yahoogroups.com, "monz" <monz@...>
> wrote:
>
> > Ellis was the first person to use the word to
> represent the
> > 32805:32768 ratio, in the first edition of his
> English
> > translation of Helmholtz which appeared in 1875,
>
>
> I should have noted that Ellis's use of both
> "skhisma" and
> "cent" are in his own very long appendix 20 of his
> translation
> of Helmholtz's book, so they are both Ellis's
> original work
> and not a translation of Helmholtz's.
>
>
> -monz
> http://tonalsoft.com
> Tonescape microtonal music software
>
>
>
>

____________________________________________________________________________________
Food fight? Enjoy some healthy debate
in the Yahoo! Answers Food & Drink Q&A.
http://answers.yahoo.com/dir/?link=list&sid=396545367

🔗monz <monz@tonalsoft.com>

Hi Mark,

--- In tuning@yahoogroups.com, Mark Rankin <markrankin95511@...> wrote:

> Finally, I'd like to thank you for explaining Yahoo's
> "wonderful features". I'm running a four year old
> version of XP Home Edition, but unfortunately I don't
> get any mention of any "Option" when I click on Links
> in the upper right corner of my screen, nor when I
> click on the >> symbol right below Links.

It sounds like you're talking about something that might
appear on your browser. I'm talking about a link that is
on the Yahoo webpage.

If you read the tuning list on the Yahoo web interface,
you'll see the message number near the top right corner of
the page -- not at the very top, a little under that,
actually just under the top line which forms the rectangular
outline around the message itself.

Under the message number you'll see the date. Under the
date you'll see a linked called "Show Message Option".
If you click that link, it will show the options, and
middle one is "Use Fixed Width Font".

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗Glenn Leider <GlennLeider@netzero.net>

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
>
> Hi Gene,
>
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@>
> wrote:
> >
> > --- In tuning@yahoogroups.com, "monz" <monz@> wrote:
> >
> > > As a theorist who has used 612-edo as a measurment unit,
> > > an analogue of cents, have you ever used a distinctive name
> > > for one degree of this tuning when used this way?
> >
> > I had a distinctive abbreviation I sometimes used:
> > 5/4 was 197 sk, 3/2 was 358 sk, and so forth.
>
>
> Hmm, i like that ... very compact.
> I think "sk" is OK as a term.
>
>
> > > If not, i suppose we could call it a "schisma" ...
> > > note that i spell this with the "c", as opposed to the
> > > JI skhisma with the "k" which was proposed by Ellis,
> > > who AFAIK coined the term for use in tuning theory.
> >
> > "Schisma" is usually taken to mean 32805/32768 also.
>
> That's exactly what Ellis meant by "skhisma". He deliberately
> spelled it with a "k" instead of a "c" because he wanted
> to avoid the religious connotation of the Great Schism.
>
>
> > But Ellis had another definition: a fifth minus 700 cents.
>
> That is a "grad".
>
>
> > We have:
> >
> > cents(32805/32768) = 1.95372
>
> skhisma
>
>
> > cents(3/2) - 700 = 1.95500
>
> grad
>
>
> > 1200/612 = 1.96078
>
> sk
>
>
>
> -monz
> http://tonalsoft.com
> Tonescape microtonal music software
>
=====================================
I'm happy to see this discussion on the 612-edo schisma. I would
propose calling this the "tempered schisma" with abbreviation "$".
It's derived by narrowing (or tempering) the logarithm for the
32805:32768 schisma by about 0.36%, as shown below.

For the octave, 2:1 = 1200 c = 612 $.
Each 12-edo semitone is 100 c = 51 $.

What's even nicer is that Pythagorean and Just scale intervals work
out very well when converted to tempered schismas. For example:

Perfect intervals:
Fourth - 4:3 = 498.045 c = 254 (254.003) $
Fifths - 3:2 = 701.955 c = 358 (357.997) $

Major and Minor intervals:
Minor 3rd - 6:5 = 315.641 c = 161 (160.977) $
Major 3rd - 5:4 = 386.314 c = 197 (197.020) $
Minor 6th - 8:5 = 813.686 c = 415 (414.980) $
Major 6th - 5:3 = 884.359 c = 451 (451.023) $

Very small intervals:
Schisma - 01.9537 c = 01 (00.9964) $ (Hence my "tempered" schisma)
Diasch. - 19.5526 c = 10 (09.9718) $
S comma - 21.5063 c = 11 (10.9682) $
D comma - 23.4600 c = 12 (11.9646) $
Diesis -- 41.0589 c = 21 (20.9400) $

The various tones and semitones come out nicely, too.

Use of tempered schismas virtually eliminates round-off errors that
occur when using cents when combining Pythagorean and Just
intervals. Alas, such is not the case for septimal (involving 7's)
and mean-tone intervals, and other divisions of the octave; but then
cents don't do all that well here either.

🔗Gene Ward Smith <genewardsmith@sbcglobal.net>

--- In tuning@yahoogroups.com, "Glenn Leider" <GlennLeider@...> wrote:

> I'm happy to see this discussion on the 612-edo schisma. I would
> propose calling this the "tempered schisma" with abbreviation "$".

I don't like that abbreviation, as it ought to be larger, not
smaller, than a cent.

I've suggested calling an absolute logarithmic pitch standard
the "dollar", incidentally: a pitch of d dollars is

f = 440*2^((d-69)/12) Hertz

People have been using this system anyway so it ought to have
a name.

> Use of tempered schismas virtually eliminates round-off errors that
> occur when using cents when combining Pythagorean and Just
> intervals. Alas, such is not the case for septimal (involving 7's)
> and mean-tone intervals, and other divisions of the octave; but
then
> cents don't do all that well here either.

When relative errors for 612 are compared to other
systems of about the same size, 612 does extremely
well in the 5, 7, 9 and 11 limits. There's no sense
in asking for the impossible.

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning@yahoogroups.com, "Glenn Leider" <GlennLeider@...> wrote:
>
> --- In tuning@yahoogroups.com, "monz" <monz@> wrote:
> >
> > Hi Gene,
> >
> > --- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@>
> > wrote:
> > >
> > > --- In tuning@yahoogroups.com, "monz" <monz@> wrote:
> > >
> > > > As a theorist who has used 612-edo as a measurment unit,
> > > > an analogue of cents, have you ever used a distinctive name
> > > > for one degree of this tuning when used this way?
> > >
> > > I had a distinctive abbreviation I sometimes used:
> > > 5/4 was 197 sk, 3/2 was 358 sk, and so forth.
> >
> >
> > Hmm, i like that ... very compact.
> > I think "sk" is OK as a term.
> >
> >
> > > > If not, i suppose we could call it a "schisma" ...
> > > > note that i spell this with the "c", as opposed to the
> > > > JI skhisma with the "k" which was proposed by Ellis,
> > > > who AFAIK coined the term for use in tuning theory.
> > >
> > > "Schisma" is usually taken to mean 32805/32768 also.
> >
> > That's exactly what Ellis meant by "skhisma". He deliberately
> > spelled it with a "k" instead of a "c" because he wanted
> > to avoid the religious connotation of the Great Schism.
> >
> >
> > > But Ellis had another definition: a fifth minus 700 cents.
> >
> > That is a "grad".
> >
> >
> > > We have:
> > >
> > > cents(32805/32768) = 1.95372
> >
> > skhisma
> >
> >
> > > cents(3/2) - 700 = 1.95500
> >
> > grad
> >
> >
> > > 1200/612 = 1.96078
> >
> > sk
> >
> >
> > -monz
> > http://tonalsoft.com
> > Tonescape microtonal music software
> >
> =====================================
> I'm happy to see this discussion on the 612-edo schisma. I would
> propose calling this the "tempered schisma" with abbreviation "$".
> It's derived by narrowing (or tempering) the logarithm for the
> 32805:32768 schisma by about 0.36%, as shown below.
>
> For the octave, 2:1 = 1200 c = 612 $.
> Each 12-edo semitone is 100 c = 51 $.
>
> What's even nicer is that Pythagorean and Just scale intervals work
> out very well when converted to tempered schismas. For example:
>
> Perfect intervals:
> Fourth - 4:3 = 498.045 c = 254 (254.003) $
> Fifths - 3:2 = 701.955 c = 358 (357.997) $
>
> Major and Minor intervals:
> Minor 3rd - 6:5 = 315.641 c = 161 (160.977) $
> Major 3rd - 5:4 = 386.314 c = 197 (197.020) $
> Minor 6th - 8:5 = 813.686 c = 415 (414.980) $
> Major 6th - 5:3 = 884.359 c = 451 (451.023) $
>
> Very small intervals:
> Schisma - 01.9537 c = 01 (00.9964) $ (Hence my "tempered" schisma)
> Diasch. - 19.5526 c = 10 (09.9718) $
> S comma - 21.5063 c = 11 (10.9682) $
> D comma - 23.4600 c = 12 (11.9646) $
> Diesis -- 41.0589 c = 21 (20.9400) $
>
> The various tones and semitones come out nicely, too.
>
> Use of tempered schismas virtually eliminates round-off errors that
> occur when using cents when combining Pythagorean and Just
> intervals. Alas, such is not the case for septimal (involving 7's)
> and mean-tone intervals, and other divisions of the octave; but
then
> cents don't do all that well here either.

An increasingly inportant measuring unit (first suggested by Gene
Ward Smith, also a multiple of 12, and *not* in Monz's list) that
will virtually eliminate JI and 1/4-comma meantone rounding errors
through the 27-limit is a single degree of 2460-EDO, ~0.4 cents (or
alternatively, 1/233rd of an apotome). Dave Keenan and I call this
a "mina" (short for "schismina"), and we've used it extensively for
calculating the number of units a Sagittal accidental will alter.
(Note that 2460 is also 41-EDO subdivided 60-fold.) You'll need this
much precision to make a distinction between three 5-commas (80:81)
and a large 13-diesis (26:27), which differ by 2 minas
(255879:256000), or between a 5-comma-plus-7-comma (35:36) and a
medium 13-diesis (1024:1053), which differ by 1 mina (4095:4096).

But if you don't need that much precision, then the 612-EDO schisma
would be an excellent choice for a unit of measure.

--George

🔗Ozan Yarman <ozanyarman@ozanyarman.com>

🔗Glenn Leider <GlennLeider@netzero.net>

🔗monz <monz@tonalsoft.com>

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
>
> Hi George,
>
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@> wrote:
>
> > An increasingly inportant measuring unit (first suggested
> > by Gene Ward Smith, also a multiple of 12, and *not* in
> > Monz's list) that will virtually eliminate JI and 1/4-comma
> > meantone rounding errors through the 27-limit is a single
> > degree of 2460-EDO, ~0.4 cents (or alternatively, 1/233rd
> > of an apotome). Dave Keenan and I call this a "mina"
> > (short for "schismina"), and we've used it extensively for
> > calculating the number of units a Sagittal accidental will
> > alter.
>
> Thanks for that.
>
> I've added it to my list:
>
> http://tonalsoft.com/enc/u/unit-of-interval-measurement.aspx
>
> and made a new page just for it:
>
> http://tonalsoft.com/enc/m/mina.aspx
>
> -monz
> http://tonalsoft.com
> Tonescape microtonal music software

Hi Joe,

Thanks for doing that!

You might want to correct the page to indicate that Gene *suggested*
(rather than advocated) the 2460 division to Dave & me (who use &
*advocate* it). Also, it would be good to modify the 2nd sentence to
read, "... = 2,460 equal parts, which very closely (and consistently)
approximates just intonation to the 27 limit."

--George

🔗Gene Ward Smith <genewardsmith@sbcglobal.net>

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:

> An increasingly inportant measuring unit (first suggested by Gene
> Ward Smith, also a multiple of 12, and *not* in Monz's list) that
> will virtually eliminate JI and 1/4-comma meantone rounding errors
> through the 27-limit is a single degree of 2460-EDO, ~0.4 cents (or
> alternatively, 1/233rd of an apotome). Dave Keenan and I call this
> a "mina" (short for "schismina"), and we've used it extensively for
> calculating the number of units a Sagittal accidental will alter.

Mina is an excellent unit; divisible by 12, it gets the 27-limit
diamond with an error of less than 1/6 cent, and is consistent
through the 27-limit. Since the 27 odd limit is a good cutoff
(otherwise you get 29 and 31, and even 23 is pretty much of a
stretch) I think there's a fine case for saying that 2460 level
accuracy is all the majority of microtonalists need.

It's an "atomic" system, tempering out the atom, and it also
eliminates the glum comma (|91 -12 -31>), the landscape comma,
the gauss comma (9801/9800) and on and on of course.

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> Do JI fanatics need any more precision than 2460-EDO?

I hope not. But just in case, there's always 6079-EDO (29-limit
consistent, but not a multiple of 12), which Sagittal just might be
able to notate (with a few tricks).

Beyond that, there's 11664-EDO (27-limit consistency that's better
than 2460), which is a multiple of 12 (and also 72), but I have no
idea how I would notate it (which doesn't really matter, since I
figure the above is a rhetorical question. ;-)

All kidding aside, Sagittal *actually does* offer more precision than
2460-EDO for those who need it. For example, 98:99 and 99:100 equate
to the same number of minas (36), but they may be notated with
separate symbols, if necessary.

Also, 45:46 and 1664:1701 (both 78 minas) differ by ~0.0226 cents,
but they can also be notated with separate symbols.

Finally, 40960:41553 and 6561:6656 (both 51 minas) differ by only
~0.0033 cents, but they also can have separate symbols.

These distinctions are more useful than they may seem, inasmuch as
these pairs of intervals will not always map to the same number of
degrees in a given division of the octave. The 51-mina pair, e.g.,
maps differently in each of the following EDOs: 217, 270, 301, 311,
342, 364, 525, 653, and 742.

Dave & I hope to release a complete list of Sagittal symbol
definitions by midyear.

--George

🔗George D. Secor <gdsecor@yahoo.com>

🔗Gene Ward Smith <genewardsmith@sbcglobal.net>

🔗Ozan Yarman <ozanyarman@ozanyarman.com>

If mina is so excellent a unit, then let's see to it that a hardware tuning
table has this resolution per note per channel in polyphonic fashion.

Oz.

----- Original Message -----
From: "George D. Secor" <gdsecor@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: 11 Nisan 2007 ï¿½arï¿½amba 22:04
Subject: [tuning] mina (was: 612-edo schisma as interval measurement unit)

> --- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
> wrote:
> >
> > --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@> wrote:
> >
> > > An increasingly inportant measuring unit (first suggested by Gene
> > > Ward Smith, ... is a single degree of 2460-EDO ...
>
> > Mina is an excellent unit; divisible by 12, it gets the 27-limit
> > diamond with an error of less than 1/6 cent, and is consistent
> > through the 27-limit. Since the 27 odd limit is a good cutoff
> > (otherwise you get 29 and 31, and even 23 is pretty much of a
> > stretch) I think there's a fine case for saying that 2460 level
> > accuracy is all the majority of microtonalists need.
> >
> > It's an "atomic" system, tempering out the atom, and it also
> > eliminates the glum comma (|91 -12 -31>), the landscape comma,
> > the gauss comma (9801/9800) and on and on of course.
>
> So I guess you do *advocate* it, after all. :-) (Monz, take note.)
>
> --George
>
>

🔗Ozan Yarman <ozanyarman@ozanyarman.com>

----- Original Message -----
From: "George D. Secor" <gdsecor@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: 11 Nisan 2007 ï¿½arï¿½amba 21:56
Subject: [tuning] Re: 612-edo schisma as interval measurement unit

> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
> >
> > Do JI fanatics need any more precision than 2460-EDO?
>
> I hope not. But just in case, there's always 6079-EDO (29-limit
> consistent, but not a multiple of 12), which Sagittal just might be
> able to notate (with a few tricks).
>
> Beyond that, there's 11664-EDO (27-limit consistency that's better
> than 2460), which is a multiple of 12 (and also 72), but I have no
> idea how I would notate it (which doesn't really matter, since I
> figure the above is a rhetorical question. ;-)
>

Talk about overkill!

> All kidding aside, Sagittal *actually does* offer more precision than
> 2460-EDO for those who need it. For example, 98:99 and 99:100 equate
> to the same number of minas (36), but they may be notated with
> separate symbols, if necessary.
>
> Also, 45:46 and 1664:1701 (both 78 minas) differ by ~0.0226 cents,
> but they can also be notated with separate symbols.
>

Let me know if anyone does in fact stoop "so low".

> Finally, 40960:41553 and 6561:6656 (both 51 minas) differ by only
> ~0.0033 cents, but they also can have separate symbols.
>
> These distinctions are more useful than they may seem, inasmuch as
> these pairs of intervals will not always map to the same number of
> degrees in a given division of the octave. The 51-mina pair, e.g.,
> maps differently in each of the following EDOs: 217, 270, 301, 311,
> 342, 364, 525, 653, and 742.
>
> Dave & I hope to release a complete list of Sagittal symbol
> definitions by midyear.
>

Excellent.

> --George
>
>

Oz.

🔗Kraig Grady <kraiggrady@anaphoria.com>

🔗monz <monz@tonalsoft.com>

Hi George,

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:

> All kidding aside, Sagittal *actually does* offer more
> precision than 2460-EDO for those who need it. For example,
> 98:99 and 99:100 equate to the same number of minas (36),
> but they may be notated with separate symbols, if necessary.
>
> Also, 45:46 and 1664:1701 (both 78 minas) differ by ~0.0226
> cents, but they can also be notated with separate symbols.
>
> Finally, 40960:41553 and 6561:6656 (both 51 minas) differ
> by only ~0.0033 cents, but they also can have separate symbols.
>
> These distinctions are more useful than they may seem,
> inasmuch as these pairs of intervals will not always map to
> the same number of degrees in a given division of the octave.
> The 51-mina pair, e.g., maps differently in each of the
> following EDOs: 217, 270, 301, 311, 342, 364, 525, 653,
> and 742.

I found this very interesting as it led me to some further
investigations.

I quoted your post on my updated "mina" page:

http://tonalsoft.com/enc/m/mina.aspx

and in doing so, also wanted to provide 14mu measurements
for your intervals, since 14mu (196608-edo) is the smallest
interval measurement unit currently being used to any extent.

And what i found so interesting is that even at that level
of resolution, consistency (or lack of it) can produce
strange results.

Here is a tabulation of the three examples you provided,
giving ratio, prime-factorization, minas, and 14mus, with
the 14mus calculated in "bingo-card" fashion:

99/98 = 3^2 * 7^-2 * 11^1 = 36 minas = 2880 14mus
100/99 = 3^-2 * 5^2 * 11^-1 = 36 minas = 2852 14mus

1701/1664 = 3^5 * 7^1 * 13^-1 = 78 minas = 6236 14mus
46/45 = 3^-2 * 5^-1 * 23^1 = 78 minas = 6234 14mus

6656/6561 = 3^-8 * 13^1 = 51 minas = 4080 14mus
41553/40960 = 3^7 * 5^-1 * 19^1 = 51 minas = 4075 14mus

These 14mu values come from the way the primes actually
map to 196608-edo. But if i simply use 14mus as an
accurate floating-point value, as you did with cents,
it produces these values:

(99/98)/(100/99) = ~28.941930192 14mus
(1701/1664)/(46/45) = ~3.705627270 14mus
(6656/6561)/(41553/40960) = ~0.548294006 14mus

There's not much discrepancy in the first example:
the bingo-card difference is 28 14mus.

The second example has a bit more discrepancy: the
bingo-card difference is only 2 14mus.

It's the third example which came out weird: the
floating-point calculation gives a difference of less
than one 14mu, but the bingo-card difference is 5 14mus!

The point of all this: it shows why it's valuable to
consider consistency in determining which small unit
to use for purposes of interval measurement. I say the
fewer places needed after the decimal-point, the better.

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗monz <monz@tonalsoft.com>

Hi George (and probably Gene too),

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
>
> Hi Ozan,
>
>
> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@> wrote:
> >
> > From: "George D. Secor" <gdsecor@>
>
> > > Beyond that, there's 11664-EDO (27-limit consistency
> > > that's better than 2460), which is a multiple of 12
> > > (and also 72), but I have no idea how I would notate it
> > > (which doesn't really matter, since I figure the above
> > > is a rhetorical question. ;-)
> > >
> >
> >
> > Talk about overkill!
>
>
> Well, not really. Take a look at my latest message:
>
> /tuning/topicId_70649.html#71145
>
>
> Even using 196608-edo gives discrepancies which might matter.

In fact, i think we should give 1 degree of 11664-edo
a name, since it can be useful as a unit of interval measurement.

Since i started using it this morning, i needed a name for
it, and the first thing i thought of is "trijot", since
it is equal to 2.58 floating-point jots ... i already
know that a jot is 1/30103 of an octave and could see that
this was somewhat close to 3 of those.

It turns out that 1/11664 is quite close to 4 flus,
so perhaps "quatroflu" is OK. But i really don't like
either of those names, since they imply a precise multiple
of the smaller unit.

Can you tell us about some more EDOs which give very
close and consistent approximations to high-limit JI,
which also could be used for interval measurement?

Gene recommends "flu" (1/46032 octave) as a replacement
for tuning-unit, but he only talks about how well it
approximates 5-limit JI.

My goal here is to find some units which are good for
31-limit JI, at sizes of approximately 1/1000, 1/10000,
1/100000, and 1/200000 octave.

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗Gene Ward Smith <genewardsmith@sbcglobal.net>

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:

> Can you tell us about some more EDOs which give very
> close and consistent approximations to high-limit JI,
> which also could be used for interval measurement?

Are you requiring that they be divisible by 12?
And is 31 set in stone?

> Gene recommends "flu" (1/46032 octave) as a replacement
> for tuning-unit, but he only talks about how well it
> approximates 5-limit JI.

It has only one purpose--to do the exact same job
as TU, only better. It tempers out the atom, and in
fact is poptimal for atomic. It gives the Didymus
comma as 825 flus and the Pythagoras comma as
900 flus, so that the schisma is 75 flus. All of
this can be used to scrutinize circulating 5-limit
temperaments of 12 notes in just the same way as
tuning units, but with the advantage that the
5-limit is completely specified; so, in particular,
you know how many units the octave is.

There's no reason to use it for high limit stuff.

> My goal here is to find some units which are good for
> 31-limit JI, at sizes of approximately 1/1000, 1/10000,
> 1/100000, and 1/200000 octave.

Why 31? 11664 is not even 31-limit consistent, by the
way, and is really a 7-limit temperament which tempers
out the atom, is (therefore) divisible by 12, and is
at least consistent through the 27 limt, which still
strikes me as a better cutoff than 31.

The first 31-limit consistent et is the ineluctable
311, by the way. The list goes 311, 388, 1600, 2554,
2619, 4380, 4460, 4501 ... Going up higher, a clear
stand-out is 16808 = 2^3 * 11 * 191, which is very
strong through the 35-limit. Sadly, if you are a
fan of 37, it isn't 37-limit consistent.

By the way, the 311&388 temperament, with a generator
of 19/12, is open for business if anyone wants it.

🔗Gene Ward Smith <genewardsmith@sbcglobal.net>

🔗Ozan Yarman <ozanyarman@ozanyarman.com>

I do not see 159 in there.

----- Original Message -----
From: "Gene Ward Smith" <genewardsmith@sbcglobal.net>
To: <tuning@yahoogroups.com>
Sent: 13 Nisan 2007 Cuma 2:57
Subject: [tuning] Re: searching for some small interval measurement units

> --- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
>
> > My goal here is to find some units which are good for
> > 31-limit JI, at sizes of approximately 1/1000, 1/10000,
> > 1/100000, and 1/200000 octave.
>
> Not quite up to 200000-et (why did you add that?) but
> I recommend the following:
>
> 311-et
> 1600-et
> 16808-et
> 92782-et
>
> Plus, special mention should be made of 31920, which
> factors as 2^3*3*5*7*19, and hence has terrific
> divisibility properties, including of course 12. Here
> is a list of divisors of 31920:
>
> 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 16, 19, 20, 21, 24, 28, 30,
> 35, 38, 40, 42, 48, 56, 57, 60, 70, 76, 80, 84, 95, 105, 112, 114,
> 120, 133, 140, 152, 168, 190, 210, 228, 240, 266, 280, 285, 304, 336,
> 380, 399, 420, 456, 532, 560, 570, 665, 760, 798, 840, 912, 1064,
> 1140, 1330, 1520, 1596, 1680, 1995, 2128, 2280, 2660, 3192, 3990,
> 4560, 5320, 6384, 7980, 10640, 15960, 31920
>
>

🔗Gene Ward Smith <genewardsmith@sbcglobal.net>

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "monz" <monz@> wrote:
>
> > My goal here is to find some units which are good for
> > 31-limit JI, at sizes of approximately 1/1000, 1/10000,
> > 1/100000, and 1/200000 octave.
>
> Not quite up to 200000-et (why did you add that?) but
> I recommend the following:
>
> 311-et
> 1600-et
> 16808-et
> 92782-et
>
> Plus, special mention should be made of 31920, which
> factors as 2^3*3*5*7*19, and hence has terrific
> divisibility properties, including of course 12. Here
> is a list of divisors of 31920:
>
> 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 16, 19, 20, 21, 24, 28, 30,
> 35, 38, 40, 42, 48, 56, 57, 60, 70, 76, 80, 84, 95, 105, 112, 114,
> 120, 133, 140, 152, 168, 190, 210, 228, 240, 266, 280, 285, 304,
336,
> 380, 399, 420, 456, 532, 560, 570, 665, 760, 798, 840, 912, 1064,
> 1140, 1330, 1520, 1596, 1680, 1995, 2128, 2280, 2660, 3192, 3990,
> 4560, 5320, 6384, 7980, 10640, 15960, 31920

I forgot to add that 31920 is consistent, and strong,
though the 41 limit. It's a hell of a notation system,
so I think Monz can put me down as advocating it. One
cent is 26.6 of these units exactly, incidentally.

Also on my list, 311 is, of course, consistent and
strong through the 41 limit. 1600 is consistent
though the 37 limit, and strong through the 45
limit. 16808 and 92782 are both consistent and
strong through the 35 limit, 16808 counting as
very strong.

🔗monz <monz@tonalsoft.com>

Hi Gene,

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@>
> wrote:
> >
> > --- In tuning@yahoogroups.com, "monz" <monz@> wrote:
> >
> > > My goal here is to find some units which are good for
> > > 31-limit JI, at sizes of approximately 1/1000, 1/10000,
> > > 1/100000, and 1/200000 octave.
> >
> > Not quite up to 200000-et (why did you add that?) but
> > I recommend the following:
> >
> > 311-et
> > 1600-et
> > 16808-et
> > 92782-et
> >
> > Plus, special mention should be made of 31920, which
> > factors as 2^3*3*5*7*19, and hence has terrific
> > divisibility properties, including of course 12.
> > <snip>
>
> I forgot to add that 31920 is consistent, and strong,
> though the 41 limit. It's a hell of a notation system,
> so I think Monz can put me down as advocating it. One
> cent is 26.6 of these units exactly, incidentally.
>
> Also on my list, 311 is, of course, consistent and
> strong through the 41 limit. 1600 is consistent
> though the 37 limit, and strong through the 45
> limit. 16808 and 92782 are both consistent and
> strong through the 35 limit, 16808 counting as
> very strong.

Somehow i knew that *you* would take this particular
ball and run with it! ;-)

Thanks much for all of those candidates, and in advance
for any others to come that you may provide.

Since you posted several responses to me already
before i had a chance to read them all, i'll reply
to all of them here now:

Thanks for clarifying the purpose for using flus,
before i jumped the gun and tried to see it as a
good measurement for higher-limit JI. I can see why
you like it so much for 5-limit JI and its temperaments.

Yes, for me divisibility by 12 is desired, simply
because the MIDI standard is based on 12-edo and
so that property makes things nice when dealing with
MIDI. I also like it because it relates calculations
to the tuning that so many folks know and hate. ;-P

Actually, i have my own personal reasons for liking
to stop at 27-limit, so i guess 11664-edo would work OK
for most of my purposes.

The only special reason i had for *going up* to 31-limit
was that Ben Johnston increased the prime-limit in his
compositions until he reached 31, then he stopped there
and (IIRC) used it for several pieces. But i have no
particular reason for *stopping* there ... and in fact
going to 37-limit makes sense because Ezra Sims bases his
music on a theoretical ideal of 37-limit JI. 41-limit
is fine with me.

The only reason i added 1/200000-octave to the list
is because it's close to the MTS 196608-edo, which as
i pointed out in message 41175 fails miserably with
consistency for one pair of Herman's ratios. I investigated
that, and found that it's because 196608-edo's mapping
of prime-factor 3 is about 1/3 of a 14mu too large,
and because the exponents of 3 are so high in that
pair of ratios (-8 and 7) that the accumulated error
winds up being 5 14mus.

Perhaps, because of the pythagoreanism usually involved
in tuning systems, we should weight the prime-factors
accordingly in our search for these measurement units.
I would say that 3 needs to have the lowest error by far,
and that the acceptable error can increase a bit for 5
and a bit more for 7, and that for primes 11 and above
it probably doesn't matter as much as long as it's still
relatively low, since those primes are less likely to be
used with exponents higher than 1.

If others disagree with that, fine ... but those are my
criteria. By that measure:

* 31920-edo looks incredibly good for the 41-limit, and
for me its divisibility properties are just a bonus;

* for 31-limit, 16808-edo is terrific;

* for 27-limit with a lower-cardinality EDO, it looks
like 2460 can't be beat.

* 11664-edo is fantastic for 7-limit, as you pointed out.

* 612-edo is awesome for 5-limit, and still not bad for
11-limit.

Those unit sizes i provided (1/1000, 1/10000, etc.)
were really just general guidelines -- i'm glad that you
found some others that were in-between. By all means,
please post *any* good ones that you find, regardless
of what sizes they are.

And last but not least, we need *names* for these things
if they really deserve a shot at being used as interval
measurements. (I know, i know -- complaints about more
jargon are on the way ...)

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗Gene Ward Smith <genewardsmith@sbcglobal.net>

Pushing the search out to 350000-et, I found the
following:

131610 = 2 * 3 * 5 * 41 * 107
Strong through the 35 limit, consistent through the
39 limit

148418 = 2 * 74209
Strong through the 35 limit, consistent through the
39 limit

241200 = 2^4 * 3^2 * 5^2 * 67
Strong and consistent through the 39 limit

324296 = 2^3 * 7 * 5791
Strong and consistent through the 59 limit!

131610 and 148418 are both closer to being ten times
16808 if that is a consideration. In terms of 131610
units, a cent is exactly 109.675 units, and a step
of 2460 is exactly 53.5 units.

In terms of 241200 units, a cent is exactly 201 units.

If we were to tear up midi and start over, we might
try dividing 100 cents into something more imaginative
than a power of two, as I don't see that the power
of two nets you anything. For instance. midi pitch
bend divides 100 cents into 4096 parts. We could just
as well divide it into 2660 parts, giving greater
accuracy. Pitch bend ends up using 22.585 bits to
determine a note; just 22 bits instead with this
system would actually give more range. 32 bit numbers,
using 324296 to divide the octave would give
13.693 octaves as a range, though of course a system
such as 241200 (14.12 octaves) with exact 100 cent
semitones would probably be preferred.

🔗Gene Ward Smith <genewardsmith@sbcglobal.net>

🔗monz <monz@tonalsoft.com>

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:

> And last but not least, we need *names* for these things
> if they really deserve a shot at being used as interval
> measurements. (I know, i know -- complaints about more
> jargon are on the way ...)

You all know that lately i've been looking into the various
proposals for interval measurement units which are strong and
consistent thru the 41-limit.

Despite the fact that the main point of this search was
to find alternatives to "cents" (1200-edo) and "14mus"
(196608-edo) and a few sizes between those, i've been
amazed over and over again by 311-edo.

Not only is it excellent for 41-limit JI, but it's also
awesome for 1/6-comma meantone, which is a tuning i've
been extremely interested in.

I propose that we give one unit of 311-edo the name "gene",
both in honor of Gene Ward Smith (who AFAIK was the first
person to notice its outstanding properties, in 1969) and
also because of the connotation of the word with the small
unit from the field of genetics. Any objections?

(BTW ... i have a nagging feeling that i've proposed the
use of this word before, for the same reasons, but maybe
not in connection with 311-edo -- but i've done a search
and can't find anything.)

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗Graham Breed <gbreed@gmail.com>

🔗monz <monz@tonalsoft.com>

Hi Graham,

Woah, things are getting strange ...

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> monz wrote:
>
> > I propose that we give one unit of 311-edo the name "gene",
> > both in honor of Gene Ward Smith (who AFAIK was the first
> > person to notice its outstanding properties, in 1969) and
> > also because of the connotation of the word with the small
> > unit from the field of genetics. Any objections?
>
> Units of 1244-edo might be worth a name because they're
> closer to cents.

I had exactly the same thought this morning while i was
in the shower! But i forgot about it by the time i wrote
my post, and just went back to focusing on 311 itself.

I guess i figured that 311-edo is so good at approximating
such a high-limit JI with its relatively low cardinality
that we adopt the benefits of the lower cardinality and
just use 311 without worrying about relating it to cents.

> > (BTW ... i have a nagging feeling that i've proposed the
> > use of this word before, for the same reasons, but maybe
> > not in connection with 311-edo -- but i've done a search
> > and can't find anything.)
>
> The business of combining equal temperaments to get regular
> temperaments has been called "breeding". Obviously I can't
> comment on that. But the equal temperament mappings (what
> Gene calls "vals") are the units of inheritance. So it
> would make sense to call them "genes" if you wanted a
> special name. As Gene does, but he obviously can't comment
> on this proposal.

Hmm ... so is there some irony here, or were you and Gene
predestined from birth to be the two guys who work on this
stuff? ;-)

I don't think there are too many theorists using the term
"val" yet, and Gene is the only one i know who uses it
extensively. So now's the time to change it if we're going to.

... But still leaves me with wanting names for these units,
and for some of the other EDOs that would make good units.

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗Gene Ward Smith <genewardsmith@sbcglobal.net>

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:

> Not only is it excellent for 41-limit JI, but it's also
> awesome for 1/6-comma meantone, which is a tuning i've
> been extremely interested in.

In fact, it's a semiconvergent for 1/6 comma meantone.
It's also possible to use it for other temperaments,
such as sensi, amity, flattone, garibaldi, etc.

> I propose that we give one unit of 311-edo the name "gene",
> both in honor of Gene Ward Smith (who AFAIK was the first
> person to notice its outstanding properties, in 1969) and
> also because of the connotation of the word with the small
> unit from the field of genetics. Any objections?

Um...is anyone going to actually use it as a unit?
It's a prime number, and it's on the small side for
use as a unit. But it obviously has a lot going for
it if we could only find out what to use it for.

Here's a 41-limit comma basis for anyone who finds
inspiration in that:

[595/594, 625/624, 697/696, 703/702, 714/713, 760/759, 784/783,
820/819, 833/832, 875/874, 900/899, 931/930]

Smaller limit commas it tempers out include the amity
comma, 1600000/1594323; 2401/2400, 4000/3993, 65625/65536
and 3025/3024.

🔗Gene Ward Smith <genewardsmith@sbcglobal.net>

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:

> > Units of 1244-edo might be worth a name because they're
> > closer to cents.

> I had exactly the same thought this morning while i was
> in the shower! But i forgot about it by the time i wrote
> my post, and just went back to focusing on 311 itself.

I can't see you get any bang for your buck in this way.
What about 1600? It's a nice round number, at least,
even if not divisible by 3. A number with good divisibility
properties, aside from the already-discussed 1578 and 2460
is 1848 = 2^3 * 3 * 7 * 11. It's consistent to the 15 limit
and quite strong in the 11 limit.

> I guess i figured that 311-edo is so good at approximating
> such a high-limit JI with its relatively low cardinality
> that we adopt the benefits of the lower cardinality and
> just use 311 without worrying about relating it to cents.

That would be the way to use it, all right. If you could remember
the complete mapping up to 41 you could quickly work out about
how big most any small ratio will be. Maybe the thing to do with
311, though, is use it for music; I just haven't been able to
get into the idea of high-limit music myself. It gets the entire
41-limit tonality diamond, all 357 notes of it, within two cents
of error, the biggest error coming on 31/25.

🔗monz <monz@tonalsoft.com>

Hi Gene,

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "monz" <monz@> wrote:
>
> > > Units of 1244-edo might be worth a name because they're
> > > closer to cents.
>
> > I had exactly the same thought this morning while i was
> > in the shower! But i forgot about it by the time i wrote
> > my post, and just went back to focusing on 311 itself.
>
> I can't see you get any bang for your buck in this way.
> What about 1600? It's a nice round number, at least,
> even if not divisible by 3. A number with good divisibility
> properties, aside from the already-discussed 1578 and 2460
> is 1848 = 2^3 * 3 * 7 * 11. It's consistent to the 15 limit
> and quite strong in the 11 limit.

I think that since the mina (2460-edo) is already somewhat
established, we don't really need another one near that
cardinality, and it's so close to 1/2 cent that we don't
really need another replacement for cents either.

The main reason i like 311 so much is that it does
such a good job with such a low cardinality.

> > I guess i figured that 311-edo is so good at approximating
> > such a high-limit JI with its relatively low cardinality
> > that we adopt the benefits of the lower cardinality and
> > just use 311 without worrying about relating it to cents.
>
> That would be the way to use it, all right. If you could
> remember the complete mapping up to 41 you could quickly
> work out about how big most any small ratio will be.

Exactly!

I see 311 being better than cents (1200) for two reasons:
it's more accurate, *and* it's a lower cardinality, so
the numbers you have to remember are smaller by one digit.

For the record, i thought it good to mention that 311 is also
pretty close to the historical measurements of heptamerides
(301) and savarts (300). I don't think heptamerides ever got
much use, but i've seen savarts in some of the French tuning
literature ... even in 20th-century treatises in which you'd
think the author would use cents.

> Maybe the thing to do with 311, though, is use it for music;
> I just haven't been able to get into the idea of high-limit
> music myself. It gets the entire 41-limit tonality diamond,
> all 357 notes of it, within two cents of error, the biggest
> error coming on 31/25.

There's no reason 311 can't be used both as a measurement unit
and as a tuning which approximates 41-limit JI.

Unfortunately, we built Tonescape to deal with "only"
7-dimensional tunings, so the full 41-limit is out, as
it would require 13 dimensions.

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗Gene Ward Smith <genewardsmith@sbcglobal.net>

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
>
> Hi George (and probably Gene too),

Hi, Monz. I see that this thread is still going strong since I left
over the weekend. I'm responding to this, in the event you're still
interested in a name for 11664.

> > > ...
> > > From: "George D. Secor" <gdsecor@>
> >
> > > > Beyond that, there's 11664-EDO (27-limit consistency
> > > > that's better than 2460), which is a multiple of 12
> > > > (and also 72), but I have no idea how I would notate it
> > > > (which doesn't really matter, since I figure the above
> > > > is a rhetorical question. ;-)
> > > ...
>
> In fact, i think we should give 1 degree of 11664-edo
> a name, since it can be useful as a unit of interval measurement.
>
> Since i started using it this morning, i needed a name for
> it, and the first thing i thought of is "trijot", since
> it is equal to 2.58 floating-point jots ... i already
> know that a jot is 1/30103 of an octave and could see that
> this was somewhat close to 3 of those.
>
> It turns out that 1/11664 is quite close to 4 flus,
> so perhaps "quatroflu" is OK. But i really don't like
> either of those names, since they imply a precise multiple
> of the smaller unit.

11664 has also been of interest to Dave & myself as a unit of measure
in investigating commas to notate in Sagittal: A single degree of
11664 happens to correspond to 4 minas, allowing that the JI
definition of "mina" refers to either of the two smallest intervals
(4095:4096 and 4374:4375) notated in Sagittal, by |', a right-
accent. We also define the double-right accent |'' (of 2 minas) as
either 2079:2080 or 255879:256000 (depending on the symbol core it's
modifying), which corresponds to 8deg11664.

Therefore, a name denoting 1/4-mina would be appropriate. I don't
know if I would like "quartina", but maybe you can think of something
similar.

BTW, the tool I use to evaluate consistency is this spreadsheet:
/tuning-
math/files/secor/Constncy.xls
which gives the error of each odd harmonic not only in cents (in col.
D), but also as % of a degree (in col. E). The latter figure weights
accuracy against complexity, which enables one to compare one
division against another equitably. N-limit consistency is
determined by the last row in which there's no figure displayed in
either column F or G.

In evaluating all of those divisions that have been popping up, I
agree that the lowest primes should have very low degree-%
deviations. For 3, this should be significantly lower than 5%, since
at that point a chain of 10 fifths (i.e., 3^10) results in a 50%
deviation, which is midway between degrees. Likewise, it's highly
desirable that 5 and 7 each be below 10%, since it's not unusual for
those primes to be taken to the 4th or 5th power in notational commas.

You'll see that 2460 and 11664 fare quite well under these
circumstances. (I also see that 324296 is exceptional.)

--George

🔗monz <monz@tonalsoft.com>

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:

> By the way, if you want to add something to 2460
> on the high end, 31920 remains a strong choice.
> It's also almost a hundred (102.6) times bigger
> than 311. About in the middle is 3125, but I guess
> that's squashed by competition from 2460.

Yes, actually i always liked De Morgan's unit called "jot"
(= 30103-edo) because, until "mu"s (MIDI-units) came along,
it filled a niche in interval measurement that gave much
higher resolution than cents. So 31920 would be a great
replacement for that ... and again, it needs a name.

As far as the mus go, the only ones that we really seem to
need to pay attention to are:

* 6mu (= 768-edo), which uses all bits available for
pitch-bend data (plus or minus the default MIDI-note pitch)
in one byte;

* 12mu (= 49152-edo), which uses all bits available for
pitch-bend data (plus or minus the default MIDI-note pitch)
in two bytes; and

* 14mu (= 196608-edo), which uses all bits available for
tuning data in a two-byte MIDI tuning command.

The other mus aren't really used.

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗Gene Ward Smith <genewardsmith@sbcglobal.net>

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@>
> wrote:
>
> > Why do you say it is more accurate?
>
> Just for fun, and using patent tunings, here are
> the smallest ets which beat 1200 in various limits
> in terms of max error:
>
> 3: 253
> 5: 171
> 7: 171
> 9: 171
> 11: 342
> 13: 494
> 15: 494
> 17: 581
> 19: 581
> 21: 581
> 23: 718
> 25: 525
> 27: 624

By way of comparison, here is 311:

3: 53
5: 118
7: 171
9: 171
11: 270
13: 270
15: 311
17: 311
19: 311
etc etc

Here's 72:

3: 29
5: 53
7: 72
9: 72
11: 72
13: 72
15: 72
etc etc

99:

3: 41
5: 53
7: 99
9: 99
11: 58
13: 58
15: 58
17: 58
19: 62
21: 62
23: 62
25: 80
27: 80
29: 80
31: 87

And no list would be complete without 159:

3: 159 (53)
5: 118
7: 99
9: 99
11: 152
13: 159
15: 159
17: 159
19: 159
etc etc

And I suppose we must do 12:

3: 12
5: 12
7: 12
9: 12
11: 12
13: 9
15: 9
17: 10
19: 12
21: 12
etc etc

🔗Gene Ward Smith <genewardsmith@sbcglobal.net>

🔗Ozan Yarman <ozanyarman@ozanyarman.com>

----- Original Message -----
From: "Gene Ward Smith" <genewardsmith@sbcglobal.net>
To: <tuning@yahoogroups.com>
Sent: 16 Nisan 2007 Pazartesi 23:28
Subject: [tuning] Re: proposed name: "gene", for one degree of 311-edo

> --- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
> wrote:
> >
> > --- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@>
> > wrote:
> >
> > > Why do you say it is more accurate?
> >
> > Just for fun, and using patent tunings, here are
> > the smallest ets which beat 1200 in various limits
> > in terms of max error:
> >
> > 3: 253
> > 5: 171
> > 7: 171
> > 9: 171
> > 11: 342
> > 13: 494
> > 15: 494
> > 17: 581
> > 19: 581
> > 21: 581
> > 23: 718
> > 25: 525
> > 27: 624
>
> By way of comparison, here is 311:
>
> 3: 53
> 5: 118
> 7: 171
> 9: 171
> 11: 270
> 13: 270
> 15: 311
> 17: 311
> 19: 311
> etc etc
>
> Here's 72:
>
> 3: 29
> 5: 53
> 7: 72
> 9: 72
> 11: 72
> 13: 72
> 15: 72
> etc etc
>
> 99:
>
> 3: 41
> 5: 53
> 7: 99
> 9: 99
> 11: 58
> 13: 58
> 15: 58
> 17: 58
> 19: 62
> 21: 62
> 23: 62
> 25: 80
> 27: 80
> 29: 80
> 31: 87
>
> And no list would be complete without 159:
>
> 3: 159 (53)
> 5: 118
> 7: 99
> 9: 99
> 11: 152
> 13: 159
> 15: 159
> 17: 159
> 19: 159
> etc etc
>

SNIP

That's right.

Oz.

🔗Ozan Yarman <ozanyarman@ozanyarman.com>

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@> wrote:
>
> > In evaluating all of those divisions that have been popping up, I
> > agree that the lowest primes should have very low degree-%
> > deviations. For 3, this should be significantly lower than 5%,
since
> > at that point a chain of 10 fifths (i.e., 3^10) results in a 50%
> > deviation, which is midway between degrees. Likewise, it's
highly
> > desirable that 5 and 7 each be below 10%, since it's not unusual
for
> > those primes to be taken to the 4th or 5th power in notational
commas.
> >
> > You'll see that 2460 and 11664 fare quite well under these
> > circumstances. (I also see that 324296 is exceptional.)
>
> 324296 would be just about perfect if only it was
> divisible by 12. A 12-divisible system which also
> does well on 3 and 5 is 31920--have you given any
> thought to the potential uses of that?

No, because:

1) It isn't good enough for 7, failing at 7^3 and 5*7^2;
2) 27-limit consistency is adequate for most purposes;
3) 11664 has 0.10-cent resolution, which is around the limit of
audibity, accurate enough, I would think, for even the most demanding
JI fanatic.

Would we really need more precision than that for an alternative MIDI
standard?

--George

🔗Charles Lucy <lucy@harmonics.com>

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning@yahoogroups.com, Charles Lucy <lucy@...> wrote:
>
> Some people make all sorts of rash statements about being able to
> hear intervals, yet the real test seems to me to be hearing the
> beating as pitches sound simultaneously.
>
> So I want to get the best precision possible.
>
> Charles Lucy lucy@...

Even to the point of overkill?

Consider a just major 3rd with C=264 as the bottom tone, widened 0.1
cents. When the two tones are sounded simultaneously, the 4th
harmonic of C (1320 Hz) beats against the 5th harmonic of E\ (1320.08
Hz). That 0.08 Hz difference amounts to one beat every 12 seconds.
You'd never be able to hear it (unless you're living in a dream
house ;-).

--George

> > [I wrote:]
> > No, because:
> >
> > 1) It isn't good enough for 7, failing at 7^3 and 5*7^2;
> > 2) 27-limit consistency is adequate for most purposes;
> > 3) 11664 has 0.10-cent resolution, which is around the limit of
> > audibity, accurate enough, I would think, for even the most
demanding
> > JI fanatic.
> >
> > Would we really need more precision than that for an alternative
MIDI
> > standard?
> >
> > --George
> >
> > .
> >
> >
>

612-edo schisma as interval measurement unit

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