Bach, Werckmeister & Co.

Kurt wrote:

>However, some months back there were discussions >of "gleichschwebenden temperatur" in which it came
>out that this German phase may mean either "equal
>temperament" or "equal beating temperament". However,
>I was not ever aware of the english phrase "equal
>temperament" ever having but one meaning.

Hi there, the literal translation of "gleichschwebend" is in our context "equal beating", which would define a temperature in which e.g. any fifth, be it "a-d" or "e#-c" or "d flat-g#" have the same beating. Which would consequently mean that all these intervals are absolutely equal. So that's 12 tet, isn't it ?
This is what brought our system of keys to an end in the last consequence...
:-D But we are again not going into exploration of new tunings but revising history.
bye
Werner

--- In tuning@yahoogroups.com, Wernerlinden@a... wrote:

> Hi there, the literal translation of "gleichschwebend" is in our
context "equal beating", which would define a temperature in which
e.g. any fifth, be it "a-d" or "e#-c" or "d flat-g#" have the same
beating. Which would consequently mean that all these intervals are
absolutely equal. So that's 12 tet, isn't it ?

Hallo Werner:

In an equal temperament, the beating rate of a given interval class
varies with the absolute frequency. So, by definition, an equal
beating temperament cannot be an equal (interval) temperament. I
believe, though, there is some confusion over this in history, so you
must read descriptions of tunings closely to see if they are equal
beating or equal (interval) temperaments.

This is one reason whay it is hard to tune equal temperaments -- you
must figure out the beating rates for each individual interval, not
each interval class.

Gabor

On Friday 12 December 2003 07:59 am, alternativetuning wrote:
> --- In tuning@yahoogroups.com, Wernerlinden@a... wrote:
> > Hi there, the literal translation of "gleichschwebend" is in our
>
> context "equal beating", which would define a temperature in which
> e.g. any fifth, be it "a-d" or "e#-c" or "d flat-g#" have the same
> beating. Which would consequently mean that all these intervals are
> absolutely equal. So that's 12 tet, isn't it ?
>
>
> Hallo Werner:
>
> In an equal temperament, the beating rate of a given interval class
> varies with the absolute frequency. So, by definition, an equal
> beating temperament cannot be an equal (interval) temperament.

True, however, if one does an equal-beating pythagorean, the result is so
close to equal temperament as to be indistinguishable to the ear...still,
this is very painstaking to do by ear without an accurate way of measuring
the sought after beat rate...

-Aaron.

>
>
>
>
>
> You do not need web access to participate. You may subscribe through
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--- In tuning@yahoogroups.com, Wernerlinden@a... wrote:
> Kurt wrote:
>
> >However, some months back there were discussions
>of "gleichschwebenden temperatur" in which it came
> >out that this German phase may mean either "equal
> >temperament" or "equal beating temperament". However,
> >I was not ever aware of the english phrase "equal
> >temperament" ever having but one meaning.
>
> Hi there, the literal translation of "gleichschwebend" is in our
context "equal beating", which would define a temperature in which
e.g. any fifth, be it "a-d" or "e#-c" or "d flat-g#" have the same
beating. Which would consequently mean that all these intervals are
absolutely equal. So that's 12 tet, isn't it ?
> This is what brought our system of keys to an end in the last
consequence...
> :-D But we are again not going into exploration of new tunings but
revising history.
> bye
> Werner

Quite. I believe that interval recognition in general is a matter of
the difference tones.

Although the supremacy of the octave interval has been recognized for
many thousands of years I myself seem to be the first to have
observed that the reason for this is that the difference tone is
exactly equal to the bass note. If anyone here knows of an earlier
statement of the same by someone else I would be pleased to know
about it.

Peter

--- In tuning@yahoogroups.com, "alternativetuning"
<alternativetuning@y...> wrote:
> --- In tuning@yahoogroups.com, Wernerlinden@a... wrote:
>
> > Hi there, the literal translation of "gleichschwebend" is in our
> context "equal beating", which would define a temperature in which
> e.g. any fifth, be it "a-d" or "e#-c" or "d flat-g#" have the same
> beating. Which would consequently mean that all these intervals are
> absolutely equal. So that's 12 tet, isn't it ?
>
>
> Hallo Werner:
>
> In an equal temperament, the beating rate of a given interval class
> varies with the absolute frequency. So, by definition, an equal
> beating temperament cannot be an equal (interval) temperament. I
> believe, though, there is some confusion over this in history, so
you
> must read descriptions of tunings closely to see if they are equal
> beating or equal (interval) temperaments.
>
> This is one reason whay it is hard to tune equal temperaments --
you
> must figure out the beating rates for each individual interval, not
> each interval class.
>
> Gabor

Yes but Bach lacked an oscilloscope so all he could do was to
equalize the beats, by ear. It would not matter what the actual
precise beat rate was so long as it matched, by ear, the other beat
rates (in the major tonic triad). Bach's purpose in this could only
have been to minimize conflicting beats.

hi Peter,

--- In tuning@yahoogroups.com, "peter_wakefield_sault" <sault@c...>
wrote:

> Although the supremacy of the octave interval has
> been recognized for many thousands of years I myself
> seem to be the first to have observed that the reason
> for this is that the difference tone is exactly equal
> to the bass note. If anyone here knows of an earlier
> statement of the same by someone else I would be pleased
> to know about it.

Ezra Sims presented the recognition of difference tones
as the basis of his harmonic practice in one of his papers.
it was either one of these two, i think it was the
_Perspectives_ article:

Ezra Sims, "Yet Another 72-Noter",
in _Computer Music Journal_ 12, no. 4 [winter 1988], p 28 - 45.

Sims, Ezra. "Reflections on This and That (Perhaps a Polemic)."
_Perspectives of New Music_ 29 (1991): 258-263.

and of course, Helmholtz discussed difference tones quite
a bit. ref. Helmholtz, Hermann. _On the Sensations of Tone..._,
english trans. Alexander Ellis, 1875; Dover reprint readily
available.

i'm not sure that either of them stated exactly what you
say about the 8ve, but even if they did not, it's obvious
by extrapolation from what they did say.

-monz

-----Urspr�ngliche Nachricht-----
Von: Wernerlinden@aol.com [mailto:Wernerlinden@aol.com]
Gesendet: Freitag, 12. Dezember 2003 12:42
An: tuning@yahoogroups.com
Betreff: [tuning] Bach, Werckmeister & Co.

Kurt wrote:

>However, some months back there were discussions >of "gleichschwebenden
temperatur" in which it came
>out that this German phase may mean either "equal
>temperament" or "equal beating temperament". However,
>I was not ever aware of the english phrase "equal
>temperament" ever having but one meaning.

Hi there, the literal translation of "gleichschwebend" is in our context
"equal beating", which would define a temperature in which e.g. any fifth,
be it "a-d" or "e#-c" or "d flat-g#" have the same beating. Which would
consequently mean that all these intervals are absolutely equal. So that's
12 tet, isn't it ?
This is what brought our system of keys to an end in the last
consequence...
:-D But we are again not going into exploration of new tunings but
revising history.
bye
Werner

The term "gleichschwebend" or "equal beating" is a fiction or a nonsense.
This idea cannot become achieved by any temperament. An equal tempered
fifth, e. g. C1 - G1, beats one octave higher as C2 - G2 with the double
rapidity. D1 - A1 in E.T. beats quicker than C1 - G1, E1 - B(H)1 quicker
than D1 - A1 ... and so on.
"Gleichstufig" which means "Identical steps" is in German the actual and
better term for E.T.

Werner Mohrlok

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-----Urspr�ngliche Nachricht-----
Von: Aaron K. Johnson [mailto:akjmicro@comcast.net]
Gesendet: Freitag, 12. Dezember 2003 15:35
An: tuning@yahoogroups.com
Betreff: Re: [tuning] Re: Bach, Werckmeister & Co.

On Friday 12 December 2003 07:59 am, alternativetuning wrote:
> --- In tuning@yahoogroups.com, Wernerlinden@a... wrote:
> > Hi there, the literal translation of "gleichschwebend" is in our
>
> context "equal beating", which would define a temperature in which
> e.g. any fifth, be it "a-d" or "e#-c" or "d flat-g#" have the same
> beating. Which would consequently mean that all these intervals are
> absolutely equal. So that's 12 tet, isn't it ?
>
>
> Hallo Werner:
>
> In an equal temperament, the beating rate of a given interval class
> varies with the absolute frequency. So, by definition, an equal
> beating temperament cannot be an equal (interval) temperament.

True, however, if one does an equal-beating pythagorean, the result is so
close to equal temperament as to be indistinguishable to the ear...still,
this is very painstaking to do by ear without an accurate way of measuring
the sought after beat rate...

-Aaron.

No, Aaron (sorry) - this is somehow a sophisticated answer. You know
as well as I: In the pythagorean model always one of the fifths beats very
hard,
as the pythagorean comma strikes back.
Besides, the term "equal-beating" was created for tuning models
considering fifths and (!) thirds intervals.
And with pythagorean tuning the thirds beat with deep tones quick and
with higher tones extremely quick.

I repeat: Equal beating is a fiction.

Werner Mohrlok

> You do not need web access to participate. You may subscribe through
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-----Urspr�ngliche Nachricht-----
Von: Peter Wakefield Sault [mailto:sault@cyberware.co.uk]
Gesendet: Samstag, 13. Dezember 2003 04:41
An: tuning@yahoogroups.com
Betreff: [tuning] Re: Bach, Werckmeister & Co.

--- In tuning@yahoogroups.com, "alternativetuning"
<alternativetuning@y...> wrote:
> --- In tuning@yahoogroups.com, Wernerlinden@a... wrote:
>
> > Hi there, the literal translation of "gleichschwebend" is in our
> context "equal beating", which would define a temperature in which
> e.g. any fifth, be it "a-d" or "e#-c" or "d flat-g#" have the same
> beating. Which would consequently mean that all these intervals are
> absolutely equal. So that's 12 tet, isn't it ?
>
>
> Hallo Werner:
>
> In an equal temperament, the beating rate of a given interval class
> varies with the absolute frequency. So, by definition, an equal
> beating temperament cannot be an equal (interval) temperament. I
> believe, though, there is some confusion over this in history, so
you
> must read descriptions of tunings closely to see if they are equal
> beating or equal (interval) temperaments.
>
> This is one reason whay it is hard to tune equal temperaments --
you
> must figure out the beating rates for each individual interval, not
> each interval class.
>
> Gabor

Yes but Bach lacked an oscilloscope so all he could do was to
equalize the beats, by ear. It would not matter what the actual
precise beat rate was so long as it matched, by ear, the other beat
rates (in the major tonic triad). Bach's purpose in this could only
have been to minimize conflicting beats.

Nobody knows Bach's idea exactly. But in all probability he followed the
same idea as many others of his contemporaries: Tuning the thirds of
C-major, F-major. G-major better to than equal temperament and as a result
of this (if one follows the idea of a "closen system") the thirds of others
e.g. B-major, F#-major worse.
This means an approximate meantone line of the fifths C-G-D-A-E and an
approximate pythagoran line of the fifths E-B-F#-C#-G#(Ab)-Eb-Bb-F-C.
If you don't understand this principle, please go to our website:

www.hermode.com
and there to the "historical" part. You will find "easy-to-understand"
diagrams including sound examples.

A typical example of such a temperament is "Kirnberger III" and a well
educated Harpsichord player can tune such a temperament by controlling it
only by the ears.

Once again: Where are teh harbsichord players beyond us

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-----Urspr�ngliche Nachricht-----
Von: peter_wakefield_sault [mailto:sault@cyberware.co.uk]
Gesendet: Samstag, 13. Dezember 2003 04:03
An: tuning@yahoogroups.com
Betreff: [tuning] Re: Bach, Werckmeister & Co.

--- In tuning@yahoogroups.com, Wernerlinden@a... wrote:
> Kurt wrote:
>
> >However, some months back there were discussions
>of "gleichschwebenden temperatur" in which it came
> >out that this German phase may mean either "equal
> >temperament" or "equal beating temperament". However,
> >I was not ever aware of the english phrase "equal
> >temperament" ever having but one meaning.
>
> Hi there, the literal translation of "gleichschwebend" is in our
context "equal beating", which would define a temperature in which
e.g. any fifth, be it "a-d" or "e#-c" or "d flat-g#" have the same
beating. Which would consequently mean that all these intervals are
absolutely equal. So that's 12 tet, isn't it ?
> This is what brought our system of keys to an end in the last
consequence...
> :-D But we are again not going into exploration of new tunings but
revising history.
> bye
> Werner

Quite. I believe that interval recognition in general is a matter of
the difference tones.

Although the supremacy of the octave interval has been recognized for
many thousands of years I myself seem to be the first to have
observed that the reason for this is that the difference tone is
exactly equal to the bass note. If anyone here knows of an earlier
statement of the same by someone else I would be pleased to know
about it.

Peter

Hi Peter,

on our websites
www.hermode.com
you will find the same idea. by a diagram. But I never feeled that this
could be an invention of mine. It is an "old hat" and if you want to look by
yourself to earlier statements: I am sure, you will find more than one
publication with the same idea.
Nevertehelss: Some authors answer say "no" to this, at it is not confirm
with the "Ortstheorie" which is one of different models (and as to my
opinion a wrong one) for the explanation why and how we hear music.
And please be not dissapointed. It is better to learn something by own
reflections than by books.

Who can translate "Ortstheorie"?
"Gabor" perhaps?
Thanks in advance.

Werner Mohrlok

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on 12/12/03 8:52 PM, Werner Mohrlok <wmohrlok@hermode.com> wrote:

>
>
> -----Ursprüngliche Nachricht-----
> Von: Aaron K. Johnson [mailto:akjmicro@comcast.net]
> Gesendet: Freitag, 12. Dezember 2003 15:35
> An: tuning@yahoogroups.com
> Betreff: Re: [tuning] Re: Bach, Werckmeister & Co.
>
>
> On Friday 12 December 2003 07:59 am, alternativetuning wrote:
>> --- In tuning@yahoogroups.com, Wernerlinden@a... wrote:
>>> Hi there, the literal translation of "gleichschwebend" is in our
>>
>> context "equal beating", which would define a temperature in which
>> e.g. any fifth, be it "a-d" or "e#-c" or "d flat-g#" have the same
>> beating. Which would consequently mean that all these intervals are
>> absolutely equal. So that's 12 tet, isn't it ?
>>
>>
>> Hallo Werner:
>>
>> In an equal temperament, the beating rate of a given interval class
>> varies with the absolute frequency. So, by definition, an equal
>> beating temperament cannot be an equal (interval) temperament.
>
> True, however, if one does an equal-beating pythagorean, the result is so
> close to equal temperament as to be indistinguishable to the ear...still,
> this is very painstaking to do by ear without an accurate way of measuring
> the sought after beat rate...
>
> -Aaron.
>
> No, Aaron (sorry) - this is somehow a sophisticated answer. You know
> as well as I: In the pythagorean model always one of the fifths beats very
> hard,
> as the pythagorean comma strikes back.
> Besides, the term "equal-beating" was created for tuning models
> considering fifths and (!) thirds intervals.
> And with pythagorean tuning the thirds beat with deep tones quick and
> with higher tones extremely quick.
>
> I repeat: Equal beating is a fiction.
>
> Werner Mohrlok

True enough, but language develops in spite of literal truth. Besides,
language often abbreviates the truth and so fails it literally, while doing
no damage to those who use it. I can't speak of what might have happenned
historically or in Germany, but I have so far had the notion in my mind that
equal beating meant "equal beating within the octave". Equal beating within
the octave also achieves equal or power-of-two-multiple beating across
octaves. To me this does not rule out meantone tunings truncated to a
12-tone system simply because it would in that be understood in practice in
the craft (?) of tuning that the wolf interval would be an exception to the
equal-beating pattern. Nonetheless a "good" equal-beating tuning might be
one in which the predominant beating of the wolf has a "good" multiple
relationship to the beating of the other 5ths.

However, this last point is based on reference to another value judgement
which does not have universal acceptance, but which has acceptance in some
circles, namely, that equal-beating has value for its own sake (that is,
aside from its use to approximate an equal-temperament lacking better
methods). Again literal equal-beating and exact-multiple-beating fit
together into a pattern which is referred to as a whole by the term "equal
beating" for the simple convenience of it, regardless of issues of literal
truth. The difference from truth is simply understood, as is the case with
much usage in naturally occurring langauges, but made more obvious in the
context of languages of the "crafts". Such language is semi-technical but
take such liberties all the time.

An academic treatment of musical language must make allowances for actual
usage, in which case literal failure of meaning must also be understood and
accepted.

-Kurt

>

Hello Peter!
Related but not exactly i have continually stated over the years that consonance is base on differeance tones and/or the coincidences they produce.
Being the case 12 ET is not good because the differance tones make it impossible to tune by ear. Samples from all over the world show that human being prefer scales with unequal size steps. None has designed equal size scale steps by ear.

>
> From: "peter_wakefield_sault" <sault@cyberware.co.uk>
> Subject: Re: Bach, Werckmeister & Co.
>
> -
>
> Quite. I believe that interval recognition in general is a matter of
> the difference tones.
>
> Although the supremacy of the octave interval has been recognized for
> many thousands of years I myself seem to be the first to have
> observed that the reason for this is that the difference tone is
> exactly equal to the bass note. If anyone here knows of an earlier
> statement of the same by someone else I would be pleased to know
> about it.
>
> Peter
>
>

-- -Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com
The Wandering Medicine Show
KXLU 88.9 FM WED 8-9PM PST

On Friday 12 December 2003 10:52 pm, Werner Mohrlok wrote:
> -----Ursprüngliche Nachricht-----
> Von: Aaron K. Johnson [mailto:akjmicro@comcast.net]
> Gesendet: Freitag, 12. Dezember 2003 15:35
> An: tuning@yahoogroups.com
> Betreff: Re: [tuning] Re: Bach, Werckmeister & Co.
>
> On Friday 12 December 2003 07:59 am, alternativetuning wrote:
> > --- In tuning@yahoogroups.com, Wernerlinden@a... wrote:
> > > Hi there, the literal translation of "gleichschwebend" is in our
> >
> > context "equal beating", which would define a temperature in which
> > e.g. any fifth, be it "a-d" or "e#-c" or "d flat-g#" have the same
> > beating. Which would consequently mean that all these intervals are
> > absolutely equal. So that's 12 tet, isn't it ?
> >
> >
> > Hallo Werner:
> >
> > In an equal temperament, the beating rate of a given interval class
> > varies with the absolute frequency. So, by definition, an equal
> > beating temperament cannot be an equal (interval) temperament.
>
> True, however, if one does an equal-beating pythagorean, the result is so
> close to equal temperament as to be indistinguishable to the ear...still,
> this is very painstaking to do by ear without an accurate way of measuring
> the sought after beat rate...
>
> -Aaron.
>
> No, Aaron (sorry) - this is somehow a sophisticated answer. You know
> as well as I: In the pythagorean model always one of the fifths beats very
> hard,
> as the pythagorean comma strikes back.
> Besides, the term "equal-beating" was created for tuning models
> considering fifths and (!) thirds intervals.
> And with pythagorean tuning the thirds beat with deep tones quick and
> with higher tones extremely quick.
>
> I repeat: Equal beating is a fiction.
>

No Werner, you misunderstand--

Withing the octave, you can have all fifths beat at the same rate. Do it in
scala as equal beating pythagorean:

|
1: 100.034 cents 100.034
2: 199.519 cents 199.519
3: 299.800 cents 299.800
4: 399.516 cents 399.516
5: 500.017 cents 500.017
6: 599.941 cents 599.941
7: 699.322 cents 699.322
8: 799.504 cents 799.504
9: 899.127 cents 899.127
10: 999.540 cents 999.540
11: 1099.381 cents 1099.381
12: 2/1 1200.000 octave
|
Base frequency : 261.6256 Hertz
Beat frequencies of 3/2
0: 0.000: -1.1930
1: 100.034: -1.1930
2: 199.519: -1.1930
3: 299.800: -1.1930
4: 399.516: -1.1930
5: 500.017: -1.1930
6: 599.941: -1.1930
7: 699.322: -1.1930
8: 799.504: -1.1930
9: 899.127: -1.1930
10: 999.540: -1.1930
11: 1099.381: -1.1930
12: 1200.000: -2.3859
Total abs. beats : 14.3157
Average abs. beats: 1.1930
Highest abs. beats: 1.1930

-Aaron.

-----Urspr�ngliche Nachricht-----
Von: Kurt Bigler [mailto:kkb@breathsense.com]
Gesendet: Samstag, 13. Dezember 2003 07:17
An: tuning@yahoogroups.com
Betreff: Re: AW: [tuning] Re: Bach, Werckmeister & Co.

on 12/12/03 8:52 PM, Werner Mohrlok <wmohrlok@hermode.com> wrote:

>
>
> -----Urspr�ngliche Nachricht-----
> Von: Aaron K. Johnson [mailto:akjmicro@comcast.net]
> Gesendet: Freitag, 12. Dezember 2003 15:35
> An: tuning@yahoogroups.com
> Betreff: Re: [tuning] Re: Bach, Werckmeister & Co.
>
>
> On Friday 12 December 2003 07:59 am, alternativetuning wrote:
>> --- In tuning@yahoogroups.com, Wernerlinden@a... wrote:
>>> Hi there, the literal translation of "gleichschwebend" is in our
>>
>> context "equal beating", which would define a temperature in which
>> e.g. any fifth, be it "a-d" or "e#-c" or "d flat-g#" have the same
>> beating. Which would consequently mean that all these intervals are
>> absolutely equal. So that's 12 tet, isn't it ?
>>
>>
>> Hallo Werner:
>>
>> In an equal temperament, the beating rate of a given interval class
>> varies with the absolute frequency. So, by definition, an equal
>> beating temperament cannot be an equal (interval) temperament.
>
> True, however, if one does an equal-beating pythagorean, the result is
so
> close to equal temperament as to be indistinguishable to the
ear...still,
> this is very painstaking to do by ear without an accurate way of
measuring
> the sought after beat rate...
>
> -Aaron.
>
> No, Aaron (sorry) - this is somehow a sophisticated answer. You know
> as well as I: In the pythagorean model always one of the fifths beats
very
> hard,
> as the pythagorean comma strikes back.
> Besides, the term "equal-beating" was created for tuning models
> considering fifths and (!) thirds intervals.
> And with pythagorean tuning the thirds beat with deep tones quick and
> with higher tones extremely quick.
>
> I repeat: Equal beating is a fiction.
>
> Werner Mohrlok

True enough, but language develops in spite of literal truth. Besides,
language often abbreviates the truth and so fails it literally, while
doing
no damage to those who use it. I can't speak of what might have happenned
historically or in Germany, but I have so far had the notion in my mind
that
equal beating meant "equal beating within the octave". Equal beating
within
the octave also achieves equal or power-of-two-multiple beating across
octaves. To me this does not rule out meantone tunings truncated to a
12-tone system simply because it would in that be understood in practice
in
the craft (?) of tuning that the wolf interval would be an exception to
the
equal-beating pattern. Nonetheless a "good" equal-beating tuning might be
one in which the predominant beating of the wolf has a "good" multiple
relationship to the beating of the other 5ths.

However, this last point is based on reference to another value judgement
which does not have universal acceptance, but which has acceptance in some
circles, namely, that equal-beating has value for its own sake (that is,
aside from its use to approximate an equal-temperament lacking better
methods). Again literal equal-beating and exact-multiple-beating fit
together into a pattern which is referred to as a whole by the term "equal
beating" for the simple convenience of it, regardless of issues of literal
truth. The difference from truth is simply understood, as is the case
with
much usage in naturally occurring langauges, but made more obvious in the
context of languages of the "crafts". Such language is semi-technical but
take such liberties all the time.

An academic treatment of musical language must make allowances for actual
usage, in which case literal failure of meaning must also be understood
and
accepted.

-Kurt

Kurt,

You are right. Nevertheless I prefer the term "Gleichstufig".

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--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> hi Peter,
>
>
> --- In tuning@yahoogroups.com, "peter_wakefield_sault" <sault@c...>
> wrote:
>
>
> > Although the supremacy of the octave interval has
> > been recognized for many thousands of years I myself
> > seem to be the first to have observed that the reason
> > for this is that the difference tone is exactly equal
> > to the bass note. If anyone here knows of an earlier
> > statement of the same by someone else I would be pleased
> > to know about it.
>
>
> Ezra Sims presented the recognition of difference tones
> as the basis of his harmonic practice in one of his papers.
> it was either one of these two, i think it was the
> _Perspectives_ article:
>
> Ezra Sims, "Yet Another 72-Noter",
> in _Computer Music Journal_ 12, no. 4 [winter 1988], p 28 - 45.
>
> Sims, Ezra. "Reflections on This and That (Perhaps a Polemic)."
> _Perspectives of New Music_ 29 (1991): 258-263.
>
>
> and of course, Helmholtz discussed difference tones quite
> a bit. ref. Helmholtz, Hermann. _On the Sensations of Tone..._,
> english trans. Alexander Ellis, 1875; Dover reprint readily
> available.
>
>
>
> i'm not sure that either of them stated exactly what you
> say about the 8ve, but even if they did not, it's obvious
> by extrapolation from what they did say.
>
>
>
> -monz

Well now - everything's obvious *after* you've heard it, isn't it?
That's always the way.

Fact is that Helmholtz did not say it, neither did Tartini who was
the first to point to the existence of difference tones and neither
did Aristotle who just baldly identified the octave as supreme,
following Pythagoras who called it the Universal Concord, again
without giving a reason.

Until and unless someone can point me to a prior statement of "the
obvious" then my claim to be the first to state it in words still
stands. Hey - pop my balloon why doncha? But I want the exact
reference.

It's almost like my discovery that no one knows the precise azimuth
of the Parthenon. I spoke to all the world's acknowledged experts on
the matter and they all thought that it should be recorded somewhere
or that someone should know. After all, it's such an "obvious" thing
to measure, is it not? We all discovered that nobody knows (and the
Greeks won't talk to any of us Barbarians, including profs of
archeology, about their precious pile of rubble).

--- In tuning@yahoogroups.com, "Werner Mohrlok" <wmohrlok@h...> wrote:
>
> -----Ursprüngliche Nachricht-----
> Von: peter_wakefield_sault [mailto:sault@c...]
> Gesendet: Samstag, 13. Dezember 2003 04:03
> An: tuning@yahoogroups.com
> Betreff: [tuning] Re: Bach, Werckmeister & Co.
>
>
> --- In tuning@yahoogroups.com, Wernerlinden@a... wrote:
> > Kurt wrote:
> >
> > >However, some months back there were discussions
> >of "gleichschwebenden temperatur" in which it came
> > >out that this German phase may mean either "equal
> > >temperament" or "equal beating temperament". However,
> > >I was not ever aware of the english phrase "equal
> > >temperament" ever having but one meaning.
> >
> > Hi there, the literal translation of "gleichschwebend" is in our
> context "equal beating", which would define a temperature in which
> e.g. any fifth, be it "a-d" or "e#-c" or "d flat-g#" have the same
> beating. Which would consequently mean that all these intervals
are
> absolutely equal. So that's 12 tet, isn't it ?
> > This is what brought our system of keys to an end in the last
> consequence...
> > :-D But we are again not going into exploration of new tunings
but
> revising history.
> > bye
> > Werner
>
> Quite. I believe that interval recognition in general is a matter
of
> the difference tones.
>
> Although the supremacy of the octave interval has been recognized
for
> many thousands of years I myself seem to be the first to have
> observed that the reason for this is that the difference tone is
> exactly equal to the bass note. If anyone here knows of an earlier
> statement of the same by someone else I would be pleased to know
> about it.
>
> Peter
>
> Hi Peter,
>
> on our websites
> www.hermode.com
> you will find the same idea. by a diagram. But I never feeled
that this
> could be an invention of mine. It is an "old hat" and if you want
to look by
> yourself to earlier statements: I am sure, you will find more than
one
> publication with the same idea.
> Nevertehelss: Some authors answer say "no" to this, at it is not
confirm
> with the "Ortstheorie" which is one of different models (and as to
my
> opinion a wrong one) for the explanation why and how we hear music.
> And please be not dissapointed. It is better to learn something
by own
> reflections than by books.
>
> Who can translate "Ortstheorie"?
> "Gabor" perhaps?
> Thanks in advance.
>
> Werner Mohrlok

I have looked and cannot find. All I have ever come across if the oft
repeated refrain - "nobody knows...". Unless someone else can find an
earlier specific statement of the same then the credit is mine, all
mine. Like I already said elsewhere, if someone wants to pop my
ballon then I want the reference that is their pin.

Peter

--- In tuning@yahoogroups.com, kraig grady <kraiggrady@a...> wrote:
> >
>
> Hello Peter!
> Related but not exactly i have continually stated over the
years that consonance is base on differeance tones and/or the
coincidences they produce.
> Being the case 12 ET is not good because the differance tones
make it impossible to tune by ear. Samples from all over the world
show that human being prefer scales with unequal size steps. None has
designed equal size scale steps by ear.
>

I agree. ET is a rationalization that requires a certain level of
technology to achieve with precision. That technology has existed
since the 16th century, when the Spaniards started making ET guitars.

I myself have replaced a guitar fingerboard and calculated the fret
positions by the crudest of methods, back in the days before
calculators (are you, too, old enough to remember those days?). It's
called iteration. You take a guess, say 1.06 and multiply it by
itself, on paper, using long multiplication. If the result is too
high then reduce the trial number a little bit. If the result is too
low, then increase the trial number a little bit. By this means,
working at it 16 hours a day for two weeks, I obtained the 12th root
of two to 10 decimal places. The rest is straightforward.

--- In tuning@yahoogroups.com, "Aaron K. Johnson" <akjmicro@c...>
wrote:
> On Friday 12 December 2003 10:52 pm, Werner Mohrlok wrote:
> > -----Ursprüngliche Nachricht-----
> > Von: Aaron K. Johnson [mailto:akjmicro@c...]
> > Gesendet: Freitag, 12. Dezember 2003 15:35
> > An: tuning@yahoogroups.com
> > Betreff: Re: [tuning] Re: Bach, Werckmeister & Co.
> >
> > On Friday 12 December 2003 07:59 am, alternativetuning wrote:
> > > --- In tuning@yahoogroups.com, Wernerlinden@a... wrote:
> > > > Hi there, the literal translation of "gleichschwebend" is in
our
> > >
> > > context "equal beating", which would define a temperature in
which
> > > e.g. any fifth, be it "a-d" or "e#-c" or "d flat-g#" have the
same
> > > beating. Which would consequently mean that all these intervals
are
> > > absolutely equal. So that's 12 tet, isn't it ?
> > >
> > >
> > > Hallo Werner:
> > >
> > > In an equal temperament, the beating rate of a given interval
class
> > > varies with the absolute frequency. So, by definition, an equal
> > > beating temperament cannot be an equal (interval) temperament.
> >
> > True, however, if one does an equal-beating pythagorean, the
result is so
> > close to equal temperament as to be indistinguishable to the
ear...still,
> > this is very painstaking to do by ear without an accurate way of
measuring
> > the sought after beat rate...
> >
> > -Aaron.
> >
> > No, Aaron (sorry) - this is somehow a sophisticated answer. You
know
> > as well as I: In the pythagorean model always one of the fifths
beats very
> > hard,
> > as the pythagorean comma strikes back.
> > Besides, the term "equal-beating" was created for tuning models
> > considering fifths and (!) thirds intervals.
> > And with pythagorean tuning the thirds beat with deep tones quick
and
> > with higher tones extremely quick.
> >
> > I repeat: Equal beating is a fiction.
> >
>
> No Werner, you misunderstand--
>
> Withing the octave, you can have all fifths beat at the same rate.
Do it in
> scala as equal beating pythagorean:
>
> |
> 1: 100.034 cents 100.034
> 2: 199.519 cents 199.519
> 3: 299.800 cents 299.800
> 4: 399.516 cents 399.516
> 5: 500.017 cents 500.017
> 6: 599.941 cents 599.941
> 7: 699.322 cents 699.322
> 8: 799.504 cents 799.504
> 9: 899.127 cents 899.127
> 10: 999.540 cents 999.540
> 11: 1099.381 cents 1099.381
> 12: 2/1 1200.000 octave
> |
> Base frequency : 261.6256 Hertz
> Beat frequencies of 3/2
> 0: 0.000: -1.1930
> 1: 100.034: -1.1930
> 2: 199.519: -1.1930
> 3: 299.800: -1.1930
> 4: 399.516: -1.1930
> 5: 500.017: -1.1930
> 6: 599.941: -1.1930
> 7: 699.322: -1.1930
> 8: 799.504: -1.1930
> 9: 899.127: -1.1930
> 10: 999.540: -1.1930
> 11: 1099.381: -1.1930
> 12: 1200.000: -2.3859
> Total abs. beats : 14.3157
> Average abs. beats: 1.1930
> Highest abs. beats: 1.1930
>
> -Aaron.

Hi Aaron

Would explain the workings of that for us please? What are the steps
to arrive at the number -1.1930? And what does it represent?

Peter

on 12/12/03 11:48 PM, Peter Wakefield Sault <sault@cyberware.co.uk> wrote:

> --- In tuning@yahoogroups.com, "Werner Mohrlok" <wmohrlok@h...> wrote:
>>
>> -----Ursprüngliche Nachricht-----
>> Von: peter_wakefield_sault [mailto:sault@c...]
>> Gesendet: Samstag, 13. Dezember 2003 04:03
>> An: tuning@yahoogroups.com
>> Betreff: [tuning] Re: Bach, Werckmeister & Co.
>>
>>
>> --- In tuning@yahoogroups.com, Wernerlinden@a... wrote:
>>> Kurt wrote:
>>>
>>>> However, some months back there were discussions
>>> of "gleichschwebenden temperatur" in which it came
>>>> out that this German phase may mean either "equal
>>>> temperament" or "equal beating temperament". However,
>>>> I was not ever aware of the english phrase "equal
>>>> temperament" ever having but one meaning.
>>>
>>> Hi there, the literal translation of "gleichschwebend" is in our
>> context "equal beating", which would define a temperature in which
>> e.g. any fifth, be it "a-d" or "e#-c" or "d flat-g#" have the same
>> beating. Which would consequently mean that all these intervals
> are
>> absolutely equal. So that's 12 tet, isn't it ?
>>> This is what brought our system of keys to an end in the last
>> consequence...
>>> :-D But we are again not going into exploration of new tunings
> but
>> revising history.
>>> bye
>>> Werner
>>
>> Quite. I believe that interval recognition in general is a matter
> of
>> the difference tones.
>>
>> Although the supremacy of the octave interval has been recognized
> for
>> many thousands of years I myself seem to be the first to have
>> observed that the reason for this is that the difference tone is
>> exactly equal to the bass note. If anyone here knows of an earlier
>> statement of the same by someone else I would be pleased to know
>> about it.
>>
>> Peter
>>
>> Hi Peter,
>>
>> on our websites
>> www.hermode.com
>> you will find the same idea. by a diagram. But I never feeled
> that this
>> could be an invention of mine. It is an "old hat" and if you want
> to look by
>> yourself to earlier statements: I am sure, you will find more than
> one
>> publication with the same idea.
>> Nevertehelss: Some authors answer say "no" to this, at it is not
> confirm
>> with the "Ortstheorie" which is one of different models (and as to
> my
>> opinion a wrong one) for the explanation why and how we hear music.
>> And please be not dissapointed. It is better to learn something
> by own
>> reflections than by books.
>>
>> Who can translate "Ortstheorie"?
>> "Gabor" perhaps?
>> Thanks in advance.
>>
>> Werner Mohrlok
>
>
> I have looked and cannot find. All I have ever come across if the oft
> repeated refrain - "nobody knows...". Unless someone else can find an
> earlier specific statement of the same then the credit is mine, all
> mine. Like I already said elsewhere, if someone wants to pop my
> ballon then I want the reference that is their pin.
>
> Peter

If no one pops your balloon, you will never land on the earth. I will pray
for your pin.

-Kurt

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> on 12/12/03 11:48 PM, Peter Wakefield Sault <sault@c...> wrote:
>
> > --- In tuning@yahoogroups.com, "Werner Mohrlok" <wmohrlok@h...>
wrote:
> >>
> >> -----Ursprüngliche Nachricht-----
> >> Von: peter_wakefield_sault [mailto:sault@c...]
> >> Gesendet: Samstag, 13. Dezember 2003 04:03
> >> An: tuning@yahoogroups.com
> >> Betreff: [tuning] Re: Bach, Werckmeister & Co.
> >>
> >>
> >> --- In tuning@yahoogroups.com, Wernerlinden@a... wrote:
> >>> Kurt wrote:
> >>>
> >>>> However, some months back there were discussions
> >>> of "gleichschwebenden temperatur" in which it came
> >>>> out that this German phase may mean either "equal
> >>>> temperament" or "equal beating temperament". However,
> >>>> I was not ever aware of the english phrase "equal
> >>>> temperament" ever having but one meaning.
> >>>
> >>> Hi there, the literal translation of "gleichschwebend" is in our
> >> context "equal beating", which would define a temperature in
which
> >> e.g. any fifth, be it "a-d" or "e#-c" or "d flat-g#" have the
same
> >> beating. Which would consequently mean that all these intervals
> > are
> >> absolutely equal. So that's 12 tet, isn't it ?
> >>> This is what brought our system of keys to an end in the last
> >> consequence...
> >>> :-D But we are again not going into exploration of new tunings
> > but
> >> revising history.
> >>> bye
> >>> Werner
> >>
> >> Quite. I believe that interval recognition in general is a matter
> > of
> >> the difference tones.
> >>
> >> Although the supremacy of the octave interval has been recognized
> > for
> >> many thousands of years I myself seem to be the first to have
> >> observed that the reason for this is that the difference tone is
> >> exactly equal to the bass note. If anyone here knows of an
earlier
> >> statement of the same by someone else I would be pleased to know
> >> about it.
> >>
> >> Peter
> >>
> >> Hi Peter,
> >>
> >> on our websites
> >> www.hermode.com
> >> you will find the same idea. by a diagram. But I never feeled
> > that this
> >> could be an invention of mine. It is an "old hat" and if you want
> > to look by
> >> yourself to earlier statements: I am sure, you will find more
than
> > one
> >> publication with the same idea.
> >> Nevertehelss: Some authors answer say "no" to this, at it is not
> > confirm
> >> with the "Ortstheorie" which is one of different models (and as
to
> > my
> >> opinion a wrong one) for the explanation why and how we hear
music.
> >> And please be not dissapointed. It is better to learn something
> > by own
> >> reflections than by books.
> >>
> >> Who can translate "Ortstheorie"?
> >> "Gabor" perhaps?
> >> Thanks in advance.
> >>
> >> Werner Mohrlok
> >
> >
> > I have looked and cannot find. All I have ever come across if the
oft
> > repeated refrain - "nobody knows...". Unless someone else can
find an
> > earlier specific statement of the same then the credit is mine,
all
> > mine. Like I already said elsewhere, if someone wants to pop my
> > ballon then I want the reference that is their pin.
> >
> > Peter
>
> If no one pops your balloon, you will never land on the earth. I
will pray
> for your pin.
>
> -Kurt

Thankyou Kurt. But why should I be denied my tiny little bit of
glory? It's not as though I want credit for the atom bomb (and what
sane man would?)

Peter.

on 12/13/03 12:56 AM, Peter Wakefield Sault <sault@cyberware.co.uk> wrote:

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>> on 12/12/03 11:48 PM, Peter Wakefield Sault <sault@c...> wrote:
>>
>>> --- In tuning@yahoogroups.com, "Werner Mohrlok" <wmohrlok@h...>
> wrote:
>>>>
>>>> -----Ursprüngliche Nachricht-----
>>>> Von: peter_wakefield_sault [mailto:sault@c...]
>>>> Gesendet: Samstag, 13. Dezember 2003 04:03
>>>> An: tuning@yahoogroups.com
>>>> Betreff: [tuning] Re: Bach, Werckmeister & Co.
>>>>
>>>>
>>>> --- In tuning@yahoogroups.com, Wernerlinden@a... wrote:
>>>>> Kurt wrote:
>>>>>
>>>>>> However, some months back there were discussions
>>>>> of "gleichschwebenden temperatur" in which it came
>>>>>> out that this German phase may mean either "equal
>>>>>> temperament" or "equal beating temperament". However,
>>>>>> I was not ever aware of the english phrase "equal
>>>>>> temperament" ever having but one meaning.
>>>>>
>>>>> Hi there, the literal translation of "gleichschwebend" is in our
>>>> context "equal beating", which would define a temperature in
> which
>>>> e.g. any fifth, be it "a-d" or "e#-c" or "d flat-g#" have the
> same
>>>> beating. Which would consequently mean that all these intervals
>>> are
>>>> absolutely equal. So that's 12 tet, isn't it ?
>>>>> This is what brought our system of keys to an end in the last
>>>> consequence...
>>>>> :-D But we are again not going into exploration of new tunings
>>> but
>>>> revising history.
>>>>> bye
>>>>> Werner
>>>>
>>>> Quite. I believe that interval recognition in general is a matter
>>> of
>>>> the difference tones.
>>>>
>>>> Although the supremacy of the octave interval has been recognized
>>> for
>>>> many thousands of years I myself seem to be the first to have
>>>> observed that the reason for this is that the difference tone is
>>>> exactly equal to the bass note. If anyone here knows of an
> earlier
>>>> statement of the same by someone else I would be pleased to know
>>>> about it.
>>>>
>>>> Peter
>>>>
>>>> Hi Peter,
>>>>
>>>> on our websites
>>>> www.hermode.com
>>>> you will find the same idea. by a diagram. But I never feeled
>>> that this
>>>> could be an invention of mine. It is an "old hat" and if you want
>>> to look by
>>>> yourself to earlier statements: I am sure, you will find more
> than
>>> one
>>>> publication with the same idea.
>>>> Nevertehelss: Some authors answer say "no" to this, at it is not
>>> confirm
>>>> with the "Ortstheorie" which is one of different models (and as
> to
>>> my
>>>> opinion a wrong one) for the explanation why and how we hear
> music.
>>>> And please be not dissapointed. It is better to learn something
>>> by own
>>>> reflections than by books.
>>>>
>>>> Who can translate "Ortstheorie"?
>>>> "Gabor" perhaps?
>>>> Thanks in advance.
>>>>
>>>> Werner Mohrlok
>>>
>>>
>>> I have looked and cannot find. All I have ever come across if the
> oft
>>> repeated refrain - "nobody knows...". Unless someone else can
> find an
>>> earlier specific statement of the same then the credit is mine,
> all
>>> mine. Like I already said elsewhere, if someone wants to pop my
>>> ballon then I want the reference that is their pin.
>>>
>>> Peter
>>
>> If no one pops your balloon, you will never land on the earth. I
> will pray
>> for your pin.
>>
>> -Kurt
>
> Thankyou Kurt. But why should I be denied my tiny little bit of
> glory? It's not as though I want credit for the atom bomb (and what
> sane man would?)
>
> Peter.

Yea, maybe you're just more honest about your desire for glory than I am.
At the same time knowing my weaknesses I see glory as something that would
make me ill rather quickly, and so I value it less. And I also believe that
in spite of desire for glory, and never mind illness, in the end it isn't
worth anything, is a distraction, and reduces our capacity for relationship.
But that's just my opinion.

-Kurt

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> on 12/13/03 12:56 AM, Peter Wakefield Sault <sault@c...> wrote:
>
> > --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> >> on 12/12/03 11:48 PM, Peter Wakefield Sault <sault@c...> wrote:
> >>
> >>> --- In tuning@yahoogroups.com, "Werner Mohrlok" <wmohrlok@h...>
> > wrote:
> >>>>
> >>>> -----Ursprüngliche Nachricht-----
> >>>> Von: peter_wakefield_sault [mailto:sault@c...]
> >>>> Gesendet: Samstag, 13. Dezember 2003 04:03
> >>>> An: tuning@yahoogroups.com
> >>>> Betreff: [tuning] Re: Bach, Werckmeister & Co.
> >>>>
> >>>>
> >>>> --- In tuning@yahoogroups.com, Wernerlinden@a... wrote:
> >>>>> Kurt wrote:
> >>>>>
> >>>>>> However, some months back there were discussions
> >>>>> of "gleichschwebenden temperatur" in which it came
> >>>>>> out that this German phase may mean either "equal
> >>>>>> temperament" or "equal beating temperament". However,
> >>>>>> I was not ever aware of the english phrase "equal
> >>>>>> temperament" ever having but one meaning.
> >>>>>
> >>>>> Hi there, the literal translation of "gleichschwebend" is in
our
> >>>> context "equal beating", which would define a temperature in
> > which
> >>>> e.g. any fifth, be it "a-d" or "e#-c" or "d flat-g#" have the
> > same
> >>>> beating. Which would consequently mean that all these intervals
> >>> are
> >>>> absolutely equal. So that's 12 tet, isn't it ?
> >>>>> This is what brought our system of keys to an end in the last
> >>>> consequence...
> >>>>> :-D But we are again not going into exploration of new tunings
> >>> but
> >>>> revising history.
> >>>>> bye
> >>>>> Werner
> >>>>
> >>>> Quite. I believe that interval recognition in general is a
matter
> >>> of
> >>>> the difference tones.
> >>>>
> >>>> Although the supremacy of the octave interval has been
recognized
> >>> for
> >>>> many thousands of years I myself seem to be the first to have
> >>>> observed that the reason for this is that the difference tone
is
> >>>> exactly equal to the bass note. If anyone here knows of an
> > earlier
> >>>> statement of the same by someone else I would be pleased to
know
> >>>> about it.
> >>>>
> >>>> Peter
> >>>>
> >>>> Hi Peter,
> >>>>
> >>>> on our websites
> >>>> www.hermode.com
> >>>> you will find the same idea. by a diagram. But I never feeled
> >>> that this
> >>>> could be an invention of mine. It is an "old hat" and if you
want
> >>> to look by
> >>>> yourself to earlier statements: I am sure, you will find more
> > than
> >>> one
> >>>> publication with the same idea.
> >>>> Nevertehelss: Some authors answer say "no" to this, at it is
not
> >>> confirm
> >>>> with the "Ortstheorie" which is one of different models (and as
> > to
> >>> my
> >>>> opinion a wrong one) for the explanation why and how we hear
> > music.
> >>>> And please be not dissapointed. It is better to learn something
> >>> by own
> >>>> reflections than by books.
> >>>>
> >>>> Who can translate "Ortstheorie"?
> >>>> "Gabor" perhaps?
> >>>> Thanks in advance.
> >>>>
> >>>> Werner Mohrlok
> >>>
> >>>
> >>> I have looked and cannot find. All I have ever come across if
the
> > oft
> >>> repeated refrain - "nobody knows...". Unless someone else can
> > find an
> >>> earlier specific statement of the same then the credit is mine,
> > all
> >>> mine. Like I already said elsewhere, if someone wants to pop my
> >>> ballon then I want the reference that is their pin.
> >>>
> >>> Peter
> >>
> >> If no one pops your balloon, you will never land on the earth. I
> > will pray
> >> for your pin.
> >>
> >> -Kurt
> >
> > Thankyou Kurt. But why should I be denied my tiny little bit of
> > glory? It's not as though I want credit for the atom bomb (and
what
> > sane man would?)
> >
> > Peter.
>
> Yea, maybe you're just more honest about your desire for glory than
I am.
> At the same time knowing my weaknesses I see glory as something
that would
> make me ill rather quickly, and so I value it less. And I also
believe that
> in spite of desire for glory, and never mind illness, in the end it
isn't
> worth anything, is a distraction, and reduces our capacity for
relationship.
> But that's just my opinion.
>
> -Kurt

It's not as easy as that, Kurt. I am not seeking glory particularly
but reacting (again) against those who would diminish my achievement,
even if it is only to state the obvious (and hey - the *wheel* was
obvious only *after* someone had invented it). If it is the first
time that it has been stated then by rights I should get the credit
for doing that. And again, I argue for my glory not because I want
the glory so much as because I want the argument. In the final
analysis it does not matter to me. I would hope that those who would
deny me it learn a little about themselves in the process.

Peter

hi Peter,

--- In tuning@yahoogroups.com, "Peter Wakefield Sault" <sault@c...>
wrote:

> I agree. ET is a rationalization that requires a certain
> level of technology to achieve with precision. That
> technology has existed since the 16th century, when the
> Spaniards started making ET guitars.

the technology has existed for as long as people have been
able to write. more below ...

> I myself have replaced a guitar fingerboard and calculated
> the fret positions by the crudest of methods, back in the
> days before calculators (are you, too, old enough to remember
> those days?). It's called iteration. You take a guess, say
> 1.06 and multiply it by itself, on paper, using long
> multiplication. If the result is too high then reduce the
> trial number a little bit. If the result is too low,
> then increase the trial number a little bit. By this means,
> working at it 16 hours a day for two weeks, I obtained the
> 12th root of two to 10 decimal places. The rest is
> straightforward.

i show how it could have been done by the Sumerians 5000 years
ago, on two of my webpages. the simpler approximation, which
also has graphics to illustrate, is here:

http://sonic-arts.org/monzo/sumerian/simplified-sumeriantuning.htm

... but of course, the numbers here are base-60 not base-10.

-monz

--- In tuning@yahoogroups.com, "Werner Mohrlok" <wmohrlok@h...> wrote:

/tuning/topicId_49654.html#49733

> Once again: Where are teh harbsichord players beyond us
>

***There used to be one within the last few years, but I can't
remember her name and she isn't here with us now anymore...

And our Medieval expert, Margo Schulter, apparently has gone to post
at Usenet of late...

J. Pehrson

--- In tuning@yahoogroups.com, "Peter Wakefield Sault" <sault@c...>
wrote:
> --- In tuning@yahoogroups.com, "Aaron K. Johnson" <akjmicro@c...>
> wrote:
> > On Friday 12 December 2003 10:52 pm, Werner Mohrlok wrote:
> > > -----Ursprüngliche Nachricht-----
> > > Von: Aaron K. Johnson [mailto:akjmicro@c...]
> > > Gesendet: Freitag, 12. Dezember 2003 15:35
> > > An: tuning@yahoogroups.com
> > > Betreff: Re: [tuning] Re: Bach, Werckmeister & Co.
> > >
> > > On Friday 12 December 2003 07:59 am, alternativetuning wrote:
> > > > --- In tuning@yahoogroups.com, Wernerlinden@a... wrote:
> > > > > Hi there, the literal translation of "gleichschwebend" is
in
> our
> > > >
> > > > context "equal beating", which would define a temperature in
> which
> > > > e.g. any fifth, be it "a-d" or "e#-c" or "d flat-g#" have
the
> same
> > > > beating. Which would consequently mean that all these
intervals
> are
> > > > absolutely equal. So that's 12 tet, isn't it ?
> > > >
> > > >
> > > > Hallo Werner:
> > > >
> > > > In an equal temperament, the beating rate of a given
interval
> class
> > > > varies with the absolute frequency. So, by definition, an
equal
> > > > beating temperament cannot be an equal (interval)
temperament.
> > >
> > > True, however, if one does an equal-beating pythagorean, the
> result is so
> > > close to equal temperament as to be indistinguishable to the
> ear...still,
> > > this is very painstaking to do by ear without an accurate way
of
> measuring
> > > the sought after beat rate...
> > >
> > > -Aaron.
> > >
> > > No, Aaron (sorry) - this is somehow a sophisticated answer.
You
> know
> > > as well as I: In the pythagorean model always one of the
fifths
> beats very
> > > hard,
> > > as the pythagorean comma strikes back.
> > > Besides, the term "equal-beating" was created for tuning models
> > > considering fifths and (!) thirds intervals.
> > > And with pythagorean tuning the thirds beat with deep tones
quick
> and
> > > with higher tones extremely quick.
> > >
> > > I repeat: Equal beating is a fiction.
> > >
> >
> > No Werner, you misunderstand--
> >
> > Withing the octave, you can have all fifths beat at the same
rate.
> Do it in
> > scala as equal beating pythagorean:
> >
> > |
> > 1: 100.034 cents 100.034
> > 2: 199.519 cents 199.519
> > 3: 299.800 cents 299.800
> > 4: 399.516 cents 399.516
> > 5: 500.017 cents 500.017
> > 6: 599.941 cents 599.941
> > 7: 699.322 cents 699.322
> > 8: 799.504 cents 799.504
> > 9: 899.127 cents 899.127
> > 10: 999.540 cents 999.540
> > 11: 1099.381 cents 1099.381
> > 12: 2/1 1200.000 octave
> > |
> > Base frequency : 261.6256 Hertz
> > Beat frequencies of 3/2
> > 0: 0.000: -1.1930
> > 1: 100.034: -1.1930
> > 2: 199.519: -1.1930
> > 3: 299.800: -1.1930
> > 4: 399.516: -1.1930
> > 5: 500.017: -1.1930
> > 6: 599.941: -1.1930
> > 7: 699.322: -1.1930
> > 8: 799.504: -1.1930
> > 9: 899.127: -1.1930
> > 10: 999.540: -1.1930
> > 11: 1099.381: -1.1930
> > 12: 1200.000: -2.3859
> > Total abs. beats : 14.3157
> > Average abs. beats: 1.1930
> > Highest abs. beats: 1.1930
> >
> > -Aaron.
>
> Hi Aaron
>
> Would explain the workings of that for us please? What are the
steps
> to arrive at the number -1.1930? And what does it represent?
>
> Peter

First, calculating these values in cents into absolute pitches from
C = 261.6256, then substracting each of them from the absolute pitch
value a musical fifth below multiplied by 3/2 yields a close
approximation of this value, except I get only one half the value,
somewhere around - .5965, starting in the indicated octave, and then
I get the right value in the next octave. What am I missing? Maybe
beats are heard at half the difference, see what I know. Anyway, as
you say, it's not very practical to count beats that carefully, but
still a fascinating fact about equal temperament and equal beating
fifths. --JeffO

--- In tuning@yahoogroups.com, "Jeff Olliff" <jolliff@d...> wrote:

/tuning/topicId_49654.html#49846

> --- In tuning@yahoogroups.com, "Peter Wakefield Sault" <sault@c...>
> wrote:
> > --- In tuning@yahoogroups.com, "Aaron K. Johnson" <akjmicro@c...>
> > wrote:
> > > On Friday 12 December 2003 10:52 pm, Werner Mohrlok wrote:
> > > > -----Ursprüngliche Nachricht-----
> > > > Von: Aaron K. Johnson [mailto:akjmicro@c...]
> > > > Gesendet: Freitag, 12. Dezember 2003 15:35
> > > > An: tuning@yahoogroups.com
> > > > Betreff: Re: [tuning] Re: Bach, Werckmeister & Co.
> > > >
> > > > On Friday 12 December 2003 07:59 am, alternativetuning wrote:
> > > > > --- In tuning@yahoogroups.com, Wernerlinden@a... wrote:
> > > > > > Hi there, the literal translation of "gleichschwebend" is
> in
> > our
> > > > >
> > > > > context "equal beating", which would define a temperature
in
> > which
> > > > > e.g. any fifth, be it "a-d" or "e#-c" or "d flat-g#" have
> the
> > same
> > > > > beating. Which would consequently mean that all these
> intervals
> > are
> > > > > absolutely equal. So that's 12 tet, isn't it ?
> > > > >
> > > > >
> > > > > Hallo Werner:
> > > > >
> > > > > In an equal temperament, the beating rate of a given
> interval
> > class
> > > > > varies with the absolute frequency. So, by definition, an
> equal
> > > > > beating temperament cannot be an equal (interval)
> temperament.
> > > >
> > > > True, however, if one does an equal-beating pythagorean, the
> > result is so
> > > > close to equal temperament as to be indistinguishable to the
> > ear...still,
> > > > this is very painstaking to do by ear without an accurate way
> of
> > measuring
> > > > the sought after beat rate...
> > > >
> > > > -Aaron.
> > > >
> > > > No, Aaron (sorry) - this is somehow a sophisticated answer.
> You
> > know
> > > > as well as I: In the pythagorean model always one of the
> fifths
> > beats very
> > > > hard,
> > > > as the pythagorean comma strikes back.
> > > > Besides, the term "equal-beating" was created for tuning
models
> > > > considering fifths and (!) thirds intervals.
> > > > And with pythagorean tuning the thirds beat with deep tones
> quick
> > and
> > > > with higher tones extremely quick.
> > > >
> > > > I repeat: Equal beating is a fiction.
> > > >
> > >
> > > No Werner, you misunderstand--
> > >
> > > Withing the octave, you can have all fifths beat at the same
> rate.
> > Do it in
> > > scala as equal beating pythagorean:
> > >
> > > |
> > > 1: 100.034 cents 100.034
> > > 2: 199.519 cents 199.519
> > > 3: 299.800 cents 299.800
> > > 4: 399.516 cents 399.516
> > > 5: 500.017 cents 500.017
> > > 6: 599.941 cents 599.941
> > > 7: 699.322 cents 699.322
> > > 8: 799.504 cents 799.504
> > > 9: 899.127 cents 899.127
> > > 10: 999.540 cents 999.540
> > > 11: 1099.381 cents 1099.381
> > > 12: 2/1 1200.000 octave
> > > |
> > > Base frequency : 261.6256 Hertz
> > > Beat frequencies of 3/2
> > > 0: 0.000: -1.1930
> > > 1: 100.034: -1.1930
> > > 2: 199.519: -1.1930
> > > 3: 299.800: -1.1930
> > > 4: 399.516: -1.1930
> > > 5: 500.017: -1.1930
> > > 6: 599.941: -1.1930
> > > 7: 699.322: -1.1930
> > > 8: 799.504: -1.1930
> > > 9: 899.127: -1.1930
> > > 10: 999.540: -1.1930
> > > 11: 1099.381: -1.1930
> > > 12: 1200.000: -2.3859
> > > Total abs. beats : 14.3157
> > > Average abs. beats: 1.1930
> > > Highest abs. beats: 1.1930
> > >
> > > -Aaron.
> >
> > Hi Aaron
> >
> > Would explain the workings of that for us please? What are the
> steps
> > to arrive at the number -1.1930? And what does it represent?
> >
> > Peter
>
> First, calculating these values in cents into absolute pitches from
> C = 261.6256, then substracting each of them from the absolute
pitch
> value a musical fifth below multiplied by 3/2 yields a close
> approximation of this value, except I get only one half the value,
> somewhere around - .5965, starting in the indicated octave, and
then
> I get the right value in the next octave. What am I missing?
Maybe
> beats are heard at half the difference, see what I know. Anyway,
as
> you say, it's not very practical to count beats that carefully, but
> still a fascinating fact about equal temperament and equal beating
> fifths. --JeffO

***Does this make any sense?? since I think that anything close to
12-equal is *not* going to have equally beating fifths throughout the
octave... Where's Paul Erlich when we need him...

J. Pehrson

>> > Hi Aaron
>> >
>> > Would explain the workings of that for us please? What are the
>> > steps to arrive at the number -1.1930? And what does it represent?
>> >
>> > Peter
>>
>> First, calculating these values in cents into absolute pitches from
>> C = 261.6256, then substracting each of them from the absolute
>> pitch value a musical fifth below multiplied by 3/2 yields a close
>> approximation of this value, except I get only one half the value,
>> somewhere around - .5965, starting in the indicated octave, and
>> then I get the right value in the next octave. What am I missing?
//
>> --JeffO
>
>***Does this make any sense?? since I think that anything close to
>12-equal is *not* going to have equally beating fifths throughout the
>octave... Where's Paul Erlich when we need him...

It is possible to have equal-beating fifths throughout an octave
in a temperament like this. And yes, JeffO, the rate doubles/halves
when you change octaves. From the sound of it you did the calculation
correctly.

-Carl

monz wrote:

> > i show how it could have been done by the Sumerians 5000 years
> ago, on two of my webpages. the simpler approximation, which
> also has graphics to illustrate, is here:
> > http://sonic-arts.org/monzo/sumerian/simplified-sumeriantuning.htm
> > ... but of course, the numbers here are base-60 not base-10.
> ... and you leave out the practical aspect entirely. How could the poor Sumerians measure ratios? The typical pitched instrument of the day was the harp (I seem to remember pictures of horn or trumpet players; let's disregard them since there music was probably separate from the rest). They most likely tuned in fifths, using the third harmonic. But there were no monochords, let alone regular fretted instruments.

The only practical way to tune 12et i've heard of is the chinese way, and it works by an arithmetic as i like it: fill a pipe with grains, count 'em, divide by half for the octave and by 12 (i.e. 24) for the semitones, then put the required number back in again. Equally simple for every number system, but you need your organ type instrument.

klaus

on 12/14/03 11:48 AM, Joseph Pehrson <jpehrson@rcn.com> wrote:

> --- In tuning@yahoogroups.com, "Jeff Olliff" <jolliff@d...> wrote:
>
> /tuning/topicId_49654.html#49846
>
>> --- In tuning@yahoogroups.com, "Peter Wakefield Sault" <sault@c...>
>> wrote:
>>> --- In tuning@yahoogroups.com, "Aaron K. Johnson" <akjmicro@c...>
>>> wrote:
>>>> On Friday 12 December 2003 10:52 pm, Werner Mohrlok wrote:
>>>>> -----Ursprüngliche Nachricht-----
>>>>> Von: Aaron K. Johnson [mailto:akjmicro@c...]
>>>>> Gesendet: Freitag, 12. Dezember 2003 15:35
>>>>> An: tuning@yahoogroups.com
>>>>> Betreff: Re: [tuning] Re: Bach, Werckmeister & Co.
>>>>>
>>>>> On Friday 12 December 2003 07:59 am, alternativetuning wrote:
>>>>>> --- In tuning@yahoogroups.com, Wernerlinden@a... wrote:
>>>>>>> Hi there, the literal translation of "gleichschwebend" is
>> in
>>> our
>>>>>>
>>>>>> context "equal beating", which would define a temperature
> in
>>> which
>>>>>> e.g. any fifth, be it "a-d" or "e#-c" or "d flat-g#" have
>> the
>>> same
>>>>>> beating. Which would consequently mean that all these
>> intervals
>>> are
>>>>>> absolutely equal. So that's 12 tet, isn't it ?
>>>>>>
>>>>>>
>>>>>> Hallo Werner:
>>>>>>
>>>>>> In an equal temperament, the beating rate of a given
>> interval
>>> class
>>>>>> varies with the absolute frequency. So, by definition, an
>> equal
>>>>>> beating temperament cannot be an equal (interval)
>> temperament.
>>>>>
>>>>> True, however, if one does an equal-beating pythagorean, the
>>> result is so
>>>>> close to equal temperament as to be indistinguishable to the
>>> ear...still,
>>>>> this is very painstaking to do by ear without an accurate way
>> of
>>> measuring
>>>>> the sought after beat rate...
>>>>>
>>>>> -Aaron.
>>>>>
>>>>> No, Aaron (sorry) - this is somehow a sophisticated answer.
>> You
>>> know
>>>>> as well as I: In the pythagorean model always one of the
>> fifths
>>> beats very
>>>>> hard,
>>>>> as the pythagorean comma strikes back.
>>>>> Besides, the term "equal-beating" was created for tuning
> models
>>>>> considering fifths and (!) thirds intervals.
>>>>> And with pythagorean tuning the thirds beat with deep tones
>> quick
>>> and
>>>>> with higher tones extremely quick.
>>>>>
>>>>> I repeat: Equal beating is a fiction.
>>>>>
>>>>
>>>> No Werner, you misunderstand--
>>>>
>>>> Withing the octave, you can have all fifths beat at the same
>> rate.
>>> Do it in
>>>> scala as equal beating pythagorean:
>>>>
>>>> |
>>>> 1: 100.034 cents 100.034
>>>> 2: 199.519 cents 199.519
>>>> 3: 299.800 cents 299.800
>>>> 4: 399.516 cents 399.516
>>>> 5: 500.017 cents 500.017
>>>> 6: 599.941 cents 599.941
>>>> 7: 699.322 cents 699.322
>>>> 8: 799.504 cents 799.504
>>>> 9: 899.127 cents 899.127
>>>> 10: 999.540 cents 999.540
>>>> 11: 1099.381 cents 1099.381
>>>> 12: 2/1 1200.000 octave
>>>> |
>>>> Base frequency : 261.6256 Hertz
>>>> Beat frequencies of 3/2
>>>> 0: 0.000: -1.1930
>>>> 1: 100.034: -1.1930
>>>> 2: 199.519: -1.1930
>>>> 3: 299.800: -1.1930
>>>> 4: 399.516: -1.1930
>>>> 5: 500.017: -1.1930
>>>> 6: 599.941: -1.1930
>>>> 7: 699.322: -1.1930
>>>> 8: 799.504: -1.1930
>>>> 9: 899.127: -1.1930
>>>> 10: 999.540: -1.1930
>>>> 11: 1099.381: -1.1930
>>>> 12: 1200.000: -2.3859
>>>> Total abs. beats : 14.3157
>>>> Average abs. beats: 1.1930
>>>> Highest abs. beats: 1.1930
>>>>
>>>> -Aaron.
>>>
>>> Hi Aaron
>>>
>>> Would explain the workings of that for us please? What are the
>> steps
>>> to arrive at the number -1.1930? And what does it represent?
>>>
>>> Peter
>>
>> First, calculating these values in cents into absolute pitches from
>> C = 261.6256, then substracting each of them from the absolute
> pitch
>> value a musical fifth below multiplied by 3/2 yields a close
>> approximation of this value, except I get only one half the value,
>> somewhere around - .5965, starting in the indicated octave, and
> then
>> I get the right value in the next octave. What am I missing?
> Maybe
>> beats are heard at half the difference, see what I know. Anyway,
> as
>> you say, it's not very practical to count beats that carefully, but
>> still a fascinating fact about equal temperament and equal beating
>> fifths. --JeffO
>
>
> ***Does this make any sense?? since I think that anything close to
> 12-equal is *not* going to have equally beating fifths throughout the
> octave... Where's Paul Erlich when we need him...
>
> J. Pehrson

The numerical example is one of an equal-beating pythagorean, to demonstrate
to Werner that equal-beating *exists*. It has nothing to do with 12-ET.

-Kurt

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

/tuning/topicId_49654.html#49874

>
> The numerical example is one of an equal-beating pythagorean, to
demonstrate
> to Werner that equal-beating *exists*. It has nothing to do with
12-ET.
>
> -Kurt

***Well, maybe this is a self-evident question, but then how do you
fit 12 equal steps of them in an octave? Isn't that where the
problem with the Pythagorean comma comes in, or am I totally missing
something??

Thanks!

JP

on 12/14/03 1:33 PM, Joseph Pehrson <jpehrson@rcn.com> wrote:

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>
> /tuning/topicId_49654.html#49874
>
>>
>> The numerical example is one of an equal-beating pythagorean, to
> demonstrate
>> to Werner that equal-beating *exists*. It has nothing to do with
> 12-ET.
>>
>> -Kurt
>
> ***Well, maybe this is a self-evident question, but then how do you
> fit 12 equal steps of them in an octave? Isn't that where the
> problem with the Pythagorean comma comes in, or am I totally missing
> something??
>
> Thanks!
>
> JP

Missing something, I guess. 12 equal steps is not the topic here.
Equal-beating means unequal steps.

-Kurt

on 12/14/03 1:41 PM, Kurt Bigler <kkb@breathsense.com> wrote:

> on 12/14/03 1:33 PM, Joseph Pehrson <jpehrson@rcn.com> wrote:
>
>> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>>
>> /tuning/topicId_49654.html#49874
>>
>>>
>>> The numerical example is one of an equal-beating pythagorean, to
>> demonstrate
>>> to Werner that equal-beating *exists*. It has nothing to do with
>> 12-ET.
>>>
>>> -Kurt
>>
>> ***Well, maybe this is a self-evident question, but then how do you
>> fit 12 equal steps of them in an octave? Isn't that where the
>> problem with the Pythagorean comma comes in, or am I totally missing
>> something??
>>
>> Thanks!
>>
>> JP
>
> Missing something, I guess. 12 equal steps is not the topic here.
> Equal-beating means unequal steps.
>
> -Kurt

Further clarification: we are not talking exact 3/2 pythagorean here.
Pythagoran is used as a general class of tunings also, in spite of the monz
dictionary definition appearing to only recognize the exact 3/2 and EDO
approximations thereof (unless I missed something). Maybe the correct term
for this is pythagorean temperament? I seem to recall it working something
like this: 12edo is the line between meantone on the one side and
pythagorean on the other, and as someone recently stated, 12edo is both a
meantone and a pythagorean.

Someone else can clear up the details when they arrive, but I think that's
the general picture.

-Kurt

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> Further clarification: we are not talking exact 3/2 pythagorean
here.
> Pythagoran is used as a general class of tunings also, in spite of
the monz
> dictionary definition appearing to only recognize the exact 3/2 and
EDO
> approximations thereof (unless I missed something). Maybe the
correct term
> for this is pythagorean temperament? I seem to recall it working
something
> like this: 12edo is the line between meantone on the one side and
> pythagorean on the other, and as someone recently stated, 12edo is
both a
> meantone and a pythagorean.
>
> Someone else can clear up the details when they arrive, but I think
that's
> the general picture.

I think you may mean schismic vs meantone temperaments. In 12edo the
major third is both four fifths up (meantone) and eight fifths down
(schismic), where in each case we reduce to the octave. More
characteristic meantones than 12 are 31 or 50, more characteristic
schismic divivisons are 53 or 118.
> -Kurt

on 12/14/03 2:05 PM, Gene Ward Smith <gwsmith@svpal.org> wrote:

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>
>> Further clarification: we are not talking exact 3/2 pythagorean
> here.
>> Pythagoran is used as a general class of tunings also, in spite of
> the monz
>> dictionary definition appearing to only recognize the exact 3/2 and
> EDO
>> approximations thereof (unless I missed something). Maybe the
> correct term
>> for this is pythagorean temperament? I seem to recall it working
> something
>> like this: 12edo is the line between meantone on the one side and
>> pythagorean on the other, and as someone recently stated, 12edo is
> both a
>> meantone and a pythagorean.
>>
>> Someone else can clear up the details when they arrive, but I think
> that's
>> the general picture.
>
> I think you may mean schismic vs meantone temperaments. In 12edo the
> major third is both four fifths up (meantone) and eight fifths down
> (schismic), where in each case we reduce to the octave. More
> characteristic meantones than 12 are 31 or 50, more characteristic
> schismic divivisons are 53 or 118.

I must have been confused by some usage I was exposed to.

So then Aaron's use of the term "equal beating pythagorean" must refer to an
approximation to pythagorean?

-Kurt

hi Kurt,

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> on 12/14/03 1:41 PM, Kurt Bigler <kkb@b...> wrote:
>
> > on 12/14/03 1:33 PM, Joseph Pehrson <jpehrson@r...> wrote:
> >
> >> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> >>
> >> /tuning/topicId_49654.html#49874
> >>
> >>>
> >>> The numerical example is one of an equal-beating
> >>> pythagorean, to demonstrate to Werner that equal-beating
> >>> *exists*. It has nothing to do with 12-ET.
> >>>
> >>> -Kurt
> >>
> >> ***Well, maybe this is a self-evident question,
> >> but then how do you fit 12 equal steps of them in
> >> an octave? Isn't that where the problem with the
> >> Pythagorean comma comes in, or am I totally missing
> >> something??
> >>
> >> Thanks!
> >>
> >> JP
> >
> > Missing something, I guess. 12 equal steps is not
> > the topic here. Equal-beating means unequal steps.
> >
> > -Kurt
>
> Further clarification: we are not talking exact 3/2
> pythagorean here. Pythagoran is used as a general class
> of tunings also, in spite of the monz dictionary definition
> appearing to only recognize the exact 3/2 and EDO
> approximations thereof (unless I missed something).
> Maybe the correct term for this is pythagorean temperament?
> I seem to recall it working something like this: 12edo
> is the line between meantone on the one side and pythagorean
> on the other, and as someone recently stated, 12edo is both
> a meantone and a pythagorean.
>
> Someone else can clear up the details when they arrive,
> but I think that's the general picture.

i think you did miss something, unless i'm not following.

we have coined the term "aristoxenean" to describe the
family of temperaments which tempers out the Pythagorean
comma.

http://tonalsoft.com/enc/aristox.htm

is that what you mean?

-monz

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

/tuning/topicId_49654.html#49877

> on 12/14/03 1:33 PM, Joseph Pehrson <jpehrson@r...> wrote:
>
> > --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> >
> > /tuning/topicId_49654.html#49874
> >
> >>
> >> The numerical example is one of an equal-beating pythagorean, to
> > demonstrate
> >> to Werner that equal-beating *exists*. It has nothing to do with
> > 12-ET.
> >>
> >> -Kurt
> >
> > ***Well, maybe this is a self-evident question, but then how do
you
> > fit 12 equal steps of them in an octave? Isn't that where the
> > problem with the Pythagorean comma comes in, or am I totally
missing
> > something??
> >
> > Thanks!
> >
> > JP
>
> Missing something, I guess. 12 equal steps is not the topic here.
> Equal-beating means unequal steps.
>
> -Kurt

***Thanks, Kurt! I figures "something's got to give..."

But, when you have a chain of all "pure" Pythagorean intervals, don't
you end up with a B# that is *higher* than C, rather than an octave
at 1200 cents, or am I *again* missing something??

JP

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

/tuning/topicId_49654.html#49878

>>
> Further clarification: we are not talking exact 3/2 pythagorean
here.

***Oh! But, wouldn't it have to be exact in order to have no
beating??

Signed,

Confused...

JP

On Sunday 14 December 2003 12:59 am, Jeff Olliff wrote:

Aaron wrote:
> > > Withing the octave, you can have all fifths beat at the same
>
> rate.
>
> > Do it in
> >
> > > scala as equal beating pythagorean:
> > >
> > >
> > > 1: 100.034 cents 100.034
> > > 2: 199.519 cents 199.519
> > > 3: 299.800 cents 299.800
> > > 4: 399.516 cents 399.516
> > > 5: 500.017 cents 500.017
> > > 6: 599.941 cents 599.941
> > > 7: 699.322 cents 699.322
> > > 8: 799.504 cents 799.504
> > > 9: 899.127 cents 899.127
> > > 10: 999.540 cents 999.540
> > > 11: 1099.381 cents 1099.381
> > > 12: 2/1 1200.000 octave
> > >
> > > Base frequency : 261.6256 Hertz
> > > Beat frequencies of 3/2
> > > 0: 0.000: -1.1930
> > > 1: 100.034: -1.1930
> > > 2: 199.519: -1.1930
> > > 3: 299.800: -1.1930
> > > 4: 399.516: -1.1930
> > > 5: 500.017: -1.1930
> > > 6: 599.941: -1.1930
> > > 7: 699.322: -1.1930
> > > 8: 799.504: -1.1930
> > > 9: 899.127: -1.1930
> > > 10: 999.540: -1.1930
> > > 11: 1099.381: -1.1930
> > > 12: 1200.000: -2.3859
> > > Total abs. beats : 14.3157
> > > Average abs. beats: 1.1930
> > > Highest abs. beats: 1.1930
> > >
> > > -Aaron.
> >
> > Hi Aaron
> >
> > Would explain the workings of that for us please? What are the
>
> steps
>
> > to arrive at the number -1.1930? And what does it represent?
> >
> > Peter
>
> First, calculating these values in cents into absolute pitches from
> C = 261.6256, then substracting each of them from the absolute pitch
> value a musical fifth below multiplied by 3/2 yields a close
> approximation of this value, except I get only one half the value,
> somewhere around - .5965, starting in the indicated octave, and then
> I get the right value in the next octave. What am I missing? Maybe
> beats are heard at half the difference, see what I know. Anyway, as
> you say, it's not very practical to count beats that carefully, but
> still a fascinating fact about equal temperament and equal beating
> fifths. --JeffO

Hello all-

A fact about beating: the overtone you are interesting in testing the beat
rate of minus one will be what you multiply the hertz difference by to get
the beat rate.

An example suffices to explain. Let's say I have two sawtooth waves with
harmonic spectra, one 3:2 plus one hertz apart:

first wave: 200 400 600 800 1000 1200, etc.
second ": 301 602 903 1204 , etc.

we see that a 1 hz detuned perfect fifth would beat between the 600hz and
602hz partials. Thus, the 3:2 ratio has a beat multiple factor of 2.

Now lets do a major third:

200 400 600 800 1000 1200
251 502 753 1004 1255

here, we see that a 5:4 has a beat multiple factor of 4.

In general, if the upper of two-voiced dyad moves, the beat ratio multiplier
is the denominator. If the lower moves, it's the numerator.

That's why you got -.5965 - you need to multiply by 2, because the taget beat
ratio multiplier is 2 -- then you get the correct answer -1.193

Ok?

Best,
Aaron.

on 12/14/03 4:03 PM, Joseph Pehrson <jpehrson@rcn.com> wrote:

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>
> /tuning/topicId_49654.html#49878
>
>>>
>> Further clarification: we are not talking exact 3/2 pythagorean
> here.
>
> ***Oh! But, wouldn't it have to be exact in order to have no
> beating??

Exactly. But we are talking about equal-beating here, not no beating,
right?

I'm confused too at the use of pythagorean here. Gene and Monz sorted out
my misuse of "pythagorean" but now I don't understand at all why the term is
being used in this equal-beating example.

-Kurt

>
> Signed,
>
> Confused...
>
> JP

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

/tuning/topicId_49654.html#49926

> on 12/14/03 4:03 PM, Joseph Pehrson <jpehrson@r...> wrote:
>
> > --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> >
> > /tuning/topicId_49654.html#49878
> >
> >>>
> >> Further clarification: we are not talking exact 3/2 pythagorean
> > here.
> >
> > ***Oh! But, wouldn't it have to be exact in order to have no
> > beating??
>
> Exactly. But we are talking about equal-beating here, not no
beating,
> right?
>

***Thanks, Kurt!

Actually, that thought came to me away from the list today, but
thanks for confirming it!

Joseph