Wide 5ths

Despite all the quotes Andreas provided about the wide 5ths not sounding good, I'll say that they sound just fine. After years of playing and composing with 34 tone equal tempered guitars in many different styles of music, there's no problems that I can hear. Don't always trust what you read about a tuning; it's only how it sounds with real music that matters...best...Hstick
myspace.com/microstick

--- In tuning@yahoogroups.com, <microstick@...> wrote:
>
Dear Neil,
>Despite all the quotes Andreas provided about the wide 5ths not
>sounding good, I'll say that they sound just fine.
yours opinion agrees partially with:
http://ptg.org/pipermail/pianotech/2000-November/074561.html
judgement.

But anyhow conversely to Brad,
that author distincts too also clearly
inbetween the wide and narrow version
as each others different sounding in his ears.

Psychoacustical experiments show statistically more
tolerance for the underbeating 5ths
rather than for the same amount overbeating,
but consistent perference in the few deviating cases
of subjects that do love the wide version more than the narrow.
Maybe that explains the exceptions from the general
observed trend, as for example:

http://www-personal.umich.edu/~bpl/larips/errata.html
"The "wide 5th Bb-F" that many readers reject or misunderstand
This is not an error in the article or at LaripS.com! It becomes an
error of comprehension, however, when readers misunderstand and
misquote it, or try to "improve" the temperament by eliminating it."
Does he wants to convince us by that statement:
Only barely Lehman can "improve" his own private tuning?
alike he deems al other squiggle before and after him as
"unsatisfactory" compared to his own selfpraised versions?

he continues:
"A wide "5th" is apparently an anomaly to some readers who cannot
believe that any "well temperament" (a 20th century English term, and
ungrammatical, and based on theoretical expectations invented long
after Bach's death!) "

That's literally translated from the german term:
"Wohl-temperiert"
originally coined by Werckmeister and
for the first time overtaken by Bach,
as referecne to W. in order to honour W.
by that dedication in the title of his composition:
the WTC:
W's tuning enabled Bach to use all 24 major and minor keys.

Dr. L. premise:
"could ever have even the most slightly wide 5th in it."

but there is rather no wide 5th in Weckmeister's #3:

C ~ G ~ D ~ A E B ~ F# C# G# D# Bb F C

that divides the PC into 4 subparts inbetween C~G~D~A and B~F#,
with all other eight 5ths pure.

Where on earth inbetween that is there space for an
waste wide 5th?

Werckmeister's definition of "well" includes
inherently what Gene labeles as "authentic"-well:
meaning all 5ths smaller or equal than pure,
rahter than dispensable wide.

Brad points out his own personal problem:
"This problem has already generated a number of confused, angry, or
"helpful" letters to "correct" this point on which I have been accused
(some in public) as bewilderingly ignorant."

There's nothing to object against that attitude.

in
http://www-personal.umich.edu/~bpl/larips/bachtemps.html
he admits at least:
"my own first preferred reading of the Bach diagram (spring 2004) was
as shown here: the syntonic comma interpretation, with an arbitrarily
pure fifth at Bb-F. I developed this in the first draft of the
article. But, discussion with Ross Duffin and Debra Nagy helped to
convince me to focus on the Pythagorean comma interpretation more
centrally for all the later rounds of the work."
Probably he overtook his "wide-5th" nonsene from there.

But I do consider at least his fist 'syntonic-comma' 2004 attempt
as reasonable, because there lacks the disturbing broade 5th,
that appeared one year later in his ill-bred wayward failure
reattempt.

but back to Brad's bollocks again:
"The presence of a wide 5th evidently upsets some readers so much--at
least in their conceptual understanding of the temperament--that they
must try to "correct" it in practice, or in argument against the
printed material..."

That's anyway an progress:
At least he recognizes the secpticism caused by his wide 5th.

"even though this particular interval is so nearly pure that it's not
noticeable from even two meters away from a harpsichord!"

Apology, in order to excuse Brad's ears:
That's his harpsichord's fault,
or even caused by the 2 meters distance to the instrument?

he continues the bosh:
"The A#-F diminished 6th in this temperament is exactly that: a
diminished 6th, not a 5th. "
Quest:
Who else in that group here percieves an interval of
~704Cents as 'dim. 6th'?

"(Every 12-note keyboard temperament must have an enharmonic
diminished 6th in it somewhere!)"

Am I really wrong if I'm telling in lectures to my students,
that an well-temperament consists in a dozen more
less tempered 5ths in order to compensate the PC?

but back to Dr. Lehman's drivel:
" The A#-F interval in this proposed Bach temperament is incidentally
almost a pure Bb-F 5th, but it happens not to be one; and this is
based directly on Bach's diagram as source."

May we all conclude from that daffing:
Brad's alleged "discovery" consists that his claim:
He has found out conversely to all other Bach-scholars,
presuming JSB would follow his questionable opinion of his
"dim. 6th" premise inbetween: (B#=Bb) and F ???

Who in that grup here agree's with his weird point of view
that JSB would had labeled the 5th Bb-F really as an "dim-6th"?

> After years of playing and composing with 34 tone equal tempered
>guitars in many different styles of music,
that's 2*17-EDO

> there's no problems that I can hear.
Agreed.
as long as you don't foist that to members of Bach-family,
in order to claim to rewrite the history of tuning anew.

> Don't always trust what you read about a tuning;
especially in that journal:
http://muse.jhu.edu/journals/early_music/v033/33.1knighton.html
"Editorial
Over the last 30 years a number of exciting discoveries have appeared
in the pages of Early music: many of these have helped to shape
thinking on questions of performance practice, and have thus been
influential on the way we perform and hear early music today. So I am
very happy to present a further discovery, made last year by
harpsichordist and scholar Bradley Lehman, whose fascinating article
clearly establishes the temperament Bach had in mind when he composed
Das wohltemperirte Clavier, and, very probably, much of the rest of
his Å"uvre.
Lehman has risen to the challenge offered by Malcolm Boyd in his
discussion of performance practice in Bach's music. 'The quest', wrote
Boyd in his biography of Bach, 'continues, not for the complete
"authenticity" in performance which is unattainable and probably
undesirable, but to establish what conditions are indispensable for
any performance that hopes to capture the true essence of Bach's art.
Possibly the most revealing discoveries still to be made await a more
thorough application than has so far been attempted of the
temperaments (tunings) in use during Bach's time.' I will leave Lehman
to describe and substantiate the evidence for the temperament he has
discoveredâ€"or more accurately, perhaps, uncovered, for the story
begins with a brilliant piece of detective work or, again more
accurately, decodingâ€"in this and the next issue. Based on a unique
blend of three different 'equal' scales, the temperament works equally
well in any key, or genre. This discovery should indeed afford further
insight into the 'true essence' of Bach's art."

Refers that especially the alleged "dim-6th" inbetween Bb-F ?

> it's only how it sounds with real music that matters...

that's true,
even when an ~704Cents 5th is confused with an "dim-6th"
Whoever here is able to follw Brad in his personal misconceptions?

A.S.

My dear Andreas, what language... Drivel? Daffing? Are these words accepted
as legitimate within the Bach academia? What would Bach's tortured spirit
say?

Cordially,
Oz.

----- Original Message -----
From: "Andreas Sparschuh" <a_sparschuh@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: 19 Haziran 2007 Sal� 22:55
Subject: [tuning] Re: Wide 5ths

--- In tuning@yahoogroups.com, <microstick@...> wrote:
>
Dear Neil,
>Despite all the quotes Andreas provided about the wide 5ths not
>sounding good, I'll say that they sound just fine.
yours opinion agrees partially with:
http://ptg.org/pipermail/pianotech/2000-November/074561.html
judgement.

But anyhow conversely to Brad,
that author distincts too also clearly
inbetween the wide and narrow version
as each others different sounding in his ears.

Psychoacustical experiments show statistically more
tolerance for the underbeating 5ths
rather than for the same amount overbeating,
but consistent perference in the few deviating cases
of subjects that do love the wide version more than the narrow.
Maybe that explains the exceptions from the general
observed trend, as for example:

http://www-personal.umich.edu/~bpl/larips/errata.html
"The "wide 5th Bb-F" that many readers reject or misunderstand
This is not an error in the article or at LaripS.com! It becomes an
error of comprehension, however, when readers misunderstand and
misquote it, or try to "improve" the temperament by eliminating it."
Does he wants to convince us by that statement:
Only barely Lehman can "improve" his own private tuning?
alike he deems al other squiggle before and after him as
"unsatisfactory" compared to his own selfpraised versions?

he continues:
"A wide "5th" is apparently an anomaly to some readers who cannot
believe that any "well temperament" (a 20th century English term, and
ungrammatical, and based on theoretical expectations invented long
after Bach's death!) "

That's literally translated from the german term:
"Wohl-temperiert"
originally coined by Werckmeister and
for the first time overtaken by Bach,
as referecne to W. in order to honour W.
by that dedication in the title of his composition:
the WTC:
W's tuning enabled Bach to use all 24 major and minor keys.

Dr. L. premise:
"could ever have even the most slightly wide 5th in it."

but there is rather no wide 5th in Weckmeister's #3:

C ~ G ~ D ~ A E B ~ F# C# G# D# Bb F C

that divides the PC into 4 subparts inbetween C~G~D~A and B~F#,
with all other eight 5ths pure.

Where on earth inbetween that is there space for an
waste wide 5th?

Werckmeister's definition of "well" includes
inherently what Gene labeles as "authentic"-well:
meaning all 5ths smaller or equal than pure,
rahter than dispensable wide.

Brad points out his own personal problem:
"This problem has already generated a number of confused, angry, or
"helpful" letters to "correct" this point on which I have been accused
(some in public) as bewilderingly ignorant."

There's nothing to object against that attitude.

in
http://www-personal.umich.edu/~bpl/larips/bachtemps.html
he admits at least:
"my own first preferred reading of the Bach diagram (spring 2004) was
as shown here: the syntonic comma interpretation, with an arbitrarily
pure fifth at Bb-F. I developed this in the first draft of the
article. But, discussion with Ross Duffin and Debra Nagy helped to
convince me to focus on the Pythagorean comma interpretation more
centrally for all the later rounds of the work."
Probably he overtook his "wide-5th" nonsene from there.

But I do consider at least his fist 'syntonic-comma' 2004 attempt
as reasonable, because there lacks the disturbing broade 5th,
that appeared one year later in his ill-bred wayward failure
reattempt.

but back to Brad's bollocks again:
"The presence of a wide 5th evidently upsets some readers so much--at
least in their conceptual understanding of the temperament--that they
must try to "correct" it in practice, or in argument against the
printed material..."

That's anyway an progress:
At least he recognizes the secpticism caused by his wide 5th.

"even though this particular interval is so nearly pure that it's not
noticeable from even two meters away from a harpsichord!"

Apology, in order to excuse Brad's ears:
That's his harpsichord's fault,
or even caused by the 2 meters distance to the instrument?

he continues the bosh:
"The A#-F diminished 6th in this temperament is exactly that: a
diminished 6th, not a 5th. "
Quest:
Who else in that group here percieves an interval of
~704Cents as 'dim. 6th'?

"(Every 12-note keyboard temperament must have an enharmonic
diminished 6th in it somewhere!)"

Am I really wrong if I'm telling in lectures to my students,
that an well-temperament consists in a dozen more
less tempered 5ths in order to compensate the PC?

but back to Dr. Lehman's drivel:
" The A#-F interval in this proposed Bach temperament is incidentally
almost a pure Bb-F 5th, but it happens not to be one; and this is
based directly on Bach's diagram as source."

May we all conclude from that daffing:
Brad's alleged "discovery" consists that his claim:
He has found out conversely to all other Bach-scholars,
presuming JSB would follow his questionable opinion of his
"dim. 6th" premise inbetween: (B#=Bb) and F ???

Who in that grup here agree's with his weird point of view
that JSB would had labeled the 5th Bb-F really as an "dim-6th"?

> After years of playing and composing with 34 tone equal tempered
>guitars in many different styles of music,
that's 2*17-EDO

> there's no problems that I can hear.
Agreed.
as long as you don't foist that to members of Bach-family,
in order to claim to rewrite the history of tuning anew.

> Don't always trust what you read about a tuning;
especially in that journal:
http://muse.jhu.edu/journals/early_music/v033/33.1knighton.html
"Editorial
Over the last 30 years a number of exciting discoveries have appeared
in the pages of Early music: many of these have helped to shape
thinking on questions of performance practice, and have thus been
influential on the way we perform and hear early music today. So I am
very happy to present a further discovery, made last year by
harpsichordist and scholar Bradley Lehman, whose fascinating article
clearly establishes the temperament Bach had in mind when he composed
Das wohltemperirte Clavier, and, very probably, much of the rest of
his �"uvre.
Lehman has risen to the challenge offered by Malcolm Boyd in his
discussion of performance practice in Bach's music. 'The quest', wrote
Boyd in his biography of Bach, 'continues, not for the complete
"authenticity" in performance which is unattainable and probably
undesirable, but to establish what conditions are indispensable for
any performance that hopes to capture the true essence of Bach's art.
Possibly the most revealing discoveries still to be made await a more
thorough application than has so far been attempted of the
temperaments (tunings) in use during Bach's time.' I will leave Lehman
to describe and substantiate the evidence for the temperament he has
discovered�"or more accurately, perhaps, uncovered, for the story
begins with a brilliant piece of detective work or, again more
accurately, decoding�"in this and the next issue. Based on a unique
blend of three different 'equal' scales, the temperament works equally
well in any key, or genre. This discovery should indeed afford further
insight into the 'true essence' of Bach's art."

Refers that especially the alleged "dim-6th" inbetween Bb-F ?

> it's only how it sounds with real music that matters...

that's true,
even when an ~704Cents 5th is confused with an "dim-6th"
Whoever here is able to follw Brad in his personal misconceptions?

A.S.

Dear Andreas,

Bravo! I do agree with your basic points. Bach said he used a well
temperament. Utrecht-based Rudolph Rasch published that well temperament is "An
English-language term introduced by Jorgensen for certain types of circular
temperaments." I agree with you, Andreas. I'd call this poppycock, but that's
me. I clearly see the connections between Bach and Werckmeister.

Alas, there is so much going on besides simply finding the truth. Sorry,
Neil, but saying you like sharp fifths doesn't add a thing to what temperament
Bach intended for his music. And I like all sorts of fifths, and other
intervals, too! :)

This argument is where Irregular Temperament and Well Temperament split off;
irregular emphasizes the central keys, while well emphasizes the peripheral.
This is the reason why Fischer only wrote in 20 keys (irregular tuning),
while Bach wrote in 24 keys.

From a just intonation perspective, I like the idea of a just interval being
tempered in only one direction within a single temperament. To do otherwise
provides a waste basket of several keys.

Ozan, drivel is drivel. But you rooting and tooting seems unseemly. Please
let us each speak in our own voice.

Johnny

************************************** See what's free at http://www.aol.com.

Rooting and tooting eh? One could expect no less from you my dear Johnny Reinhard.

Cordially,
Oz.
----- Original Message -----
From: Afmmjr@aol.com
To: tuning@yahoogroups.com
Sent: 20 Haziran 2007 Çarşamba 0:40
Subject: Re: [tuning] Re: Wide 5ths

Dear Andreas,

Bravo! I do agree with your basic points. Bach said he used a well temperament. Utrecht-based Rudolph Rasch published that well temperament is "An English-language term introduced by Jorgensen for certain types of circular temperaments." I agree with you, Andreas. I'd call this poppycock, but that's me. I clearly see the connections between Bach and Werckmeister.

Alas, there is so much going on besides simply finding the truth. Sorry, Neil, but saying you like sharp fifths doesn't add a thing to what temperament Bach intended for his music. And I like all sorts of fifths, and other intervals, too! :)

This argument is where Irregular Temperament and Well Temperament split off; irregular emphasizes the central keys, while well emphasizes the peripheral. This is the reason why Fischer only wrote in 20 keys (irregular tuning), while Bach wrote in 24 keys.

From a just intonation perspective, I like the idea of a just interval being tempered in only one direction within a single temperament. To do otherwise provides a waste basket of several keys.

Ozan, drivel is drivel. But you rooting and tooting seems unseemly. Please let us each speak in our own voice.

Johnny

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
Dear Ozan,
> My dear Andreas, what language...
Drivel?
What about the earlier usage of that word here in that group?:
/tuning/msearch?query=drivel&submit=Search&charset=UTF-8

> Daffing?
http://www.wordwebonline.com/en/DAFF
"Verb: daff
Usage: archaic
1. To cast aside; to put off; to doff
Derived forms: daffed, daffs, daffing"
http://www.thefreedictionary.com/cast%20aside
/tuning/msearch?query=daft&submit=Search&charset=UTF-8
You asked there in using just that expression once yourself in:
/tuning/topicId_64793.html#64837
"Who would be DAFT enough to claim that there is no modulation when
you play solo?"
/tuning/topicId_64269.html#64757
"Therefore, it would be nothing less than a DAFT suggestion to speak
of unison as a musical interval here..."

Can you still remember yours private applications of "daft" there?

How do you want to explain us yours own usage of "daft" there,
before citizing others alike Mark Lindley and me about the same
spelling?

Before bashing others about yours own bad habits
just take look into the mirror and grab yours own nose!

People who live in glass houses shouldn't throw stones.

> Are these words accepted
> as legitimate within the Bach academia?
within the:
http://www.bachakademie.de/
they are translated back into:

http://dict.leo.org/ende?lp=ende&lang=de&searchLoc=0&cmpType=relaxedB'Hdr=on&spellToler=on&search=daffing&relink=on
http://dict.leo.org/ende?lp=ende&lang=de&searchLoc=0&cmpType=relaxedB'Hdr=on&spellToler=on&search=daft&relink=on
http://dict.leo.org/ende?lp=ende&lang=de&searchLoc=0&cmpType=relaxedB'Hdr=on&spellToler=on&search=drivel&relink=on

> What would Bach's tortured spirit
> say?
About Lehman's "dim-6th" theory concerning in Lehman's recordings?
he would probably adjudicate:
http://www.bach-cantatas.com/BWV78-D.htm
"To this recording Bach would have said: "es werde des Auctoris
Dreckohr gereiniget, und zur Anhörung der Musik geschickter gemacht
werden." 10.12.1749 In this case (Harnoncourt "will have to get his
dirty ears cleaned out so that he, in time, can then apply his ears to
hear music properly.")"

http://www.bach-cantatas.com/Topics/Freedom.htm
Harnoncourt: If you want to hear what Mattheson is talking about when
he refers to the "Schreihälse" ("those who scream instead of really
trying to sing,") you have an example of this here

http://dict.leo.org/ende?lp=ende&p=/oHL..&search=Schreihals

BWV 94
I am quite certain (just as certain as Harnoncourt was about these
recordings) that Bach would have restated his words (although in the
quote he had directed his statement against a music critic, it fits
for those who follow the Harnoncourt Doctrine without relying on their
own musical instincts and an ear for that which sounds good): "...so
zweifle nicht, es werde des Auctoris Dreckohr gereiniget, und zur
Anhörung der Musik geschickter gemacht werden." 10.12.1749 {"don't
ever doubt that he will get his dirty ears cleaned out so that he, in
time, can then apply his ears to hear music properly.}

ttp://dict.leo.org/ende?lp=ende&lang=de&searchLoc=0&cmpType=relaxedB'Hdr=on&spellToler=on&search=dreck&relink=on

***This guy probably worked on MTV some years ago.Do you remember when
they use to cover female nipples, featured in some videos, with black
strips? He uses the same hypocritical and bigot style : "Dreckor" is
German for "ear full of s***" or "s***y ear", it's not a "clear" dirty
ears. Bach dictated this letter to his ex-pupil Georg Friedrich
Einicke (then Cantor in Frankenhausen) ; after Bach's death Mattheson
harshly condemned what happened : "expression basse & dègoutante,
indigne d'un maitre de chapelle...pauvre allusion au mot: Rector"
[Dreckor-Rector : Bach was writing against Freiberg Rector Biedermann].
See at : Bach-Dokumente, BAND I, 1963, page 53 + comment.***
or in
http://www.media.bachakademie.de/media/pdf/2004061_bachweltlich.pdf -
(contains JSB's popular quote fully in the german original)

Sorry, but i can't take people serious as piano-tuners,
that are not able to discern an ~704Cent wide 5th
versus an "dim-6th" rangeing approx: ~760-~840 Cents.
That problem appears to be deeper hard-wired in the brains author,
rather than barely Biedermann's "dirt-in-the-ears" case.

Conclusion:
hence i'm feeling just confirmed in my right of freedom
to choose the above adequate terms
when considering good old JSB's verbal tradition :-)

Yours Sincerely
A.S.
p.s:
In agreeing with Voltaire's statement:
http://web.archive.org/web/20021020063641/http:/www.riaa.com/Freedom-History.cfm
"I disapprove of what you say, but I will defend to the death your
right to say it."

The great variety of Slendro scales use very wide 5ths. It must be one of the most beautiful scales in the world to my ear.
--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/index.html>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main/index.asp> 88.9 FM Wed 8-9 pm Los Angeles

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
>
> The great variety of Slendro scales
http://en.wikipedia.org/wiki/Slendro
>use very wide 5ths.
http://en.wikipedia.org/wiki/Gamelan

> It must be one
> of the most beautiful scales in the world to my ear.

due to
http://en.wikipedia.org/wiki/Metallophone
's and
http://en.wikipedia.org/wiki/Gong
's
inharmonicity bias effects
resulting in an tuning
with 5ths wider than pure.

J.S.Bach composed so profoundly and robust,
that his music even tolerates to be performed
succesfully in any possible slendro tunings,
not to mention the 'rosetta-stone'd wide 5th: Bb-F ~704 Cents.

A.S.

But my dear Andreas, resorting to disintering the unforgivably juvenile and
sarcastic past utterances of a frustrated doctorand in excuse of your usage
of same in what ought to be a prestigious scholarship?

Granted, I am a Tartuffian wretch for being so inconsistent and fallible in
my behaviour as a human being (hopefully I'll improve), and I am covered in
shame from head to toe (may Allah show mercy upon my soul), is that any
reason for you to take shelter behind the deplorable actions of my poor
self?

Now, I know for a fact that to drivel is to "speak like an idiot" and
daffing is "playing the fool". My earnest apologies and deepest regrets for
ever having spelled the word daft, which means "silly" or "insane". Such
hurtful language surely invites disaster into any academia. One should know
better than to indulge in such words.

Now that I am about to earn a degree in doctorate of musicology, I am
inclined to curb my desires to that effect. Is that not required of any
scholarship?

In the case of Bach's phraseology, I think he was just being humourous,
witty ol' Bach!

However, Bach scholarship should not be constrained to imitating the exact
wording of J.S. Bach in order to denounce convictions against one's liking,
don't you agree?

Cordially,
Oz, with a bulbous nose.

----- Original Message -----
From: "Andreas Sparschuh" <a_sparschuh@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: 20 Haziran 2007 �ar�amba 18:06
Subject: [tuning] Freedom to choose colloquial language words alike JSB did,
was Re: Wide 5ths

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
Dear Ozan,
> My dear Andreas, what language...
Drivel?
What about the earlier usage of that word here in that group?:
/tuning/msearch?query=drivel&submit=Sear
ch&charset=UTF-8

> Daffing?
http://www.wordwebonline.com/en/DAFF
"Verb: daff
Usage: archaic
1. To cast aside; to put off; to doff
Derived forms: daffed, daffs, daffing"
http://www.thefreedictionary.com/cast%20aside
/tuning/msearch?query=daft&submit=Search
&charset=UTF-8
You asked there in using just that expression once yourself in:
/tuning/topicId_64793.html#64837
"Who would be DAFT enough to claim that there is no modulation when
you play solo?"
/tuning/topicId_64269.html#64757
"Therefore, it would be nothing less than a DAFT suggestion to speak
of unison as a musical interval here..."

Can you still remember yours private applications of "daft" there?

How do you want to explain us yours own usage of "daft" there,
before citizing others alike Mark Lindley and me about the same
spelling?

Before bashing others about yours own bad habits
just take look into the mirror and grab yours own nose!

People who live in glass houses shouldn't throw stones.

> Are these words accepted
> as legitimate within the Bach academia?
within the:
http://www.bachakademie.de/
they are translated back into:

http://dict.leo.org/ende?lp=ende&lang=de&searchLoc=0&cmpType=relaxed§Hdr
=on&spellToler=on&search=daffing&relink=on
http://dict.leo.org/ende?lp=ende&lang=de&searchLoc=0&cmpType=relaxed§Hdr
=on&spellToler=on&search=daft&relink=on
http://dict.leo.org/ende?lp=ende&lang=de&searchLoc=0&cmpType=relaxed§Hdr
=on&spellToler=on&search=drivel&relink=on

> What would Bach's tortured spirit
> say?
About Lehman's "dim-6th" theory concerning in Lehman's recordings?
he would probably adjudicate:
http://www.bach-cantatas.com/BWV78-D.htm
"To this recording Bach would have said: "es werde des Auctoris
Dreckohr gereiniget, und zur Anh�rung der Musik geschickter gemacht
werden." 10.12.1749 In this case (Harnoncourt "will have to get his
dirty ears cleaned out so that he, in time, can then apply his ears to
hear music properly.")"

http://www.bach-cantatas.com/Topics/Freedom.htm
Harnoncourt: If you want to hear what Mattheson is talking about when
he refers to the "Schreih�lse" ("those who scream instead of really
trying to sing,") you have an example of this here

http://dict.leo.org/ende?lp=ende&p=/oHL..&search=Schreihals

BWV 94
I am quite certain (just as certain as Harnoncourt was about these
recordings) that Bach would have restated his words (although in the
quote he had directed his statement against a music critic, it fits
for those who follow the Harnoncourt Doctrine without relying on their
own musical instincts and an ear for that which sounds good): "...so
zweifle nicht, es werde des Auctoris Dreckohr gereiniget, und zur
Anh�rung der Musik geschickter gemacht werden." 10.12.1749 {"don't
ever doubt that he will get his dirty ears cleaned out so that he, in
time, can then apply his ears to hear music properly.}

ttp://dict.leo.org/ende?lp=ende&lang=de&searchLoc=0&cmpType=relaxed§Hdr=
on&spellToler=on&search=dreck&relink=on

***This guy probably worked on MTV some years ago.Do you remember when
they use to cover female nipples, featured in some videos, with black
strips? He uses the same hypocritical and bigot style : "Dreckor" is
German for "ear full of s***" or "s***y ear", it's not a "clear" dirty
ears. Bach dictated this letter to his ex-pupil Georg Friedrich
Einicke (then Cantor in Frankenhausen) ; after Bach's death Mattheson
harshly condemned what happened : "expression basse & d�goutante,
indigne d'un maitre de chapelle...pauvre allusion au mot: Rector"
[Dreckor-Rector : Bach was writing against Freiberg Rector Biedermann].
See at : Bach-Dokumente, BAND I, 1963, page 53 + comment.***
or in
http://www.media.bachakademie.de/media/pdf/2004061_bachweltlich.pdf -
(contains JSB's popular quote fully in the german original)

Sorry, but i can't take people serious as piano-tuners,
that are not able to discern an ~704Cent wide 5th
versus an "dim-6th" rangeing approx: ~760-~840 Cents.
That problem appears to be deeper hard-wired in the brains author,
rather than barely Biedermann's "dirt-in-the-ears" case.

Conclusion:
hence i'm feeling just confirmed in my right of freedom
to choose the above adequate terms
when considering good old JSB's verbal tradition :-)

Yours Sincerely
A.S.
p.s:
In agreeing with Voltaire's statement:
http://web.archive.org/web/20021020063641/http:/www.riaa.com/Freedom-History
.cfm
"I disapprove of what you say, but I will defend to the death your
right to say it."

--- In tuning@yahoogroups.com, "Andreas Sparschuh" <a_sparschuh@...>
wrote:
> Sorry, but i can't take people serious as piano-tuners,
> that are not able to discern an ~704Cent wide 5th
> versus an "dim-6th" rangeing approx: ~760-~840 Cents.
> That problem appears to be deeper hard-wired in the brains author,
> rather than barely Biedermann's "dirt-in-the-ears" case.

Apparently you don't understand what a diminished 6th is, or its
typical sizes. In 12-ET the diminished 6th is (obviously) 700 cents,
the same as the 5ths are. In regular 1/6 comma the diminished 6th is
up to about 718 cents. In 1/4 comma meantone it is up to 737. In
none of these cases is it anywhere near your alleged 760-840 range.

And the more important point, as I mentioned last week, is: at least
on harpsichords, a 704-cent interval and a 700-cent interval sound
alike, very difficult to tell one from the other. They are both
beating by the tiniest bit from a pure 702-cent 5th (only 1/12th comma
of impurity here), and merely beating in the opposite direction from
one another.

Brad Lehman

Andreas Sparschuh wrote:
> Who in that group here agree's with his weird point of view that JSB > would had labeled the 5th Bb-F really as an "dim-6th"?

The point is not how it would be spelled in the music; rather, what it is spelled in relation to the chain of 5ths in the temperament.

You certainly have heard of the concept of enharmonic intervals? e.g. in 12-edo an Ab-C (Maj 3rd) and G#-C (dim 4th) are the same sound.

There are plenty of examples of temperaments where enharmonic intervals from a circle of 5ths point-of-view function as usable consonant intervals which are otherwise 'misspelled'

-A.

While i imagine that there are 17 different pitches used in Slendro tunings by vocalist relative to a single example, the idea presented that it relates to the shruti system might be an extreme stretch of the imagination. One question off the top is where is the other 5. maybe it has to with tones in between that each have two options which has been mentioned. In this way i can think of one way the idea could work But i would have to see what exactly the paper being quoted actually says. Bach's rhythm makes his music logically transposable into anything and is not a good evaluation of much. Perhaps a better point, if we are going to obsess on his tuning is how bad Slendro music sounds in any of the tunings put forth. For Slendro tunings, one might be better to look at Kunst as opposed to wikipedia. I have often thought of Bach as a Lutheran gamelan composer. Lou Harrison , Who was anything but shabby when it came to counterpoint, and a major part of his music, mentioned he much preferred to write for gamelan cause he had more parts available. Due to balance issues he would often run out of instruments with the former.
--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/index.html>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main/index.asp> 88.9 FM Wed 8-9 pm Los Angeles

We live with the luxury, or curse, of viewing well-temperaments in
the frame of before and after.

I have a number of opinions about well-temperaments, observations
which may very well be greeted with scorn but so what?

First of all, it's obvious to the ear, and spending some time
calculating and listening to critical band interactions will
confirm this, there is more tempering room above 5/4 than below it.
Listen to 81/64 and its mirror against 5/4, 100/81,
and check out the critical band interactions, and you'll know what I
mean. Fifths, in triadic music, must reflect and be
affected by this, therefore controversy over the size of fifths is
inevitable, and good.

Secondly, when it comes to triadic music, what 12-tET actually
attempts to "equalize", in practice, is not really the whole
gamut (haha), nor are the thirds actually always supposed to be 6/5
and 5/4. What 300 and 400 cents actually do, pretty well,
is split 7/6 and 6/5, and 5/4 and 9/7. If, in acoustic 12tET
practice or in some keys of a well-temperament, 12-tET is itself
tempered, so to speak, to have thirds of 13/11 and 14/11, ie, sing M
high and m low, this is even more "ET" than ET, in this
respect, in addition to acquiring a consonance of its own a couple
steps up the harmonic series.

To step back, and to put it all another way, I believe that one
source of Romantic music was the love of the more distant
keys (das Zimmermaedchen and die Schwarze Gredel, for crying out
loud), an idea tied into class and nationalistic warfare of
the late 18th and the 19th centuries.

If the distant keys were to have been virtually indistinguishable
from the near keys, this would not have happened. The fact
that we have inherited the idea of near and distant keys reflects
more than ordeals of keyboard fingering (the fingers of many
untrained pianists can fall naturally into keys that bristle like
porcupines on paper, cf Irving Berlin and F# Major): it reflects
something that must have once been clearly audible.

Audible, how? The wide variety of tunings throughout the coupla
centuries preceding the 19th defies the idea that specific
key colors were drilled into everyone's head by sheer rote. Yet
there must have been some red thread to a vast number of
tunings throughout the Europes, for some length of time, if there is
any merit to the placing of the sound of the "distant
keys" in the geneology of Romanticism.

I propose that we need look no further than the overtone series to
find an audible and dare I say "natural" red thread in
this case: thirds in the near keys of the well-tempered era focused
on lower partials, and the distant keys on higher. A more
or less infinite number of possible temperaments, in different
places and times, could and I believe did conform to this
general, and audible, idea.

What are we looking at? Basically 5/4-9/7 and their corresponding
minors, with a real-life realization, all things
considered, which has a range of more or less "5/4 + a little"
to "about-14/11", with corresponding minors.

In light of what came after Bach, for example, can we honestly
dismiss a well- temperament because it has Pythagorean or
higher thirds in distant keys? I don't think so, because whether you
call an 81/64, a 14/11 or even, may I be so bold, a 9/7,
"hard", "hysterical", "glaring", or "bright", "bold", "yearning",
you could hardly deny that those kind of sound values, for
better or worse, are increasingly popular in the music of the
Europes, following the time of Bach.

The argument that the inclusion of distant keys in Romantic music is
all about modulations won't fly in my book, bogged down
as it is by things like Wagner dipping the gilded lily into the
laudanum of a single chord for enough time to go out and piss, buy a
beer and come back. No: the keys in the 19th century had distinct
colors, and 12tET wasn't about dictating a grid for the orchestra,
but about neutrality vis-a-vis what were in practice
pretty extreme thirds and other variations between keys. Once this
neutral-sounding grid (12-tET)was entrenched on the
recently ubiquitous piano, it took on a life of its own, another
story (see Taruskin on Schoenberg for a thought-provoking
look at things).

So: to dismiss, say, WkIII or whatever on grounds of an 81/64 or
three is, in my opinion, ahistorical, because it completely
disconnects a well-temperament from its future, viewing only its
past, whereas all we who have birthed and buried know that
things are simply not that way.

I believe that well-temperaments of the 17th and 18th century
weren't about uniformity, but about a diversity justified by
"nature". 81/64 wasn't a crappy 5/4, but a Pythagorean third, the
thirds yet higher weren't disasterous 5/4s, but decent
14/11s, etc. What may have started out as flaws and byproducts
became desirable features at some point.

And so I believe that the search for Bach's temperament should be
focussed on getting as far AWAY from 12-tET as possible,
and that 12-tET is NOT a direct descendant of the WTs of Bach's
time, rather a whole different compromise based on achieving
as neutral a sound as possible in order to avoid clashing with the
INCREASINGLY wild and "microtonal" colors of the following
centuries, a pallette of colors which developed from the "distant
keys".

WTC is not "Giant Steps". I find it hard to believe that a musician
of Bach's (pro)creativity would not have written "Giant
Steps" kind of display pieces if his tuning had actually been as
close to 12-tET as Brad Lehman's is (not that Lehman's
tuning isn't an excellent tuning in its own right ).

I believe that we can speak with some authority about the intonation
of the late 19th century, and that it's
not a stretch to say that a century ago diatonic thinking,
chromaticism and what would today be called "microtonality"
existed together in a very colorful way, as a norm. I'm not the
oldest here and yet I have "from the horse's mouth" musical
instruction by people born in the 19th century. THAT world, as far
as I can tell, indicates a decidely non-12tET way of
thinking and hearing, and I believe its tangibly Romantic nature
goes way back. I simply connect "that" with the Baroque by
way of the "distant keys". If this connection is valid, the "distant
keys" must have been truly "distant".

Put yet another way, I know, physically, the third of the
dominant/leading tone-to-tonic as something that is in no way a 5/4,
but suspiciously similar to a 9/7, from five days a week for five
years with a teacher whose professional career was between the wars
and treasured letters from Richard Strauss himself. Rather than
ahistorically imposing sudden leaps into art, I assume evolutions
and permutations, thread this kind of intonation all the way back to
the "distant keys" of Werckmeister et al.

Anyway, carry on- the religions of 5/4 and 12-tET excommunicate all
I have said above anyway.

-Cameron Bobro

--- In tuning@yahoogroups.com, "Brad Lehman" <bpl@...> wrote:
> In 12-ET the diminished 6th is (obviously) 700 cents,
> the same as the 5ths are.
Not at all!

some 6ths:

1.Pythagoren (1 200 * ln(27 / 16)) / ln(2) = ~905.865003...Cents
2.syntonic (1 200 * ln(5 / 3)) / ln(2) = ~884.358713...Cents
3. ET 2^(9/12) =900C

The accidntial "diminsihed" variant of that 3 cases amount
all ~100Cents less of that, resulting about ~800C,
if we define the "sharp" symbol "#" as 100C exactly,
instad the older pythagorean limma: 2187/2048 := 3^7/2^11

(1 200 * ln((3^7) / (2^11))) / ln(2) = ~113.685006...Cents

Considering that:
In 12-ET any "dim-6th" corresponds enharmonic equivalent
to an augmented-5th (700C+100C)
Both do amount exactly 2^(8/12) the same 800 Cents.
The intervals (C-Ab) = (C-G#) coincide enharmonically
equivalent to each others in any circular well-tuning
without over-broade 5ths claims.

The 12-ET all the dozen 5ths of 700Cents 2^(7/12)
-without any alleged "meantonic" exception, as you try to assume-
turn out to become congruent enharmonically to:
1. the double augmented 4th: interval (C-G) = (C-F##) or
2. the double! diminshed 6th: interval (G-G) = (C-Abb)
but should never be mistaken alike in yours private terminology
an "dim-6th" of exactly 800 Cents 2^(8/12).

Do you really mean that an simple accidential 'sharp' symbol
amounts in 12-ET an whole-tone of 200Cents 2^(2/12),
instead barely the half of that value:
100Cents 2^(1/12) as common usual in 12-ET general notation.

http://www.experiencefestival.com/a/Accidental_music_-_Standard_use_of_accidentals/id/4766666
"Double accidentals raise or lower the pitch of a note by two
semitones, an innovation developed as early as 1615. An F with a
double sharp applied raises it a whole step so it is enharmonic with a G."
Respectively an dim-6th (C-Ab)
is an double-sharp 5th (C-Gx) or (C-G##),
rahter than defintive not barely an single sharp only.

Have you ever realized the difference among
'double'=200Cents versus 'single'=100Cents in 12-ET?
http://en.wikipedia.org/wiki/Accidental_(music)

Attend: When seeing that
http://en.wikipedia.org/wiki/Image:Double_accidentals.gif
symbols in yours score that means an alteration about
200Cents or 2 buttons up or down on yours harpsichord
or organ keyboard.
again more elaborated in detail :
"Double accidentals raise or lower the pitch of a note by two
semitones, an innovation developed as early as 1615. An F with a
double sharp applied raises it a whole step so it is enharmonically
equivalent to a G. Usage varies on how to notate the situation in
which a note with a double sharp is followed in the same measure by a
note with a single sharp: some publications simply use the single
accidental for the latter note, whereas others use a combination of a
natural and a sharp, with the natural being understood to apply to
only the second sharp."

Analysis:
maybe that lack of clarity in double accidentials caused yours
permanent problems in resolving such dubieties in musical notation?

> In regular 1/6 comma the diminished 6th is
> up to about 718 cents.
Wrong!
An dim-6th (C-Ab) in '1/6 comma' consists in a chain of 4 times 4ths
C-F-Bb-Eb-Ab
with each 4th about 1/6 comma wider than pure.
Distinct the 2 cases of comma:
1. Pythagorean: 800C + 4/6*~23.4...C = ~815.6...Cents
2. syntonic: 800C +4/6*~21.5...C = ~814.3...Cents

Next problem:
> In 1/4 comma meantone it is up to 737.
Nonsene:
Respectively: 800C + 4/4*SC = yielding ~821.5...Cents
analogous.

The meantonic 'dim-6th' 8/5 (C-Ab) or enh. (C-G#)
complements 3rd 5/4 (C-E) ~384C to the octave.

Hence yours private marking is misleading astray
concerning the range of 3-limit and 5-limit intervals.

> In
> none of these cases is it anywhere near your alleged 760-840 range.
Thers is nothing wrong about the inequation
760 << 814.3 < 821.5 << 840
when determening the 12-ET 'dim-6th'(C-Ab) = 800 Cents.

The correct calclution meets just into that range well fitting,
when reminding that an augmentation in 12et amounts +100C in seize,
and an diminshing about -100C respectively in the reverse direction.

Appearently you contine to confuse 800 from 700 Cents,
by insiting that there is no difference of 100 Cents among them.
Sorry,
but i know no cure for that persitent osbtinacy.
>
> And the more important point, as I mentioned last week, is: at least
> on harpsichords, a 704-cent interval and a 700-cent interval sound
> alike,
that's correct barely only in pure sine-tones, without the higher
partials above the 5th,
But when considering also the higher partials,
as experienced tuners always inherently do,
then one discerns easily inbetween ~700 and ~704 Cents
as distinct in the higher harmoincs-beating from each others,
especially just on harpsichords with an rich spectre in timbre.

> very difficult to tell one from the other.
That may well be for own yours ears:
You should learn at least that the sound of harpsichords
contains a strong 5th partial (double octaved just 3rd=5/4)
in order to improve yours basic tuning skills,
not to mention the resonant seventh 7/4 of
(1 200 C * ln(7 / 4)) / ln(2) = ~968.825906 Cents
involved in
professional tuning:
Literature reccomendation:
http://www.amazon.de/Lehrgang-Stimmkunst-Josef-Nix/dp/3923639732
http://www.allbookstores.com/book/9783920112022/Josef_Nix/Lehrgang_Der_Stimmkunst.html
http://ugenie.com/products/Lehrgang_Der_Stimmkunst_4th_edition?isbn=9783923639748&productId=BOK-13133414-1&catId=1&rank=1&click=title
http://www.amazon.co.jp/Lehrgang-Stimmkunst-Josef-Nix/dp/3923639732
in order learning to discern inbetween wide and flat 5ths
by counting beats inbetween the higher partials.

> They are both
> beating by the tiniest bit from a pure 702-cent 5th (only 1/12th >comma
> of impurity here),
Just known as the difference inbetween an just 5th and 12-ET 5th.

> and merely beating in the opposite direction from
> one another.
>
only when neglecting the higher partials than 3rd order,
barely
than (0-700) turn out to be hard distinguishable from (0-704)

first step testing that:
try out to keep apart asunder the example triads:
(0-384-700)
against versus
(0-384-704)
in order to hear that live in yours own personal experience.

happy success in trying out to undergo and proceed
in counting triadic beats that way

in never abandoning hope
A.S.

--- In tuning@yahoogroups.com, "Andreas Sparschuh" <a_sparschuh@...>
wrote:
>
> --- In tuning@yahoogroups.com, "Brad Lehman" <bpl@> wrote:
> > In 12-ET the diminished 6th is (obviously) 700 cents,
> > the same as the 5ths are.
> Not at all!
>

Huh? I learnt the intervals in 12-ET as follows:

1100 cents - major 7th
100 cents - minor 7th/augmented 6th
900 cents - major 6th/diminished 7th
800 cents - minor 6th/augmented 5th
700 cents - diminished 6th/perfect 5th
--
Hans Straub

> If the distant keys were to have been virtually indistinguishable
> from the near keys, this would not have happened. The fact
> that we have inherited the idea of near and distant keys reflects
> more than ordeals of keyboard fingering (the fingers of many
> untrained pianists can fall naturally into keys that bristle like
> porcupines on paper, cf Irving Berlin and F# Major): it reflects
> something that must have once been clearly audible... whether you
> call an 81/64, a 14/11 or even, may I be so bold, a 9/7,
> "hard", "hysterical", "glaring", or "bright", "bold", "yearning",
> you could hardly deny that those kind of sound values, for
> better or worse, are increasingly popular in the music of the
> Europes, following the time of Bach.

http://en.wikipedia.org/wiki/Ghent_Altarpiece
It says here 15th century Flemish hymnals instructed different
kinds of expressions for different notes.

Clark

--- In tuning@yahoogroups.com, "Andreas Sparschuh" <a_sparschuh@...>
wrote:
>
> --- In tuning@yahoogroups.com, "Brad Lehman" <bpl@> wrote:
> > In 12-ET the diminished 6th is (obviously) 700 cents,
> > the same as the 5ths are.
> Not at all!
>
> some 6ths:
>
> 1.Pythagoren (1 200 * ln(27 / 16)) / ln(2) = ~905.865003...Cents
> 2.syntonic (1 200 * ln(5 / 3)) / ln(2) = ~884.358713...Cents
> 3. ET 2^(9/12) =900C
>
> The accidntial "diminsihed" variant of that 3 cases amount
> all ~100Cents less of that, resulting about ~800C,
> if we define the "sharp" symbol "#" as 100C exactly,
> instad the older pythagorean limma: 2187/2048 := 3^7/2^11
>
> (1 200 * ln((3^7) / (2^11))) / ln(2) = ~113.685006...Cents
>
> Considering that:
> In 12-ET any "dim-6th" corresponds enharmonic equivalent
> to an augmented-5th (700C+100C)
> Both do amount exactly 2^(8/12) the same 800 Cents.
> The intervals (C-Ab) = (C-G#) coincide enharmonically
> equivalent to each others in any circular well-tuning
> without over-broade 5ths claims.

Yadda yadda yadda, yadda yadda yadda. And obviously you don't know
what we're referring to as a DIMINISHED 6th, which is (indeed) 700
cents in 12-ET.

Perhaps you're confusing all of this with a minor 6th? You're first
calculating major 6ths and then whacking approximately 100 cents off
them (to get pseudo-minor-6ths); but we're talking about something
entirely different and always have been: DIMINISHED 6ths, the ones
that are approximately a semitone smaller than a minor 6th.

By the way, the diminished 6th in Pythagorean is about 678 cents.
Tiny! Smaller than the 5ths!

Brad Lehman

--- In tuning@yahoogroups.com, "Brad Lehman" <bpl@...> wrote:
>

>
> Yadda yadda yadda, yadda yadda yadda. And obviously you don't know
> what we're referring to as a DIMINISHED 6th, which is (indeed) 700
> cents in 12-ET.

http://www.jdooley.com/aug_5_dim_6.htm
"The augmented fifth or diminished sixth is equal to four whole steps,
or eight frets distance between the two notes."
in 12 ET:
4* 200 Cents = 800 Cents

Who else in that group -except the famous musicologist Dr. Lehman-
confuses that with an tiny to much over-broadly 5th ~704 Cents?

extreme minimal variant in:
http://www.kylegann.com/Octave.html
"192/125 743.014 diminished sixth (8/5 x 24/25)"

but no more less than that ~743 Cents.

even in meantonics:
http://rollingball.com/images/HT3.htm
"Another characteristic
is the augmented fifth (actually a diminished sixth) at Ab, the "wolf"
fifth. In any meantone
temperament, one-third of the potential harmony was intolerable. "

Your problem appears to confuse simply the
just PURE-5th ~702Cents without any accidenti
with it's augmented(+100Cents)-5th a halftone higher.

> Perhaps you're confusing all of this with a minor 6th?
No, we had talked all the time about major,
-convention: without futher labeling means always major intervals-
but may be there's perhaps an other solution of the "dim-6th" riddle:

NP-4FKYDTN-3&_user=2717328&_coverDate=05%2F01%2F2005&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000056831&_version=1&_urlVersion=0&_userid=2717328&md5=8e991344ddcd770ba787500b8c627f8f

"Neapolitan chord (irregular, a Neapolitan is a minor subdominant with
a diminished sixth instead of a fifth). Classically, Neapolitan chords
are sixth chords, but for a proper comparison between Neapolitans and
tonics, both tonic and Neapolitan chords were presented equiprobably
in root position and as sixth chords"

http://en.wikipedia.org/wiki/Neapolitan_chord

Can i conclude from that in Brad's
broade 5th: Bb-F tuning
the major chord Bb-D-F (0~390~704Cents)
sounds in his ears not really as major chord,
but rather Neapolitan (dim-6th) chord?

Jimmy Hendrix:
http://www.songfacts.com/detail.php?id=2553
"The opening "interval" is actually a SHARP or AUGMENTENTED fifth, not
a flat fifth. You could also call it a flat or diminished SIXTH. More
correctly, it is an E7aug9 chord played with the two E strings open,
bouncing back & forth between the open bass string and the fretted
strings. I doubt Jimi knew this was Musica Diabolica; it was
considered such by the enlightend members of the church because it was
a discordant SIXTH, making it doubly demonic."

Hmm,
considering that,
Brad's "dim-6th" sounds especially in the key
F-major really
devilish "Musica-Diabolica"
apt for expressing demonic feelings
in order to reply the question:
"Where the deuce did he hide it?"

> You're first
> calculating major 6ths and then whacking approximately 100 cents off
> them
"diminsihing" means in 12-ET simply to subtract actually 100Cents
from the ET-6th of 900Cents resulting in 900-100=800 Cents
as final seize of any "dim-6th" in 12-ET.

> DIMINISHED 6ths, the ones
> that are approximately a semitone smaller than a minor 6th.

http://library.thinkquest.org/17321/data/glossary.html
"Sixth - The sixth degree of the diatonic scale. Also, the interval
formed by a given tone and the sixth tone above or below it, e.g. c up
to a, or c down to e. Intervals of the sixth may be major, minor,
diminished, or augmented."

In deed the
the minor-and-diminished sixth (C-Abb) or (C-F##)
also labeled as DOUBLE-dimished
turns out meet enharmoniclly equivalent an 12-ET
5th=700C ,
1/12 octave less than an simple single-"dim-6th"=800C,
barley one-times unique diminished only.

Is that so difficult to see?

You should correct that soon at least in yours web pages,
that you had meant the minor-dim. or double-dim. 6th,
in order to avoid further trouble.
> By the way, the diminished 6th in Pythagorean is about 678 cents.
> Tiny! Smaller than the 5ths!
What an crude logic in that an accidental 6th is
located below an pure 5th?

http://en.wikipedia.org/wiki/Pythagorean_tuning

"B 27:16 905.87 900 major sixth"

sounding in pitch a whole tone 9/8 above the

"A 3:2 701.96 700 perfect fifth"

with the pyth-dim. 6th an ~half-tone inbetween that:
pitch-name: A# or Bb @ ~800 Cents.

Probably you mean by the above interval of
~678Cents = ~702C - ~24C (C Abb)
an pure 5th diminshed by an PC (Pythagorean Comma)
from 11 times downwards 5ths or 11 upwards 4ths:

C F Bb Eb Ab Db Gb Cb Fbb Bbb Ebb Abb~678C

correctly labeld as 'double-diminished 6th' (Abb)
Conversely the simple-dim-6th (Ab) is located
barely the fourth position of 4ths in that series
about an sharp apotome
http://de.wikipedia.org/wiki/Apotome
'#'=3^7/2^11 ~112C higher than the requested Abb~678C.

to Hans Straub:
>Huh? I learnt the intervals in 12-ET as follows:

>1100 cents - major 7th
>100 cents - minor 7th/augmented 6th
>900 cents - major 6th/diminished 7th
>800 cents - minor 6th/augmented 5th
>700 cents - diminished 6th/perfect 5th

hmm, that nomenclature appears to be inconsistent with the
Pythagorean definitions of the accidentials 'b' and '#':

diminshing:
=equivalent= 1 apotome downwards, labeled by an 'b'flat sign

augmenting:
=equivalent= 1 apotome upwards, labeled by an '#'sharp sign.

Is 12-ET terminology different from that:
augmenting means there in some cases
an deviation of even +-200Cents (about an whole tone)
but also in other cases barely +-100 (half tone)?

Who in that group knows knows certain:
How much Cents in seize amounts diminishing/augmenting in ET really?

A.S.

A.S.

I agree with Andreas. If an interval is diminished, it is subtracted a
semitone, if augmented, it is added a semitone.

In the meantone or Pythagorean philosophy of the cycle of fifths, G#-Eb is a
wolf fifth, first a comma wide, the latter a comma narrow.

Oz.

SNIP

>
> diminshing:
> =equivalent= 1 apotome downwards, labeled by an 'b'flat sign
>
> augmenting:
> =equivalent= 1 apotome upwards, labeled by an '#'sharp sign.
>

SNIP

----- Original Message -----
From: "Brad Lehman" <bpl@umich.edu>
To: <tuning@yahoogroups.com>
Sent: 21 Haziran 2007 Per�embe 19:36
Subject: [tuning] Confusing "dim-6ths" with Re: Wide 5ths

> --- In tuning@yahoogroups.com, "Andreas Sparschuh" <a_sparschuh@...>
> wrote:
> >
> > --- In tuning@yahoogroups.com, "Brad Lehman" <bpl@> wrote:
> > > In 12-ET the diminished 6th is (obviously) 700 cents,
> > > the same as the 5ths are.
> > Not at all!
> >
> > some 6ths:
> >
> > 1.Pythagoren (1 200 * ln(27 / 16)) / ln(2) = ~905.865003...Cents
> > 2.syntonic (1 200 * ln(5 / 3)) / ln(2) = ~884.358713...Cents
> > 3. ET 2^(9/12) =900C
> >
> > The accidntial "diminsihed" variant of that 3 cases amount
> > all ~100Cents less of that, resulting about ~800C,
> > if we define the "sharp" symbol "#" as 100C exactly,
> > instad the older pythagorean limma: 2187/2048 := 3^7/2^11
> >
> > (1 200 * ln((3^7) / (2^11))) / ln(2) = ~113.685006...Cents
> >
> > Considering that:
> > In 12-ET any "dim-6th" corresponds enharmonic equivalent
> > to an augmented-5th (700C+100C)
> > Both do amount exactly 2^(8/12) the same 800 Cents.
> > The intervals (C-Ab) = (C-G#) coincide enharmonically
> > equivalent to each others in any circular well-tuning
> > without over-broade 5ths claims.
>
>
> Yadda yadda yadda, yadda yadda yadda. And obviously you don't know
> what we're referring to as a DIMINISHED 6th, which is (indeed) 700
> cents in 12-ET.
>
> Perhaps you're confusing all of this with a minor 6th? You're first
> calculating major 6ths and then whacking approximately 100 cents off
> them (to get pseudo-minor-6ths); but we're talking about something
> entirely different and always have been: DIMINISHED 6ths, the ones
> that are approximately a semitone smaller than a minor 6th.
>
> By the way, the diminished 6th in Pythagorean is about 678 cents.
> Tiny! Smaller than the 5ths!
>
>
> Brad Lehman
>
>

But I see what Brad means, he is correct. a diminished minor 6th in a
12-tone closed cycle is a perfect fifth.

Oz.

> > By the way, the diminished 6th in Pythagorean is about 678 cents.
> > Tiny! Smaller than the 5ths!
>
> But I see what Brad means, he is correct. a diminished minor 6th in a
> 12-tone closed cycle is a perfect fifth.

Almost everything I post is in the context of 12-note keyboard
temperaments.

And the "diminished 6th" is the point wherever we meet the circle of
5ths coming around the other side, and join ourselves with *some*
manner of non-5th as that 12th interval. Eleven 5ths of some size(s)
in one direction, meeting the enharmonic swap from the opposite
direction...for example, if we've done some cycle of
Eb-Bb-F-C-G-D-A-E-B-F#-C#-G#, our diminished 6th is that G# to Eb
non-5th resulting. We have a raised G (sharpened) and a lowered E
(flattened): that's a 6th because it's *some* G and *some* E taken
together; and it's diminished because we've brought both notes inward
by a semitone. It's a semitone smaller than whatever minor 6th the
temperament happens to have.

Or the breaking point could be between some other pair of notes, but
it always exists. D# to Bb, or A# to F, or G# to Eb, or C# to Ab, or
whatever! Seven semitones (of whatever size(s)) out of the twelve, in
the closed 12-note system.

The enharmonic swap itself buys us one Pythagorean comma, less
whatever we've burned off by narrowing any of the 5ths. In the case
of Pythagorean, since we've burned off nothing but have eleven pure
5ths, our Pythagorean comma remains...and our diminished 6th is (sure
enough) a pure 5th of ~702 less a comma of ~24, i.e. 678.

In the case of 12-ET, we've burned off 11/12th of a comma (1/12 for
each of the 11 5ths we've generated), and when we meet our diminished
6th (wherever we choose to put it, having named our 12 notes
enharmonically) we simply have that remaining 1/12th, i.e. resulting
in an interval of 702 - 2 = 700.

In the case of various meantone schemes, and some irregular systems,
we take out *more* than a total of one Pythagorean comma among those
eleven 5ths, and the diminished 6th ends up being somewhat larger than
702. So what? That's not necessarily a flaw! The thing doesn't even
*start* to sound like any manner of wolf, until (arguably) at least
707 or 708...but that's a matter of taste, of course, and even then it
depends on the musical context where it's played.

And the more we've taken out among those eleven 5ths, the smaller (on
average) all of those eight resulting major 3rds are going to be, the
eight correctly-spelled major 3rds. By compensation, the four
diminished 4ths -- the four intervals that cross the enharmonic gap --
get wider.

The single "overshoot" point allows us to have a greater range of
variety among the sizes of all twelve major 3rds -- if we're going to
pretend that the four diminished 4ths are major 3rds, or if we're just
going to go ahead and use them musically as if they were...which is
the point of having a circulating temperament in the first place.

Equal temperament (12-ET) happens to have four diminished 4ths in it,
too, technically along with its diminished 6th; but the whole thing
has been so smoothed out where they all sound the same, that the
practical distinction tends to vanish.

This is not rocket science, or even bottle-rocket science. It's
straightforward hands-on harpsichord tuning, in action. All the
numerical stuff of cents (or whatever) is just a writing-down of the
result, a convenient measurement.

Brad Lehman

Very well put. I have no objections to any of these. Kudos for you.

Oz.

----- Original Message -----
From: "Brad Lehman" <bpl@umich.edu>
To: <tuning@yahoogroups.com>
Sent: 22 Haziran 2007 Cuma 0:54
Subject: [tuning] Confusing "dim-6ths" with Re: Wide 5ths

> > > By the way, the diminished 6th in Pythagorean is about 678 cents.
> > > Tiny! Smaller than the 5ths!
> >
> > But I see what Brad means, he is correct. a diminished minor 6th in a
> > 12-tone closed cycle is a perfect fifth.
>
> Almost everything I post is in the context of 12-note keyboard
> temperaments.
>
> And the "diminished 6th" is the point wherever we meet the circle of
> 5ths coming around the other side, and join ourselves with *some*
> manner of non-5th as that 12th interval. Eleven 5ths of some size(s)
> in one direction, meeting the enharmonic swap from the opposite
> direction...for example, if we've done some cycle of
> Eb-Bb-F-C-G-D-A-E-B-F#-C#-G#, our diminished 6th is that G# to Eb
> non-5th resulting. We have a raised G (sharpened) and a lowered E
> (flattened): that's a 6th because it's *some* G and *some* E taken
> together; and it's diminished because we've brought both notes inward
> by a semitone. It's a semitone smaller than whatever minor 6th the
> temperament happens to have.
>
> Or the breaking point could be between some other pair of notes, but
> it always exists. D# to Bb, or A# to F, or G# to Eb, or C# to Ab, or
> whatever! Seven semitones (of whatever size(s)) out of the twelve, in
> the closed 12-note system.
>
> The enharmonic swap itself buys us one Pythagorean comma, less
> whatever we've burned off by narrowing any of the 5ths. In the case
> of Pythagorean, since we've burned off nothing but have eleven pure
> 5ths, our Pythagorean comma remains...and our diminished 6th is (sure
> enough) a pure 5th of ~702 less a comma of ~24, i.e. 678.
>
> In the case of 12-ET, we've burned off 11/12th of a comma (1/12 for
> each of the 11 5ths we've generated), and when we meet our diminished
> 6th (wherever we choose to put it, having named our 12 notes
> enharmonically) we simply have that remaining 1/12th, i.e. resulting
> in an interval of 702 - 2 = 700.
>
> In the case of various meantone schemes, and some irregular systems,
> we take out *more* than a total of one Pythagorean comma among those
> eleven 5ths, and the diminished 6th ends up being somewhat larger than
> 702. So what? That's not necessarily a flaw! The thing doesn't even
> *start* to sound like any manner of wolf, until (arguably) at least
> 707 or 708...but that's a matter of taste, of course, and even then it
> depends on the musical context where it's played.
>
> And the more we've taken out among those eleven 5ths, the smaller (on
> average) all of those eight resulting major 3rds are going to be, the
> eight correctly-spelled major 3rds. By compensation, the four
> diminished 4ths -- the four intervals that cross the enharmonic gap --
> get wider.
>
> The single "overshoot" point allows us to have a greater range of
> variety among the sizes of all twelve major 3rds -- if we're going to
> pretend that the four diminished 4ths are major 3rds, or if we're just
> going to go ahead and use them musically as if they were...which is
> the point of having a circulating temperament in the first place.
>
> Equal temperament (12-ET) happens to have four diminished 4ths in it,
> too, technically along with its diminished 6th; but the whole thing
> has been so smoothed out where they all sound the same, that the
> practical distinction tends to vanish.
>
> This is not rocket science, or even bottle-rocket science. It's
> straightforward hands-on harpsichord tuning, in action. All the
> numerical stuff of cents (or whatever) is just a writing-down of the
> result, a convenient measurement.
>
>
> Brad Lehman
>

--- In tuning@yahoogroups.com, "Aaron K. Johnson" <aaron@...> wrote:

> There are plenty of examples of temperaments where enharmonic
intervals
> from a circle of 5ths point-of-view function as usable consonant
> intervals which are otherwise 'misspelled'

If the meantone tuning is mild enough there isn't a heck of a lot of
difference between the wolf=diminished sixth and a fifth anyway. For
instance, 11 fifths of 158deg271 and a "wolf" of 159deg271 complete a
circle of 12 fifths. But the "wolf" of 704.059 cents is actually in
slightly better tune than the "perfect fifths" of 699.631 cents.

--- In tuning@yahoogroups.com, "Brad Lehman" <bpl@...> wrote:

> Perhaps you're confusing all of this with a minor 6th? You're
first
> calculating major 6ths and then whacking approximately 100 cents
off
> them (to get pseudo-minor-6ths); but we're talking about something
> entirely different and always have been: DIMINISHED 6ths, the ones
> that are approximately a semitone smaller than a minor 6th.

In particular, a chromatic semitone, not a diatonic semitone.
It is whatever the (regular) temperament approximates the apotome of
2187/2048 as; or if we assume 81/80 is tempered out, which we are
implicitly doing, we can call it whatever 25/24 is approximated hy,

The confusion comes since (16/15)/(25/24) = 128/125, and if we temper
out both 81/80 and 128/125, then the two are the same. But only 12-et
does this!

> By the way, the diminished 6th in Pythagorean is about 678 cents.
> Tiny! Smaller than the 5ths!

Here's the calculation: suppose we call the Pythagorean major third
(9/8)^2 = 81/64, so that the Pythagorean minor sixth is 128/81. Take
that down an apotome and you get 262144/177147, 678.495 cents. I putr
in a "suppose" there because there's the diminished fourth theory of
what a major third ought to be: 4/3 down by an apotome is 8192/6561,
which happens to be a much better major third. The
corresponding "minor sixth" would be 6561/4096, and taking that down
by an apotome gives--wait for it--3/2.

--- In tuning@yahoogroups.com, "Andreas Sparschuh" <a_sparschuh@...>
wrote:

> http://www.jdooley.com/aug_5_dim_6.htm
> "The augmented fifth or diminished sixth is equal to four whole
steps,
> or eight frets distance between the two notes."
> in 12 ET:
> 4* 200 Cents = 800 Cents
>
> Who else in that group -except the famous musicologist Dr. Lehman-
> confuses that with an tiny to much over-broadly 5th ~704 Cents?

The definition you quote is wrong. This can happen when you cull
information from the Web.

> extreme minimal variant in:
> http://www.kylegann.com/Octave.html
> "192/125 743.014 diminished sixth (8/5 x 24/25)"
>
> but no more less than that ~743 Cents.

That's a 5-limit version of a diminished sixth, since it is a 5-limit
minor sixth of 8/5 taken down a 5-limit chromatic semitone of 24/25.
The result is that you can call whatever this interval gets
approximated by in any meantone system a "diminished sixth":

12: 700 cents
55: 720 cents
50: 744 cents

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> I agree with Andreas. If an interval is diminished, it is subtracted a
> semitone, if augmented, it is added a semitone.

No, it is required that what is added or substracted is a *chromatic*
semitone. Diatonic semitones do not count, unless (as in 12-et) they
are the same as chromatic semitones.

Precisely.

----- Original Message -----
From: "Gene Ward Smith" <genewardsmith@sbcglobal.net>
To: <tuning@yahoogroups.com>
Sent: 22 Haziran 2007 Cuma 4:46
Subject: [tuning] Confusing "dim-6ths" with Re: Wide 5ths

> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
> >
> > I agree with Andreas. If an interval is diminished, it is subtracted a
> > semitone, if augmented, it is added a semitone.
>
> No, it is required that what is added or substracted is a *chromatic*
> semitone. Diatonic semitones do not count, unless (as in 12-et) they
> are the same as chromatic semitones.
>

--- In tuning@yahoogroups.com, "Brad Lehman" <bpl@...> wrote:

> Yadda yadda yadda, yadda yadda yadda. And obviously you
> don't know what we're referring to as a DIMINISHED 6th,
> which is (indeed) 700 cents in 12-ET.
>
> Perhaps you're confusing all of this with a minor 6th?
> You're first calculating major 6ths and then whacking
> approximately 100 cents off them (to get pseudo-minor-6ths);
> but we're talking about something entirely different and
> always have been: DIMINISHED 6ths, the ones that are
> approximately a semitone smaller than a minor 6th.
>
> By the way, the diminished 6th in Pythagorean is about
> 678 cents. Tiny! Smaller than the 5ths!

Just for the sake of comparison, here are the cents values
for diminished-6ths of various tunings, found in any 12-tone
tuning between for example G# on the bottom and Eb on top:

pythagorean ......... 678.49
1/8-skhisma ......... 681.18
12-edo .............. 700
1/6-comma meantone .. 717.92
1/4-comma meantone .. 737.64
golden meantone ..... 741.64
19-edo .............. 757.89

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

A hypothesis about this:

The augmenteds are derived from steps of fifths;

whereas for the (particularly demented) diminisheds - they are from steps of fourths.

i.e. the two directions around the spiral of fourths and fifths.

Charles Lucy lucy@lucytune.com

----- Promoting global harmony through LucyTuning -----

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On 22 Jun 2007, at 03:14, Ozan Yarman wrote:

> Precisely.
>
> ----- Original Message -----
> From: "Gene Ward Smith" <genewardsmith@sbcglobal.net>
> To: <tuning@yahoogroups.com>
> Sent: 22 Haziran 2007 Cuma 4:46
> Subject: [tuning] Confusing "dim-6ths" with Re: Wide 5ths
>
>
>> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>>>
>>> I agree with Andreas. If an interval is diminished, it is >>> subtracted a
>>> semitone, if augmented, it is added a semitone.
>>
>> No, it is required that what is added or substracted is a *chromatic*
>> semitone. Diatonic semitones do not count, unless (as in 12-et) they
>> are the same as chromatic semitones.
>>
>
>

--- In tuning@yahoogroups.com "Brad Lehman" wrote:
>
> > Eb-Bb-F-C-G-D-A-E-B-F#-C#-G#, our diminished 6th is that G# to Eb
> > non-5th resulting. We have a raised G (sharpened) and a lowered E
> > (flattened): that's a
DOUBLE-diminshed, because you have to count at first the one sharp
"#"-symbol and then additional another "b"-symbol too,
hence you obtain a TWICE-diminsihed sixt (=enharmonic 5th)
instead yours hypothesis of a barely SINGLE dimished-6th.

>> 6th because it's *some* G and *some* E taken
> > together;
add both that 2 accidentials all together,
the resulting interval is enharmonic equivalent in seize to
(G#-Eb)=(C-Abb)=(C-G) but defintively not
an SINGLE "dim-6th":(G#-E)=(G-Eb)=(C-Ab)=(C-G#)
located an semitone higher.

Is it really so diffcult to admit that
you simply forgot to count the second accidential shift downwards
from barely SINGLE to DOUBLE accidential down-alterrated?

>> and it's diminished because we've brought
emphsis on
> both
"BOTH" means here twice or double
> notes inward
> > by a semitone. It's a semitone smaller than whatever minor 6th >>the
> > temperament happens to have.

Ok,
one can explain the one "b"-accidential down
accordingly by the term "minor":

resulting in:
"minor-dim.-6th" = "double-dim.-6th"

Then you sould replace the ambigous (major or minor?)
"dim.-6th" in that inprecise terminology more exactly
by "minor-dim.-6th" in order to prevent further confusion.

A.S.

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

>
> http://en.wikipedia.org/wiki/Ghent_Altarpiece
> It says here 15th century Flemish hymnals instructed different
> kinds of expressions for different notes.

That's really cool- I guess you'd have to stick your tongue out and
cross your eyes for the tritonus, which would clear up the mystery of
Gene Simmon's anachronistic appearances in 15th century paintings! :-)

Andreas is right, for G-Eb truly IS a diminished sixth and thus G#-Eb
becomes double diminished. But G-Eb could be considered a minor sixth
instead in its own right, in which case Brad is correct to say that it is a
diminished sixth.

So end this meaningless feud already.

Oz.

----- Original Message -----
From: "Andreas Sparschuh" <a_sparschuh@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: 22 Haziran 2007 Cuma 15:08
Subject: [tuning] Confusing "dim-6ths" with Re: Wide 5ths

> --- In tuning@yahoogroups.com "Brad Lehman" wrote:
> >
> > > Eb-Bb-F-C-G-D-A-E-B-F#-C#-G#, our diminished 6th is that G# to Eb
> > > non-5th resulting. We have a raised G (sharpened) and a lowered E
> > > (flattened): that's a
> DOUBLE-diminshed, because you have to count at first the one sharp
> "#"-symbol and then additional another "b"-symbol too,
> hence you obtain a TWICE-diminsihed sixt (=enharmonic 5th)
> instead yours hypothesis of a barely SINGLE dimished-6th.
>
> >> 6th because it's *some* G and *some* E taken
> > > together;
> add both that 2 accidentials all together,
> the resulting interval is enharmonic equivalent in seize to
> (G#-Eb)=(C-Abb)=(C-G) but defintively not
> an SINGLE "dim-6th":(G#-E)=(G-Eb)=(C-Ab)=(C-G#)
> located an semitone higher.
>
> Is it really so diffcult to admit that
> you simply forgot to count the second accidential shift downwards
> from barely SINGLE to DOUBLE accidential down-alterrated?
>
> >> and it's diminished because we've brought
> emphsis on
> > both
> "BOTH" means here twice or double
> > notes inward
> > > by a semitone. It's a semitone smaller than whatever minor 6th >>the
> > > temperament happens to have.
>
> Ok,
> one can explain the one "b"-accidential down
> accordingly by the term "minor":
>
> resulting in:
> "minor-dim.-6th" = "double-dim.-6th"
>
> Then you sould replace the ambigous (major or minor?)
> "dim.-6th" in that inprecise terminology more exactly
> by "minor-dim.-6th" in order to prevent further confusion.
>
> A.S.
>

Hi Ozan and Andreas,

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> Andreas is right, for G-Eb truly IS a diminished sixth
> and thus G#-Eb becomes double diminished. But G-Eb could
> be considered a minor sixth instead in its own right,
> in which case Brad is correct to say that it is a
> diminished sixth.
>
> So end this meaningless feud already.
>
> Oz.
>
> ----- Original Message -----
> From: "Andreas Sparschuh" <a_sparschuh@...>
> To: <tuning@yahoogroups.com>
> Sent: 22 Haziran 2007 Cuma 15:08
> Subject: [tuning] Confusing "dim-6ths" with Re: Wide 5ths
>
> > Eb-Bb-F-C-G-D-A-E-B-F#-C#-G#, our diminished 6th
> > is that G# to Eb non-5th resulting. We have a
> > raised G (sharpened) and a lowered E (flattened):
> > that's a DOUBLE-diminshed, because you have to
> > count at first the one sharp "#"-symbol and then
> > additional another "b"-symbol too, hence you obtain
> > a TWICE-diminsihed sixt (=enharmonic 5th) instead
> > yours hypothesis of a barely SINGLE dimished-6th.

No, Andreas is incorrect -- at least for Western theory.
I can't speak for Turkish or other non-Western theory.

See my Encyclopedia page:
http://tonalsoft.com/enc/i/interval.aspx

In Western music-theory, there are two aspects to the
naming of intervals:

1) the actual name, which is an ordinal number, counting
all of the letters subtended by the interval, including
the letters of both notes in the interval: prime, 2nd,
3rd, 4th, 5th, 6th, 7th, 8ve, etc.; and

2) the quality of the interval: that is, the basic
sizes are either perfect or imperfect, with the
imperfect intervals further divided into minor (small)
and major (big). If the interval is a chromatic-semitone
larger than perfect or major, it is augmented, and if
it is a chromatic-semitone smaller than perfect or minor,
it is diminished.

You can use those two rules to find the correct interval
name for any interval that can be found in standard
theory. In diatonic tunings, it's easy enough to
count the half-steps (both diatonic and chromatic
semitones) to find the quality -- i have some tables
on my webpage showing how this works for 12-edo,
19-edo, and 31-edo.

However, note that it is not a simple matter to just
count sharps and flats, because even with the plain
nominals A B C D E F G, we already have an augmented-4th
between F:B and a diminished-5th between B:F, without
any accidentals.

This is an artifact of the logical mistake inherent
in Western musical notation: that the nominals stand
for notes of a scale (the heptatonic diatonic scale)
which has 2 different step sizes. Most of the steps
are "whole-steps", and there is no indication in the
notation where the 2 half-steps (diatonic-semitones)
are.

It's too bad it happened that way, and there have been
many proposals for a logically consistent notation
system (including several by me). But this is the
standard and we have to deal with it.

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

----- Original Message -----
From: "monz" <monz@tonalsoft.com>
To: <tuning@yahoogroups.com>
Sent: 22 Haziran 2007 Cuma 17:38
Subject: [tuning] on interval names (was: Confusing "dim-6ths" with Re: Wide
5ths)

> Hi Ozan and Andreas,
>
>
> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
> >
> > Andreas is right, for G-Eb truly IS a diminished sixth
> > and thus G#-Eb becomes double diminished. But G-Eb could
> > be considered a minor sixth instead in its own right,
> > in which case Brad is correct to say that it is a
> > diminished sixth.
> >
> > So end this meaningless feud already.
> >
> > Oz.
> >
> > ----- Original Message -----
> > From: "Andreas Sparschuh" <a_sparschuh@...>
> > To: <tuning@yahoogroups.com>
> > Sent: 22 Haziran 2007 Cuma 15:08
> > Subject: [tuning] Confusing "dim-6ths" with Re: Wide 5ths
> >
> > > Eb-Bb-F-C-G-D-A-E-B-F#-C#-G#, our diminished 6th
> > > is that G# to Eb non-5th resulting. We have a
> > > raised G (sharpened) and a lowered E (flattened):
> > > that's a DOUBLE-diminshed, because you have to
> > > count at first the one sharp "#"-symbol and then
> > > additional another "b"-symbol too, hence you obtain
> > > a TWICE-diminsihed sixt (=enharmonic 5th) instead
> > > yours hypothesis of a barely SINGLE dimished-6th.
>
>
> No, Andreas is incorrect -- at least for Western theory.
> I can't speak for Turkish or other non-Western theory.
>
> See my Encyclopedia page:
> http://tonalsoft.com/enc/i/interval.aspx
>
> In Western music-theory, there are two aspects to the
> naming of intervals:
>
> 1) the actual name, which is an ordinal number, counting
> all of the letters subtended by the interval, including
> the letters of both notes in the interval: prime, 2nd,
> 3rd, 4th, 5th, 6th, 7th, 8ve, etc.; and
>

Prime is not an interval! (unless augmented or diminished) It is the lack
thereof: the so called "intervallessness". We've been through this, haven't
we?

> 2) the quality of the interval: that is, the basic
> sizes are either perfect or imperfect, with the
> imperfect intervals further divided into minor (small)
> and major (big). If the interval is a chromatic-semitone
> larger than perfect or major, it is augmented, and if
> it is a chromatic-semitone smaller than perfect or minor,
> it is diminished.
>

Ok.

> You can use those two rules to find the correct interval
> name for any interval that can be found in standard
> theory. In diatonic tunings, it's easy enough to
> count the half-steps (both diatonic and chromatic
> semitones) to find the quality -- i have some tables
> on my webpage showing how this works for 12-edo,
> 19-edo, and 31-edo.
>

Ok.

> However, note that it is not a simple matter to just
> count sharps and flats, because even with the plain
> nominals A B C D E F G, we already have an augmented-4th
> between F:B and a diminished-5th between B:F, without
> any accidentals.
>

Ok.

> This is an artifact of the logical mistake inherent
> in Western musical notation: that the nominals stand
> for notes of a scale (the heptatonic diatonic scale)
> which has 2 different step sizes. Most of the steps
> are "whole-steps", and there is no indication in the
> notation where the 2 half-steps (diatonic-semitones)
> are.
>

Right.

> It's too bad it happened that way, and there have been
> many proposals for a logically consistent notation
> system (including several by me). But this is the
> standard and we have to deal with it.
>

Sadly.

Nevertheless, G-Eb could be considered both a diminished major sixth, and a
minor sixth, which would make G#-Eb either a double diminished major sixth
or a diminished minor sixth.

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

Oz.

Well said, Joe and Oz.

And here are several other standard texts explaining the normal
English-language nomenclature in music theory (which is *not* "minor
diminished 6th" as Andreas would have us switch to, but
simply "diminished 6th"!):

_The Harvard Brief Dictionary of Music_ (Apel/Daniel), in
entry "Intervals" : "The fourth, fifth, and octave exist in three
varieties, _diminished, perfect, and augmented_, while each of the
other intervals has four varieties, _diminished, minor, major, and
augmented_."

_Webster's New World Dictionary of Music_ (Slonimsky/Kassel), in
entry "diminished interval" : "Perfect or minor interval contracted
by a chromatic semitone."

Looks perfectly clear to me, and it jives with my university music-
theory classes of 20+ years ago, too. Start with either a perfect
interval (4th, 5th, or octave), or with a minor interval (minor 3rd,
minor 6th, minor 7th, etc), shrink it inward by one chromatic
semitone, and you've got a diminished interval. In the case of
6ths, "diminished 6th" is one of the four normal varieties!

Brad Lehman

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
> > ----- Original Message -----
> > From: "Andreas Sparschuh" <a_sparschuh@>
> > To: <tuning@yahoogroups.com>
> > Sent: 22 Haziran 2007 Cuma 15:08
> > Subject: [tuning] Confusing "dim-6ths" with Re: Wide 5ths
> >
> > > Eb-Bb-F-C-G-D-A-E-B-F#-C#-G#, our diminished 6th
> > > is that G# to Eb non-5th resulting. We have a
> > > raised G (sharpened) and a lowered E (flattened):
> > > that's a DOUBLE-diminshed, because you have to
> > > count at first the one sharp "#"-symbol and then
> > > additional another "b"-symbol too, hence you obtain
> > > a TWICE-diminsihed sixt (=enharmonic 5th) instead
> > > yours hypothesis of a barely SINGLE dimished-6th.
>
>
> No, Andreas is incorrect -- at least for Western theory.
> I can't speak for Turkish or other non-Western theory.
>
> See my Encyclopedia page:
> http://tonalsoft.com/enc/i/interval.aspx
>
> In Western music-theory, there are two aspects to the
> naming of intervals:
>
> 1) the actual name, which is an ordinal number, counting
> all of the letters subtended by the interval, including
> the letters of both notes in the interval: prime, 2nd,
> 3rd, 4th, 5th, 6th, 7th, 8ve, etc.; and
>
> 2) the quality of the interval: that is, the basic
> sizes are either perfect or imperfect, with the
> imperfect intervals further divided into minor (small)
> and major (big). If the interval is a chromatic-semitone
> larger than perfect or major, it is augmented, and if
> it is a chromatic-semitone smaller than perfect or minor,
> it is diminished.
>
> You can use those two rules to find the correct interval
> name for any interval that can be found in standard
> theory. In diatonic tunings, it's easy enough to
> count the half-steps (both diatonic and chromatic
> semitones) to find the quality -- i have some tables
> on my webpage showing how this works for 12-edo,
> 19-edo, and 31-edo.
>
> However, note that it is not a simple matter to just
> count sharps and flats, because even with the plain
> nominals A B C D E F G, we already have an augmented-4th
> between F:B and a diminished-5th between B:F, without
> any accidentals.
>
> This is an artifact of the logical mistake inherent
> in Western musical notation: that the nominals stand
> for notes of a scale (the heptatonic diatonic scale)
> which has 2 different step sizes. Most of the steps
> are "whole-steps", and there is no indication in the
> notation where the 2 half-steps (diatonic-semitones)
> are.
>
> It's too bad it happened that way, and there have been
> many proposals for a logically consistent notation
> system (including several by me). But this is the
> standard and we have to deal with it.
>
>
> -monz
> http://tonalsoft.com
> Tonescape microtonal music software
>

On 6/22/07, Brad Lehman <bpl@umich.edu> wrote:
> Well said, Joe and Oz.
>
> And here are several other standard texts explaining the normal
> English-language nomenclature in music theory (which is *not* "minor
> diminished 6th" as Andreas would have us switch to, but
> simply "diminished 6th"!):
>
> _The Harvard Brief Dictionary of Music_ (Apel/Daniel), in
> entry "Intervals" : "The fourth, fifth, and octave exist in three
> varieties, _diminished, perfect, and augmented_, while each of the
> other intervals has four varieties, _diminished, minor, major, and
> augmented_."
>
> _Webster's New World Dictionary of Music_ (Slonimsky/Kassel), in
> entry "diminished interval" : "Perfect or minor interval contracted
> by a chromatic semitone."
>
>
> Looks perfectly clear to me, and it jives with my university music-
> theory classes of 20+ years ago, too. Start with either a perfect
> interval (4th, 5th, or octave), or with a minor interval (minor 3rd,
> minor 6th, minor 7th, etc), shrink it inward by one chromatic
> semitone, and you've got a diminished interval. In the case of
> 6ths, "diminished 6th" is one of the four normal varieties!

Exactly. Reading Andreas's posts really confused me, because I think
"that's the wrong name for that interval; it's probably just a
mistake", but then I see the same mistake 3 or 4 more times and I
start to wonder...

For reference, here are all the Western named intervals within an
octave, ignoring doubly or more -diminished or -augmented intervals,
and ignoring the unison for Ozan's sake. =P

Name 12-EDO 4-cent meantone (50-EDO)
Diminished second 0 48
Minor second 100 120
Major second 200 192
Diminished third 200 240
Augmented second 300 264
Minor third 300 312
Major third 400 384
Diminished fourth 400 432
Augmented third 500 456
Perfect fourth 500 504
Augmented fourth 600 576
Diminished fifth 600 624
Perfect fifth 700 696
Diminished sixth 700 744
Augmented fifth 800 768
Minor sixth 800 816
Major sixth 900 888
Diminished seventh 900 936
Augmented sixth 1000 960
Minor seventh 1000 1008
Major seventh 1100 1080
Diminished octave 1100 1128
Augmented seventh 1200 1152
Perfect octave 1200 1200
Augmented octave 1300 1272

I will not argue with the veracity of your posted namings as you list them.

I first came across this naming system, aged about four.
(My mother had studied at the Royal College of Music, here in London)
and even at that age I realised that it was ludicrous; as any smart kid would.

It is amazing that it has persisted so long; and may even now be being passed on
blindly to confuse future generations.

"Whether a scale is major or minor seems to be determined by whether the third (if one is present) is a natural or a flattened note, respectively."
seems to be the whole logic behind this system or is it?
Maybe it is ----
"If there is a flat third interval in the scale, it is a minor; otherwise it's not a minor?"

Some tunaniks may appreciate the subtle difference.

I am open to being educated otherwise.

In the meantime I shall avoid this naming system like the plague,

and continue to describe meantone-type intervals as # or b, + number (Arabic or Roman) and multiple #'s and b's:

(religiously avoiding the terms "minor" "diminished" and "augmented" except when naming chords, as in pop guitar, and jazz traditions for which I have yet to think of better names -
scalecoding?)
Maybe we already have the solution at:

http://www.lucytune.com/new_to_lt/pitch_05.html

Charles Lucy lucy@lucytune.com

----- Promoting global harmony through LucyTuning -----

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

LucyTuned Lullabies (from around the world):
http://www.lullabies.co.uk

Skype user = lucytune

http://www.myspace.com/lucytuning

On 22 Jun 2007, at 18:08, Keenan Pepper wrote:

> On 6/22/07, Brad Lehman <bpl@umich.edu> wrote:
> > Well said, Joe and Oz.
> >
> > And here are several other standard texts explaining the normal
> > English-language nomenclature in music theory (which is *not* "minor
> > diminished 6th" as Andreas would have us switch to, but
> > simply "diminished 6th"!):
> >
> > _The Harvard Brief Dictionary of Music_ (Apel/Daniel), in
> > entry "Intervals" : "The fourth, fifth, and octave exist in three
> > varieties, _diminished, perfect, and augmented_, while each of the
> > other intervals has four varieties, _diminished, minor, major, and
> > augmented_."
> >
> > _Webster's New World Dictionary of Music_ (Slonimsky/Kassel), in
> > entry "diminished interval" : "Perfect or minor interval contracted
> > by a chromatic semitone."
> >
> >
> > Looks perfectly clear to me, and it jives with my university music-
> > theory classes of 20+ years ago, too. Start with either a perfect
> > interval (4th, 5th, or octave), or with a minor interval (minor 3rd,
> > minor 6th, minor 7th, etc), shrink it inward by one chromatic
> > semitone, and you've got a diminished interval. In the case of
> > 6ths, "diminished 6th" is one of the four normal varieties!
>
> Exactly. Reading Andreas's posts really confused me, because I think
> "that's the wrong name for that interval; it's probably just a
> mistake", but then I see the same mistake 3 or 4 more times and I
> start to wonder...
>
> For reference, here are all the Western named intervals within an
> octave, ignoring doubly or more -diminished or -augmented intervals,
> and ignoring the unison for Ozan's sake. =P
>
> Name 12-EDO 4-cent meantone (50-EDO)
> Diminished second 0 48
> Minor second 100 120
> Major second 200 192
> Diminished third 200 240
> Augmented second 300 264
> Minor third 300 312
> Major third 400 384
> Diminished fourth 400 432
> Augmented third 500 456
> Perfect fourth 500 504
> Augmented fourth 600 576
> Diminished fifth 600 624
> Perfect fifth 700 696
> Diminished sixth 700 744
> Augmented fifth 800 768
> Minor sixth 800 816
> Major sixth 900 888
> Diminished seventh 900 936
> Augmented sixth 1000 960
> Minor seventh 1000 1008
> Major seventh 1100 1080
> Diminished octave 1100 1128
> Augmented seventh 1200 1152
> Perfect octave 1200 1200
> Augmented octave 1300 1272
>
>

Hi Oz,

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> See my Encyclopedia page:
> http://tonalsoft.com/enc/i/interval.aspx
>
> > In Western music-theory, there are two aspects to the
> > naming of intervals:
> >
> > 1) the actual name, which is an ordinal number, counting
> > all of the letters subtended by the interval, including
> > the letters of both notes in the interval: prime, 2nd,
> > 3rd, 4th, 5th, 6th, 7th, 8ve, etc.; and
>
>
> Prime is not an interval! (unless augmented or diminished)
> It is the lack thereof: the so called "intervallessness".
> We've been through this, haven't we?

Yes, and i disagreed with you every step of the way.

"Prime" is simply the Latin word for "first", just
as "octave" is the Latin word for "eighth". The words
themselves are no more significant than that. We use
the English words (or in my case, the abbreviation
based on the numeral) for all others: 2nd, 3rd, 4th,
etc.

The word which really means "intervallessness" is
"unison", from the Latin for "one sound". However,
even this word is usually classified as an interval.
See, for example:

http://www.dolmetsch.com/defsu1.htm

(a website, BTW, which has frequently lifted material
out of my Encyclopedia articles ... which is something
i guess i should be proud of ...?)

So while musicians ordinarily learn that the interval
between two sound of the same pitch is a "unison",
the situation is really backwards. That term for the
interval is properly "prime" because of the way it
fits into the numerical categorization and naming system
as i outlined above. "Unison" has not other real
counterpart in music-theory terminology, and thus
*it* is the term which should be reserved for referring
to the "intervallessness" of such a sonority.

> Nevertheless, G-Eb could be considered both a diminished
> major sixth, and a minor sixth, which would make G#-Eb
> either a double diminished major sixth or a diminished
> minor sixth.

Having missed part of the discussion (mainly because i
find much of what Andreas posts to be a bit difficult
to understand, and i haven't had the proper time to sit
and ponder what he wrote), i'm guessing that there must
be some other classification of intervals which allows
the use of *both* diminished and minor/major in the
same name. Standard Western theory doesn't. An interval
is either one of the three basic sizes (perfect, minor,
or major), or it is augmented or diminished, or doubly ...
etc.

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

Hi Keenan,

-- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:

> For reference, here are all the Western named intervals within an
> octave, ignoring doubly or more -diminished or -augmented intervals,
> and ignoring the unison for Ozan's sake. =P
>
> Name 12-EDO 4-cent meantone (50-EDO)
> Diminished second 0 48
> Minor second 100 120
> <etc. -- snip>

I give a table of these on my Encyclopedia "1/4-comma meantone"
page, listing every example which occurs within a 12-tone
version of 1/4-comma meantone, and its equivalent in 31-edo:

http://www.tonalsoft.com/enc/number/1-4cmt.aspx

(Unfortunately, the website is temporarily down right now,
but will be back up again later today.)

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

--- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@...>
wrote:

> For reference, here are all the Western named intervals within an
> octave, ignoring doubly or more -diminished or -augmented intervals,
> and ignoring the unison for Ozan's sake. =P

Here are corresponding positions on the chain of fifths and septimal
intervals:

> Name 12-EDO 4-cent meantone (50-EDO)

Perfect unison 0 0 0 1, 81/80, 126/125, 225/224
> Diminished second 0 48 -12 36/35, 50/49, 64/63
Augmented unison 100 72 +7 21/20, 25/24, 28/27
> Minor second 100 120 -5 15/14, 16/15, 27/25
> Major second 200 192 +2 9/8, 10/9
> Diminished third 200 240 -10 8/7
> Augmented second 300 264 +9 7/6
> Minor third 300 312 -3 6/5
> Major third 400 384 +4 5/4
> Diminished fourth 400 432 -8 9/7
> Augmented third 500 456 +11 21/16, 35/27
> Perfect fourth 500 504 -1 4/3
> Augmented fourth 600 576 +6 7/5
> Diminished fifth 600 624 -6 10/7
> Perfect fifth 700 696 +1 3/2
> Diminished sixth 700 744 -11 32/21, 54/35
> Augmented fifth 800 768 +8 14/9, 25/16
> Minor sixth 800 816 -4 8/5
> Major sixth 900 888 +3 5/3
> Diminished seventh 900 936 -9 12/7
> Augmented sixth 1000 960 +10 7/4
> Minor seventh 1000 1008 -2 9/5, 16/9, 25/14
> Major seventh 1100 1080 +5 15/8, 28/15, 50/27
> Diminished octave 1100 1128 -7 27/14, 40/21, 48/25
> Augmented seventh 1200 1152 +12 35/18, 40/25, 63/32

This may not be the best system of nomenclature ever devised, but
note we get a name for all 25 intervals from -12 to +12 fifths. If we
are willing to adopt 31-et equivalencies, we can extend this to 31
notes with a few doubly diminished or augmented names, and get a
complete cicle, though a tad awkwardly. But -15 to +15 will do the
trick.

An alternative, sticking to 31, is to come up with some 31-specific
names. Half of a minor third, rather than being doubly diminished or
augmented, could just be a semithird or hemithird or some such, which
gives us a +14. We could call -13 an 11/8, and give it an 11-bsed
name. Etc.

Gene Ward Smith wrote:

> This may not be the best system of nomenclature ever devised, but > note we get a name for all 25 intervals from -12 to +12 fifths. If we > are willing to adopt 31-et equivalencies, we can extend this to 31 > notes with a few doubly diminished or augmented names, and get a > complete cicle, though a tad awkwardly. But -15 to +15 will do the > trick.
> > An alternative, sticking to 31, is to come up with some 31-specific > names. Half of a minor third, rather than being doubly diminished or > augmented, could just be a semithird or hemithird or some such, which > gives us a +14. We could call -13 an 11/8, and give it an 11-bsed > name. Etc.

I've got different lists of names from 31 notes here:

http://x31eq.com/31eq.htm

and Dave Keenan has a similar scheme that works for JI:

http://dkeenan.com/Music/IntervalNaming.htm

Graham

----- Original Message -----
From: "monz" <monz@tonalsoft.com>
To: <tuning@yahoogroups.com>
Sent: 23 Haziran 2007 Cumartesi 0:46
Subject: [tuning] on interval names (was: Confusing "dim-6ths" with Re: Wide
5ths)

> Hi Oz,
>
>

Hi monz,

> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
> >
> > See my Encyclopedia page:
> > http://tonalsoft.com/enc/i/interval.aspx
> >
> > > In Western music-theory, there are two aspects to the
> > > naming of intervals:
> > >
> > > 1) the actual name, which is an ordinal number, counting
> > > all of the letters subtended by the interval, including
> > > the letters of both notes in the interval: prime, 2nd,
> > > 3rd, 4th, 5th, 6th, 7th, 8ve, etc.; and
> >
> >
> > Prime is not an interval! (unless augmented or diminished)
> > It is the lack thereof: the so called "intervallessness".
> > We've been through this, haven't we?
>
>
> Yes, and i disagreed with you every step of the way.
>

Possibly. What I claim is contraversial.

> "Prime" is simply the Latin word for "first", just
> as "octave" is the Latin word for "eighth". The words
> themselves are no more significant than that. We use
> the English words (or in my case, the abbreviation
> based on the numeral) for all others: 2nd, 3rd, 4th,
> etc.
>

I am aware of that.

> The word which really means "intervallessness" is
> "unison", from the Latin for "one sound". However,
> even this word is usually classified as an interval.
> See, for example:
>
> http://www.dolmetsch.com/defsu1.htm
>
> (a website, BTW, which has frequently lifted material
> out of my Encyclopedia articles ... which is something
> i guess i should be proud of ...?)
>

Unison is an interval only if it is sounded at the interval of equivalance,
which is the octave by default.

Is there an audible interval between the tricordi strings of a piano tuned
to a unison?

> So while musicians ordinarily learn that the interval
> between two sound of the same pitch is a "unison",
> the situation is really backwards. That term for the
> interval is properly "prime" because of the way it
> fits into the numerical categorization and naming system
> as i outlined above. "Unison" has not other real
> counterpart in music-theory terminology, and thus
> *it* is the term which should be reserved for referring
> to the "intervallessness" of such a sonority.
>

There is no interval in either the prime or the unison, simple as that.

>
> > Nevertheless, G-Eb could be considered both a diminished
> > major sixth, and a minor sixth, which would make G#-Eb
> > either a double diminished major sixth or a diminished
> > minor sixth.
>
>
> Having missed part of the discussion (mainly because i
> find much of what Andreas posts to be a bit difficult
> to understand, and i haven't had the proper time to sit
> and ponder what he wrote), i'm guessing that there must
> be some other classification of intervals which allows
> the use of *both* diminished and minor/major in the
> same name. Standard Western theory doesn't. An interval
> is either one of the three basic sizes (perfect, minor,
> or major), or it is augmented or diminished, or doubly ...
> etc.
>

How do you decide which sixth (major/minor) has the priority? Both can be
augmented or diminished.

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

Very good of you.

SNIP

>
> For reference, here are all the Western named intervals within an
> octave, ignoring doubly or more -diminished or -augmented intervals,
> and ignoring the unison for Ozan's sake. =P
>
> Name 12-EDO 4-cent meantone (50-EDO)
> Diminished second 0 48
> Minor second 100 120
> Major second 200 192
> Diminished third 200 240
> Augmented second 300 264
> Minor third 300 312
> Major third 400 384
> Diminished fourth 400 432
> Augmented third 500 456
> Perfect fourth 500 504
> Augmented fourth 600 576
> Diminished fifth 600 624
> Perfect fifth 700 696
> Diminished sixth 700 744
> Augmented fifth 800 768
> Minor sixth 800 816
> Major sixth 900 888
> Diminished seventh 900 936
> Augmented sixth 1000 960
> Minor seventh 1000 1008
> Major seventh 1100 1080
> Diminished octave 1100 1128
> Augmented seventh 1200 1152
> Perfect octave 1200 1200
> Augmented octave 1300 1272
>
>

Ozan Yarman schrieb:

> Is there an audible interval between the tricordi strings of a piano tuned
> to a unison?

Is there any way to play an interval with one key? (Yes there is, depending on the instrument. It's called coupling. So what's vox humana? But would you call coupled fifths and tenths intervals instead of just a tone with a specific registration?)

Think of counterpoint. Counterpoint produces intervals. What becomes of counterpoint if two parts meet in unison? I say it's still counterpoint, and the interval is a prime.

What becomes of a number when you substract the same number? The result is 0, and most people agree that the sizelessness of 0 has no impact on its being a number. Medieval and colloquial 1 is often equivalent to modern, and logical, 0: http://en.wikipedia.org/wiki/Fencepost_problem.)

klaus

In that case, you are totally right and justified in your case Brad.

Now that the confusion is cleared, we can hopefully agree that any
diminished interval is a chromatic semitone less than perfect or minor.

Oz.

----- Original Message -----
From: "Brad Lehman" <bpl@umich.edu>
To: <tuning@yahoogroups.com>
Sent: 22 Haziran 2007 Cuma 18:11
Subject: [tuning] on interval names (was: Confusing "dim-6ths" with Re: Wide
5ths)

> Well said, Joe and Oz.
>
> And here are several other standard texts explaining the normal
> English-language nomenclature in music theory (which is *not* "minor
> diminished 6th" as Andreas would have us switch to, but
> simply "diminished 6th"!):
>
> _The Harvard Brief Dictionary of Music_ (Apel/Daniel), in
> entry "Intervals" : "The fourth, fifth, and octave exist in three
> varieties, _diminished, perfect, and augmented_, while each of the
> other intervals has four varieties, _diminished, minor, major, and
> augmented_."
>
> _Webster's New World Dictionary of Music_ (Slonimsky/Kassel), in
> entry "diminished interval" : "Perfect or minor interval contracted
> by a chromatic semitone."
>
>
> Looks perfectly clear to me, and it jives with my university music-
> theory classes of 20+ years ago, too. Start with either a perfect
> interval (4th, 5th, or octave), or with a minor interval (minor 3rd,
> minor 6th, minor 7th, etc), shrink it inward by one chromatic
> semitone, and you've got a diminished interval. In the case of
> 6ths, "diminished 6th" is one of the four normal varieties!
>
>
> Brad Lehman
>
>

----- Original Message -----
From: "Klaus Schmirler" <KSchmir@online.de>
To: <tuning@yahoogroups.com>
Sent: 23 Haziran 2007 Cumartesi 17:06
Subject: Re: [tuning] on interval names

> Ozan Yarman schrieb:
>
>
> > Is there an audible interval between the tricordi strings of a piano
tuned
> > to a unison?
>
> Is there any way to play an interval with one key? (Yes there is,
> depending on the instrument. It's called coupling. So what's vox
> humana? But would you call coupled fifths and tenths intervals instead
> of just a tone with a specific registration?)
>

You can couple a 260hz tone with another of the same frequency, and the
difference being the zero would equate to a lack of any interval no matter
the change of timbre. Saying 0 distance and no interval are the same thing.

> Think of counterpoint. Counterpoint produces intervals. What becomes
> of counterpoint if two parts meet in unison? I say it's still
> counterpoint, and the interval is a prime.
>

The interval is zero, hence there is no interval.

> What becomes of a number when you substract the same number? The
> result is 0, and most people agree that the sizelessness of 0 has no
> impact on its being a number. Medieval and colloquial 1 is often
> equivalent to modern, and logical, 0:
> http://en.wikipedia.org/wiki/Fencepost_problem.)
>

A zero of any quantity means the absence of that quantity.

> klaus
>
>

Oz.

Ozan Yarman schrieb:

> > You can couple a 260hz tone with another of the same frequency, and the
> difference being the zero would equate to a lack of any interval no matter
> the change of timbre. Saying 0 distance and no interval are the same thing.
> And you are hearing the coupling as one tone. But there are contexts (like counterpoint) where you'd know there are many tones being played, aven at the same pitch.

> > A zero of any quantity means the absence of that quantity.

0 apples menas the absence of _apples,_ but 0, 1, 2, or 3 apples are all _numbers_ of apples.

klaus

----- Original Message -----
From: "Klaus Schmirler" <KSchmir@online.de>
To: <tuning@yahoogroups.com>
Sent: 23 Haziran 2007 Cumartesi 17:51
Subject: Re: [tuning] on interval names

> Ozan Yarman schrieb:
>
> >
> > You can couple a 260hz tone with another of the same frequency, and the
> > difference being the zero would equate to a lack of any interval no
matter
> > the change of timbre. Saying 0 distance and no interval are the same
thing.
> >
>
> And you are hearing the coupling as one tone. But there are contexts
> (like counterpoint) where you'd know there are many tones being
> played, aven at the same pitch.
>

That is made distinct by timbre, phase, vibrato, etc..., not the musical
interval, which could be anything in size but zero.

> >
> > A zero of any quantity means the absence of that quantity.
>
> 0 apples menas the absence of _apples,_ but 0, 1, 2, or 3 apples are
> all _numbers_ of apples.
>

I agree that 0 is a number by which all counts start (for there could be
less than 1 item to be counted), and hence, is a unit of interval denoting
the absence of an interval.

For an interval to exist, there needs to be a measurable gap. How do you
measure 0?

And who can claim to hear the unison as an interval with a keyboard?

> klaus
>

Oz.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> This may not be the best system of nomenclature ever devised,...
>
Here some 'helpful' defintions out of relevant dictionaries:

#1:
New Grove (2001) Vol.7 p.361 defines
DIMINISHED INTERVAL as
"
A perfect or minor INTERVAL from which a semitone has been subtracted.
The perfect 5thth C-G is made into a diminshed 5th by raising C or
lowering G (i.e. C#-G or C-Gb). The diminished 3rd (e.g. F#-Ab) is the
inversion of the augmented 6th. A doubly diminsihed interval is made
by subtracting two chromatic semitones from a perfect or minor
interval, for ex. G-Fbb, G#-Fb & Gx-F are all doubly diminished 7ths
from the minor 7th G-F."

#2:
The OXFORD companion to music; OUP (2002) p.612

"...a minor 6th becomes a diminished 6th in either the froms
C-Abb or C#-Ab. Very occasionally augmented or diminshed
intervals are reduced by a further semitone, to become
'double augmented' or 'double diminished'.
Ith should be noted that, while C-Ab and C-G# are identical
intervals on modern keyboard intruments, they none the less
have different names:
C-Ab is a minor 6th (six note-names embraced) and
G-G# an augmented 5th (five note names).
Although on the keyboard G# and Ab are the same note,
they are acoustically distinct; such intervals as that inbetween G#
and Ab are called *enharmonic...."

Attend:
The (brackets) deliver the information on the nomenclauture of:
How to count intervals properly?

Let's try to apply that method on an old popular tuning,
mentioned almost half an millemium ago by:
http://en.wikipedia.org/wiki/Arnolt_Schlick
"Spiegel der Orgelmacher und Organisten"
published @ Mainz 1511
reprint ed. Peter Williams, Buren 1980 on p.87
S.'s critics:
"darumb vff yren eilff augen beharren wolten"
'Therefore they want to persist with their 11-eyes"

referring to the traditional popular
"gothic"-scale of 11-pure pythagorean just 5ths.
That extended the antique pythagorean-heptatonics

4/3 : 1 : 3/2 : 9/8 : 27/16 : 81/64 : 243/128 == F:C:G:D:A:E:B
==>
1 : 9/8 : 81/64 : 4/3 : 3/2 : 27/16 : 243/128 : 2 == C:D:E:F:G:A:B:C'

to an advanced chain of 11 consecutive 5ths,
for yielding dodecaphonic pythagorean 12-tonality
by using the 2 accidentials '#' and 'b':

1.
'#':= 3^7/2^11 = 2187/2048 for the apotome=major-semitone
as sharp-symbol in upwards direction;
and reverse corresponding

2.
'b':= 1/# = 2048/2187 respectively for flattening
about an major-semitone downwards in pitch.

note-names:

3^0: a=Bbb
3^1: e=Fb
3^2: b=Cb
3^3: f#=Gb
3^4: c#=Db
3^5: g#=Ab
3^6: d#=Eb
3^7: a#=Bb
3^8: e#=F
3^9: b#=C
3^10 fx=G
3^11 cx=D

short (without the enharmonic equivalences):

start: a - e - b - (f#-c#-g#-Eb-Fb) - F - C - G - D :terminate

The lower-case letters mark pitches
that reside about an PC = SC*schisma
below their corresponding UPPER-case
enharmonic equal counterparts.

In that sequence of 11 times 5ths: the final
conclding pythagorean wolf-5th
is located inbetween: D-a or cx-Bbb amounting
~678Cents := ~702Cents -~24Cents in seize,
for returning back to the initial a=Bbb,
due to the limitation of 12 notes/oct. in keyboards.

The interval D-a consists -correctly spoken- in:
*** An enharmonically double-dim-4th! ***
because the pyth.-wolf interval *** D-a *** does
'embrance' barely 4 heptationc 'note-name' steps :
*** D-e-F-G-a ***
when applying properly above OXFORD's-companion advise:
Count the 'number of embraced tone-steps'
involved in the interval, that you want to investigate.

That particular enharmonic double augmented-4th: D-a
can also be labeled as (cx-a) or (Ebb-a) or (D-Bbb) too
with 'x' := ## = #^2 = 3^14/2^22 = (9/8) * PC
as an about 'pythagorean-comma elevated whole-tone'
or commatic-second?

Hence Brad's "dim-6th" claims that any ~678 Cents pyth.-wolf 5th
/tuning/msearch?query=678&submit=Search&charset=ISO-8859-1
would always consist in an pretended
"dimisihed-6th" := "embraced by 6 heptatonic note-names"
is disproved by the above gothic '11-eyes' counter-example 5th @
D-a
or OXFORDian spoken: an enh. 'double-augmented-4th'

Kinrnberger's #1 syntonic-comma version of that works similar:
http://groenewald-berlin.de/text/text_T001.html
The ratio amounts the "syntonic"-wolf D-a double augmented-4th:

40/27 := (3/2)(80/81) = (5th)/(SC)
about
1200 Cents * ln(40/27))/ln(2) = ~680.448711...Cents
with SC = PC/schisma

11-5ths do appear also in:
http://www.music.indiana.edu/som/piano_repair/temperaments/pythagorean.html
53-tone extension:
http://en.wikipedia.org/wiki/53_equal_temperament
"... the dominant seventh chord would be spelled C-Fb-G-Bb, but the
otonal tetrad is C-Fb-G-Cbb, and C-Fb-G-A# is still another seventh
chord...."
(4: 5/s : 6 : ~7)

The gothic '11-eyes' extension of 7-tone pyth.
3-lim. hepatatonics contains 3 preferred "schismic-triads":

(1 : 8192/6561 : 3/2) = (4 : 5/s : 6)
logarithmically: (0 ~384C ~702C)

with 's' := 5*3^8/2^15 = 32805/32768 the schisma.

F-a-C, C-e-G, G-b-D;

F : a : C : e : G : b : D

Conclusion:
The schismic-heptationic scale:
C : D : e : F : G : a : b : C'
had replaced ~1500 the good old pythagorean
C-D-E-F-G-A-B-C' heptatonics by
C-D-e-F-G-a-b-C'
via the accidential core f#-c#-g#-Eb-Fb
by an linking the ends via 11 just 5ths:

a - e - b - (f#-c#-g#-Eb-Fb) - F - C - G - D

That procedure charges the residual worse wolf-5th:
D-a about on PC flattend.
Kirnbergers's #1 yields an similar result by
doing that with the SC in the same position.

Attend the 702Cents "dim-6th" interval:
g#-Eb of the just 5th ratio
3/2 = 1.5 in the middle of that 11-eyes chain :-)

I.m.h.o:
It's about time to revise at least the confusion in
labeling of 3-limit interval teminology, arisen due to
beeing trapped in heptatonic nomenclauture
since already more than 1/2 millenium.

A.S.

Ozan Yarman wrote:
> interval[s]...[sic] could be anything in size but zero.
>

A unison is considered an interval to you, is it not?
>>> A zero of any quantity means the absence of that quantity.
>>> >> 0 apples menas the absence of _apples,_ but 0, 1, 2, or 3 apples are
>> all _numbers_ of apples.
>>
>> Agreed. Seems right to me.
> I agree that 0 is a number by which all counts start (for there could be
> less than 1 item to be counted), and hence, is a unit of interval denoting
> the absence of an interval.
>
> For an interval to exist, there needs to be a measurable gap. How do you
> measure 0?
>
> And who can claim to hear the unison as an interval with a keyboard?
>
> Take an upper voice descending: c' b a g
against a lower voice ascending: d e f g

they land on "g" at the same time, and we hear a unison.

-A.

----- Original Message -----
From: "Aaron K. Johnson" <aaron@akjmusic.com>
To: <tuning@yahoogroups.com>
Sent: 23 Haziran 2007 Cumartesi 23:59
Subject: Re: [tuning] on interval names

> Ozan Yarman wrote:
> > interval[s]...[sic] could be anything in size but zero.
> >
>
> A unison is considered an interval to you, is it not?

Nope.

> >>> A zero of any quantity means the absence of that quantity.
> >>>
> >> 0 apples menas the absence of _apples,_ but 0, 1, 2, or 3 apples are
> >> all _numbers_ of apples.
> >>
> >>
>
> Agreed. Seems right to me.
> > I agree that 0 is a number by which all counts start (for there could be
> > less than 1 item to be counted), and hence, is a unit of interval
denoting
> > the absence of an interval.
> >
> > For an interval to exist, there needs to be a measurable gap. How do you
> > measure 0?
> >
> > And who can claim to hear the unison as an interval with a keyboard?
> >
> >
>
> Take an upper voice descending: c' b a g
> against a lower voice ascending: d e f g
>
> they land on "g" at the same time, and we hear a unison.
>

You hear a unison alright, but as an interval?

> -A.
>
>

Hi Oz,

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:

> There is no interval in either the prime or the unison,
> simple as that.

Your dogmatic stance that "prime" is not an interval
is one thing that contributes to the confusion.

As i wrote, prime (= "1st") is indeed an interval,
just like all the others which are named by the
ordinal numbers. It just happens to be the interval
of zero pitch distance.

So do you maintain that zero is not a number?
It was this way of thinking which held back mathematics
for thousands of years until mathematicians realized
the importance and value of a symbol to represent nothing.

If it's important for you to have a word which means
"no interval" or "intervalessness", then i suggest you
use "unison" for that purpose. Leave "prime" alone and
let it stay where it belongs, with its other interval
siblings.

> > > Nevertheless, G-Eb could be considered both a diminished
> > > major sixth, and a minor sixth, which would make G#-Eb
> > > either a double diminished major sixth or a diminished
> > > minor sixth.
> >
> >
> > Having missed part of the discussion (mainly because i
> > find much of what Andreas posts to be a bit difficult
> > to understand, and i haven't had the proper time to sit
> > and ponder what he wrote), i'm guessing that there must
> > be some other classification of intervals which allows
> > the use of *both* diminished and minor/major in the
> > same name. Standard Western theory doesn't. An interval
> > is either one of the three basic sizes (perfect, minor,
> > or major), or it is augmented or diminished, or doubly ...
> > etc.
> >
>
> How do you decide which sixth (major/minor) has
> the priority? Both can be augmented or diminished.

You're not getting what i wrote. The point is that
an interval is either perfect or imperfect, and the
augmented and diminished are alterations of *that*.

There is no decision to be made about whether major
or minor has priority, because neither of them do.

major lowered a chromatic-semitone = minor
minor raised a chromatic-semitone = major

By definition they are "imperfect" because of this.

perfect lowered a chromatic-semitone = diminished
imperfect (minor) lowered a chromatic-semitone = diminished

perfect raised a chromatic-semitone = augmented
imperfect (major) raised a chromatic-semitone = augmented

That's it in a nutshell.

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

Hi Oz,

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:

> For an interval to exist, there needs to be a
> measurable gap. How do you measure 0?

Umm ... how about with the number 0.

If you ask someone to measure the distance from one point
to another point which is in the same place, is there
hence "no distance". Of course not. The distance is zero.

An interval is by definition the pitch distance between
two pitches. If those pitches are the same, then that
does not mean that there is no interval, is just means
that the pitch distance is zero, so the interval is thus
a prime.

My goodness, this feels like deja-vu ...

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

----- Original Message -----
From: "monz" <monz@tonalsoft.com>
To: <tuning@yahoogroups.com>
Sent: 24 Haziran 2007 Pazar 1:15
Subject: [tuning] Re: on interval names

> Hi Oz,
>
>
> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> > For an interval to exist, there needs to be a
> > measurable gap. How do you measure 0?
>
>
> Umm ... how about with the number 0.
>

You can only measure with what you have, not with what you don't have.

> If you ask someone to measure the distance from one point
> to another point which is in the same place, is there
> hence "no distance". Of course not. The distance is zero.
>

No distance and zero distance are the same thing. What a confusion!

> An interval is by definition the pitch distance between
> two pitches.

Exactly. There must be a pitch distance. and 0 pitch distance means no pitch
distance.

If those pitches are the same, then that
> does not mean that there is no interval, is just means
> that the pitch distance is zero, so the interval is thus
> a prime.
>

If the pitch distance between two tones sounded together is zero, it means
that there is no interval to speak of. You can name this condition prime, to
which I have no objections.

> My goodness, this feels like deja-vu ...
>

Indeed!

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

Oz.

----- Original Message -----
From: "monz" <monz@tonalsoft.com>
To: <tuning@yahoogroups.com>
Sent: 24 Haziran 2007 Pazar 1:05
Subject: [tuning] on interval names (was: Confusing "dim-6ths" with Re: Wide
5ths)

> Hi Oz,
>
>
> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> > There is no interval in either the prime or the unison,
> > simple as that.
>
>
> Your dogmatic stance that "prime" is not an interval
> is one thing that contributes to the confusion.
>

No it does not.

> As i wrote, prime (= "1st") is indeed an interval,
> just like all the others which are named by the
> ordinal numbers. It just happens to be the interval
> of zero pitch distance.
>

I have made my arguments in the other post.

> So do you maintain that zero is not a number?

Not at all. It is a number by which all counts begin, for there could be
less than 1 item in what you count.

> It was this way of thinking which held back mathematics
> for thousands of years until mathematicians realized
> the importance and value of a symbol to represent nothing.
>

What is this babbling about? I am not holding back anything.

> If it's important for you to have a word which means
> "no interval" or "intervalessness", then i suggest you
> use "unison" for that purpose. Leave "prime" alone and
> let it stay where it belongs, with its other interval
> siblings.
>

I shan't leave it alone just because you say so.

SNIP

> >
> > How do you decide which sixth (major/minor) has
> > the priority? Both can be augmented or diminished.
>
>
> You're not getting what i wrote. The point is that
> an interval is either perfect or imperfect, and the
> augmented and diminished are alterations of *that*.
>
> There is no decision to be made about whether major
> or minor has priority, because neither of them do.
>
> major lowered a chromatic-semitone = minor
> minor raised a chromatic-semitone = major
>
> By definition they are "imperfect" because of this.
>
>
> perfect lowered a chromatic-semitone = diminished
> imperfect (minor) lowered a chromatic-semitone = diminished
>
> perfect raised a chromatic-semitone = augmented
> imperfect (major) raised a chromatic-semitone = augmented
>
>
> That's it in a nutshell.
>

Very good then. So you side with Andreas?

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

Oz.

Hi Oz,

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
>
> ----- Original Message -----
> From: "monz" <monz@...>
> To: <tuning@yahoogroups.com>
> Sent: 24 Haziran 2007 Pazar 1:15
> Subject: [tuning] Re: on interval names
>
>
> > --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@> wrote:
> >
> > > For an interval to exist, there needs to be a
> > > measurable gap. How do you measure 0?
> >
> >
> > Umm ... how about with the number 0.
> >
>
>
> You can only measure with what you have, not with what
> you don't have.

If what is to be measured is nothing, then you must
use the number 0 to measure it.

(this discussion is beginning to feel like a zen experience)

> > An interval is by definition the pitch distance between
> > two pitches.
>
>
> Exactly. There must be a pitch distance. and 0 pitch
> distance means no pitch distance.

You could look at it that way. But if someone asks you
what is that interval, and if there is indeed a term
which describes an interval of that size, then your
answer is "prime".

> > If those pitches are the same, then that
> > does not mean that there is no interval, is just means
> > that the pitch distance is zero, so the interval is thus
> > a prime.
> >
>
>
> If the pitch distance between two tones sounded together
> is zero, it means that there is no interval to speak of.

No it doesn't. It simply means that the interval describes
a pitch distance of zero.

> You can name this condition prime, to which I have no
> objections.

That's good. But the condition is "unison".
The *interval* name is "prime".

And anyway i certainly didn't name it. The interval
names are so old that in fact they come from the time
before zero was understood as a number. This is exactly
why the unison condition is called "prime" (= "1st").

Logically, the it would be best to overhaul the whole
system so that the condition of unison is called the
interval "zeroth" and each successively larger interval
would have a name which is one ordinal number less than
what they currently are.

So tones and semitones would be "1st" or "prime" (instead
of "2nd" as we already call it), the interval spanning
two tones would be "2nd" (instead of "3rd" as we already
call it), etc., and the usual equivalence-interval would
be "7th" instead of "8ve".

This would be great, because then interval mathematics
would work out they way they're supposed to, instead of
always having to subtract "1" from the total each time
another interval is added. But it's highly unlikely that
it will ever be adopted.

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

Ok, I give in to your arguments and wish that your proposal could come to
life someday.

Oz.

----- Original Message -----
From: "monz" <monz@tonalsoft.com>
To: <tuning@yahoogroups.com>
Sent: 24 Haziran 2007 Pazar 2:39
Subject: [tuning] Re: on interval names

> Hi Oz,
>
>
> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
> >
> >
> > ----- Original Message -----
> > From: "monz" <monz@...>
> > To: <tuning@yahoogroups.com>
> > Sent: 24 Haziran 2007 Pazar 1:15
> > Subject: [tuning] Re: on interval names
> >
> >
> > > --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@> wrote:
> > >
> > > > For an interval to exist, there needs to be a
> > > > measurable gap. How do you measure 0?
> > >
> > >
> > > Umm ... how about with the number 0.
> > >
> >
> >
> > You can only measure with what you have, not with what
> > you don't have.
>
>
> If what is to be measured is nothing, then you must
> use the number 0 to measure it.
>
> (this discussion is beginning to feel like a zen experience)
>
>
> > > An interval is by definition the pitch distance between
> > > two pitches.
> >
> >
> > Exactly. There must be a pitch distance. and 0 pitch
> > distance means no pitch distance.
>
>
> You could look at it that way. But if someone asks you
> what is that interval, and if there is indeed a term
> which describes an interval of that size, then your
> answer is "prime".
>
>
> > > If those pitches are the same, then that
> > > does not mean that there is no interval, is just means
> > > that the pitch distance is zero, so the interval is thus
> > > a prime.
> > >
> >
> >
> > If the pitch distance between two tones sounded together
> > is zero, it means that there is no interval to speak of.
>
> No it doesn't. It simply means that the interval describes
> a pitch distance of zero.
>
>
> > You can name this condition prime, to which I have no
> > objections.
>
> That's good. But the condition is "unison".
> The *interval* name is "prime".
>
> And anyway i certainly didn't name it. The interval
> names are so old that in fact they come from the time
> before zero was understood as a number. This is exactly
> why the unison condition is called "prime" (= "1st").
>
> Logically, the it would be best to overhaul the whole
> system so that the condition of unison is called the
> interval "zeroth" and each successively larger interval
> would have a name which is one ordinal number less than
> what they currently are.
>
> So tones and semitones would be "1st" or "prime" (instead
> of "2nd" as we already call it), the interval spanning
> two tones would be "2nd" (instead of "3rd" as we already
> call it), etc., and the usual equivalence-interval would
> be "7th" instead of "8ve".
>
> This would be great, because then interval mathematics
> would work out they way they're supposed to, instead of
> always having to subtract "1" from the total each time
> another interval is added. But it's highly unlikely that
> it will ever be adopted.
>
>
> -monz
> http://tonalsoft.com
> Tonescape microtonal music software
>

Hi Oz,

> > [monz:]
> > So do you maintain that zero is not a number?
>
> [Oz:]
> Not at all. It is a number by which all counts begin,
> for there could be less than 1 item in what you count.

So then if you are counting the number of letters between
two pitches, and there is less than one letter between
them (i.e., zero), then what is the big deal recognizing
the existence of an interval of that size?

I really don't get it. If you want to have a term for
the non-existence of interval, then you can use "unison".
Prime will continue to be what it always was: the interval
which describes the unison.

> > You're not getting what i wrote. The point is that
> > an interval is either perfect or imperfect, and the
> > augmented and diminished are alterations of *that*.
> >
> > There is no decision to be made about whether major
> > or minor has priority, because neither of them do.
> >
> > major lowered a chromatic-semitone = minor
> > minor raised a chromatic-semitone = major
> >
> > By definition they are "imperfect" because of this.
> >
> >
> > perfect lowered a chromatic-semitone = diminished
> > imperfect (minor) lowered a chromatic-semitone = diminished
> >
> > perfect raised a chromatic-semitone = augmented
> > imperfect (major) raised a chromatic-semitone = augmented
> >
> >
> > That's it in a nutshell.
> >
>
>
> Very good then. So you side with Andreas?

The last time i posted something mentioning Andreas's name,
it was to say that he was incorrect. So i'm not sure what
you're referring to now.

In any case, i've stated everything i have for this
argument and see no point in going on, so i don't
plan to have any more responses in this thread, unless
there are further questions that i might be able to answer.

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

Hi Oz,

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
> >
> > Logically, the it would be best to overhaul the whole
> > system so that the condition of unison is called the
> > interval "zeroth" and each successively larger interval
> > would have a name which is one ordinal number less than
> > what they currently are.
> >
> > So tones and semitones would be "1st" or "prime" (instead
> > of "2nd" as we already call it), the interval spanning
> > two tones would be "2nd" (instead of "3rd" as we already
> > call it), etc., and the usual equivalence-interval would
> > be "7th" instead of "8ve".

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> Ok, I give in to your arguments and wish that your proposal
> could come to life someday.

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
>
> In any case, i've stated everything i have for this
> argument and see no point in going on, so i don't
> plan to have any more responses in this thread, unless
> there are further questions that i might be able to answer.

It looks like we both got tired of the argument at the
same time! :-D

Anyway, yes, revamping the interval names, starting them
with zero, so that they make mathematical sense, would
make things so much easier.

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

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:

> How do you decide which sixth (major/minor) has the priority? Both
can be
> augmented or diminished.

If you say "augumented sixth", then it is major, since the major sixth
is sharper than the minor. If you say "diminised sixth" then it is
diminished from a minor sixth.

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> Gene Ward Smith wrote:

> > An alternative, sticking to 31, is to come up with some 31-
specific
> > names. Half of a minor third, rather than being doubly diminished
or
> > augmented, could just be a semithird or hemithird or some such,
which
> > gives us a +14. We could call -13 an 11/8, and give it an 11-bsed
> > name. Etc.
>
> I've got different lists of names from 31 notes here:

I like this. Graham and Dave are proposing that we can extend the 25
names of standard meantone nomenclature by adding a neutral second,
third, sixth and seventh, plus a subfifth and superfourth interval.
That gives us 31.

This post of yours seems to have been greeted with silence so far?

--- In tuning@yahoogroups.com, "Andreas Sparschuh" <a_sparschuh@...>
wrote:
>
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@>
> wrote:
> >
> > This may not be the best system of nomenclature ever devised,...
> >
> Here some 'helpful' defintions out of relevant dictionaries:
>
> #1:
> New Grove (2001) Vol.7 p.361 defines
> DIMINISHED INTERVAL as
> "
> A perfect or minor INTERVAL from which a semitone has been
subtracted.
> The perfect 5thth C-G is made into a diminshed 5th by raising C or
> lowering G (i.e. C#-G or C-Gb). The diminished 3rd (e.g. F#-Ab) is
the
> inversion of the augmented 6th. A doubly diminsihed interval is
made
> by subtracting two chromatic semitones from a perfect or minor
> interval, for ex. G-Fbb, G#-Fb & Gx-F are all doubly diminished
7ths
> from the minor 7th G-F."
>
> #2:
> The OXFORD companion to music; OUP (2002) p.612
>
> "...a minor 6th becomes a diminished 6th in either the froms
> C-Abb or C#-Ab. Very occasionally augmented or diminshed
> intervals are reduced by a further semitone, to become
> 'double augmented' or 'double diminished'.
> Ith should be noted that, while C-Ab and C-G# are identical
> intervals on modern keyboard intruments, they none the less
> have different names:
> C-Ab is a minor 6th (six note-names embraced) and
> G-G# an augmented 5th (five note names).
> Although on the keyboard G# and Ab are the same note,
> they are acoustically distinct; such intervals as that inbetween G#
> and Ab are called *enharmonic...."
>
> Attend:
> The (brackets) deliver the information on the nomenclauture of:
> How to count intervals properly?
>
> Let's try to apply that method on an old popular tuning,
> mentioned almost half an millemium ago by:
> http://en.wikipedia.org/wiki/Arnolt_Schlick
> "Spiegel der Orgelmacher und Organisten"
> published @ Mainz 1511
> reprint ed. Peter Williams, Buren 1980 on p.87
> S.'s critics:
> "darumb vff yren eilff augen beharren wolten"
> 'Therefore they want to persist with their 11-eyes"
>
> referring to the traditional popular
> "gothic"-scale of 11-pure pythagorean just 5ths.
> That extended the antique pythagorean-heptatonics
>
> 4/3 : 1 : 3/2 : 9/8 : 27/16 : 81/64 : 243/128 == F:C:G:D:A:E:B
> ==>
> 1 : 9/8 : 81/64 : 4/3 : 3/2 : 27/16 : 243/128 : 2 ==
C:D:E:F:G:A:B:C'
>
> to an advanced chain of 11 consecutive 5ths,
> for yielding dodecaphonic pythagorean 12-tonality
> by using the 2 accidentials '#' and 'b':
>
> 1.
> '#':= 3^7/2^11 = 2187/2048 for the apotome=major-semitone
> as sharp-symbol in upwards direction;
> and reverse corresponding
>
> 2.
> 'b':= 1/# = 2048/2187 respectively for flattening
> about an major-semitone downwards in pitch.
>
> note-names:
>
> 3^0: a=Bbb
> 3^1: e=Fb
> 3^2: b=Cb
> 3^3: f#=Gb
> 3^4: c#=Db
> 3^5: g#=Ab
> 3^6: d#=Eb
> 3^7: a#=Bb
> 3^8: e#=F
> 3^9: b#=C
> 3^10 fx=G
> 3^11 cx=D
>
> short (without the enharmonic equivalences):
>
> start: a - e - b - (f#-c#-g#-Eb-Fb) - F - C - G - D :terminate
>
> The lower-case letters mark pitches
> that reside about an PC = SC*schisma
> below their corresponding UPPER-case
> enharmonic equal counterparts.
>
> In that sequence of 11 times 5ths: the final
> conclding pythagorean wolf-5th
> is located inbetween: D-a or cx-Bbb amounting
> ~678Cents := ~702Cents -~24Cents in seize,
> for returning back to the initial a=Bbb,
> due to the limitation of 12 notes/oct. in keyboards.
>
> The interval D-a consists -correctly spoken- in:
> *** An enharmonically double-dim-4th! ***
> because the pyth.-wolf interval *** D-a *** does
> 'embrance' barely 4 heptationc 'note-name' steps :
> *** D-e-F-G-a ***
> when applying properly above OXFORD's-companion advise:
> Count the 'number of embraced tone-steps'
> involved in the interval, that you want to investigate.
>
> That particular enharmonic double augmented-4th: D-a
> can also be labeled as (cx-a) or (Ebb-a) or (D-Bbb) too
> with 'x' := ## = #^2 = 3^14/2^22 = (9/8) * PC
> as an about 'pythagorean-comma elevated whole-tone'
> or commatic-second?
>
> Hence Brad's "dim-6th" claims that any ~678 Cents pyth.-wolf 5th
> /tuning/msearch?
query=678&submit=Search&charset=ISO-8859-1
> would always consist in an pretended
> "dimisihed-6th" := "embraced by 6 heptatonic note-names"
> is disproved by the above gothic '11-eyes' counter-example 5th @
> D-a
> or OXFORDian spoken: an enh. 'double-augmented-4th'
>
> Kinrnberger's #1 syntonic-comma version of that works similar:
> http://groenewald-berlin.de/text/text_T001.html
> The ratio amounts the "syntonic"-wolf D-a double augmented-4th:
>
> 40/27 := (3/2)(80/81) = (5th)/(SC)
> about
> 1200 Cents * ln(40/27))/ln(2) = ~680.448711...Cents
> with SC = PC/schisma
>
> 11-5ths do appear also in:
>
http://www.music.indiana.edu/som/piano_repair/temperaments/pythagorea
n.html
> 53-tone extension:
> http://en.wikipedia.org/wiki/53_equal_temperament
> "... the dominant seventh chord would be spelled C-Fb-G-Bb, but the
> otonal tetrad is C-Fb-G-Cbb, and C-Fb-G-A# is still another seventh
> chord...."
> (4: 5/s : 6 : ~7)
>
> The gothic '11-eyes' extension of 7-tone pyth.
> 3-lim. hepatatonics contains 3 preferred "schismic-triads":
>
> (1 : 8192/6561 : 3/2) = (4 : 5/s : 6)
> logarithmically: (0 ~384C ~702C)
>
> with 's' := 5*3^8/2^15 = 32805/32768 the schisma.
>
> F-a-C, C-e-G, G-b-D;
>
> F : a : C : e : G : b : D
>
> Conclusion:
> The schismic-heptationic scale:
> C : D : e : F : G : a : b : C'
> had replaced ~1500 the good old pythagorean
> C-D-E-F-G-A-B-C' heptatonics by
> C-D-e-F-G-a-b-C'
> via the accidential core f#-c#-g#-Eb-Fb
> by an linking the ends via 11 just 5ths:
>
> a - e - b - (f#-c#-g#-Eb-Fb) - F - C - G - D
>
> That procedure charges the residual worse wolf-5th:
> D-a about on PC flattend.
> Kirnbergers's #1 yields an similar result by
> doing that with the SC in the same position.
>
> Attend the 702Cents "dim-6th" interval:
> g#-Eb of the just 5th ratio
> 3/2 = 1.5 in the middle of that 11-eyes chain :-)
>
> I.m.h.o:
> It's about time to revise at least the confusion in
> labeling of 3-limit interval teminology, arisen due to
> beeing trapped in heptatonic nomenclauture
> since already more than 1/2 millenium.
>
> A.S.
>

Quoting monz:

> Anyway, yes, revamping the interval names, starting them
> with zero, so that they make mathematical sense, would
> make things so much easier.

I agree and I think it's absurd that we have had to call them their
historical names for so long. It makes it an extra hurdle to teach intervals for sure.

Like you said, telling a student that a third plus a third is a fifth,
when 3+3 != 5, is a pure pain in the @&!. Obviously, it makes more
sense to update the nomenclature as far as the staff is concerned,
too: we see an interval '4 steps up' or '4 steps down', and yet, we
call it a '5th'. The other aspect which is crazy is that odd numbers should have the property of reversing the 'line or space' condition on
the staff, NOT even numbers. But we notice, by standard nomenclature,
that a 'space to space' or a 'line to line' interval on the staff is
always an odd interval, exactly the opposite of what intuition tells
us the case should be in a system with two choices: line or space.

It's a didactic nightmare, but it also shows us the power of tradition
that it hasn't changed, and like you say, probably never will.....

-A.

--- In tuning@yahoogroups.com, Aaron Krister Johnson <aaron@...> wrote:

> It's a didactic nightmare, but it also shows us the power of
tradition
> that it hasn't changed, and like you say, probably never will.....

It's like AD/BC--a holdover from an era before the invention of zero
which really makes no sense.

--- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...> wrote:
>
> This post of yours seems to have been greeted with silence so far?
Dear Cameron,
by the historical '11-eyes' example i intended to illustrate:
The traditional-grown crude polysemy in counting interval-names
leads obviously to problematic ambiguities
due to lack in clarity about the real seize of the actual interval,
caused by appearent inconsistency in labeling the pitch properly
by an numerically indication of value rate.

Corrective relief/remedy to workaround that inherent defict:

If one really intends to stipulate an interval clearly and precisely:
Indicate the amount additional togehter with its concrete
arithmetically proportion represented in an
corresponding fractional term
expressed by 2 integral numbers:

Numerator over denominator.

Otherwise the floodgates are open to speculate about:
What intensional ratio could really be meant indeed
by the the absence of such an concrete indication?
especially
when reading tiresome tellings about gibberish long rigmaroles:
on pretended claims that foist "dim.-6ths" upon J.S. Bach
allegedly imputed inbetween A#-F, howsoever, but not authentic.

Remember again C.P.E. Bach's statement in the necrology of his father:
"Nobody was able to tune his instrument to his satisfaction."
How would JSB judge about the modern abuse of and in his name?

A.S.

--- In tuning@yahoogroups.com, Aaron Krister Johnson <aaron@...> wrote:
>
> Quoting monz:
>
> > Anyway, yes, revamping the interval names, starting them
> > with zero, so that they make mathematical sense, would
> > make things so much easier.
>
> I agree and I think it's absurd that we have had to call them their
> historical names for so long. It makes it an extra hurdle to teach
> intervals for sure.
>
> Like you said, telling a student that a third plus a third is a fifth,
> when 3+3 != 5, is a pure pain in the @&!. Obviously, it makes more
> sense to update the nomenclature as far as the staff is concerned,
> too: we see an interval '4 steps up' or '4 steps down', and yet, we
> call it a '5th'. The other aspect which is crazy is that odd numbers
> should have the property of reversing the 'line or space' condition on
> the staff, NOT even numbers. But we notice, by standard nomenclature,
> that a 'space to space' or a 'line to line' interval on the staff is
> always an odd interval, exactly the opposite of what intuition tells
> us the case should be in a system with two choices: line or space.
>
> It's a didactic nightmare, but it also shows us the power of tradition
> that it hasn't changed, and like you say, probably never will.....
>
> -A.
>

Counting (number theory = natural) numbers are used for diatonic
intervals. Two pitches are involved. We don't count them 'zero, one'.
People generally don't start counting with zero; they count
starting with 1. Interval numbers come from staff distace, which comes
from 7 diatonic nominals, which make perfect sense when distances
between two points begin by counting 1 at the first point. Even though
distance isn't usually measured this way, this isn't just a distance; it's
a structure within a scale, and the degrees are what are being counted.
Explaining why an interval and its inversion add up to 9 is then not
really so tough. The staff shows pitch height for 7 diatonic naturals.
Simple permutations of these 7 tones give 13 tones on the
chain which represent all the basic diatonic intervals from the
augmented fouth to the diminished fifth, and accidentals are not
necessary to show this.

Yours,
Aaron Hunt
H-Pi Instruments

Aaron Krister Johnson schrieb:

> Like you said, telling a student that a third plus a third is a fifth, > when 3+3 != 5, is a pure pain in the @&!. Obviously, it makes more > sense to update the nomenclature as far as the staff is concerned, > too: we see an interval '4 steps up' or '4 steps down', and yet, we > call it a '5th'. The other aspect which is crazy is that odd numbers > should have the property of reversing the 'line or space' condition on > the staff, NOT even numbers. But we notice, by standard nomenclature, > that a 'space to space' or a 'line to line' interval on the staff is > always an odd interval, exactly the opposite of what intuition tells > us the case should be in a system with two choices: line or space.
> > It's a didactic nightmare, but it also shows us the power of tradition > that it hasn't changed, and like you say, probably never will.....
> > -A.

Don't blame it all on note names, it happens everywhere. From among the numbers from 1 to 10, where's the middle? (The question itself is the first thing that's wrong here.)

The following problem is taken from http://en.wikipedia.org/wiki/Fencepost_error

You want to build a fence that is 100 feet long. There should be fence post every 10 feet. How many posts do you need? 100/10, that's 10 posts, right?

So why did you dig 11 holes?

But I think Ozan's problem is really Zenon's: 0 simply can't be reached: http://www.geocities.com/igoryu66/zenon-jar.htm

klaus

Heh heh.

Oz.

----- Original Message -----
From: "monz" <monz@tonalsoft.com>
To: <tuning@yahoogroups.com>
Sent: 25 Haziran 2007 Pazartesi 3:41
Subject: [tuning] Re: on interval names

> Hi Aaron (Johnson),
>
>
> > --- In tuning@yahoogroups.com, Aaron Krister Johnson <aaron@> wrote:
> > >
> > > Quoting monz:
> > >
> > > > Anyway, yes, revamping the interval names, starting them
> > > > with zero, so that they make mathematical sense, would
> > > > make things so much easier.
> >
> >
> > Counting (number theory = natural) numbers are used
> > for diatonic intervals. Two pitches are involved. We don't
> > count them 'zero, one'. People generally don't start
> > counting with zero; they count starting with 1. Interval
> > numbers come from staff distace, which comes from 7
> > diatonic nominals, which make perfect sense when
> > distances between two points begin by counting 1 at
> > the first point. Even though distance isn't usually
> > measured this way, this isn't just a distance; it's
> > a structure within a scale, and the degrees are what
> > are being counted.
>
>
> Mostly, what you wrote here is correct. But actually,
> the counting numbers are the cardinal numbers
> (1, 2, 3, ...), and i think the dead giveaway with this
> situation is that the interval names are *not* cardinal
> numbers, but *ordinal* numbers (1st, 2nd, 3rd, ...).
>
> While my argument is against the illogicality of
> interval names because they start from 1 instead of 0,
> it *is* entirely logical to call the starting pitch
> of a distance the "first". The illogicality comes in
> because the interval names count the nominal
> (A, B, C, etc.) degree steps, and the starting pitch
> is not a step. As Ozan says, it's nothing.
>
> So there, now it sounds like i'm agreeing with Ozan
> after all the arguing i've done against what he wrote.
> Shows hows ridiculous the whole thing is.
>
> Maybe if we argue long enough and loudly enough,
> we can convince the musical establishment at large
> to switch to the 0th, 1st, 2nd, ... system.
>
>
>
> -monz
> http://tonalsoft.com
> Tonescape microtonal music software
>
>

Gene Ward Smith wrote:
> --- In tuning@yahoogroups.com, Aaron Krister Johnson <aaron@...> wrote:
>
> >> It's a didactic nightmare, but it also shows us the power of >> > tradition > >> that it hasn't changed, and like you say, probably never will.....
>> >
> It's like AD/BC--a holdover from an era before the invention of zero > which really makes no sense.
> You lost me.

-A.

Aaron Andrew Hunt wrote:
> --- In tuning@yahoogroups.com, Aaron Krister Johnson <aaron@...> wrote:
> >> Quoting monz:
>>
>> >>> Anyway, yes, revamping the interval names, starting them
>>> with zero, so that they make mathematical sense, would
>>> make things so much easier.
>>> >> I agree and I think it's absurd that we have had to call them their >> historical names for so long. It makes it an extra hurdle to teach >> intervals for sure.
>>
>> Like you said, telling a student that a third plus a third is a fifth, >> when 3+3 != 5, is a pure pain in the @&!. Obviously, it makes more >> sense to update the nomenclature as far as the staff is concerned, >> too: we see an interval '4 steps up' or '4 steps down', and yet, we >> call it a '5th'. The other aspect which is crazy is that odd numbers >> should have the property of reversing the 'line or space' condition on >> the staff, NOT even numbers. But we notice, by standard nomenclature, >> that a 'space to space' or a 'line to line' interval on the staff is >> always an odd interval, exactly the opposite of what intuition tells >> us the case should be in a system with two choices: line or space.
>>
>> It's a didactic nightmare, but it also shows us the power of tradition >> that it hasn't changed, and like you say, probably never will.....
>>
>> -A.
>>
>> >
>
> Counting (number theory = natural) numbers are used for diatonic > intervals. Two pitches are involved. We don't count them 'zero, one'.
> So, by your logic, a unison ought to be called a 2nd if we count the pitches: "C, C: 1, 2"?

> People generally don't start counting with zero; they count > starting with 1.

Yes, and we can and should, if we were a bit smarter, start at the step away from the unison: one step away, two steps away.

> Interval numbers come from staff distace, which comes > from 7 diatonic nominals, which make perfect sense when distances > between two points begin by counting 1 at the first point.
Of course it makes perfect sense when starting from that premise. I just think that premise is stupid. Plus, it is mathematically dumb:

5+4=8, 5+5+5=13, 8+2=9 etc.

...which means we have to learn an extra math algorithm: find the number of pitches contributing to the stack, and subtract that number, then add one to get the 'real' (fake) interval name. As if kids don't hate math enough, in general, we musicians add to the mess by making them learn a really pointless exception to their basic and important
arithmetic skills.

> Even though > distance isn't usually measured this way, this isn't just a distance; it's > a structure within a scale, and the degrees are what are being counted. > Why not have a distance always be measured the way one expects? The logic of scale structure still remains, just the names change for clarity.
Why should scale degree names have to come into play at all?

> Explaining why an interval and its inversion add up to 9 is then not > really so tough.
Really, you are right, but for less than perfectly bright students, which are rare these days, it is an unnecessary confusion. My students are never confused when I ask "How many steps away is this?" or even "what interval is this---remember, add 1 to the steps away, or start counting at one on the note", but the more complex things get with stacking, the more math exceptions have to be done, and why not make it the math they learn in school? It just serves to slow things down, I think completely unnecessarily.

I *do* however see the counter argument---this is what musicians have called things for centuries, and they need to know that.

*Sigh*---what a ridiculous race humanity is.

> The staff shows pitch height for 7 diatonic naturals. > Simple permutations of these 7 tones give 13 tones on the > chain which represent all the basic diatonic intervals from the > augmented fouth to the diminished fifth, and accidentals are not > necessary to show this.
> This has nothing to do with what Monz or I were saying, but I'm glad you've taken the course 'Staff 101'.

-A.

monz wrote:
> Hi Aaron (Johnson),
>
> Monz! You should not be directing this to me......I AGREE with you, remember?...it's Aaron ANDREW HUNT who is an interval conservative, maybe even a neo-con ;),
and disagrees with us.

So take back what you said to me and say it to him....I didn't say that stuff below. You are suffering from too-many-Aaron-itis.

>>> Quoting monz:
>>>
>>> >>>> Anyway, yes, revamping the interval names, starting them
>>>> with zero, so that they make mathematical sense, would
>>>> make things so much easier.
>>>> >> Aaron ANDREW HUNT said: Counting (number theory = natural) numbers are used
>> for diatonic intervals. Two pitches are involved. We don't
>> count them 'zero, one'. People generally don't start
>> counting with zero; they count starting with 1. Interval
>> numbers come from staff distace, which comes from 7
>> diatonic nominals, which make perfect sense when
>> distances between two points begin by counting 1 at
>> the first point. Even though distance isn't usually
>> measured this way, this isn't just a distance; it's >> a structure within a scale, and the degrees are what
>> are being counted.
>> >
>
> Mostly, what you wrote here is correct. But actually,
> the counting numbers are the cardinal numbers
> (1, 2, 3, ...), and i think the dead giveaway with this
> situation is that the interval names are *not* cardinal
> numbers, but *ordinal* numbers (1st, 2nd, 3rd, ...).
>
> While my argument is against the illogicality of
> interval names because they start from 1 instead of 0,
> it *is* entirely logical to call the starting pitch
> of a distance the "first". The illogicality comes in
> because the interval names count the nominal > (A, B, C, etc.) degree steps, and the starting pitch
> is not a step. As Ozan says, it's nothing.
>
> So there, now it sounds like i'm agreeing with Ozan
> after all the arguing i've done against what he wrote.
> Shows hows ridiculous the whole thing is.
>
> Maybe if we argue long enough and loudly enough,
> we can convince the musical establishment at large
> to switch to the 0th, 1st, 2nd, ... system.
>
>
>
> -monz
> http://tonalsoft.com
> Tonescape microtonal music software
>
>
> >

Hi Aaron JOHNSON :-),

--- In tuning@yahoogroups.com, "Aaron K. Johnson" <aaron@...> wrote:
>
> Gene Ward Smith wrote:
>
> > --- In tuning@yahoogroups.com, Aaron Krister Johnson
> > <aaron@> wrote:
> >
> > > It's a didactic nightmare, but it also shows us
> > > the power of tradition that it hasn't changed,
> > > and like you say, probably never will.....
> >
> >
> > It's like AD/BC--a holdover from an era before the
> > invention of zero which really makes no sense.
> >
>
> You lost me.

For a minute i didn't know what he was talking about
either. AD = _Anno Domini_, BC = Before Christ.

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

Hi Aaron JOHNSON,

--- In tuning@yahoogroups.com, "Aaron K. Johnson" <aaron@...> wrote:
>
> monz wrote:
>
> Monz! You should not be directing this to me......I AGREE
> with you, remember?...it's Aaron ANDREW HUNT who is an
> interval conservative, maybe even a neo-con ;),
> and disagrees with us.
>
> So take back what you said to me and say it to him....
> I didn't say that stuff below. You are suffering from
> too-many-Aaron-itis.

Damn ... i was replying to each of you in separate posts,
and really thought i had it straight. Oh well, i guess
i have to remember "Hunt: 1, Johnson: 0". :-P

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

Greetings, AKJ!

If you're not satisfied with what makes fine muscial
sense already, you can rename all the intervals by subtracting one,
perform simple modular arithmetic, and tell everyone you've
got the better way! To make your results intelligible to educated
musicians and relevant to musical discourse of the past ages you are
superceding, just add 1; it's like Magic!

Actually, I suppose understand your hangup with interval numbers.
I used to be annoyed with them, before I taught undergraduate music
theory and decided to really dig into these things so that I could teach
them well. They make sense, not just historically, but logically; no need
to change them. Expand them systematically, yes, but not change them.

Sorry if I sound pedantic. It's a teacher's curse.

Yours,
Aaron Hunt
H-Pi Instruments

--- In tuning@yahoogroups.com, "Aaron K. Johnson" <aaron@...> wrote:
>
> Aaron Andrew Hunt wrote:
> > --- In tuning@yahoogroups.com, Aaron Krister Johnson <aaron@> wrote:
> >
> >> Quoting monz:
> >>
> >>
> >>> Anyway, yes, revamping the interval names, starting them
> >>> with zero, so that they make mathematical sense, would
> >>> make things so much easier.
> >>>
> >> I agree and I think it's absurd that we have had to call them their
> >> historical names for so long. It makes it an extra hurdle to teach
> >> intervals for sure.
> >>
> >> Like you said, telling a student that a third plus a third is a fifth,
> >> when 3+3 != 5, is a pure pain in the @&!. Obviously, it makes more
> >> sense to update the nomenclature as far as the staff is concerned,
> >> too: we see an interval '4 steps up' or '4 steps down', and yet, we
> >> call it a '5th'. The other aspect which is crazy is that odd numbers
> >> should have the property of reversing the 'line or space' condition on
> >> the staff, NOT even numbers. But we notice, by standard nomenclature,
> >> that a 'space to space' or a 'line to line' interval on the staff is
> >> always an odd interval, exactly the opposite of what intuition tells
> >> us the case should be in a system with two choices: line or space.
> >>
> >> It's a didactic nightmare, but it also shows us the power of tradition
> >> that it hasn't changed, and like you say, probably never will.....
> >>
> >> -A.
> >>
> >>
> >
> >
> > Counting (number theory = natural) numbers are used for diatonic
> > intervals. Two pitches are involved. We don't count them 'zero, one'.
> >
> So, by your logic, a unison ought to be called a 2nd if we count the
> pitches: "C, C: 1, 2"?
>
> > People generally don't start counting with zero; they count
> > starting with 1.
>
> Yes, and we can and should, if we were a bit smarter, start at the step
> away from the unison: one step away, two steps away.
>
> > Interval numbers come from staff distace, which comes
> > from 7 diatonic nominals, which make perfect sense when distances
> > between two points begin by counting 1 at the first point.
> Of course it makes perfect sense when starting from that premise. I just
> think that premise is stupid. Plus, it is mathematically dumb:
>
> 5+4=8, 5+5+5=13, 8+2=9 etc.
>
> ...which means we have to learn an extra math algorithm: find the number
> of pitches contributing to the stack, and subtract that number, then add
> one to get the 'real' (fake) interval name. As if kids don't hate math
> enough, in general, we musicians add to the mess by making them learn a
> really pointless exception to their basic and important
> arithmetic skills.
>
> > Even though
> > distance isn't usually measured this way, this isn't just a distance; it's
> > a structure within a scale, and the degrees are what are being counted.
> >
> Why not have a distance always be measured the way one expects? The
> logic of scale structure still remains, just the names change for clarity.
> Why should scale degree names have to come into play at all?
>
> > Explaining why an interval and its inversion add up to 9 is then not
> > really so tough.
> Really, you are right, but for less than perfectly bright students,
> which are rare these days, it is an unnecessary confusion. My students
> are never confused when I ask "How many steps away is this?" or even
> "what interval is this---remember, add 1 to the steps away, or start
> counting at one on the note", but the more complex things get with
> stacking, the more math exceptions have to be done, and why not make it
> the math they learn in school? It just serves to slow things down, I
> think completely unnecessarily.
>
> I *do* however see the counter argument---this is what musicians have
> called things for centuries, and they need to know that.
>
> *Sigh*---what a ridiculous race humanity is.
>
> > The staff shows pitch height for 7 diatonic naturals.
> > Simple permutations of these 7 tones give 13 tones on the
> > chain which represent all the basic diatonic intervals from the
> > augmented fouth to the diminished fifth, and accidentals are not
> > necessary to show this.
> >
>
> This has nothing to do with what Monz or I were saying, but I'm glad
> you've taken the course 'Staff 101'.
>
> -A.
>

--- In tuning@yahoogroups.com, "Aaron K. Johnson" <aaron@...> wrote:
> > The staff shows pitch height for 7 diatonic naturals.
> > Simple permutations of these 7 tones give 13 tones on the
> > chain which represent all the basic diatonic intervals from the
> > augmented fouth to the diminished fifth, and accidentals are not
> > necessary to show this.
> >
>
> This has nothing to do with what Monz or I were saying, but I'm glad
> you've taken the course 'Staff 101'.
>
> -A.
>

In fact it has everything to do with what you are talking about.
Interval distances are named according to steps in a diatonic scale,
which are represented on a staff. That isn't arbitrary, isn't illogical,
and is completely relevant. And, thanks for the kudos but I didn't
take the course, I taught the course.

Yours,
Aaron Hunt
H-Pi Instruments

Hi monz. In number theory, the 'natural numbers' are the counting
numbers; no need to correct me there, thanks. It isn't illogical to call a
unison a unison. You guys want to call a unison a 'nothing'? Go back to
the beginning. In Western music theory, tones are represented by notes
which potentially become pitches, and because of this an interval
means many things; it exists between two tones (what's mathematical),
two notes (what's written), and two pitches (what's heard). The tones are
too complicated; nobody wants to worry about fractions and big numbers.
The names come from the notes, which is why I worte about the staff earlier,
and we use the same names to identify pitches by ear. An interval distance
corresponds to staff positions spanned. It only secondarily corresponds
to scale steps traversed. Staff positions are counted, 1, 2, 3, etc. If there
is only one position present, then only 1 is counted. Children count starting
with 1, and they get this immediately. Look at the staff, point at the notes,
count positions. Done. It isn't illogical and it isn't tough to comprehend.

Yours,
Aaron Hunt
H-Pi Instruments

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
>
> Hi Aaron (Johnson),
>
>
> > --- In tuning@yahoogroups.com, Aaron Krister Johnson <aaron@> wrote:
> > >
> > > Quoting monz:
> > >
> > > > Anyway, yes, revamping the interval names, starting them
> > > > with zero, so that they make mathematical sense, would
> > > > make things so much easier.
> >
> >
> > Counting (number theory = natural) numbers are used
> > for diatonic intervals. Two pitches are involved. We don't
> > count them 'zero, one'. People generally don't start
> > counting with zero; they count starting with 1. Interval
> > numbers come from staff distace, which comes from 7
> > diatonic nominals, which make perfect sense when
> > distances between two points begin by counting 1 at
> > the first point. Even though distance isn't usually
> > measured this way, this isn't just a distance; it's
> > a structure within a scale, and the degrees are what
> > are being counted.
>
>
> Mostly, what you wrote here is correct. But actually,
> the counting numbers are the cardinal numbers
> (1, 2, 3, ...), and i think the dead giveaway with this
> situation is that the interval names are *not* cardinal
> numbers, but *ordinal* numbers (1st, 2nd, 3rd, ...).
>
> While my argument is against the illogicality of
> interval names because they start from 1 instead of 0,
> it *is* entirely logical to call the starting pitch
> of a distance the "first". The illogicality comes in
> because the interval names count the nominal
> (A, B, C, etc.) degree steps, and the starting pitch
> is not a step. As Ozan says, it's nothing.
>
> So there, now it sounds like i'm agreeing with Ozan
> after all the arguing i've done against what he wrote.
> Shows hows ridiculous the whole thing is.
>
> Maybe if we argue long enough and loudly enough,
> we can convince the musical establishment at large
> to switch to the 0th, 1st, 2nd, ... system.
>
>
>
> -monz
> http://tonalsoft.com
> Tonescape microtonal music software
>

Hi Aaron,

Hoping to clarify a few things:

--- In tuning@yahoogroups.com, "Aaron Andrew Hunt" <aahunt@...> wrote:
>
> Hi monz. In number theory, the 'natural numbers' are
> the counting numbers; no need to correct me there, thanks.

I wasn't saying that you made an error concerning
number theory ... the error was that interval names
are not the cardinal numbers, which are the counting
numbers -- they are the ordinal numbers. I think
there's a significance to that: the interval names
do not actually count the steps, they count the pitches.

> It isn't illogical to call a unison a unison.
> You guys want to call a unison a 'nothing'?

I'm simply arguing to Ozan that if he wants a term
which means "lack of interval", a good term for that
is "unison", which describes the fusing of many sounds
into one.

Ozan kept saying that "prime" is not an interval,
because it describes "lack of interval", and i countered
that that is not the case. "Prime" is Latin for "1st",
which is an ordinal number just like all the other
interval names, so therefore it fits perfectly into
the interval naming system.

> <snip ...> An interval distance corresponds to staff
> positions spanned. It only secondarily corresponds to
> scale steps traversed.

Right, that's what i just said above. The interval
names count the notes, not the steps between the notes.
We moderns *want* them to count the steps, but that's
not where the names came from.

This is really no big deal, because if we want intervals
which measure the steps we can just use the zero convention
instead. Sure people naturally begin counting with 1,
but today all literate people are OK with the idea of
zero being a number just like all the rest.

IMO much more confusion results from the lack of
distinction between the 2 different step sizes of
the diatonic scale. I will always support something
which i think is more logical. One suggestion i read
about once was to use a pair of integers, one representing
the number of diatonic steps and the other representing
the number of 12-edo semitones. This was a good idea,
but a bit awkward.

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

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

> One suggestion [for a new system of interval names] i read
> about once was to use a pair of integers, one representing
> the number of diatonic steps and the other representing
> the number of 12-edo semitones. This was a good idea,
> but a bit awkward.

To illustrate how this works, here is a list of all
the intervals found in the standard Eb ... G# chain-of-5ths,
tuned in 12-edo:

(turn on the "Option|Use Fixed Width Font" link if viewing
this on the stupid Yahoo web interface)

12-edo intervals, 12-tone chain-of-5ths:

interval name . generator . cents . pair . algebra
--------------+-----------+-------+------+---------
perfect-8ve ........ 0 .. 1200 .. (7,12) .. 5t + 2s
diminished-8ve .... -7 .. 1200 .. (7,11) .. 4t + 3s
major-7th ......... +5 .. 1100 .. (6,11) .. 5t + s
minor-7th ......... -2 .. 1000 .. (6,10) .. 4t + 2s
augmented-6th .... +10 ...1000 .. (5,10) .. 5t
diminshed-7th ..... -9 ... 900 .. (6, 9) .. 3t + 3s
major-6th ......... +3 ... 900 .. (5, 9) .. 4t + s
minor-6th ......... -4 ... 800 .. (5, 8) .. 3t + 2s
augmented-5th ..... +8 ... 800 .. (4, 8) .. 4t
diminished-6th.... -11 ... 700 .. (5, 7) .. 2t + 3s
perfect-5th ....... +1 ... 700 .. (4, 7) .. 3t + s
diminished-5th .... -6 ... 600 .. (4, 6) .. 2t + 2s
augmented-4th ..... +6 ... 600 .. (3, 6) .. 3t
perfect-4th ....... -1 ... 500 .. (3, 5) .. 2t + s
augmented-3rd .... +11 ... 500 .. (2, 5) .. 2t + s'
diminished-4th .... -8 ... 400 .. (3, 4) .. t + 2s
major-3rd ......... +4 ... 400 .. (2, 4) .. 2t
minor-3rd ......... -3 ... 300 .. (2, 3) .. t + s
augmented-2nd ..... +9 ... 300 .. (1, 3) .. t + s'
diminished-3rd ... -10 ... 200 .. (2, 2) .. 2s
major-2nd ......... +2 ... 200 .. (1, 2) .. t
minor-2nd ......... -5 ... 100 .. (1, 1) .. s
augmented-prime ... +7 ... 100 .. (0, 1) .. s'
prime (unison) ..... 0 ... 000 .. (0, 0) .. 0

In the "algebra" column, i use these variable names:

t = tone
s = diatonic-semitone
s' = chromatic-semitone

It's very easy to see how:

1) the interval name correlates with the first integer
of the pair, the name always one digit larger; and

2) the cents value corresponds exactly to the second
integer of the pair (multiplied by 100).

3) replacing the variables in the "algebra" column
with the integer-pair represented by each of those
variables, always gives the correct mathematical
result using vector-addition.

Now, one really nice thing about this system is that
while the first integer of the pair always designates
a diatonic-scale interval, the second integer can
be adjusted to fit any EDO. Thus, for example, the
list for the same chain-of-5ths, but tuned in 31-edo,
is as follows:

31-edo intervals, 12-tone chain-of-5ths:

interval name . generator . cents . pair . algebra
--------------+-----------+-------+------+---------
perfect-8ve ........ 0 .. 1200 .. (7,31) .. 5t + 2s
diminished-8ve .... -7 .. 1124 .. (7,29) .. 4t + 3s
major-7th ......... +5 .. 1083 .. (6,28) .. 5t + s
minor-7th ......... -2 .. 1007 .. (6,26) .. 4t + 2s
augmented-6th .... +10 ... 966 .. (5,25) .. 5t
diminshed-7th ..... -9 ... 931 .. (6,24) .. 3t + 3s
major-6th ......... +3 ... 890 .. (5,23) .. 4t + s
minor-6th ......... -4 ... 814 .. (5,21) .. 3t + 2s
augmented-5th ..... +8 ... 773 .. (4,20) .. 4t
diminished-6th.... -11 ... 738 .. (5,19) .. 2t + 3s
perfect-5th ....... +1 ... 697 .. (4,18) .. 3t + s
diminished-5th .... -6 ... 621 .. (4,16) .. 2t + 2s
augmented-4th ..... +6 ... 579 .. (3,15) .. 3t
perfect-4th ....... -1 ... 503 .. (3,13) .. 2t + s
augmented-3rd .... +11 ... 462 .. (2,12) .. 2t + s'
diminished-4th .... -8 ... 427 .. (3,11) .. t + 2s
major-3rd ......... +4 ... 386 .. (2,10) .. 2t
minor-3rd ......... -3 ... 310 .. (2, 8) .. t + s
augmented-2nd ..... +9 ... 269 .. (1, 7) .. t + s'
diminished-3rd ... -10 ... 234 .. (2, 6) .. 2s
major-2nd ......... +2 ... 193 .. (1, 5) .. t
minor-2nd ......... -5 ... 117 .. (1, 3) .. s
augmented-prime ... +7 ... 076 .. (0, 2) .. s'
prime (unison) ..... 0 ... 000 .. (0, 0) .. 0

Note that the vector-addition still works perfectly
here too. In fact, this is exactly the kind of scheme
one must use when writing software that must deal with
musical intervals.

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

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
>
> --- In tuning@yahoogroups.com, "monz" <monz@> wrote:
>
> > One suggestion [for a new system of interval names] i read
> > about once was to use a pair of integers, one representing
> > the number of diatonic steps and the other representing
> > the number of 12-edo semitones. This was a good idea,
> > but a bit awkward.
>
> <snip>
>
> It's very easy to see how:
>
> 1) the interval name correlates with the first integer
> of the pair, the name always one digit larger; and
>
> 2) the cents value corresponds exactly to the second
> integer of the pair (multiplied by 100).
>
> 3) replacing the variables in the "algebra" column
> with the integer-pair represented by each of those
> variables, always gives the correct mathematical
> result using vector-addition.
>
> Now, one really nice thing about this system is that
> while the first integer of the pair always designates
> a diatonic-scale interval, the second integer can
> be adjusted to fit any EDO.
>
> <snip>
>
> Note that the vector-addition still works perfectly
> here too. In fact, this is exactly the kind of scheme
> one must use when writing software that must deal with
> musical intervals.

And it should be pointed out that, just as the second
integer in the vector pair can be scaled to any EDO,
the first integer of the pair can be scaled to any
MOS/DES.

So, for example, Graham Breed's decimal notation idea
for the miracle scales can be applied here, so that
the first integer is a number from 0 to 9 ... at least
i think it would work OK. I can't take the time now
to figure it out and check it, but maybe someone else
will do it.

Anyway, it seems to me that this provides a very neat
way of designating intervals for all the various families
of tunings, in a way which relates to the structure
of those tunings and is thus more meaningful.

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

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

> And it should be pointed out that, just as the second
> integer in the vector pair can be scaled to any EDO,
> the first integer of the pair can be scaled to any
> MOS/DES.
>
> So, for example, Graham Breed's decimal notation idea
> for the miracle scales can be applied here, so that
> the first integer is a number from 0 to 9 ... at least
> i think it would work OK. I can't take the time now
> to figure it out and check it, but maybe someone else
> will do it.

Oops, my bad ... i had meant to post the link to the
Encyclopedia page on Graham's decimal notation:

http://tonalsoft.com/enc/d/decimal.aspx

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

monz schrieb:

> For a minute i didn't know what he was talking about
> either. AD = _Anno Domini_, BC = Before Christ.

Except that this does make sense, since you're counting years, but the zero moment is _not_ a year, but, well, a moment, which has a year before and a year after. I'm with Oz here; zero's not a year.

To make it more logical, I propose using a duration for zero reference and simultaneously doing away with these ridiculous astronomical units and replace them with something (in the age of the fridge) closer to the human experience. Take Mary's pregnancy for period 0.

Klaus
INTERVALLO AB GESTATIONE DOMINI MMDCLXXXVII

Take prophet Muhammed's (sallallahualeyh) flight from Mecca to Madina as
period 0.

Oz.

----- Original Message -----
From: "Klaus Schmirler" <KSchmir@online.de>
To: <tuning@yahoogroups.com>
Sent: 25 Haziran 2007 Pazartesi 13:11
Subject: Re: [tuning] Re: on interval names

> monz schrieb:
>
> > For a minute i didn't know what he was talking about
> > either. AD = _Anno Domini_, BC = Before Christ.
>
> Except that this does make sense, since you're counting years, but the
> zero moment is _not_ a year, but, well, a moment, which has a year
> before and a year after. I'm with Oz here; zero's not a year.
>
> To make it more logical, I propose using a duration for zero reference
> and simultaneously doing away with these ridiculous astronomical units
> and replace them with something (in the age of the fridge) closer to
> the human experience. Take Mary's pregnancy for period 0.
>
> Klaus
> INTERVALLO AB GESTATIONE DOMINI MMDCLXXXVII
>
>
> You can configure your subscription by sending an empty email to one
> of these addresses (from the address at which you receive the list):
> tuning-subscribe@yahoogroups.com - join the tuning group.
> tuning-unsubscribe@yahoogroups.com - leave the group.
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> tuning-help@yahoogroups.com - receive general help information.
>
> Yahoo! Groups Links
>
>
>

Ozan Yarman schrieb:
> Take prophet Muhammed's (sallallahualeyh) flight from Mecca to Madina as
> period 0.
>

26 days?

Then we're having the 19,470andthensometh hadj.

k

> Oz.
> > ----- Original Message -----
> From: "Klaus Schmirler" <KSchmir@online.de>
> To: <tuning@yahoogroups.com>
> Sent: 25 Haziran 2007 Pazartesi 13:11
> Subject: Re: [tuning] Re: on interval names
> > >> monz schrieb:
>>
>>> For a minute i didn't know what he was talking about
>>> either. AD = _Anno Domini_, BC = Before Christ.
>> Except that this does make sense, since you're counting years, but the
>> zero moment is _not_ a year, but, well, a moment, which has a year
>> before and a year after. I'm with Oz here; zero's not a year.
>>
>> To make it more logical, I propose using a duration for zero reference
>> and simultaneously doing away with these ridiculous astronomical units
>> and replace them with something (in the age of the fridge) closer to
>> the human experience. Take Mary's pregnancy for period 0.
>>
>> Klaus
>> INTERVALLO AB GESTATIONE DOMINI MMDCLXXXVII
>>
>>
>> You can configure your subscription by sending an empty email to one
>> of these addresses (from the address at which you receive the list):
>> tuning-subscribe@yahoogroups.com - join the tuning group.
>> tuning-unsubscribe@yahoogroups.com - leave the group.
>> tuning-nomail@yahoogroups.com - turn off mail from the group.
>> tuning-digest@yahoogroups.com - set group to send daily digests.
>> tuning-normal@yahoogroups.com - set group to send individual emails.
>> tuning-help@yahoogroups.com - receive general help information.
>>
>> Yahoo! Groups Links
>>
>>
>>
> > > > You can configure your subscription by sending an empty email to one
> of these addresses (from the address at which you receive the list):
> tuning-subscribe@yahoogroups.com - join the tuning group.
> tuning-unsubscribe@yahoogroups.com - leave the group.
> tuning-nomail@yahoogroups.com - turn off mail from the group.
> tuning-digest@yahoogroups.com - set group to send daily digests.
> tuning-normal@yahoogroups.com - set group to send individual emails.
> tuning-help@yahoogroups.com - receive general help information.
> > Yahoo! Groups Links
> > >

No my dear, the moment of flight itself.

BTW, pilgrimage is Allah's right on every human being. ;)

Oz.

----- Original Message -----
From: "Klaus Schmirler" <KSchmir@online.de>
To: <tuning@yahoogroups.com>
Sent: 25 Haziran 2007 Pazartesi 13:44
Subject: Re: [tuning] Re: on interval names

> Ozan Yarman schrieb:
> > Take prophet Muhammed's (sallallahualeyh) flight from Mecca to Madina as
> > period 0.
> >
>
> 26 days?
>
> Then we're having the 19,470andthensometh hadj.
>
> k

Except for the ludicrous names I agree in principle yet.....

You don't need three different units.
You can do it with two, as your chromatic semitone is the difference between a tone and a diatonic semitone.

t-s=s'

So we come right back to the 5 Large + 2 small = One Octave.

BTW
The new LucyTuned song "Sometimes" is released in UK today on the "Ghosts" album.

http://www.siobhandonaghy.com/index.php

Any purchases, plugs to reviewers, DJ's, etc. would be much appreciated, so that we can get the first microtuned song into the UK, US and other top 20 pop charts.

OK, so it does only have two chords, but "Sometimes" is summery-catchy, and the total album is absolutely amazing.

Charles Lucy lucy@lucytune.com

----- Promoting global harmony through LucyTuning -----

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

LucyTuned Lullabies (from around the world):
http://www.lullabies.co.uk

Skype user = lucytune

http://www.myspace.com/lucytuning

On 25 Jun 2007, at 10:06, monz wrote:

> --- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
>
> > One suggestion [for a new system of interval names] i read
> > about once was to use a pair of integers, one representing
> > the number of diatonic steps and the other representing
> > the number of 12-edo semitones. This was a good idea,
> > but a bit awkward.
>
> To illustrate how this works, here is a list of all
> the intervals found in the standard Eb ... G# chain-of-5ths,
> tuned in 12-edo:
>
> (turn on the "Option|Use Fixed Width Font" link if viewing
> this on the stupid Yahoo web interface)
>
> 12-edo intervals, 12-tone chain-of-5ths:
>
> interval name . generator . cents . pair . algebra
> --------------+-----------+-------+------+---------
> perfect-8ve ........ 0 .. 1200 .. (7,12) .. 5t + 2s
> diminished-8ve .... -7 .. 1200 .. (7,11) .. 4t + 3s
> major-7th ......... +5 .. 1100 .. (6,11) .. 5t + s
> minor-7th ......... -2 .. 1000 .. (6,10) .. 4t + 2s
> augmented-6th .... +10 ...1000 .. (5,10) .. 5t
> diminshed-7th ..... -9 ... 900 .. (6, 9) .. 3t + 3s
> major-6th ......... +3 ... 900 .. (5, 9) .. 4t + s
> minor-6th ......... -4 ... 800 .. (5, 8) .. 3t + 2s
> augmented-5th ..... +8 ... 800 .. (4, 8) .. 4t
> diminished-6th.... -11 ... 700 .. (5, 7) .. 2t + 3s
> perfect-5th ....... +1 ... 700 .. (4, 7) .. 3t + s
> diminished-5th .... -6 ... 600 .. (4, 6) .. 2t + 2s
> augmented-4th ..... +6 ... 600 .. (3, 6) .. 3t
> perfect-4th ....... -1 ... 500 .. (3, 5) .. 2t + s
> augmented-3rd .... +11 ... 500 .. (2, 5) .. 2t + s'
> diminished-4th .... -8 ... 400 .. (3, 4) .. t + 2s
> major-3rd ......... +4 ... 400 .. (2, 4) .. 2t
> minor-3rd ......... -3 ... 300 .. (2, 3) .. t + s
> augmented-2nd ..... +9 ... 300 .. (1, 3) .. t + s'
> diminished-3rd ... -10 ... 200 .. (2, 2) .. 2s
> major-2nd ......... +2 ... 200 .. (1, 2) .. t
> minor-2nd ......... -5 ... 100 .. (1, 1) .. s
> augmented-prime ... +7 ... 100 .. (0, 1) .. s'
> prime (unison) ..... 0 ... 000 .. (0, 0) .. 0
>
> In the "algebra" column, i use these variable names:
>
> t = tone
> s = diatonic-semitone
> s' = chromatic-semitone
>
> It's very easy to see how:
>
> 1) the interval name correlates with the first integer
> of the pair, the name always one digit larger; and
>
> 2) the cents value corresponds exactly to the second
> integer of the pair (multiplied by 100).
>
> 3) replacing the variables in the "algebra" column
> with the integer-pair represented by each of those
> variables, always gives the correct mathematical
> result using vector-addition.
>
> Now, one really nice thing about this system is that
> while the first integer of the pair always designates
> a diatonic-scale interval, the second integer can
> be adjusted to fit any EDO. Thus, for example, the
> list for the same chain-of-5ths, but tuned in 31-edo,
> is as follows:
>
> 31-edo intervals, 12-tone chain-of-5ths:
>
> interval name . generator . cents . pair . algebra
> --------------+-----------+-------+------+---------
> perfect-8ve ........ 0 .. 1200 .. (7,31) .. 5t + 2s
> diminished-8ve .... -7 .. 1124 .. (7,29) .. 4t + 3s
> major-7th ......... +5 .. 1083 .. (6,28) .. 5t + s
> minor-7th ......... -2 .. 1007 .. (6,26) .. 4t + 2s
> augmented-6th .... +10 ... 966 .. (5,25) .. 5t
> diminshed-7th ..... -9 ... 931 .. (6,24) .. 3t + 3s
> major-6th ......... +3 ... 890 .. (5,23) .. 4t + s
> minor-6th ......... -4 ... 814 .. (5,21) .. 3t + 2s
> augmented-5th ..... +8 ... 773 .. (4,20) .. 4t
> diminished-6th.... -11 ... 738 .. (5,19) .. 2t + 3s
> perfect-5th ....... +1 ... 697 .. (4,18) .. 3t + s
> diminished-5th .... -6 ... 621 .. (4,16) .. 2t + 2s
> augmented-4th ..... +6 ... 579 .. (3,15) .. 3t
> perfect-4th ....... -1 ... 503 .. (3,13) .. 2t + s
> augmented-3rd .... +11 ... 462 .. (2,12) .. 2t + s'
> diminished-4th .... -8 ... 427 .. (3,11) .. t + 2s
> major-3rd ......... +4 ... 386 .. (2,10) .. 2t
> minor-3rd ......... -3 ... 310 .. (2, 8) .. t + s
> augmented-2nd ..... +9 ... 269 .. (1, 7) .. t + s'
> diminished-3rd ... -10 ... 234 .. (2, 6) .. 2s
> major-2nd ......... +2 ... 193 .. (1, 5) .. t
> minor-2nd ......... -5 ... 117 .. (1, 3) .. s
> augmented-prime ... +7 ... 076 .. (0, 2) .. s'
> prime (unison) ..... 0 ... 000 .. (0, 0) .. 0
>
> Note that the vector-addition still works perfectly
> here too. In fact, this is exactly the kind of scheme
> one must use when writing software that must deal with
> musical intervals.
>
> -monz
> http://tonalsoft.com
> Tonescape microtonal music software
>
>
>

Hi Charles,

--- In tuning@yahoogroups.com, Charles Lucy <lucy@...> wrote:
>
> Except for the ludicrous names I agree in principle

Well, the whole point of using the integer-pair vector
is to do away with the "ludicrous names" and still provide
the same information. The names are directly translatable
into the vector.

> yet.....
>
> On 25 Jun 2007, at 10:06, monz wrote:
>
> > --- In tuning@yahoogroups.com, "monz" <monz@> wrote:
> >
> > > One suggestion [for a new system of interval names] i read
> > > about once was to use a pair of integers, one representing
> > > the number of diatonic steps and the other representing
> > > the number of 12-edo semitones. This was a good idea,
> > > but a bit awkward.
> >
> > To illustrate how this works, here is a list of all
> > the intervals found in the standard Eb ... G# chain-of-5ths,
> > tuned in 12-edo:
> >
> > (turn on the "Option|Use Fixed Width Font" link if viewing
> > this on the stupid Yahoo web interface)
> >
>
>
> You don't need three different units.
> You can do it with two, as your chromatic semitone
> is the difference between a tone and a diatonic semitone.
>
> t-s=s'
>
> So we come right back to the 5 Large + 2 small = One Octave.

Right you are, and thanks for pointing that out.

I used s' because my purpose was to show how any
interval can be found by *adding* the various
combinations of tones and the two different kinds
of semitones.

Using t-s instead of s' would force the description
of some intervals to use subtraction. So here it is
for the 12-edo example:

12-edo intervals, 12-tone chain-of-5ths:

interval name . generator . cents . pair . algebra
--------------+-----------+-------+------+---------
perfect-8ve ........ 0 .. 1200 .. (7,12) .. 5t + 2s
diminished-8ve .... -7 .. 1200 .. (7,11) .. 4t + 3s
major-7th ......... +5 .. 1100 .. (6,11) .. 5t + s
minor-7th ......... -2 .. 1000 .. (6,10) .. 4t + 2s
augmented-6th .... +10 ...1000 .. (5,10) .. 5t
diminshed-7th ..... -9 ... 900 .. (6, 9) .. 3t + 3s
major-6th ......... +3 ... 900 .. (5, 9) .. 4t + s
minor-6th ......... -4 ... 800 .. (5, 8) .. 3t + 2s
augmented-5th ..... +8 ... 800 .. (4, 8) .. 4t
diminished-6th.... -11 ... 700 .. (5, 7) .. 2t + 3s
perfect-5th ....... +1 ... 700 .. (4, 7) .. 3t + s
diminished-5th .... -6 ... 600 .. (4, 6) .. 2t + 2s
augmented-4th ..... +6 ... 600 .. (3, 6) .. 3t
perfect-4th ....... -1 ... 500 .. (3, 5) .. 2t + s
augmented-3rd .... +11 ... 500 .. (2, 5) .. 3t - s
diminished-4th .... -8 ... 400 .. (3, 4) .. t + 2s
major-3rd ......... +4 ... 400 .. (2, 4) .. 2t
minor-3rd ......... -3 ... 300 .. (2, 3) .. t + s
augmented-2nd ..... +9 ... 300 .. (1, 3) .. 2t - s
diminished-3rd ... -10 ... 200 .. (2, 2) .. 2s
major-2nd ......... +2 ... 200 .. (1, 2) .. t
minor-2nd ......... -5 ... 100 .. (1, 1) .. s
augmented-prime ... +7 ... 100 .. (0, 1) .. t - s
prime (unison) ..... 0 ... 000 .. (0, 0) .. 0

I only had to substitute t-s for s' in three places.
After doing that, i was thinking it would be good to
redo the whole algebra column by using addition or
subtraction depending on whether the ludicrous name
was augmented or diminished respectively ... but i
realized right away how complicated that would be,
because right at the beginning the "augmented-prime"
uses subtraction.

Actually, it might be best to render the whole
algebra column in terms of s and s', and get rid
of the t:

12-edo intervals, 12-tone chain-of-5ths:

interval name . generator . cents . pair . algebra
--------------+-----------+-------+------+---------
perfect-8ve ........ 0 .. 1200 .. (7,12) .. 7s + 5s'
diminished-8ve .... -7 .. 1200 .. (7,11) .. 7s + 4s'
major-7th ......... +5 .. 1100 .. (6,11) .. 6s + 5s'
minor-7th ......... -2 .. 1000 .. (6,10) .. 6s + 4s'
augmented-6th .... +10 ...1000 .. (5,10) .. 5s + 5s'
diminshed-7th ..... -9 ... 900 .. (6, 9) .. 6s + 3s'
major-6th ......... +3 ... 900 .. (5, 9) .. 5s + 4s'
minor-6th ......... -4 ... 800 .. (5, 8) .. 5s + 3s'
augmented-5th ..... +8 ... 800 .. (4, 8) .. 4s + 4s'
diminished-6th.... -11 ... 700 .. (5, 7) .. 5s + 2s'
perfect-5th ....... +1 ... 700 .. (4, 7) .. 4s + 3s'
diminished-5th .... -6 ... 600 .. (4, 6) .. 4s + 2s'
augmented-4th ..... +6 ... 600 .. (3, 6) .. 3s + 3s'
perfect-4th ....... -1 ... 500 .. (3, 5) .. 3s + 2s'
augmented-3rd .... +11 ... 500 .. (2, 5) .. 2s + 3s'
diminished-4th .... -8 ... 400 .. (3, 4) .. 3s + s'
major-3rd ......... +4 ... 400 .. (2, 4) .. 2s + 2s'
minor-3rd ......... -3 ... 300 .. (2, 3) .. 2s + s'
augmented-2nd ..... +9 ... 300 .. (1, 3) .. s + 2s'
diminished-3rd ... -10 ... 200 .. (2, 2) .. 2s
major-2nd ......... +2 ... 200 .. (1, 2) .. s + s'
minor-2nd ......... -5 ... 100 .. (1, 1) .. s
augmented-prime ... +7 ... 100 .. (0, 1) .. s'
prime (unison) ..... 0 ... 000 .. (0, 0) .. 0

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

Thanks, monz, for the clarification; looks like there is really no
disagreement after all.

This is just another way to say what you've already said...
The rationale for calling the unison a prime hinges on the
difference between measuring distances with a ruler and
measuing musical interval distances. When a distance is
measured with a ruler, if there is no distance, there is nothing
as a result, so the answer is 0. Measuring musical intervals is
not the same, because it's the 'sound' that is the object in
question, and if there is 'nothing' then there is no sound. The
two sounds are two points = notes, tones, pitches, in a diatonic
scale. If only one sound is made by the two points, then it's 1,
'one-sound' uni-sonos = unison, also called the prime - same
thing said in a different way. Easy and it does make sense.

I like your system using number pairs in parentheses,
but as you say the second number becomes kind
of superfluous, since it is a given that you have to use the lesser
values and then add one to make the results musically correct,
and it isn't a big deal to ask the reader to remember to add one,
or explain why you are using mod values. In acedemia at least,
mod was used on 12 tones long before 7. I wonder when the
first time modular arithmetic was used on diatonic intervals.
I remember the first time I saw it in a music journal in my youth,
and it wasn't a big deal to me then. I vaguely recall some authors
putting parenthases around the interval value to show that it was
a mod value and the reader should add one to it. I mean if the
interval was a seventh, it was written (6). I think I recall another
author used a different font for the mod values. Brackets or
something else would also work, I guess the problem there is
that brackets and such already have special mathematical
meanings, but authors always just state at the outset what their
notation means and everything is fine.

Yours lists of intervals with the steps addition reminds one
of the ancients (Aristoxenous) as well as the moderns (Martin Vogel,
Easley Blackwood). Useful stuff and well done!

Yours,
Aaron Hunt
H-Pi Instruments

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
>
> Hi Aaron,
>
>
> Hoping to clarify a few things:
>
>
> --- In tuning@yahoogroups.com, "Aaron Andrew Hunt" <aahunt@> wrote:
> >
> > Hi monz. In number theory, the 'natural numbers' are
> > the counting numbers; no need to correct me there, thanks.
>
>
> I wasn't saying that you made an error concerning
> number theory ... the error was that interval names
> are not the cardinal numbers, which are the counting
> numbers -- they are the ordinal numbers. I think
> there's a significance to that: the interval names
> do not actually count the steps, they count the pitches.
>
>
> > It isn't illogical to call a unison a unison.
> > You guys want to call a unison a 'nothing'?
>
>
> I'm simply arguing to Ozan that if he wants a term
> which means "lack of interval", a good term for that
> is "unison", which describes the fusing of many sounds
> into one.
>
> Ozan kept saying that "prime" is not an interval,
> because it describes "lack of interval", and i countered
> that that is not the case. "Prime" is Latin for "1st",
> which is an ordinal number just like all the other
> interval names, so therefore it fits perfectly into
> the interval naming system.
>
> > <snip ...> An interval distance corresponds to staff
> > positions spanned. It only secondarily corresponds to
> > scale steps traversed.
>
>
> Right, that's what i just said above. The interval
> names count the notes, not the steps between the notes.
> We moderns *want* them to count the steps, but that's
> not where the names came from.
>
> This is really no big deal, because if we want intervals
> which measure the steps we can just use the zero convention
> instead. Sure people naturally begin counting with 1,
> but today all literate people are OK with the idea of
> zero being a number just like all the rest.
>
> IMO much more confusion results from the lack of
> distinction between the 2 different step sizes of
> the diatonic scale. I will always support something
> which i think is more logical. One suggestion i read
> about once was to use a pair of integers, one representing
> the number of diatonic steps and the other representing
> the number of 12-edo semitones. This was a good idea,
> but a bit awkward.
>
>
> -monz
> http://tonalsoft.com
> Tonescape microtonal music software
>

--- In tuning@yahoogroups.com, "Aaron Andrew Hunt" <aahunt@...> wrote:
> I like your system using number pairs in parentheses,
> but as you say the second number becomes kind
> of superfluous, since it is a given that you have to use the lesser
> values and then add one to make the results musically correct,
> and it isn't a big deal to ask the reader to remember to add one,
> or explain why you are using mod values.

Oops, sorry I don't know where I got that, since your second numbers
are non-diatonic scale degrees and are not at all superfluous. I like your
system; it makes good sense.

Yours,
Aaron Hunt
H-Pi Instruments

Many years ago I thought about this and chose different letters:

L=Large s=small and d=difference.

[That was in the (last century) - LucyScaleDevelopments Lsd days]

Then I realised that I only needed the L and the s, as the d was reDunDant, for d=L-s.

Hence we both now conform, except that I used L for Large instead of t for Tone.

Back to the sound design and soundtracks ;-)

We are currently working on a short, directed by Stacy Harrison - earlier film here:

http://www.seattlefilm.org/festival/film/detail.aspx?id=21295&FID=33

http://uk.youtube.com/watch?v=WZR9yd_k2rs&mode=related&search=

and a feature:

http://www.weekendloverthefilm.com

Busy, busy, work, work......

Charles Lucy lucy@lucytune.com

----- Promoting global harmony through LucyTuning -----

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

LucyTuned Lullabies (from around the world):
http://www.lullabies.co.uk

Skype user = lucytune

http://www.myspace.com/lucytuning

On 25 Jun 2007, at 16:21, monz wrote:

> Hi Charles,
>
> --- In tuning@yahoogroups.com, Charles Lucy <lucy@...> wrote:
> >
> > Except for the ludicrous names I agree in principle
>
> Well, the whole point of using the integer-pair vector
> is to do away with the "ludicrous names" and still provide
> the same information. The names are directly translatable
> into the vector.
>
> > yet.....
> >
> > On 25 Jun 2007, at 10:06, monz wrote:
> >
> > > --- In tuning@yahoogroups.com, "monz" <monz@> wrote:
> > >
> > > > One suggestion [for a new system of interval names] i read
> > > > about once was to use a pair of integers, one representing
> > > > the number of diatonic steps and the other representing
> > > > the number of 12-edo semitones. This was a good idea,
> > > > but a bit awkward.
> > >
> > > To illustrate how this works, here is a list of all
> > > the intervals found in the standard Eb ... G# chain-of-5ths,
> > > tuned in 12-edo:
> > >
> > > (turn on the "Option|Use Fixed Width Font" link if viewing
> > > this on the stupid Yahoo web interface)
> > >
> >
> >
> > You don't need three different units.
> > You can do it with two, as your chromatic semitone
> > is the difference between a tone and a diatonic semitone.
> >
> > t-s=s'
> >
> > So we come right back to the 5 Large + 2 small = One Octave.
>
> Right you are, and thanks for pointing that out.
>
> I used s' because my purpose was to show how any
> interval can be found by *adding* the various
> combinations of tones and the two different kinds
> of semitones.
>
> Using t-s instead of s' would force the description
> of some intervals to use subtraction. So here it is
> for the 12-edo example:
>
> 12-edo intervals, 12-tone chain-of-5ths:
>
> interval name . generator . cents . pair . algebra
> --------------+-----------+-------+------+---------
> perfect-8ve ........ 0 .. 1200 .. (7,12) .. 5t + 2s
> diminished-8ve .... -7 .. 1200 .. (7,11) .. 4t + 3s
> major-7th ......... +5 .. 1100 .. (6,11) .. 5t + s
> minor-7th ......... -2 .. 1000 .. (6,10) .. 4t + 2s
> augmented-6th .... +10 ...1000 .. (5,10) .. 5t
> diminshed-7th ..... -9 ... 900 .. (6, 9) .. 3t + 3s
> major-6th ......... +3 ... 900 .. (5, 9) .. 4t + s
> minor-6th ......... -4 ... 800 .. (5, 8) .. 3t + 2s
> augmented-5th ..... +8 ... 800 .. (4, 8) .. 4t
> diminished-6th.... -11 ... 700 .. (5, 7) .. 2t + 3s
> perfect-5th ....... +1 ... 700 .. (4, 7) .. 3t + s
> diminished-5th .... -6 ... 600 .. (4, 6) .. 2t + 2s
> augmented-4th ..... +6 ... 600 .. (3, 6) .. 3t
> perfect-4th ....... -1 ... 500 .. (3, 5) .. 2t + s
> augmented-3rd .... +11 ... 500 .. (2, 5) .. 3t - s
> diminished-4th .... -8 ... 400 .. (3, 4) .. t + 2s
> major-3rd ......... +4 ... 400 .. (2, 4) .. 2t
> minor-3rd ......... -3 ... 300 .. (2, 3) .. t + s
> augmented-2nd ..... +9 ... 300 .. (1, 3) .. 2t - s
> diminished-3rd ... -10 ... 200 .. (2, 2) .. 2s
> major-2nd ......... +2 ... 200 .. (1, 2) .. t
> minor-2nd ......... -5 ... 100 .. (1, 1) .. s
> augmented-prime ... +7 ... 100 .. (0, 1) .. t - s
> prime (unison) ..... 0 ... 000 .. (0, 0) .. 0
>
> I only had to substitute t-s for s' in three places.
> After doing that, i was thinking it would be good to
> redo the whole algebra column by using addition or
> subtraction depending on whether the ludicrous name
> was augmented or diminished respectively ... but i
> realized right away how complicated that would be,
> because right at the beginning the "augmented-prime"
> uses subtraction.
>
> Actually, it might be best to render the whole
> algebra column in terms of s and s', and get rid
> of the t:
>
> 12-edo intervals, 12-tone chain-of-5ths:
>
> interval name . generator . cents . pair . algebra
> --------------+-----------+-------+------+---------
> perfect-8ve ........ 0 .. 1200 .. (7,12) .. 7s + 5s'
> diminished-8ve .... -7 .. 1200 .. (7,11) .. 7s + 4s'
> major-7th ......... +5 .. 1100 .. (6,11) .. 6s + 5s'
> minor-7th ......... -2 .. 1000 .. (6,10) .. 6s + 4s'
> augmented-6th .... +10 ...1000 .. (5,10) .. 5s + 5s'
> diminshed-7th ..... -9 ... 900 .. (6, 9) .. 6s + 3s'
> major-6th ......... +3 ... 900 .. (5, 9) .. 5s + 4s'
> minor-6th ......... -4 ... 800 .. (5, 8) .. 5s + 3s'
> augmented-5th ..... +8 ... 800 .. (4, 8) .. 4s + 4s'
> diminished-6th.... -11 ... 700 .. (5, 7) .. 5s + 2s'
> perfect-5th ....... +1 ... 700 .. (4, 7) .. 4s + 3s'
> diminished-5th .... -6 ... 600 .. (4, 6) .. 4s + 2s'
> augmented-4th ..... +6 ... 600 .. (3, 6) .. 3s + 3s'
> perfect-4th ....... -1 ... 500 .. (3, 5) .. 3s + 2s'
> augmented-3rd .... +11 ... 500 .. (2, 5) .. 2s + 3s'
> diminished-4th .... -8 ... 400 .. (3, 4) .. 3s + s'
> major-3rd ......... +4 ... 400 .. (2, 4) .. 2s + 2s'
> minor-3rd ......... -3 ... 300 .. (2, 3) .. 2s + s'
> augmented-2nd ..... +9 ... 300 .. (1, 3) .. s + 2s'
> diminished-3rd ... -10 ... 200 .. (2, 2) .. 2s
> major-2nd ......... +2 ... 200 .. (1, 2) .. s + s'
> minor-2nd ......... -5 ... 100 .. (1, 1) .. s
> augmented-prime ... +7 ... 100 .. (0, 1) .. s'
> prime (unison) ..... 0 ... 000 .. (0, 0) .. 0
>
> -monz
> http://tonalsoft.com
> Tonescape microtonal music software
>
>
>

Hi Aaron,

--- In tuning@yahoogroups.com, "Aaron Andrew Hunt" <aahunt@...> wrote:
>
> --- In tuning@yahoogroups.com, "Aaron Andrew Hunt" <aahunt@> wrote:
>
> > I like your system using number pairs in parentheses,
> > but as you say the second number becomes kind
> > of superfluous, since it is a given that you have to
> > use the lesser values and then add one to make the results
> > musically correct, and it isn't a big deal to ask the
> > reader to remember to add one, or explain why you are
> > using mod values.
>
> Oops, sorry I don't know where I got that, since your
> second numbers are non-diatonic scale degrees and are
> not at all superfluous. I like your system; it makes
> good sense.

Yes, i was going to respond to this, but got caught up
in the historical research on pseudo-Odo's _dialogus_.

Anyway, the integer-pair vector does indeed specify
both the diatonic interval *and* its exact measurement
in EDO degrees.

I only wish it *was* "my" system ... someone else
invented it, but i don't remember now where i got it.
Maybe it was Eric Regener.

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

Hi Aaron,

--- In tuning@yahoogroups.com, "Aaron Andrew Hunt" <aahunt@...> wrote:

> <snip ...> In acedemia at least, mod was used on
> 12 tones long before 7.

Ah, not true at all!

> I wonder when the first time modular arithmetic was
> used on diatonic intervals.

You asked for it ...

Around the year 1000, when A B C D E F G were first used
to describe the diatonic scale and the same letters were
repeated in other octaves.

This appears in the _dialogus_ of an author known today
as "pseudo-Odo", in Chapter 4 "De consonantiis":

http://www.chmtl.indiana.edu/tml/9th-11th/ODODIA_TEXT.html

>> "Tres sunt praeter tonum divisiones, quae naturalem
>> vocum, quam supra dixi, positionem custodiunt.
>> Prima quaternaria est, propter quod in quatuor divisa
>> est, ut a prima A. in quartam D. et haec habet voces
>> quatuor, intervalla tria, id est, duos tonos et unum
>> semitonium. Ubicumque ergo in monochordo inter duas
>> voces duos tonos, et unum semitonium invenies, ipsarum
>> duarum vocum intervallum quaternaria divisione currere
>> ad finem usque probabis: et ideo diatessaron, id est,
>> de quatuor nomen accepit. Secunda vero ternaria est,
>> ut a prima voce A. in quintam E. in cuius spatio
>> continentur voces quinque, intervalla quatuor, id est,
>> tres toni et unum semitonium. Ubicumque ergo videris
>> inter duas voces tres tonos et unum semitonium, ipsarum
>> vocum spatium ternis ad finem passibus currit. Vocatur
>> autem diapente, id est, de quinque, eoquod voces sunt
>> in eius spatio quinque. Tertia vero divisio est, quae
>> per duo vel per medium dividitur, et dicitur diapason,
>> id est, de omnibus. Hanc, ut supra dictum est, ex
>> litterarum similitudine patenter agnoscis, ut a prima A.
>> in octavam a. Constat autem vocibus octo, intervallis
>> septem, id est, tonis quinque, semitoniis duobus.
>> Continet etiam unum diatessaron et unum diapente.
>> A prima A. in quartam D. fit diatessaron, et a quarta D.
>> in octavam a. diapente: a prima A. in octavam a. diapason
>> invenitur hoc modo: A. B. C. D. E. F. G. a."

I was trying to make an English translation of that,
but right now i have to take care of the baby ...

Anyway, it uses letters instead of numbers, but by
recognizing the octave as an equivalence-interval
and repeating the same letters, it is indeed a
modulo-7 arithmetic system.

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

Feature is:

http://www.weekendloversthefilm.com/

Ghosts - free listen is at:

http://www.siobhandonaghy.co.uk/index.php

Charles Lucy lucy@lucytune.com

----- Promoting global harmony through LucyTuning -----

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

LucyTuned Lullabies (from around the world):
http://www.lullabies.co.uk

Skype user = lucytune

http://www.myspace.com/lucytuning

monz, thanks again, but this is 'duh'. Of course it's been thought
of that way. I don't mind being called on something, but I think you
knew I meant actual modular arithmetic, as in post-Gauss.

Yours,
Aaron Hunt
H-Pi Instruments

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
>
> Hi Aaron,
>
>
> --- In tuning@yahoogroups.com, "Aaron Andrew Hunt" <aahunt@> wrote:
>
> > <snip ...> In acedemia at least, mod was used on
> > 12 tones long before 7.
>
>
> Ah, not true at all!
>
>
> > I wonder when the first time modular arithmetic was
> > used on diatonic intervals.
>
> You asked for it ...
>
> Around the year 1000, when A B C D E F G were first used
> to describe the diatonic scale and the same letters were
> repeated in other octaves.
>
> This appears in the _dialogus_ of an author known today
> as "pseudo-Odo", in Chapter 4 "De consonantiis":
>
> http://www.chmtl.indiana.edu/tml/9th-11th/ODODIA_TEXT.html
>
> >> "Tres sunt praeter tonum divisiones, quae naturalem
> >> vocum, quam supra dixi, positionem custodiunt.
> >> Prima quaternaria est, propter quod in quatuor divisa
> >> est, ut a prima A. in quartam D. et haec habet voces
> >> quatuor, intervalla tria, id est, duos tonos et unum
> >> semitonium. Ubicumque ergo in monochordo inter duas
> >> voces duos tonos, et unum semitonium invenies, ipsarum
> >> duarum vocum intervallum quaternaria divisione currere
> >> ad finem usque probabis: et ideo diatessaron, id est,
> >> de quatuor nomen accepit. Secunda vero ternaria est,
> >> ut a prima voce A. in quintam E. in cuius spatio
> >> continentur voces quinque, intervalla quatuor, id est,
> >> tres toni et unum semitonium. Ubicumque ergo videris
> >> inter duas voces tres tonos et unum semitonium, ipsarum
> >> vocum spatium ternis ad finem passibus currit. Vocatur
> >> autem diapente, id est, de quinque, eoquod voces sunt
> >> in eius spatio quinque. Tertia vero divisio est, quae
> >> per duo vel per medium dividitur, et dicitur diapason,
> >> id est, de omnibus. Hanc, ut supra dictum est, ex
> >> litterarum similitudine patenter agnoscis, ut a prima A.
> >> in octavam a. Constat autem vocibus octo, intervallis
> >> septem, id est, tonis quinque, semitoniis duobus.
> >> Continet etiam unum diatessaron et unum diapente.
> >> A prima A. in quartam D. fit diatessaron, et a quarta D.
> >> in octavam a. diapente: a prima A. in octavam a. diapason
> >> invenitur hoc modo: A. B. C. D. E. F. G. a."
>
>
> I was trying to make an English translation of that,
> but right now i have to take care of the baby ...
>
>
> Anyway, it uses letters instead of numbers, but by
> recognizing the octave as an equivalence-interval
> and repeating the same letters, it is indeed a
> modulo-7 arithmetic system.
>
>
> -monz
> http://tonalsoft.com
> Tonescape microtonal music software
>

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

> I wasn't saying that you made an error concerning
> number theory ... the error was that interval names
> are not the cardinal numbers, which are the counting
> numbers -- they are the ordinal numbers.

Which, in the case of finite numbers, are the same.

> IMO much more confusion results from the lack of
> distinction between the 2 different step sizes of
> the diatonic scale. I will always support something
> which i think is more logical. One suggestion i read
> about once was to use a pair of integers, one representing
> the number of diatonic steps and the other representing
> the number of 12-edo semitones. This was a good idea,
> but a bit awkward.

Using two vals both supporting meantone is always a fine plan, and
this would be an example--Eytan Agmon's 7-12 system, in fact. But you
can also, for instance, use vals giving octaves and fifths (the 7-12
system can be seen as giving 25/24 and 128/125 steps.)

The 7-12 system, in other words, uses the pair

[<7 11 16|, <12 19 28|]

to get a pair of numbers a and b, and the corresponding interval is
(128/125)^a * (25/24)^b tempered in meantone. Octaves and fifths uses

[<1 1 0|, <0 1 4|]

to get a pair of numbers a and b, and the corresponding interval is
2^a (3/2)^b tempered in meantone.

Charles Lucy likes the 7-5 system instead of the 7-12 system, where
you can count diatonic (from the 7) and chromatic (from the 5)
semitones. This is a good system in terms of its correspondence to
classical meantone notation, though doing the translation is still
nonobvious. 7-et assigns a value of 1 to the diatonic semitone, which
we can call 16/15, and 0 to the chromatic semitone, which we can call
25/24. 5-et assigns a value of 1 to the chromatic semitone, and 0 to
the diatonic semitone.

It should be noted that to do septimal meantone requires nonpatent
vals. The 7-et should be 19-et - 12-et, and using that, you can get a
5-et which is 12-et - 7-et, or in other words 2*12-et - 19-et. Hence
the Lucy map would be

[<7 11 16 19|, <5 8 12 15|]

Hence for example 7/5 gets (3, 3) as a value, the same as 45/32 and
25/18, and three times 9/8 which gets (1, 1). It is therefore a
tritone.

--- In tuning@yahoogroups.com, Klaus Schmirler <KSchmir@...> wrote:
>
> monz schrieb:
>
> > For a minute i didn't know what he was talking about
> > either. AD = _Anno Domini_, BC = Before Christ.
>
> Except that this does make sense, since you're counting years, but
the
> zero moment is _not_ a year, but, well, a moment, which has a year
> before and a year after. I'm with Oz here; zero's not a year.

The year 1 BC is the year 0 AD, so zero is a year.

Astronomers use Julian and Besselian epochs, which give a year number
in terms of Julian dates. In current use is the Julian2000 epoch. When
you do this, you do indeed get negative year numbers. If only
historians were as logical.

--- In tuning@yahoogroups.com, Charles Lucy <lucy@...> wrote:
>
> Except for the ludicrous names I agree in principle yet.....
>
> You don't need three different units.
> You can do it with two, as your chromatic semitone is the difference
> between a tone and a diatonic semitone.
>
> t-s=s'
>
> So we come right back to the 5 Large + 2 small = One Octave.

Oops, sorry. I was attributing the diatonic-chromatic system to you,
whereas I see you are using tone-semitone.

Hence the true Lucy mapping of a 7-limit interval will be:

[<5 8 12 15|, <2 3 4 4|]

A tone gets mapped to (1,0), a major third to (2,0), a tritone to (3,0).
A minor third is (1,1), a fifth (3,1).

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

> I only wish it *was* "my" system ... someone else
> invented it, but i don't remember now where i got it.
> Maybe it was Eric Regener.

The existence of such systems is a consequence of the fact that
meantone is a rank two system, hence you can use a pair of intervals to
notate. The first used I imagine was octaves and fifths, the 1-0
system, but they are all equivalent.

Gene Ward Smith schrieb:
> --- In tuning@yahoogroups.com, Klaus Schmirler <KSchmir@...> wrote:
>> monz schrieb:
>>
>>> For a minute i didn't know what he was talking about
>>> either. AD = _Anno Domini_, BC = Before Christ.
>> Except that this does make sense, since you're counting years, but > the >> zero moment is _not_ a year, but, well, a moment, which has a year >> before and a year after. I'm with Oz here; zero's not a year.
> > The year 1 BC is the year 0 AD, so zero is a year.

So 2 BC is -1 AD? According to astronomers?

The habit (as it seems) of calling a moment an epoch is putting me off, I must admit.

klaus

> > Astronomers use Julian and Besselian epochs, which give a year number > in terms of Julian dates. In current use is the Julian2000 epoch. When > you do this, you do indeed get negative year numbers. If only > historians were as logical.
> > > > > You can configure your subscription by sending an empty email to one
> of these addresses (from the address at which you receive the list):
> tuning-subscribe@yahoogroups.com - join the tuning group.
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> tuning-help@yahoogroups.com - receive general help information.
> > Yahoo! Groups Links
> > >

Hi klaus,

--- In tuning@yahoogroups.com, Klaus Schmirler <KSchmir@...> wrote:
>
> Gene Ward Smith schrieb:
> > --- In tuning@yahoogroups.com, Klaus Schmirler <KSchmir@> wrote:
> >> monz schrieb:
> >>
> >>> For a minute i didn't know what he was talking about
> >>> either. AD = _Anno Domini_, BC = Before Christ.
> >>
> >> Except that this does make sense, since you're
> >> counting years, but the zero moment is _not_ a year,
> >> but, well, a moment, which has a year before and a
> >> year after. I'm with Oz here; zero's not a year.
> >
> > The year 1 BC is the year 0 AD, so zero is a year.
>
> So 2 BC is -1 AD? According to astronomers?

Yes, that's correct.

Good orrery software allows you to position your
perspective at any point on the Earth's surface, at
any time in history (within the capabilities of the
software). To view the heavens from a time during the
BC era, you typically have to input the year as a
negative number.

More on orrery:
http://en.wikipedia.org/wiki/Orrery

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

--- In tuning@yahoogroups.com, Klaus Schmirler <KSchmir@...> wrote:

> So 2 BC is -1 AD? According to astronomers?

Yes, though astronomers are more likely to work directly with Julian
dates. But this stuff is important sometimes; if you want to find out
the exact date on which an eclipse of the Sun occurred, messing
around with nonsense about BC does not help.

Nor, in fact, does it help historians. It's a stupid system which
even the experts can get caught by. In 1981, there was a big
gathering of Virgil scholars, the occasion being a bimillenial
celebration--it was 2000 years since the date of Virgil's death.
Except, it wasn't. Virgil died in 19 BC, which is -18 AD. 2000-18 =
1982, so they had the wrong year. All these experts, and none of them
got it right until a reporter checked on it and asked them how come.

> The habit (as it seems) of calling a moment an epoch is putting me
> off, I must admit.

An "epoch" is a certain way of specifying a baseline for statements
about the exact date and time.

To all: I had originally addressed this response incorrectly
to Aaron Krister Johnson in message 72035. I've deleted that
post and copied it here, correctly addressed to Andrew Hunt.

-------------------

Hi Aaron (Hunt),

> --- In tuning@yahoogroups.com, Aaron Krister Johnson <aaron@> wrote:
> >
> > Quoting monz:
> >
> > > Anyway, yes, revamping the interval names, starting them
> > > with zero, so that they make mathematical sense, would
> > > make things so much easier.
>
>
> Counting (number theory = natural) numbers are used
> for diatonic intervals. Two pitches are involved. We don't
> count them 'zero, one'. People generally don't start
> counting with zero; they count starting with 1. Interval
> numbers come from staff distace, which comes from 7
> diatonic nominals, which make perfect sense when
> distances between two points begin by counting 1 at
> the first point. Even though distance isn't usually
> measured this way, this isn't just a distance; it's
> a structure within a scale, and the degrees are what
> are being counted.

Mostly, what you wrote here is correct. But actually,
the counting numbers are the cardinal numbers
(1, 2, 3, ...), and i think the dead giveaway with this
situation is that the interval names are *not* cardinal
numbers, but *ordinal* numbers (1st, 2nd, 3rd, ...).

While my argument is against the illogicality of
interval names because they start from 1 instead of 0,
it *is* entirely logical to call the starting pitch
of a distance the "first". The illogicality comes in
because the interval names count the nominal
(A, B, C, etc.) degree steps, and the starting pitch
is not a step. As Ozan says, it's nothing.

So there, now it sounds like i'm agreeing with Ozan
after all the arguing i've done against what he wrote.
Shows hows ridiculous the whole thing is.

Maybe if we argue long enough and loudly enough,
we can convince the musical establishment at large
to switch to the 0th, 1st, 2nd, ... system.

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

Hi Gene and klaus,

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, Klaus Schmirler <KSchmir@> wrote:
>
> > So 2 BC is -1 AD? According to astronomers?
>
> Yes, though astronomers are more likely to work directly
> with Julian dates. But this stuff is important sometimes;
> if you want to find out the exact date on which an eclipse
> of the Sun occurred, messing around with nonsense about
> BC does not help.
>
> Nor, in fact, does it help historians. It's a stupid
> system which even the experts can get caught by. <snip>

I totally agree with this. It's a ridiculous system.

The only thing i can say here that keeps this
on-topic is that in my extensive research into ancient
Greek and Babylonian music-theory, i'm always coming
across authors saying simply "the fifth century",
"the sixteenth century", etc., and what they *really*
mean is "the fifth century BC" etc., and it drives me
nuts.

I suppose if you're a classicist and all of the dates
you ever read are always BC, then it's no big deal.
But as someone who does a ton of research into medieval
music-theory as well as the ancient stuff, for me it's
a real pain to not have the dates spelled out precisely.

For that matter, "medieval" is also a ridiculous term,
indicating a time period which is "in the middle".
Of what? It designates the so-called "dark ages" between
the great flowering of culture in ancient Greece/Rome
and the Renaissance (literally, "rebirth") of the
16th century.

But as we know, and my arguments continually emphasize,
the medieval period was not dark at all where music-theory
is concerned. And the great achievements of Greece did
not happen in isolation: they were almost entirely a
development on the previous achievements of the Sumerians,
Egyptians, and Babylonians. The real "Naissance" of
civilization is with the Sumerians ... the evidence of
which we've been busy destroying in Iraq for the past
four years.

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

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
>
> To all: I had originally addressed this response incorrectly
> to Aaron Krister Johnson in message 72035. I've deleted that
> post and copied it here, correctly addressed to Andrew Hunt.

Oops ... Aaron Andrew Hunt.

(definitely not getting enough sleep these days ...)

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

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

> --- In tuning@yahoogroups.com, "Aaron Andrew Hunt" <aahunt@> wrote:
>
> > I wonder when the first time modular arithmetic was
> > used on diatonic intervals.
>
> You asked for it ...
>
> Around the year 1000, when A B C D E F G were first used
> to describe the diatonic scale and the same letters were
> repeated in other octaves.
>
> This appears in the _dialogus_ of an author known today
> as "pseudo-Odo", in Chapter 4 "De consonantiis":
>
> http://www.chmtl.indiana.edu/tml/9th-11th/ODODIA_TEXT.html
>
> >> <Latin text of chapter 4 snipped>

Actually, this chapter 4 "De Consonatiis" (Of the Consonances)
explains how "besides the division of the tone [into
semitones, explained in chapter 3], there are three
divisions which govern the natural position of sounds"
-- and those are the quaternary (4:3 ratio), ternary (3:2)
and "per duo vel per medium" ("by 2, or in the middle", 2:1).

Actually, the place where pseudo-Odo explains the placement
of the 7 nominal letters and their octave-equivalents
by monochord division is in chapter 2:

>> In primo capite monochordi ad punctum, quem superius
>> diximus, [Gamma]. litteram, id est G. graecum pone,
>> (quae, quoniam raro est in usu, a multis non habetur).
>> Ab ipsa [Gamma]. usque ad punctum, quem in fine posuimus,
>> per novem diligenter divide, et ubi prima nona pars
>> fecerit finem, (prope [Gamma].) A litteram scribe, et
>> haec dicetur vox prima. Ab eadem prima littera A.
>> similiter per novem partire usque ad finem, et in nona
>> parte B. litteram pro voce secunda appone. Dein ad caput
>> revertere, et divide per quatuor a [Gamma]. et pro voce
>> tertia C. litteram scribe. A prima A. similiter per
>> quatuor divide, et pro voce quarta D. litteram scribe.
>> Eodem modo B. dividens per quatuor, invenies quintam E.
>> Tertia quoque C. insinuat sextam litteram F. Post haec
>> ad [Gamma]. revertere, et ab ipsa, et ab aliis, quae
>> sequuntur per ordinem, praedictam lineam in duas partes,
>> id est, per medium divide, usque dum habeas voces
>> quatuordecim vel quindecim absque [Gamma]. et dum voces
>> per medium diviseris, dissimiles easdem facere debebis.
>> Verbi gratia, dum a [Gamma]. per medium dividis, pro
>> [Gamma]. scribe G. pro A. mediata pone a. et pro B.
>> aliam [sqb]. et pro C. aliam c. et pro D. aliam d. et
>> pro E. aliam e. et pro F. aliam f. et pro G. aliam g.
>> pro a aliam aa ut a medietate monochordi in antea eaedem
>> sint litterae, quae sunt et in prima parte. Praeterea
>> a voce sexta F. per quatuor divide, et retro [sqb].
>> aliam b. rotundam pone: quae ambae pro una voce accipiuntur,
>> et una dicitur nona secunda, et utraque in eodem cantu
>> regulariter non invenietur. Figurae autem et voces et
>> litterae per ordinem ita ponuntur. [Gamma]. I. A. II.
>> B. III. C. IIII. D. V. E. VI. F. VII. G. VIII. a.
>> IX. I. b. IX. II. [sqb]. X. c. XI. d. XII. e. XIII. f.
>> XIIII. g. XV aa.

Rather than provide the English translation here, i'll
make a webpage out of this for the Encyclopedia.

Suffice here to point out the pertinent information as
it relates to this thread: the final sentence gives a
list of all the letters with the *steps* marked *between*
the letters.

Note also that b and [sqb] (which is the ASCII way of
differentiating between the round and square versions
of the letter b) are both considered to be a "9th step":
"b" is 9.1 and "[sqb]" is 9.2. Pseudo-Odo explains in
chapter 3 that a melody will not use both of these, but
only one or the other.

This is a throwback to the ancient Greek PIS
(Perfect Immutable System), where the Greater System
has a disjunct tetrachord above A _mese_ giving a
step of a tone between A and B, and the Lesser System
has a conjunct tetrachord above A _mese_ giving a
step of a semitone between A and Bb. See my webpage:

http://tonalsoft.com/enc/p/pis.aspx

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

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
>
> --- In tuning@yahoogroups.com, "monz" <monz@> wrote:
>
> > --- In tuning@yahoogroups.com, "Aaron Andrew Hunt" <aahunt@> wrote:
> >
> > > I wonder when the first time modular arithmetic was
> > > used on diatonic intervals.
> >
> > Around the year 1000, when A B C D E F G were first used
> > to describe the diatonic scale and the same letters were
> > repeated in other octaves.
> >
> > This appears in the _dialogus_ of an author known today
> > as "pseudo-Odo", <snip>
> >
> > http://www.chmtl.indiana.edu/tml/9th-11th/ODODIA_TEXT.html
>
> <snip>
>
> Actually, the place where pseudo-Odo explains the placement
> of the 7 nominal letters and their octave-equivalents
> by monochord division is in chapter 2:

Aw, what the hell ... it was easy enough to make an
approximate version of it in ASCII, so here it is, using
the English translation by Oliver Strunk in his _Source
Readings in Music History_:

(As always, to see my diagrams correctly when viewing on
the stupid Yahoo web interface, you must click the "Option"
link, then click "Use Fixed Width Font".)

The result is the standard pythagorean diatonic a-minor
scale, as it had been described since Philolaus (c.400 BC).
The unique new aspect was the introduction of only the
first seven letters of the Roman alphabet and their varied
duplication in other octaves, which is the first recognition
in notation of the octave as the equivalence-interval.

pseudo-Odo _dialogus_, chapter 2:
Of the Measurement of the Monochord

>> In primo capite monochordi ad punctum, quem superius
>> diximus, [Gamma]. litteram, id est G. graecum pone,
>> (quae, quoniam raro est in usu, a multis non habetur).

At the first end-piece of the monochord, at the point at
which we have spoken above, place the letter [Gamma],
that is, a Greek G. (This [Gamma], since it is a letter
rarely used, is by many not understood.)

step 0 = 1

[G]
|-----------------------------------------------------|
1

>> Ab ipsa [Gamma]. usque ad punctum, quem in fine posuimus,
>> per novem diligenter divide, et ubi prima nona pars fecerit
>> finem, (prope [Gamma].) A litteram scribe, et haec dicetur
>> vox prima.

Carefully divide the distance from [Gamma] to the point
placed at the other end into nine parts, and where the
first ninth from [Gamma] ends, write the letter A; we shall
call this the first step.

step 1 = 8/9

[G] .. A
.|-----|-----|-----|-----|-----|-----|-----|-----|-----|
..... 8/9

>> Ab eadem prima littera A. similiter per novem partire
>> usque ad finem, et in nona parte B. litteram pro voce
>> secunda appone.

Then, similarly, divide the distance from the first letter,
A, to the end into nine, and at the first ninth, place
the letter B for the second step.

step 2 = 8/9 * 8/9

...... A .. B
|-----|----|----|----|-----|----|----|-----|----|-----|
......... 64/81

>> Dein ad caput revertere, et divide per quatuor a
>> [Gamma]. et pro voce tertia C. litteram scribe.

Then return to the beginning, divide by four from [Gamma],
and for the third step write the letter C.

step 3 = 3/4

[G] .......... C
|-------------|------------|------------|-------------|
............. 3/4

>> A prima A. similiter per quatuor divide, et pro
>> voce quarta D. litteram scribe.

From the first letter, A, divide similarly by four,
and for the fourth step, write the letter D.

step 4 = 8/9 * 3/4

...... A ......... D
|-----|-----------|-----------|-----------|-----------|
................. 2/3

>> Eodem modo B. dividens per quatuor, invenies quintam E.

In the same way, dividing B by four, you will find the
fifth step, E.

step 5 = 64/81 * 3/4

........... B ....... E
|----------|---------|----------|----------|----------|
.................. 16/27

>> Tertia quoque C. insinuat sextam litteram F.

The third letter, C, likewise reveals the sixth step, F.

step 6 = 3/4 * 3/4

.............. C ....... F
|-------------|---------|---------|---------|---------|
....................... 9/16

>> Post haec ad [Gamma]. revertere, et ab ipsa, et ab
>> aliis, quae sequuntur per ordinem, praedictam lineam
>> in duas partes, id est, per medium divide, usque dum
>> habeas voces quatuordecim vel quindecim absque [Gamma].

Then return to [Gamma], and from it and from the other
letters that follow it in order, divide the line in two
parts, that is, in the middle, until, without [Gamma],
you have fourteen or fifteen steps.

(Note that pseudo-Odo is careful to indicate that [Gamma]
is "step 0"!)

>> et dum voces per medium diviseris, dissimiles easdem
>> facere debebis. Verbi gratia, dum a [Gamma]. per medium
>> dividis, pro [Gamma]. scribe G. pro A. mediata pone a.
>> et pro B. aliam [sqb]. et pro C. aliam c. et pro D.
>> aliam d. et pro E. aliam e. et pro F. aliam f. et
>> pro G. aliam g. pro a aliam aa ut a medietate monochordi
>> in antea eaedem sint litterae, quae sunt et in prima parte.

When you divide the sounds in the middle, you must mark
them differently. For example, when you bisect the distance
from [Gamma], instead of [Gamma], write G; for A bisected,
set down a second a; for B, a second [sqb]; for C, a second c;
for D, a second d, for E, a second e; for F, a second f;
for G, a second g; and for a, a second aa; so that from
the middle of the monochord forward, the letters will be
the same as in the first part.

step 7 = 1/2

[G] ....................... G
|--------------------------|--------------------------|
.......................... 1/2

step 8 = 8/9 * 1/2

...... A .................... a
.|-----|----------------------|------------------------|
............................ 4/9

step 9.2 = 64/81 * 1/2

........... B ................ [b]
|----------|--------------------|---------------------|
............................. 32/81

step 10 = 3/4 * 1/2
.............. C ................ c
|-------------|------------------|--------------------|
................................ 3/8

step 11 = 2/3 * 1/2
.................. D .............. d
|-----------------|----------------|------------------|
.................................. 1/3

step 12 = 16/27 * 1/2
..................... E .............. e
|--------------------|----------------|---------------|
.................................... 8/27

step 13 = 9/16 * 1/2
... .................... F ............ f
|-----------------------|--------------|--------------|
...................................... 9/32

step 14 = 1/2 * 1/2
........................... G ........... g
|--------------------------|-------------|------------|
........................................ 1/4

step 15 = 4/9 * 1/2
............................. a .......... aa
|----------------------------|------------|-----------|
......................................... 2/9

>> Praeterea a voce sexta F. per quatuor divide, et
>> retro [sqb]. aliam b. rotundam pone: quae ambae pro
>> una voce accipiuntur, et una dicitur nona secunda,
>> et utraque in eodem cantu regulariter non invenietur.

In addition, from the sixth step, F, divide into four,
and before [sqb], place a second round b; these two are
accepted as a single step, one being called the second
ninth step, and both are not regularly found in the same
melody.

step 9.1 = 9/16 * 3/4

........................ F .... b
|-----------------------|------|-------|------|-------|
............................. 27/64

>> Figurae autem et voces et litterae per ordinem ita
>> ponuntur. [Gamma]. I. A. II. B. III. C. IIII. D. V. E.
>> VI. F. VII. G. VIII. a. IX. I. b. IX. II. [sqb]. X. c.
>> XI. d. XII. e. XIII. f. XIIII. g. XV aa.

The figures, moreover, both sounds and letters, are thus
arranged in order:

[G] A B C D E F G a b. [b] c. d. e. f. g. aa = notes
... 1 2 3 4 5 6 7 8 9.1 9.2 10 11 12 13 14 15 = steps

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

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

> > > http://www.chmtl.indiana.edu/tml/9th-11th/ODODIA_TEXT.html
>
> <snip>
>
> pseudo-Odo _dialogus_, chapter 2:
> Of the Measurement of the Monochord
>
>
> >> In primo capite monochordi ad punctum, quem superius
> >> diximus, [Gamma]. litteram, id est G. graecum pone,
> >> (quae, quoniam raro est in usu, a multis non habetur).
>
> At the first end-piece of the monochord, at the point at
> which we have spoken above, place the letter [Gamma],
> that is, a Greek G. (This [Gamma], since it is a letter
> rarely used, is by many not understood.)
>
> step 0 = 1
>
> [G]
> |-----------------------------------------------------|
> 1

I forgot to mention that i use [Gamma] in the text and
[G] in the diagrams where the Greek letter Gamma actually
appears. Also ...

> step 9.2 = 64/81 * 1/2
>
> ........... B ................ [b]
> |----------|--------------------|---------------------|
> ............................. 32/81

I use [sqb] in the text and [b] in the diagrams where
square-b or a natural-sign actually appears.

The regular "b" round-b designates a Bb.

> The figures, moreover, both sounds and letters, are thus
> arranged in order:
>
> [G] A B C D E F G a b. [b] c. d. e. f. g. aa = notes
> ... 1 2 3 4 5 6 7 8 9.1 9.2 10 11 12 13 14 15 = steps

I tried my best to make a lattice of this system,
in 2-3-space, with the 2,3-monzo, letter-name,
and ratio.

................... [2 0> ...........[2>
.................. -- g ----------- aa
.....................4:1 ......... 9:2
..................... | ........... |
..................... | ........... |
[6 -3> [5 -2> [3 -1> [1 0> [0 1> [-2 2> [-3 3> [-5 4>
-- b ---- f --- c ---- G ---- d ---- a ---- e --- [b]
64:27. 32:9 . 8:3 .. 2:1 .. 3:1 .. 9:4 . 27:8 . 81:32
......... | ... | .... | .... | .... | .... | .... |
......... | ... | .... | .... | .... | .... | .... |
...... [4 -2> [2 -1> [0 0> [-1 1> [-3 2> [-4 3> [-6 4>
..... --- F ---- C -- [G] --- D ---- A ---- E ---- B
....... 16:9 . 4:3 .. 1:1 .. 3:2 .. 9:8 . 27:16. 81:64

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

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote: on
>http://www.chmtl.indiana.edu/tml/9th-11th/ODODIA_TEXT.html
> >
> > pseudo-Odo _dialogus_, chapter 2:
> > Of the Measurement of the Monochord
> >
> >
> > >> In primo capite monochordi ad punctum, quem superius
> > >> diximus, [Gamma]. litteram, id est G. graecum pone,
> > >> (quae, quoniam raro est in usu, a multis non habetur).
> >
Gamma-UT denotates in medieval theory usually the lowest
empty string on the violin GAMMA=G-D-A-E tuned in 5ths,
or the single string on the momochord, as for example:

http://www.celestialmonochord.org/log/images/celestial_monochord.jpg
or
http://www.imaginatorium.org/books/monochd.gif

> > At the first end-piece of the monochord, at the point at
> > which we have spoken above, place the letter [Gamma],
> > that is, a Greek G. (This [Gamma], since it is a letter
> > rarely used, is by many not understood.)

GAMMA indicates the initial root-pitch, the unisono,
Historically GAMMA became later the
http://upload.wikimedia.org/wikipedia/commons/f/ff/GClef.svg
in
http://en.wikipedia.org/wiki/Clef

Fludd's heptatonic scale bases also on the G-Clef too:

0. : GAMMA (becames later 'GG') (UT) unisono of the empty string
1. : A (RE = todays's DO) prime
2. : B-sqr (MI) ditone (quadratic-B)
3. : C (FA)
4. : D (SOL=sun)
5. : E (LA)
6. : F (not used in mdivial hexachord notation, later 'SI' or 'TI')
7. = 0.' : G (ut = today's do)
8. = 1.' : a (re)
9. = 2.' : b-sqr (mi)
10. = 3.' : c (fa)
11. = 4.' : d (sol)
12. = 5.' : e (la)
13. = 6.' : f (later regionally labeled as: si or ti)
14. = 7.' = 0." gg (octaved ut') double octave above GAMMA-UT

> >
> > step 0 = 1
> >
> > [G]
> > |-----------------------------------------------------|
> > 1
>
>
> I forgot to mention that i use [Gamma] in the text and
> [G] in the diagrams where the Greek letter Gamma actually
> appears. Also ...
>
that GAMMA is essential,
because all other following pitches are derived
relative from that that basic root in 5ths
in reference to the initial GAMMA:
>
> > step 9.2 = 64/81 * 1/2
> >
> > ........... B ................ [b]
> > |----------|--------------------|---------------------|
> > ............................. 32/81
>
>
> I use [sqb] in the text and [b] in the diagrams where
> square-b or a natural-sign actually appears.
>
> The regular "b" round-b designates a Bb.
>
>
> > The figures, moreover, both sounds and letters, are thus
> > arranged in order:
> >
> > [G] A B C D E F G a b. [b] c. d. e. f. g. aa = notes
following Fludd [G] means here really 'GG'

> > ... 1 2 3 4 5 6 7 8 9.1 9.2 10 11 12 13 14 15 = steps
in modern terms:
0GAMMA 1A 2B 3C 4E 5F 6G
7=0'a.
8-=1-'b_round. 8=1'b_sqr. 9=2'c. 10=3'd. 11=4'e. 12=5'f. 13=6'g.
14=7'aa

attend the two different versions of b. in different cultures
for extending hepatonics:

In middle-europe the
'b-rotundum' stayed, the simple 'B'
'b-quadratic' becamed 'h', the next letter in the alphabet.
Werckmeister defines 'H' := B#
as an sharpened accidential of B.
For yielding the german octatonic scale: A-B-H-C-D-E-F-G
from the chain of 8 times 5ths:....Eb)-B-F-C-G-D-A-E-H-(F#.....

Conversely to that local development,
in the romanic countries:
'b-rotundum' became later an accidential diminshed 'Bb', but
'b-quadratum' stayed respectively the simple diatonic
'B' leading-tone(limma) to C.
while remaining in traditional heptatonics:
A-B-C-D-E-F-G from 7 times 5ths:
....Eb-Bb)-F-C-G-D-A-E-B-(F#-C#....

Illustrating example:
While B-A-C-H consists in an vaild melody without any modulation
in octatonics,
the older heptationics considers there would be
least one change in key from F-maj to C
in order to harmonize the sequence: B-A-C-H properly.

That different developments caused many confusions
even among musicologists, unfamiliar with that
different ways of labeling pitches.

>
> I tried my best to make a lattice of this system,
> in 2-3-space, with the 2,3-monzo, letter-name,
> and ratio.
>
> ................... [2 0> ...........[2>
> .................. -- g ----------- aa
> .....................4:1 ......... 9:2
> ..................... | ........... |
> ..................... | ........... |
> [6 -3> [5 -2> [3 -1> [1 0> [0 1> [-2 2> [-3 3> [-5 4>
> -- b ---- f --- c ---- G ---- d ---- a ---- e --- [b]
> 64:27. 32:9 . 8:3 .. 2:1 .. 3:1 .. 9:4 . 27:8 . 81:32
> ......... | ... | .... | .... | .... | .... | .... |
> ......... | ... | .... | .... | .... | .... | .... |
> ...... [4 -2> [2 -1> [0 0> [-1 1> [-3 2> [-4 3> [-6 4>
> ..... --- F ---- C -- [G] --- D ---- A ---- E ---- B
> ....... 16:9 . 4:3 .. 1:1 .. 3:2 .. 9:8 . 27:16. 81:64
>
that [G]=GG == 1:1 = 3^0 definition of
marking the empty string,
agrees with my observation in:
/tuning/topicId_70684.html#71324

A.S.

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
> > --- In tuning@yahoogroups.com, "Aaron Andrew Hunt" <aahunt@> wrote:
> >
> > > I wonder when the first time modular arithmetic was
> > > used on diatonic intervals.
> >
> > You asked for it ...
> >
> > Around the year 1000, when A B C D E F G were first used
> > to describe the diatonic scale and the same letters were
> > repeated in other octaves.
> >
> > This appears in the _dialogus_ of an author known today
> > as "pseudo-Odo", in Chapter 4 "De consonantiis":

Monz, thanks for all your work, but since you open your
presentation here with my original question, just let me say
(1) I didn't 'ask for it', as if you are teaching me a lesson,
and (2) repeating letters in various octaves is an instance of
modular arithmetic, the same way that it's totally obvious
this has been the way things have worked basically since
the beginning of Western music, although the letters have
always been qualified by their case or by tick marks or such
things for register, which itself shows that this is not
what you are saying it is. Show us where in this treatise
there is a statement of modular arithmetic and you will have
a case. Something like:

4+4 = 1 mod 7

It isn't there, because as you know C.F.Gauss came up with
modular arithmetic around the turn of the 19th century. Nice
to think that octave congruences were there before Gauss, but
so was the 12 hour clock, many thousands of years before. You
can point to a clock and point to the Egyptions and say 'they
were doing modular arithmetic' and, OK, yes the concepts are
basically the same, but I don't think you'll find some ancient
source with markings like 5+4 = 2 mod 7, the same way you
don't find statements like that in this treatise.

Yours,
Aaron Hunt
H-Pi Instruments

Hi Aaron,

--- In tuning@yahoogroups.com, "Aaron Andrew Hunt" <aahunt@...> wrote:

> Monz, thanks for all your work, but since you open
> your presentation here with my original question, just
> let me say (1) I didn't 'ask for it', as if you are
> teaching me a lesson,

Apparently putting in that remark offended you, and for
that i apologize. I didn't mean to seem as if i am
teaching you a lesson, or anything else condescending.
It was really meant in a jocular way, because my
response became so long and involved. Sorry about that.

> and (2) repeating letters in various octaves is an
> instance of modular arithmetic, the same way that
> it's totally obvious this has been the way things
> have worked basically since the beginning of Western
> music,

How true that is, depends on what you consider to be
the "beginning" of Western music. My goal in quoting,
translating, and diagramming pseudo-Odo's _dialogus_
was to pinpoint exactly the moment when the modulo
system was first used with the Roman letters, which
is the standard system of nominals we still use today.

"Western" music is generally considered to originate
either with the ancient Greeks or with the medieval
Franks. There was a definite break in the theoretical
tradition during the so-called "dark ages" c.500-800 AD.
The music-theory of the Franks which emerges around
800 is clearly different from that of the ancient
Greeks, the last important description of which is
that written by Boethius c.500 AD.

The ancient Greeks did have two music-notation systems
based on using the alphabet, one "instrumental" and
one "vocal". The instrumental system seems to be much
older, as many of the symbols resemble Phoenician letters,
whereas the vocal notation is purely from the Greek
alphabet. The vocal notation does not indicate any
equivalence-interval, merely running thru the whole
alphabet from low pitch to high. But the instrumental
notation does use a tick-mark (') after some of the
symbols to indicate pitches which are an octave higher.
The main source for this is the fragments of a treatise
from Alypius ... this book is from 1891, but it's freely
available and a good source of a lot of historical info,
and here is an illustration showing the Greek notation:

http://www.gutenberg.org/files/20293/20293-h/20293-h.htm#Page_69

Boethius also uses these two Greek notations in his
description of the modal system. But he also did
indeed use Roman letters to label the bridge positions
on his division of the monochord, but without any
recognition of an equivalence-interval. His letters
were akin to the way we label points on a diagram
in geometry class, and he used them in conjunction
with the long and complicated Greek note-names.
Boethius used the entire Roman alphabet, and then
when he ran out of letters, started again at the
beginning but doubled each letter: "AA", "BB", etc.

In fact, his system of labeling proceeds thru the
alphabet in the order in which he describes his
divisions, and has nothing at all to do with pitch-height.
But of course the lowest note was labeled "A".

A diagram of Boethius's monochord division is here:
http://www.chmtl.indiana.edu/tml/6th-8th/BOEMUS4_07GF.gif

The earliest extant documents of Frankish theory,
coming right after the Carolingian Renaissance, show
theorists trying to figure out a way to classify the
modes being used in Frankish chant, and to reconcile
it with what they knew of ancient Greek theory from
Boethius. The _music enhiriadis_ is one of the most
fascinating documents in the theoretical literature,
and is one of the few books which describes "daseian"
notation, an attempt to notate Frankish music which
only lasted about a century. See:

http://tonalsoft.com/enc/d/daseian.aspx

It was Hucbald who c.900 revamped the ancient Greek PIS
(Perfect Immutable System, _systema teleion ametabolon_
in Greek) by moving the "fixed notes" (_hestotes_)
bounding the tetrachords, down a diatonic step, thus
moving the entire system down a step, to make the
finals of the Frankish modes fit into a tetrachord.

This changed the tone/semitone pattern inside the
tetrachord from s-t-t ascending (for the Greeks it
had actually been t-t-s descending) to t-s-t ascending,
and also had the effect of making the lowest note
a step below "A". Thus the problem of finding a label
for the new lowest note. Boethius had given both sets
of ancient Greek notation, and Hucbald selected one
from either set (apparently arbitrarily) to create
a simplified notation adapted to the music of his
time. See my old posts on this:

/tuning/topicId_31040.html#31040
/tuning/topicId_62290.html#62449

It was the pseudo-Odo _dialogus_ which finally
simplified the naming scheme to the first seven
letters of the Roman alphabet, including the two
alternate versions of "b" (round and square, to
represent flat and natural respectively) and
recognizing octave-equivalence in the notation.

As for the rest of your response to me: OK, agreed.
There is not an explicit statement about modulo
arithmetic in musical notation until very recently,
probably mid-1900s. But i view the A B C D E F G A
system as *almost* exactly the same as 0 1 2 3 4 5 6 0.

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

Hi Andreas,

--- In tuning@yahoogroups.com, "Andreas Sparschuh" <a_sparschuh@...>
wrote:
>
> --- In tuning@yahoogroups.com, "monz" <monz@> wrote: on
> >http://www.chmtl.indiana.edu/tml/9th-11th/ODODIA_TEXT.html
> > >
> > > pseudo-Odo _dialogus_, chapter 2:
> > > Of the Measurement of the Monochord
> > >
> > >
> > > >> In primo capite monochordi ad punctum, quem superius
> > > >> diximus, [Gamma]. litteram, id est G. graecum pone,
> > > >> (quae, quoniam raro est in usu, a multis non habetur).
> > >
> Gamma-UT denotates in medieval theory usually the lowest
> empty string on the violin GAMMA=G-D-A-E tuned in 5ths,
> or the single string on the momochord, as for example:
>
> http://www.celestialmonochord.org/log/images/celestial_monochord.jpg
> or
> http://www.imaginatorium.org/books/monochd.gif

No, that is not correct. Gamma-ut denotates the note
which we today write as the note G on the bottom line
of the bass clef. The lowest note of the violin is
an octave above that, and was called "G-sol-re-ut"
(with the Roman capital "G") in medieval theory.
You can see my explanations here:

http://tonalsoft.com/enc/h/hexachord.aspx
http://tonalsoft.com/enc/m/mutation.aspx

> > > At the first end-piece of the monochord, at the point at
> > > which we have spoken above, place the letter [Gamma],
> > > that is, a Greek G. (This [Gamma], since it is a letter
> > > rarely used, is by many not understood.)
>
> GAMMA indicates the initial root-pitch, the unisono,
> Historically GAMMA became later the
> http://upload.wikimedia.org/wikipedia/commons/f/ff/GClef.svg
> in
> http://en.wikipedia.org/wiki/Clef

That is also not correct. The note which eventually
transformed into the treble clef was another octave
higher than the lowest violin note, that is,
"g-sol-re-ut" (with the lowercase Roman "g").

> > I forgot to mention that i use [Gamma] in the text and
> > [G] in the diagrams where the Greek letter Gamma actually
> > appears. Also ...
> >
> that GAMMA is essential,
> because all other following pitches are derived
> relative from that that basic root in 5ths
> in reference to the initial GAMMA:

I didn't leave it out of my description, i just forgot
to mention that i was using "[GAMMA]" and "[G]" to
represent the letter Gamma, since i can't write it
in ASCII text here.

> > I use [sqb] in the text and [b] in the diagrams where
> > square-b or a natural-sign actually appears.
> >
> > The regular "b" round-b designates a Bb.
>
> <Snip>
>
> attend the two different versions of b. in different cultures
> for extending hepatonics:
>
> In middle-europe the
> 'b-rotundum' stayed, the simple 'B'
> 'b-quadratic' becamed 'h', the next letter in the alphabet.
> Werckmeister defines 'H' := B#
> as an sharpened accidential of B.
> For yielding the german octatonic scale: A-B-H-C-D-E-F-G
> from the chain of 8 times 5ths:....Eb)-B-F-C-G-D-A-E-H-(F#.....
>
> Conversely to that local development,
> in the romanic countries:
> 'b-rotundum' became later an accidential diminshed 'Bb', but
> 'b-quadratum' stayed respectively the simple diatonic
> 'B' leading-tone(limma) to C.
> while remaining in traditional heptatonics:
> A-B-C-D-E-F-G from 7 times 5ths:
> ....Eb-Bb)-F-C-G-D-A-E-B-(F#-C#....
>
> Illustrating example:
> While B-A-C-H consists in an vaild melody without any modulation
> in octatonics,
> the older heptationics considers there would be
> least one change in key from F-maj to C
> in order to harmonize the sequence: B-A-C-H properly.
>
> That different developments caused many confusions
> even among musicologists, unfamiliar with that
> different ways of labeling pitches.

In fact, it also resulted in an error in Roy Carter's
1978 translation of Schoenberg's 1911 _Harmonielehre_,
where a note labeled "B" in German was also labeled "B"
in English but should have been "Bb". This is during
Schoenberg's explanation of how the scale comes from
the overtones.

I did a very extensive study of Schoenberg's book, and
this error caused me an incredible amount of confusion
for years (before i knew it was an error), until i
finally got a copy of the original German edition and
saw that in fact the note was supposed to be Bb. See
my webpage:

http://sonic-arts.org/monzo/schoenberg/harm/ch-4.htm#monznote

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

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
>
> Hi Aaron,
>
> <snip>
>
> In fact, [Boethius's] system of labeling proceeds thru
> the alphabet in the order in which he describes his
> divisions, and has nothing at all to do with pitch-height.
> But of course the lowest note was labeled "A".

Just to clarify: by "lowest note" of course i mean the
whole undivided string. Using the letters to mark points,
the obvious starting point is to mark the undivided
string as "A".

> <snip ...> The _music enhiriadis_ is one of the most
> fascinating documents in the theoretical literature,
> and is one of the few books which describes "daseian"
> notation, an attempt to notate Frankish music which
> only lasted about a century. See:
>
> http://tonalsoft.com/enc/d/daseian.aspx

Oops, my bad ... i misspelled the title: it's
_musica enchiriadis_. Its companion treatise, the
_scolica enchiriadis_, also uses daseian notation.

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

Just testing to see if this works ... i don't expect
that it will, but if it does then it will look so much
better, so it's worth a try ...

If it does work, it will only work on the stupid Yahoo
web interface if you click the "Option" and "Use Fixed
With Font" links. For those getting this via email it
should be OK.

Here again is my 2-3-space lattice of the pseudo-Odo
system -- the horizontal axis is powers of 3, and the
vertical axis is powers of 2 (octaves):

[2 0> [-1 2>
-- g ----------- aa
4:1 9:2
| |
| |
[6 -3> [5 -2> [3 -1> [1 0> [0 1> [-2 2> [-3 3> [-5 4>
-- b ---- f --- c ---- G ---- d ---- a ---- e --- [b]
64:27 32:9 8:3 2:1 3:1 9:4 27:8 81:32
| | | | | | |
| | | | | | |
[4 -2> [2 -1> [0 0> [-1 1> [-3 2> [-4 3> [-6 4>
--- F ---- C -- [G] --- D ---- A ---- E ---- B
16:9 4:3 1:1 3:2 9:8 27:16 81:64

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

Hi monz, no worries. You've pieced things together nicely here.
Repeating octave labels as a basic concept was of course around
in ancient Greece. Boethius's Enchiriadis is also of course standard
fare, but I'm only familiar with one volume (in German) which
specifically delves into the Dasian notation. You may know that the
Greek writings have been summarized excellently by Andrew
Barker in a two volume set, the second of which deals with
theoretical writings. The hardback was expensive, but I see these
are now available in paperback. For anybody interested - well worth
the cash:

http://www.amazon.com/Musical-Writings-Cambridge-Readings-Literature/dp/
0521616972/ref=pd_bbs_sr_1/102-2692302-8754505?
ie=UTF8&s=books&qid=1182972790&sr=8-1

Yours,
Aaron Hunt
H-Pi Instruments

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
>
> Hi Aaron,
>
>
> --- In tuning@yahoogroups.com, "Aaron Andrew Hunt" <aahunt@> wrote:
>
> > Monz, thanks for all your work, but since you open
> > your presentation here with my original question, just
> > let me say (1) I didn't 'ask for it', as if you are
> > teaching me a lesson,
>
>
> Apparently putting in that remark offended you, and for
> that i apologize. I didn't mean to seem as if i am
> teaching you a lesson, or anything else condescending.
> It was really meant in a jocular way, because my
> response became so long and involved. Sorry about that.
>
>
> > and (2) repeating letters in various octaves is an
> > instance of modular arithmetic, the same way that
> > it's totally obvious this has been the way things
> > have worked basically since the beginning of Western
> > music,
>
>
> How true that is, depends on what you consider to be
> the "beginning" of Western music. My goal in quoting,
> translating, and diagramming pseudo-Odo's _dialogus_
> was to pinpoint exactly the moment when the modulo
> system was first used with the Roman letters, which
> is the standard system of nominals we still use today.
>
> "Western" music is generally considered to originate
> either with the ancient Greeks or with the medieval
> Franks. There was a definite break in the theoretical
> tradition during the so-called "dark ages" c.500-800 AD.
> The music-theory of the Franks which emerges around
> 800 is clearly different from that of the ancient
> Greeks, the last important description of which is
> that written by Boethius c.500 AD.
>
>
> The ancient Greeks did have two music-notation systems
> based on using the alphabet, one "instrumental" and
> one "vocal". The instrumental system seems to be much
> older, as many of the symbols resemble Phoenician letters,
> whereas the vocal notation is purely from the Greek
> alphabet. The vocal notation does not indicate any
> equivalence-interval, merely running thru the whole
> alphabet from low pitch to high. But the instrumental
> notation does use a tick-mark (') after some of the
> symbols to indicate pitches which are an octave higher.
> The main source for this is the fragments of a treatise
> from Alypius ... this book is from 1891, but it's freely
> available and a good source of a lot of historical info,
> and here is an illustration showing the Greek notation:
>
> http://www.gutenberg.org/files/20293/20293-h/20293-h.htm#Page_69
>
>
> Boethius also uses these two Greek notations in his
> description of the modal system. But he also did
> indeed use Roman letters to label the bridge positions
> on his division of the monochord, but without any
> recognition of an equivalence-interval. His letters
> were akin to the way we label points on a diagram
> in geometry class, and he used them in conjunction
> with the long and complicated Greek note-names.
> Boethius used the entire Roman alphabet, and then
> when he ran out of letters, started again at the
> beginning but doubled each letter: "AA", "BB", etc.
>
> In fact, his system of labeling proceeds thru the
> alphabet in the order in which he describes his
> divisions, and has nothing at all to do with pitch-height.
> But of course the lowest note was labeled "A".
>
> A diagram of Boethius's monochord division is here:
> http://www.chmtl.indiana.edu/tml/6th-8th/BOEMUS4_07GF.gif
>
>
> The earliest extant documents of Frankish theory,
> coming right after the Carolingian Renaissance, show
> theorists trying to figure out a way to classify the
> modes being used in Frankish chant, and to reconcile
> it with what they knew of ancient Greek theory from
> Boethius. The _music enhiriadis_ is one of the most
> fascinating documents in the theoretical literature,
> and is one of the few books which describes "daseian"
> notation, an attempt to notate Frankish music which
> only lasted about a century. See:
>
> http://tonalsoft.com/enc/d/daseian.aspx
>
>
> It was Hucbald who c.900 revamped the ancient Greek PIS
> (Perfect Immutable System, _systema teleion ametabolon_
> in Greek) by moving the "fixed notes" (_hestotes_)
> bounding the tetrachords, down a diatonic step, thus
> moving the entire system down a step, to make the
> finals of the Frankish modes fit into a tetrachord.
>
> This changed the tone/semitone pattern inside the
> tetrachord from s-t-t ascending (for the Greeks it
> had actually been t-t-s descending) to t-s-t ascending,
> and also had the effect of making the lowest note
> a step below "A". Thus the problem of finding a label
> for the new lowest note. Boethius had given both sets
> of ancient Greek notation, and Hucbald selected one
> from either set (apparently arbitrarily) to create
> a simplified notation adapted to the music of his
> time. See my old posts on this:
>
> /tuning/topicId_31040.html#31040
> /tuning/topicId_62290.html#62449
>
>
> It was the pseudo-Odo _dialogus_ which finally
> simplified the naming scheme to the first seven
> letters of the Roman alphabet, including the two
> alternate versions of "b" (round and square, to
> represent flat and natural respectively) and
> recognizing octave-equivalence in the notation.
>
>
> As for the rest of your response to me: OK, agreed.
> There is not an explicit statement about modulo
> arithmetic in musical notation until very recently,
> probably mid-1900s. But i view the A B C D E F G A
> system as *almost* exactly the same as 0 1 2 3 4 5 6 0.
>
>
> -monz
> http://tonalsoft.com
> Tonescape microtonal music software
>

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
>
Hi Monz,
>
>
> As for the rest of your response to me: OK, agreed.
> There is not an explicit statement about modulo
> arithmetic in musical notation until very recently,
> probably mid-1900s.
http://en.wikipedia.org/wiki/Sebastian_Virdung
wrote in
http://www.library.appstate.edu/music/lute/vir1511.html
http://www.library.appstate.edu/music/lute/16index/tvir11.html
about pitch-name 'modulo-arithmetics' on folio 2, p.3:

"So ist der gebrauch Gwidonis gewesen das er die ersten
sibn buchstaben / uff die claues mit grossen versalen hat beschriebe.
Als da stet:
A B C D E F G
Das ander alphabet hat er mit schlechten Buchstaben beschrieben /
Als da steht:
a b c d e f g
Das dritt hat er duplieret:
aa bb cc dd ee ff
....."
Tr:
Guido used for the notes the first 7 upper case letters:
A B(sqr) C D E F G
for the other next ones, the lower case letters
a b(sqr) c d e f g
respectively for the 3rd time:
aa bb(sqr) cc dd ee ff gg... "
.....
".../vnd so offt du eyn not in dem nächsten spacio vunder dem
GAMMAut in den gsang sichst stan / so setze für das spaciu
die note das groß GGin dye tabulatur...."

tr: replace GAMMUut by GG

after that he explains that coeval organs start
an whole tone below GG (formerly Guido's GAMMA)
namely on FF

alike already
http://en.wikipedia.org/wiki/Arnolt_Schlick
in his
"Mirror of organ-makers and organists..." also 1511
Chap. VIII

later, when the compass of the instruments were
more and more enlarged,
triple and even quadruple letter-pitches
were introduced within time, when ever needed.

> But i view the A B C D E F G A
> system as *almost* exactly the same as 0 1 2 3 4 5 6 0.
it's generally up to you in cyclic system where
to start with zero.
Due to above historically reasons, i do prefer for
labeling the ocatonic scale:
.....
-5:
-4: CC
-3: DD
-2: EE
-1: FF
0 : GG or GAMMAut
1 : A
2-: Bb or german B(round)
2 : B or german H(square)
3 : C
4 : D
5 : E
6 : F
7=0' : G
8=1' : a
9-=2-: bb or german b(round)
9=2' : b or german h(square)
10=3': c
.....
&ct.

A.S.

Yay, it works! ASCII lattices are back on the tuning list!!

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
>
> Just testing to see if this works ... i don't expect
> that it will, but if it does then it will look so much
> better, so it's worth a try ...

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

OK, since i know it works now, here is the definitive version,
with a new row which shows the diatonic step-number in pitch-height:

pseduo-Odo _dialogus_, c.1000 AD
first use of modulo-7 A B C D E F G nominals in music notation
lattice in 2,3-space

top row is 2,3-monzo
second row is letter notation, where [G]=Gamma and [b]=square-b
third row is ratio
fourth row is step number

[2 0> [-1 2>
g ------------ aa
4:1 | 9:2
14 | 15
| | |
| | |
[6 -3> [5 -2> [3 -1> [1 0> [0 1> [-2 2> [-3 3> [-5 4>
b ----- f ----- c ----- G ----- d ----- a ----- e ---- [b]
64:27 32:9 8:3 2:1 3:1 9:4 27:8 81:32
9.1 13 10 7 11 8 12 9.2
| | | | | | |
| | | | | | |
[4 -2> [2 -1> [0 0> [-1 1> [-3 2> [-4 3> [-6 4>
F ----- C ---- [G] ---- D ----- A ----- E ----- B
16:9 4:3 1:1 3:2 9:8 27:16 81:64
6 3 0 4 1 5 2

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

Hi Andreas,

--- In tuning@yahoogroups.com, "Andreas Sparschuh" <a_sparschuh@...>
wrote:

> http://en.wikipedia.org/wiki/Sebastian_Virdung
> wrote in
> http://www.library.appstate.edu/music/lute/vir1511.html
> http://www.library.appstate.edu/music/lute/16index/tvir11.html
> about pitch-name 'modulo-arithmetics' on folio 2, p.3:
>
> "So ist der gebrauch Gwidonis gewesen das er die ersten
> sibn buchstaben / uff die claues mit grossen versalen
> hat beschriebe.
> Als da stet:
> A B C D E F G
> Das ander alphabet hat er mit schlechten Buchstaben beschrieben /
> Als da steht:
> a b c d e f g
> Das dritt hat er duplieret:
> aa bb cc dd ee ff
> ....."
> Tr:
> Guido used for the notes the first 7 upper case letters:
> A B(sqr) C D E F G
> for the other next ones, the lower case letters
> a b(sqr) c d e f g
> respectively for the 3rd time:
> aa bb(sqr) cc dd ee ff gg... "
> .....

Guido d'Arezzo almost always gets the credit for being
the first to use the Roman letters, but as i showed it
was actually the pseudo-Odo _dialogus_ which did it
first, perhaps about 50 years before Guido.

Guido also gets credit for inventing the lines-and-spaces
form of staff-notation, but this also had been used
earlier, namely in the _enchiriadis_ treatises i mentioned.

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

Hi Aaron,

--- In tuning@yahoogroups.com, "Aaron Andrew Hunt" <aahunt@...> wrote:
>
> Hi monz, no worries.

Good ... i try hard not to create enemies here.

> You've pieced things together nicely here.
> Repeating octave labels as a basic concept was of
> course around in ancient Greece.

Well, as i said, octave-equivalence was only recognized
in ancient Greek instrumental notation in a few of the notes,
and not at all in either the vocal notation or the
regular long-winded note-names (hypate, paranete, etc.)
-- those names recognized perfect-4th-equivalence.

> Boethius's Enchiriadis is also of course standard
> fare, but I'm only familiar with one volume (in German)
> which specifically delves into the Dasian notation.

The _musica enchiriadis_ is not by Boethius -- it came
along 300 to 400 years later, and is anonymous. Boethius
used the two forms of Greek notation, and his own
Roman-letter geometric labeling system. The daseian
notation came later, and as i said was only used for
a short time around the 10th century.

> You may know that the Greek writings have been
> summarized excellently by Andrew Barker in a two
> volume set, the second of which deals with theoretical
> writings. The hardback was expensive, but I see these
> are now available in paperback. For anybody interested
> - well worth the cash:
>
> http://www.amazon.com/Musical-Writings-Cambridge-Readings-Literature/dp/
> 0521616972/ref=pd_bbs_sr_1/102-2692302-8754505?
> ie=UTF8&s=books&qid=1182972790&sr=8-1

Yes, good for you to put that link here.

Barker's book (primarily Volume 2) has been a mainstay
of my work on all this stuff. It wasn't until after i
had studied every page of it, that i finally dug up
copies of the Greek texts and tried translating some
of them myself.

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

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

> pseduo-Odo _dialogus_, c.1000 AD
> first use of modulo-7 A B C D E F G nominals in music notation
> lattice in 2,3-space
>
> top row is 2,3-monzo
> second row is letter notation, where [G]=Gamma and [b]=square-b
> third row is ratio
> fourth row is step number

Duh ... oops, my bad ... again, i messed it up: the
pitches were not on the proper vertical rows according
to the powers of 2.

*Here* is the definitive version:

[6 -3>
b -----+
64:27 |
9.1 |
| |
| |
| [5 -2>
+------ f
32:9
13
|
|
[4 -2>
F -----+
16:9 |
6 |
| |
| |
| [3 -1>
+----- c
8:3
10
|
|
[2 -1> [2 0>
C ----- g
4:3 4:1
3 14
|
|
[1 0>
G
2:1
7
|
|
[0 0> [0 1>
[G] ----- d
1:1 3:1
0 11
|
|
[-1 1> [-1 2>
D ----- aa
3:2 9:2
4 15
|
|
[-2 2>
a
9:4
8
|
|
[-3 2> [-3 3>
A ----- e
9:8 27:8
1 12
|
|
[-4 3>
E ------+
27:16 |
5 |
| |
| |
| [-5 4>
+----- [b]
81:32
9.2
|
|
[-6 4>
B
81:64
2

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

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
> The _musica enchiriadis_ is not by Boethius -- it came
> along 300 to 400 years later, and is anonymous.

Yikes! Sorry; how easily I confuse myself... I recall now
that the Enchiriadis is considered a partner text with
Boethius's Institutione; hence my muddling!

Yours,
Aaron Hunt
H-Pi Instruments

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote: about
>
> > > > http://www.chmtl.indiana.edu/tml/9th-11th/ODODIA_TEXT.html
> > <snip>
> > pseudo-Odo _dialogus_, chapter 2:
A more modern elaborated interpretation on:
> > Of the Measurement of the Monochord
> >
> > place the letter {Gamma},
that is, a old Greek G,
http://en.wikipedia.org/wiki/Gamma
actual that unisono became later the pitch of GG:={Gamma},
one octave below G and two octaves below g respectively,
replaced in order to get rid of the unfamiliar greek letter
for laypersons without skills in classical antique languages.

>>This {Gamma}, since it is a letter
> > rarely used, is by many not understood.
In deed, so it is!
> >
> > step 0 = 1
> > {GG} = UT
> > |-----------------------------------------------------|
> > 1
>
hence replace {Gamma} by {GG} = UT
in the diagrams where the Greek letter Gamma actually
appears.
>
....
The sacle sounds arranged in ascending letter-order:
> >
>{GG} A (B) C D E F G a (b.) [b.]=h. c. d. e. f. g. aa = notes...
>{0} 1 (2.1) 3 4 5 6 7 8 (9.1) [9.2] 10 11 12 13 14 15 = steps...
UT RE (MI-) FA SOL LA TI ut re (mi-)[mi+] fa sol la ti ut' re'...

The twofold notes: (b.) and [b.]=h.
appear here connotated as an distinct pair of pitches
within the second octave.
Attend the lacking [B]=H step in the octave below,
that would belong interposed virtually at position [2.2]
>
That 8 pitch-classes
results from the 3-limit chain of 7 times 5ths octatonics:

-g---???--- aa
4:1 -???-- 9:2 that aa belongs already to the treble octave
-ut'-???-- re'

aa is the last key on the right side in:
http://upload.wikimedia.org/wikipedia/commons/3/33/Zarlinocembalo.png
Probably may be:
Zarlinos drawing of the letters [B] and [b] illustrate
how that pitch-names converted later into 'H' and 'h' respectively.
Appearently Zarlino no more offered GG in his 19-tone harpsichord,
due to his preferred unisono=A=1:1
instead pseud-Odo's one tone below GG of that.
Conversely:
Todays modern choice of the unisono as C = 1:1
was later proposed by:
http://en.wikipedia.org/wiki/Salomon_de_Caus
in german
http://de.wikipedia.org/wiki/Salomon_de_Caus
but was never really accepted in practice:
http://en.wikipedia.org/wiki/A440
due to the conservative insistence on a4=aa=a'=A440 of the
http://en.wikipedia.org/wiki/International_Organization_for_Standardization

-- (b) ---- f --- c ---- G ---- d ---- a ---- e --- [b]=h
-- 64:27. 32:9 . 8:3 .. 2:1 .. 3:1 .. 9:4 . 27:8 . 81:32
---(9.1)----13-----10-----6------11-----8------12-----[9.2]
----(mi-)---ti-----fa-----ut-----sol----re-----la-----[mi+]

....(B) --- F ---- C -- {GG} --- D ---- A ---- E ---- [B=H]
>..32:27..16:9 . 4:3 .. 1:1 .. 3:2 .. 9:8 . 27:16. 81:64
....(2.1)--5------3------0------4------1------5------[2.2]
....(MI-)- TI-----FA-----UT----SOL-----RE-----LA-----[MI+]

Nonenclature:
I do use 'B' and 'b' only with the essential
(round) or [squared] brackets in order to avoid
the irritating confusion inbetween that 2 versions.
Replace also [B] by H and respectively [b] by h too,
for gaining direct clarness in labeling the two different notes
B and H, so that each letter occurs barely unique
within any octave.

For preventing confusion of ()&[] with
the empty-string root ptich:
GAMMA = {GG} = UT,
i do prefer to
put it into {accolade or curly} braces,
for better distinction from the
already occupied meanings of ()&[].

any further suggestions?
A.S.

Hi Andreas,

--- In tuning@yahoogroups.com, "Andreas Sparschuh" <a_sparschuh@...>
wrote:

> The sacle sounds arranged in ascending letter-order:
> > >
> >{GG} A (B) C D E F G a (b.) [b.]=h. c. d. e. f. g. aa = notes...
> >{0} 1 (2.1) 3 4 5 6 7 8 (9.1) [9.2] 10 11 12 13 14 15 = steps...
> UT RE (MI-) FA SOL LA TI ut re (mi-)[mi+] fa sol la ti ut' re'...
>
> The twofold notes: (b.) and [b.]=h.
> appear here connotated as an distinct pair of pitches
> within the second octave.
> Attend the lacking [B]=H step in the octave below,
> that would belong interposed virtually at position [2.2]

But the plain "B" occupies the pitch-space which would
have contained [B]=H if both types of B were used in
the octave below (position 2), however, they were not.

This is because the Frankish theorists of the 900s
were trying to organize their familiar pitch-space
according to the ancient Greek theory as translated
into Latin by Boethius c.500 AD.

In the ancient Greek PIS (Perfect Immutable System):

* the Greater Perfect System has a disjunct tetrachord
above _mese_ which contains the note represented by
[b]=h, and

* the Lesser Perfect System has a conjunct tetrachord
above _mese_ which contains the note represented by b.

The lower octave was composed of two conjunct tetrachords
which contained the note represented by B (natural = H)
in both the Lesser and Greater systems.

Again, i direct the reader's attention to my webpages:

http://tonalsoft.com/enc/p/pis.aspx
http://tonalsoft.com/enc/g/gps.aspx
http://tonalsoft.com/enc/l/lps.aspx

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

Andreas Sparschuh schrieb:

>> {GG} A (B) C D E F G a (b.) [b.]=h. c. d. e. f. g. aa = notes...
>> {0} 1 (2.1) 3 4 5 6 7 8 (9.1) [9.2] 10 11 12 13 14 15 = steps...
> UT RE (MI-) FA SOL LA TI ut re (mi-)[mi+] fa sol la ti ut' re'... > That's correcting the Ancients, Guido to be exact, to make them look acceptable to us Moderni. There is only one mi, invariably a semitone below fa, and there is no ti.

The hexachord do re mi fa sol la always describes to ditoni joined by a semitone. To get the seventh tone in the octave, you transposed the hexachord from C ("h. naturale") to either F ("h. molle") or G ("h. durum"). Notes were called by their letter (pitch order) and the possible or relevant solmisation syllables, equivalent to its place in the tone/semitone framework. So the lowest B _must_ be a whole tone above A and a semitone below C. An octave higher, it may belong to hexachordum molle (F fa ut, G sol re, A la mi, B fa: a semitone above A), or to hexachordum durum (G sol ut, A la re, B mi, C fa ut).

Take note how B can be either mi or fa (there is no eight tone system). Guido's system was a way to track "modulations", different intonations of a note, e.g. the note above the tenoris A in the authentic D mode. Preferably, all singers would sing it the same way ("mi contra fa, diabolus in musica") - in the notation both B mi and B fa were just a dot between A and C; b and h came much later, in German organ tablatures.

klaus

--- In tuning@yahoogroups.com, Klaus Schmirler <KSchmir@...> wrote:
>
Dears Klaus & Monz,
> >> {GG} A (B) C D E F G a (b.) [b.]=h. c. d. e. f. g. aa = notes...
> >> {0} 1 (2.1) 3 4 5 6 7 8 (9.1) [9.2] 10 11 12 13 14 15 = steps...
> > UT RE (MI-) FA SOL LA TI ut re (mi-)[mi+] fa sol la ti ut' re'...
> >
>
> That's correcting the Ancients,
i never intented to "correct" them howsoever,
But i do hope:
All i want is to understand
the historically roots of
todays modern nomenclauture,
without insisting in the
old traditional usage
as exacty as in the source texts
for gaining deeper insights in theirs original thoughts.

> Guido to be exact, to make them look
> acceptable to us Moderni. There is only one mi, invariably a
> semitone
> below fa,
agreed.

> and there is no ti.
The leading-tone "si" became later converted to "ti":
http://en.wikipedia.org/wiki/Solfege
"In Anglo-Saxon countries, "Sol" is often changed to "So", and "Si"
was changed to "Ti" by Sarah Glover in the nineteenth century so that
every syllable might begin with a different letter. "So" and "Ti" are
used in Tonic sol-fa and in the song "Do-Re-Mi"."

http://de.wikipedia.org/wiki/Solmisation
"Die siebte Note, Si, die Note ohne Namen, wurde dann aus den
Anfangsbuchstaben Sancte Iohannes (Heiliger Johannes) gebildet. Im 17.
Jahrhundert ersetzte Otto Gibelius ut durch do und si (in Europa seit
John Curwen häufig ti') wurde für die 7. Stufe ergänzt"

Attend that I.Newton in his famous drawing:

http://mto.societymusictheory.org/issues/mto.93.0.3/mto.93.0.3.lindley7.gif

extends the hexachords by an further additional 7.th step
in order to yield heptatonics.
He marked step #7 barely weak by dashed-lines '--' alone,
silently without any further literal labeling
at position round_(B)_molle 16:9 modern: Bb.
>
> b and h came much later, in German
> organ tablatures.
also agreed,
but 'H' was already common in use coeval in Newton's time.
Appearently he preferred in his 53-comma scale the pitch of
44.B='--' instead the omitted 48.H as final step back to 53==0
He illustrates that in the diagramm by his 5
concentric arranged heptatonic scales in inwards direction
at the commatic positions:
44.Bb-maj > 22.F-maj > 53.C-maj > 31.G-maj > 9.D-maj
in modulating them 5ths-wise.

with C-major scale in the middle one @ position 53=0:

53: C = ut unisono or prime 1:1
08: d = re 2nd ~10:9
17: e = mi 3rd ~5:4
22: F = fa 4th 4:3
31: G = sol 5th 3:2
39: A = la 6th ~5:3
44:(B)='--' 7th 16:9 What would be a proper hex. name of that?
53: c = ut' 8th 2:1 octave

for proper reading of the spaceing in the ASCII transcript
of N's uncoiled diagramm, in unwinding his 53 circle:
Please click under the button <show meassage option>
there than <Use Fixed Width Font>

53=0: |__|solut|fa|--| 3^0 = 1 unsisono
____: |__|__|__|__|__| 3^...
8___: |mi|la|re|__|__| 3^-10 ~ 10/9? an schisma 32805/32768 to sharp!
9___: |__|__|__|solut| 3^2=9/8
____: |__|__|__|__|__| 3^...
13__: |fa|__|__|__|__| 3^-3=32/27
____: |__|__|__|__|__| 3^...
17__: |__|__|mi|la|re| 3^-8 ~ 5/4 also schismatic sharp
____: |__|__|__|__|__| 3^...
22__: |solut|fa|--|__| 3^-1=4/3 an 4th
____: |__|__|__|__|__| 3^...
26__: |__|__|__|__|mi| 3^-6=1024/729
____: |__|__|__|__|__| 3^...
30__: |la|re|__|__|__| 3^-11 ~ 40/27? " " "
31__: |__|__|solut|fa| 3/2 an pure 5th
____: |__|__|__|__|__| 3^...
35__: |--|__|__|__|__| 3^-4=128/81
____: |__|__|__|__|__| 3^...
39__: |__|mi|la|re|__| 3^-9 ~ 5/3? " " "
40__: |__|__|__|__|sol 3^3=27/16
____: |__|__|__|__|__| 3^...
44__: |ut|fa|--|__|__| 3^-2=16/9
____: |__|__|__|__|__| 3^...
48__: |__|__|__|mi|la| 3^-7 ~ 15/8? " " "
____: |__|__|__|__|__| 3^...
52__: |re|__|__|__|__| 3^-12=2/PC ~ ?160/81=2/SC doubtful " " "!
53=0: |__|solut|fa|--| 3^0=1 modulo (any power of 2) octaved

N's heptatonic scale seems to be derived from a subset
from the schismatic chain of 11 pure just 5ths

Ebb=re-Bbb=la-Fb=mi-(Cb-Gb-Db-Ab-Eb-Bb=--)-F=fa-C=ut-G=sol

appearently with a gap inbetween the accidentials:

3^-10:Ebb d_re_ tone ~10:9
3^-9: Bbb a_la_ sixth ~5:3
3^-8: Fb _e_mi_ pure 3rd ~5:4
3^-7: Cb
3^-6: Gb
3^-5: Db
3^-4: Ab
3^-3: Eb
3^-2: Bb _--_ minor 7th 16.9
3^-1: F __fa_ 4th 4:3
3^00: C __ut_ unisono 1:1
3^+1: G __sol 5th 3:2

Quest:
Who in that group here knows anything about:
What 53-tuning had Newton really in mind when he penned down
his circular drawing?

1. All intervals barely plain Pythagorean 3-limit ?
alike http://en.wikipedia.org/wiki/Jing_Fang
's chain of 52 pure 5ths leaving the one residual 5th
an 'Mercator's-Comma' flat.
http://en.wikipedia.org/wiki/53_equal_temperament

2. 5-3 limit 53 just?

3. 53-EDO?

4. something similar as instead
http://en.wikipedia.org/wiki/Harry_Partch%27s_43-tone_scale
in an rational 53, but rather higher than even 5-limit?

A.S.

--- In tuning@yahoogroups.com, "Andreas Sparschuh" <a_sparschuh@...>
wrote:

> Who in that group here knows anything about:
> What 53-tuning had Newton really in mind when he penned down
> his circular drawing?

I would guess 53-equal. Newton also wrote "612" down somewhere, and he
could not have been thinking of this in Pythagorean terms.

Hi Andreas,

--- In tuning@yahoogroups.com, "Andreas Sparschuh" <a_sparschuh@...>
wrote:

> Attend that I.Newton in his famous drawing:
>
>
http://mto.societymusictheory.org/issues/mto.93.0.3/mto.93.0.3.lindley7.gif
>
> extends the hexachords by an further additional 7.th step
> in order to yield heptatonics.
> He marked step #7 barely weak by dashed-lines '--' alone,
> silently without any further literal labeling
> at position round_(B)_molle 16:9 modern: Bb.
> >
> > b and h came much later, in German
> > organ tablatures.
> also agreed,
> but 'H' was already common in use coeval in Newton's time.
> Appearently he preferred in his 53-comma scale the pitch of
> 44.B='--' instead the omitted 48.H as final step back to 53==0
> He illustrates that in the diagramm by his 5
> concentric arranged heptatonic scales in inwards direction
> at the commatic positions:
> 44.Bb-maj > 22.F-maj > 53.C-maj > 31.G-maj > 9.D-maj
> in modulating them 5ths-wise.
>
> with C-major scale in the middle one @ position 53=0:
>
> 53: C = ut unisono or prime 1:1
> 08: d = re 2nd ~10:9
> 17: e = mi 3rd ~5:4
> 22: F = fa 4th 4:3
> 31: G = sol 5th 3:2
> 39: A = la 6th ~5:3
> 44:(B)='--' 7th 16:9 What would be a proper hex. name of that?
> 53: c = ut' 8th 2:1 octave
>
> <table snipped>
>
> N's heptatonic scale seems to be derived from a subset
> from the schismatic chain of 11 pure just 5ths
>
> Ebb=re-Bbb=la-Fb=mi-(Cb-Gb-Db-Ab-Eb-Bb=--)-F=fa-C=ut-G=sol
>
> appearently with a gap inbetween the accidentials:
>
> 3^-10:Ebb d_re_ tone ~10:9
> 3^-9: Bbb a_la_ sixth ~5:3
> 3^-8: Fb _e_mi_ pure 3rd ~5:4
> 3^-7: Cb
> 3^-6: Gb
> 3^-5: Db
> 3^-4: Ab
> 3^-3: Eb
> 3^-2: Bb _--_ minor 7th 16.9
> 3^-1: F __fa_ 4th 4:3
> 3^00: C __ut_ unisono 1:1
> 3^+1: G __sol 5th 3:2
>
> Quest:
> Who in that group here knows anything about:
> What 53-tuning had Newton really in mind when he penned down
> his circular drawing?

I have never read any of Newton's actual writings on
music, and after an extensive search i have also not
been able to find any of it online, not even in Latin.

The only substantial reference i found is the one
in the Lindley article which accompanies the graphic
in your link:

http://www.societymusictheory.org/mto/issues/mto.93.0.3/mto.93.0.3.lindley.art

See paragraph [20]:

>> The diatonic semitones, labeled "mi-fa" in GIF 7,
>> amount to 5/53-octave; the chromatic semitones amount
>> to 4/53. Newton's harmonic system is not coherent,
>> but if he had provided for an additional pitch
>> class at "4" in the diagram, it would have made at
>> once a good Ab to his Eb (at "35") and a good G# to
>> his C# (at "26"), and thus he would have had a coherent,
>> quasi-Pythagorean system.(12)

But Newton was writing his music-theory ideas around
1700, and by then meantone was firmly entrenched as
the standard orchestral and organ tuning in Western
music, and well-temperaments for the other keyboard
instruments.

All of his contemporaries who wrote about tuning used
5-limit JI as the theoretical ideal basis of tuning.
By that time, pythagorean tuning had ceased being the
standard of tuning for over two centuries. So IMO
it's very doubtful that Newton had any pythagorean
model in mind.

Using "C" as the nominal for 1:1 and showing the
53-edo approximation to JI, the standard 5-limit
JI major scale can be latticed like this:

(use the "Option" and "Use Fixed Width Font" links
to view properly on the stupid Yahoo web interface)

A ----- E ----- B
39 17 48
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
F ----- C ----- G ----- D
22 53=0 31 9

What's interesting is that Newton's set of pitches
matches this one, but the reference (Newton's "Ut")
is in the lattice position where i have G=31. So
instead of the usual C=53 D=9 E=17 F=22 G=31 A=39 B=48,
Newton's Ut re mi fa so la (ti) is equivalent to
what i have labeled G=31 A=39 B=48 C=53 D=9 E=17 F=22.
Transposing this so the Ut=53=0, the result would be
C=53 D=8 E=17 F=22 G=31 A=39 Bb=44:

D ----- A ----- E
8 39 17
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
Bb ---- F ----- C ----- G
44 22 53=0 31

What's interesting to me is that Rameau and some other
later theorists also referred to the JI diatonic
major scale in this fashion -- that is, with the
10:9 as the first "whole-step" instead of 9:8 --
and i always wondered why. Perhaps they were following
Newton, or (more likely, since it appears that Newton
never published his music-theory speculations) perhaps
Newton himself was following an earlier example.

As Lindley says, the diatonic-semitone is 5 degrees
of 53-edo and the chromatic-semitone is 4 -- these
equate to the 5-limit ratios ratios 16:15 and 135:128.

As you (Andreas) know, 135:128 is also only a skhisma
(~2 cents) larger than the pythagorean ratio 256:243,
this ratio, however, functions in pythagorean tuning
not as the chromatic-semtione but as the diatonic-semitone
(_limma_).

This schismatic equivalence is shown in your tables, and
was indeed used in the 1400s. In the late 1800s and early
1900s the tempering-out of the skhisma resulted in the
schismatic temperaments of Helmholtz and Groven respectively,
which exploits this equivalence. But i doubt very much
that Newton around 1700 was thinking in terms of pythagorean
tuning at all. His use of 53 degrees almost certainly
refers to 53-edo, being used as an approximation to
5-limit JI.

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

I'd like to see Newton's circular drawing - does
anyone know where one could come by a photocopy of
it?

BTW, Gene is right that Newton "wrote '612' down
somewhere". It was in one of his notebooks from the
late 1660's. It's mentioned in an article or book
called "Let Newton Be" by Penelope G...something or
other, which I have somewhere.

Mark

--- Gene Ward Smith <genewardsmith@sbcglobal.net>
wrote:

> --- In tuning@yahoogroups.com, "Andreas Sparschuh"
> <a_sparschuh@...>
> wrote:
>
> > Who in that group here knows anything about:
> > What 53-tuning had Newton really in mind when he
> penned down
> > his circular drawing?
>
> I would guess 53-equal. Newton also wrote "612" down
> somewhere, and he
> could not have been thinking of this in Pythagorean
> terms.
>
>
>

____________________________________________________________________________________
Get your own web address.
Have a HUGE year through Yahoo! Small Business.
http://smallbusiness.yahoo.com/domains/?p=BESTDEAL

Andreas Sparschuh schrieb:
> --- In tuning@yahoogroups.com, Klaus Schmirler <KSchmir@...> wrote:
> Dears Klaus & Monz,
>>>> {GG} A (B) C D E F G a (b.) [b.]=h. c. d. e. f. g. aa = notes...
>>>> {0} 1 (2.1) 3 4 5 6 7 8 (9.1) [9.2] 10 11 12 13 14 15 = steps...
>>> UT RE (MI-) FA SOL LA TI ut re (mi-)[mi+] fa sol la ti ut' re'... >>>
>> That's correcting the Ancients,
> i never intented to "correct" them howsoever,
> But i do hope:
> All i want is to understand
> the historically roots of
> todays modern nomenclauture,

OK. Quite a step from Odo, though.

> with C-major scale in the middle one @ position 53=0:

Well, it's not a C-major scale, and I see not reason why Newton should single out a C-major scale. It's just the seven tones in the octave in five transpositions, labeled not according to the more practical letter scheme (A-G with mutable B), but with the hexacordal names. But he should have labeled his "--" "fa" (and his "fa" "fa ut").

> > 53: C = ut unisono or prime 1:1
> 08: d = re 2nd ~10:9
> 17: e = mi 3rd ~5:4
> 22: F = fa 4th 4:3
> 31: G = sol 5th 3:2
> 39: A = la 6th ~5:3
> 44:(B)='--' 7th 16:9 What would be a proper hex. name of that? > 53: c = ut' 8th 2:1 octave
> My guess at why he did it this way? He divides the hexachord into two sets of fifths, fa-ut-sol and re-mi-fa, one of them lowered to approximate 5:4 relations with the other tones. A symmetric and elegant division. Even more elegant is the choice of the fa incarnation of the B note: this then serves as the beginning of the regular chain of fifths and the lowered one, ca. 5/4 away.

re - 3:2 - mi - 3:2 - la
/
5:4
/
--
\
3:2
\
fa - 3:2 - ut - 3:2 - sol

klaus

> --- Gene Ward Smith <genewardsmith@...>
> wrote:
>
> > --- In tuning@yahoogroups.com, "Andreas Sparschuh"
> > <a_sparschuh@>
> > wrote:
> >
> > > Who in that group here knows anything about:
> > > What 53-tuning had Newton really in mind when he
> > penned down
> > > his circular drawing?
> >
> > I would guess 53-equal. Newton also wrote "612" down
> > somewhere, and he
> > could not have been thinking of this in Pythagorean
> > terms.

With 53-equal, you've got a just triad on the central, 12'oclock, Ut
of the drawing, the same in the outer ring, etc., right off the bat,
so at first glance Gene is probably right. In my opinion,
however, "53-equal" can be taken a little more broadly than what
we'd strictly call equal today- 77/76 for example, found right
smack in the middle of what Shaahin would call 53-ADO, is identical
to a 53-EDO step in practical reality, so we're looking at simple
ways to get what works out to 53-EDO. Therefore, even if the actual
method were rational, even mechanical, the result would be "equal"
in "real life".

-Cameron Bobro

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
>
http://mto.societymusictheory.org/issues/mto.93.0.3/mto.93.0.3.lindley7.gif
dated November 1665
Literature reference to the original autograph manuscript:
Cambridge Univ.Lib.,Ms.Add.4000,fol.105v

facsimle reprint and review by my personal friend:
http://en.wikipedia.org/wiki/Mark_Lindley
"Stimmung und Temperatur" (in F. Zaminer, ed., Geschichte der
Musiktheorie, Vol. 6: Hören, Messen und Rechnen in der Frühen Neuzeit
(Wissenschaftliche Buchgesellschaft, 1987) pp.205-210

with an
http://en.wikipedia.org/wiki/Mixolydian_mode
with an additional pyth. minor sevth (B)
in the middle of the 5 hexachordian circles:

53: GG=GAMMAut unisono or prime 1:1
08: A- re 2nd 10:9
17: B- mi 3rd 5:4
22: F fa 4th 4:3
31: C sol 5th 3:2
39: la 6th 5:3
44:(B) ti-flat '--' 7th 16:9
48:[B]=H ti 15:8
53: c = ut' 2:1 octave
> >
>
> I have never read any of Newton's actual writings on
> music, and after an extensive search i have also not
> been able to find any of it online, not even in Latin.
the only source in print that
i do know at the moment about N. is above: Lindley's.

>
http://www.societymusictheory.org/mto/issues/mto.93.0.3/mto.93.0.3.lindley.art
>
> See paragraph [20]:
>
> >> The diatonic semitones, labeled "mi-fa" in GIF 7,
> >> amount to 5/53-octave; the chromatic semitones amount
> >> to 4/53. Newton's harmonic system is not coherent,
> >> but if he had provided for an additional pitch
> >> class at "4" in the diagram, it would have made at
> >> once a good Ab to his Eb (at "35") and a good G# to
> >> his C# (at "26"), and thus he would have had a coherent,
> >> quasi-Pythagorean system.(12)
that's only one aspect of L's cirtics about N.
>
>
> But Newton was writing his music-theory ideas around
> 1700, and by then meantone was firmly entrenched as
> the standard orchestral and organ tuning in Western
> music, and well-temperaments for the other keyboard
> instruments.
Appearently N. drawing depends mostly on the similar
predecessor disk-graphics: p.208
Fig. 30a, Rene Descartes,
http://de.wikipedia.org/wiki/Ren%C3%A9_Descartes
"Musicae compendium" (1618/1650) P.35
>
> All of his contemporaries who wrote about tuning used
> 5-limit JI as the theoretical ideal basis of tuning.
> By that time, pythagorean tuning had ceased being the
> standard of tuning for over two centuries. So IMO
> it's very doubtful that Newton had any pythagorean
> model in mind.
>
L. also presents on p.206 an facsimile of N's dodecaphonic scale in
absolute and also relative "rationall" 5-limit string-lengths,
compatible to Descartes triple - hexachord system:
"
0 : 720 G 1:1 GAMMA-ut
1 : 675 * 15:16
2 : 640 A 8:9 re
3 : 600 * 5:6
4 : 576 B 4:5 mi
5 : 540 C 3:4 fa
6 : 512 * 32:45
7 : 480 D 2:3 sol
8 : 450 * 5:8
9 : 432 E 3:5 la
10: 405 F 9:16
11: 384 * 8:15
12: 360 G 1:2

> Using "C" as the nominal for 1:1
no, his unisiono was GAMMMA=GG for 1:1

> and showing the
> 53-edo approximation to JI,
that can be excluded

> the standard 5-limit
> JI major scale
no, it's 5-limit mixolydian!

> His use of 53 degrees almost certainly
> refers to 53-edo, being used as an approximation to
> 5-limit JI.
hmm, but Lindley and i do interpret N's 14 "extract" out of 53
barely as iterated 5-limit mixolydian-scales.

A.S.

--- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...> wrote:

> With 53-equal, you've got a just triad on the central, 12'oclock, Ut
> of the drawing, the same in the outer ring, etc., right off the bat,
> so at first glance Gene is probably right. In my opinion,
> however, "53-equal" can be taken a little more broadly than what
> we'd strictly call equal today- 77/76 for example, found right
> smack in the middle of what Shaahin would call 53-ADO, is identical
> to a 53-EDO step in practical reality, so we're looking at simple
> ways to get what works out to 53-EDO. Therefore, even if the actual
> method were rational, even mechanical, the result would be "equal"
> in "real life".

This is Newton we are talking about; he was far more mathematically
sophisticated than the average late-17th century scholar, and in
particular would have no difficulty with 53-equal which would require
him to use a substitute. And where is the evidence he did?

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@>
wrote:
>
> > With 53-equal, you've got a just triad on the central,
12'oclock, Ut
> > of the drawing, the same in the outer ring, etc., right off the
bat,
> > so at first glance Gene is probably right. In my opinion,
> > however, "53-equal" can be taken a little more broadly than what
> > we'd strictly call equal today- 77/76 for example, found right
> > smack in the middle of what Shaahin would call 53-ADO, is
identical
> > to a 53-EDO step in practical reality, so we're looking at
simple
> > ways to get what works out to 53-EDO. Therefore, even if the
actual
> > method were rational, even mechanical, the result would
be "equal"
> > in "real life".
>
> This is Newton we are talking about; he was far more
mathematically
> sophisticated than the average late-17th century scholar, and in
> particular would have no difficulty with 53-equal which would
require
> him to use a substitute. And where is the evidence he did?

I didn't say that Newton had to use a "substitute", nor did I say
that there's any evidence he did use a substitute, or that there
needs to be any evidence. Moreover, you're assuming that a rational
equivalent to an equal division is a "substitute", which it is not
necessarily.

The point is, in the case of 53, it doesn't matter. The point is
that Newton's tuning drawing, viewed as 53-equal, and other 53
tunings, could be applied to actual instruments by those far less
sophisticated mathematically than Newton, even taught rules-of-thumb
to craftsmen.

I suggest that you actually read what I write before making
irrelevant comments. Even better, tell me what you think the "612"
refers to.

-Cameron Bobro

--- In tuning@yahoogroups.com, Klaus Schmirler <KSchmir@...> wrote:

> >>>> {GG} A (B) C D E F G a (b.) [b.]=h. c. d. e. f. g. aa =
> OK. Quite a step from Odo, though.
Here pseudo-Odo's extented mixolydian scale with his additional
pythagorean minor-3rd GAMMA>Bb
that became denotated in todays concept of:
http://en.wikipedia.org/wiki/Solfege
as 'Lowered 2 "Ra"'
resulting totally in
8 pitch-classes
for the extended Mixolydian scale:
obtained from a chain of 7 just pure 5ths:

3^-3 <---- 2^-2 <--3^-1 <--3^0=1---> 3 --> 3^2--> 3^3 ----> 3^4

(B)b_round <-- F <--- C <--- GAMMA --> D --> A ---> E --> [B]_square

that's in N's draft sketch @ positions:

0: GG {GAMMA}Ut 1:1 2^0 unisono-prime
9: A re 9:8 tone-2nd ~2^(9/53)
13: (B)b=german_B 32:27 pythagorean minor-3rd ~2^(13/53)
18: [B]=german_H mi 81:64 ditone-3rd ~2^(18/53)
22: C fa 4:3 quart-4th ~2^(22/53)
31: D sol 3:2 quint-5th ~2^(31/53)
40: E la 27:16 sixth-6th ~2^(40/53)
44: F '--'='Te' 16:9 pyth.dim-7th ~2^(44/53)
53: G re' 2:1 octave 2^1
>
attend @ 44: F the
http://en.wikipedia.org/wiki/Solfege
"Lowered 7 Te /te18;/"
pyth.dim.7th

Results in
relative distances of pseudo-Odo's extended mixolydian scale
with an additional interposed pyth.minor-3rd inbetween A and [B]:

GG
9:8
A
256:243
(B)
2187:2048
[B]
256:243
C
9:8
D
9:8
E
256:243
F
9:8
G

! pseudo_Odo_octatonics.scl
!
pyth. 3-limit 5ths chain of 8 pitch-classes: Bb-F-C-G-D-A-E-B
8
!
! 1/1 @ GG=GAMMAut unison prime
!
9/8 ! A
32/27 ! Bb
81/64 ! B
4/3 ! C
3/2 ! D
27/16 ! E
16/9 ! F
2/1 ! G
!
!
>
but back to Newton's drawing:

> Well, it's not a C-major scale, and I see not reason why Newton
> should
> single out a C-major scale.
fully agreed,
because appearently N. overtook
pseudo-Odo's Mixolydian scale concept
for outlineing his 14 out of 53 delineation.