Equally dividing the pure fifth into 7

And extending to the octave yields the following 12-tone system:

0: 1/1 C unison, perfect prime
1: 100.279 cents
2: 200.559 cents D
3: 300.838 cents E
4: 401.117 cents
5: 501.396 cents F
6: 601.676 cents
7: 3/2 G perfect fifth
8: 802.234 cents A
9: 902.514 cents A
10: 1002.793 cents B
11: 1103.072 cents
12: 1203.351 cents C

The fifths remain pure at every degree. Octaves sound fine too.

http://aredem.online.fr/aredem/page_cordier.html

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> And extending to the octave yields the following 12-tone system:
>
>
> 0: 1/1 C unison, perfect prime
> 1: 100.279 cents
> 2: 200.559 cents D
> 3: 300.838 cents E
> 4: 401.117 cents
> 5: 501.396 cents F
> 6: 601.676 cents
> 7: 3/2 G perfect fifth
> 8: 802.234 cents A
> 9: 902.514 cents A
> 10: 1002.793 cents B
> 11: 1103.072 cents
> 12: 1203.351 cents C
>
>
> The fifths remain pure at every degree. Octaves sound fine too.
>

Hi Ozan.

May I ask how long ago it was that this idea came to your mind? I was thinking about just the same thing about four months ago but so far I haven't found a way to realize this. The interesting thing about this kind of temperament is that 7 octaves are exactly (3/2)^12.

Petr

Something like an English translation would be fairly appropriate, I think. Speaking for myself, I'm afraid a Czech translation doesn't exist.

Petr

----- Original Message -----
From: Cameron Bobro
To: tuning@yahoogroups.com
Sent: Friday, March 23, 2007 12:32 PM
Subject: [tuning] Re: Equally dividing the pure fifth into 7

http://aredem.online.fr/aredem/page_cordier.html

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> And extending to the octave yields the following 12-tone system:
>
>
> 0: 1/1 C unison, perfect prime
> 1: 100.279 cents
> 2: 200.559 cents D
> 3: 300.838 cents E
> 4: 401.117 cents
> 5: 501.396 cents F
> 6: 601.676 cents
> 7: 3/2 G perfect fifth
> 8: 802.234 cents A
> 9: 902.514 cents A
> 10: 1002.793 cents B
> 11: 1103.072 cents
> 12: 1203.351 cents C
>
>
> The fifths remain pure at every degree. Octaves sound fine too.
>

--- In tuning@yahoogroups.com, Petr Paríº¥k <p.parizek@...> wrote:
>
> Something like an English translation would be fairly appropriate, I
think. Speaking for myself, I'm afraid a Czech translation doesn't
exist.
>
> Petr

It's Cordier's temperament from the '70s. Just checked- it's in the
Scala archives. Doesn't mean it's not a groovy discovery by Ozan, but
a re-discovery.

----- Original Message ----- From: "Ozan Yarman" <ozanyarman@ozanyarman.com>
To: "Tuning List" <tuning@yahoogroups.com>
Sent: Friday, March 23, 2007 4:21 AM
Subject: [tuning] Equally dividing the pure fifth into 7

> And extending to the octave yields the following 12-tone system:
>
>
> 0: 1/1 C unison, perfect prime
> 1: 100.279 cents
> 2: 200.559 cents D
> 3: 300.838 cents E
> 4: 401.117 cents
> 5: 501.396 cents F
> 6: 601.676 cents
> 7: 3/2 G perfect fifth
> 8: 802.234 cents A
> 9: 902.514 cents A
> 10: 1002.793 cents B
> 11: 1103.072 cents
> 12: 1203.351 cents C
>
>
> The fifths remain pure at every degree. Octaves sound fine too.

And octave-stretching is recommended for pianos, so you also have that advantage.

A while ago, I recommended 19 equal divisions of a perfect twelfth, which has a step size of 100.10289 cents. But what I really like is 49 divisions of the twelfth as a replacement for 31-tet. Fifths are a little better (now 698.67735 cents), to the detriment of the fourth and of course the octave, the latter being stretched to 1203.27765 cents.

~D.

--- In tuning@yahoogroups.com, "Danny Wier" <dawiertx@...> wrote:
>
> ----- Original Message -----
> From: "Ozan Yarman" <ozanyarman@...>
> To: "Tuning List" <tuning@yahoogroups.com>
> Sent: Friday, March 23, 2007 4:21 AM
> Subject: [tuning] Equally dividing the pure fifth into 7
>
>
> > And extending to the octave yields the following 12-tone system:
> >
> >
> > 0: 1/1 C unison, perfect prime
> > 1: 100.279 cents
> > 2: 200.559 cents D
> > 3: 300.838 cents E
> > 4: 401.117 cents
> > 5: 501.396 cents F
> > 6: 601.676 cents
> > 7: 3/2 G perfect fifth
> > 8: 802.234 cents A
> > 9: 902.514 cents A
> > 10: 1002.793 cents B
> > 11: 1103.072 cents
> > 12: 1203.351 cents C
> >
> >
> > The fifths remain pure at every degree. Octaves sound fine too.
>
> And octave-stretching is recommended for pianos, so you also have
that
> advantage.
>
> A while ago, I recommended 19 equal divisions of a perfect twelfth,
which
> has a step size of 100.10289 cents. But what I really like is 49
divisions
> of the twelfth as a replacement for 31-tet. Fifths are a little
better (now
> 698.67735 cents), to the detriment of the fourth and of course the
octave,
> the latter being stretched to 1203.27765 cents.
>
> ~D.
>
http://www.piano-stopper.de/html/onlypure_tuning.html

Clark

Seems I just revamped Cordier's concert piano tuning without realizing it.
The idea came to me just today, Petr.

Oz.

----- Original Message -----
From: "Petr Par�zek" <p.parizek@chello.cz>
To: <tuning@yahoogroups.com>
Sent: 23 Mart 2007 Cuma 13:28
Subject: Re: [tuning] Equally dividing the pure fifth into 7

> Hi Ozan.
>
> May I ask how long ago it was that this idea came to your mind? I was
> thinking about just the same thing about four months ago but so far I
> haven't found a way to realize this. The interesting thing about this kind
> of temperament is that 7 octaves are exactly (3/2)^12.
>
> Petr
>
>

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> And extending to the octave yields the following 12-tone system:
>
>
> 0: 1/1 C unison, perfect prime
> 1: 100.279 cents
> 2: 200.559 cents D
> 3: 300.838 cents E
> 4: 401.117 cents
> 5: 501.396 cents F
> 6: 601.676 cents
> 7: 3/2 G perfect fifth
> 8: 802.234 cents A
> 9: 902.514 cents A
> 10: 1002.793 cents B
> 11: 1103.072 cents
> 12: 1203.351 cents C
>
>
> The fifths remain pure at every degree. Octaves sound fine too.

This is a tuning I've tried on pianos, which appreciate
octave stretch anyway. I call it the 7th-root-of-3/2
temperament. Concise, isn't it?

The problem with this tuning is that it makes major 3rds
and 10ths worse.

Some top-notch piano tuners on the usenet claim to tune a
similar kind of thing, namely 19th-root-of-3 temperament.
This one is easier on the 3rds.

-Carl

> May I ask how long ago it was that this idea came to your mind? I was
> thinking about just the same thing about four months ago but so far I
> haven't found a way to realize this. The interesting thing about this
> kind of temperament is that 7 octaves are exactly (3/2)^12.

I first learned of it in 1998 from Julius Pierce. I posted on
it here several times in the late '90s.
I set it by using a strobe tuner to tune C-B, then tuning
pure fifths by ear from there.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:

> The problem with this tuning is that it makes major 3rds
> and 10ths worse.

Not for pianos, but for situations where you might consider 12-et TOP:
tuning so that 5 is pure; that is, with steps of 5^(1/28). Especially
nice for chords which avoid octaves and don't cover too great a range.
The major triad in close root position is 0-398.044816-696.578428.
Another triad is 0-696.578428-1592.179265. You can get interesting
seventh chords too, for example 0-696.578428-1592.179265-2189.246489.

! div28.scl
Dividing 5 into 28 equal parts
28
!
99.511204
199.022408
298.533612
398.044816
497.556020
597.067224
696.578428
796.089633
895.600837
995.112041
1094.623245
1194.134449
1293.645653
1393.156857
1492.668061
1592.179265
1691.690469
1791.201673
1890.712877
1990.224081
2089.735285
2189.246489
2288.757694
2388.268898
2487.780102
2587.291306
2686.802510
5

From: "threesixesinarow" <CACCOLA@NET1PLUS.COM>
To: <tuning@yahoogroups.com>
Sent: Friday, March 23, 2007 9:28 AM
Subject: [tuning] Re: Equally dividing the pure fifth into 7

>> A while ago, I recommended 19 equal divisions of a perfect twelfth,
> which
>> has a step size of 100.10289 cents. But what I really like is 49
> divisions
>> of the twelfth as a replacement for 31-tet. Fifths are a little
> better (now
>> 698.67735 cents), to the detriment of the fourth and of course the
> octave,
>> the latter being stretched to 1203.27765 cents.
>>
>> ~D.
>>
> http://www.piano-stopper.de/html/onlypure_tuning.html
>
> Clark

I was thinking someone would've come up with the idea before me. But where does he get the name "OnlyPure"? There are fewer pure intervals in 19-edt than in 12-edo, aren't there?

~D.

> I was thinking someone would've come up with the idea before me.
> But where does he get the name "OnlyPure"? There are fewer pure
> intervals in 19-edt than in 12-edo, aren't there?
>
> ~D.

I hate to state the obvious, but anyone who brands a tuning
as obvious as this is an idiot. I'm almost as bad with
the Alaska tunings, but geez.

-Carl

My, you are a bit touchy nowadays it seems.

----- Original Message -----
From: "Carl Lumma" <clumma@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: 24 Mart 2007 Cumartesi 1:11
Subject: [tuning] Re: Equally dividing the pure fifth into 7

> > I was thinking someone would've come up with the idea before me.
> > But where does he get the name "OnlyPure"? There are fewer pure
> > intervals in 19-edt than in 12-edo, aren't there?
> >
> > ~D.
>
> I hate to state the obvious, but anyone who brands a tuning
> as obvious as this is an idiot. I'm almost as bad with
> the Alaska tunings, but geez.
>
> -Carl
>
>

From: "Carl Lumma" <clumma@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: Friday, March 23, 2007 6:11 PM
Subject: [tuning] Re: Equally dividing the pure fifth into 7

>> I was thinking someone would've come up with the idea before me.
>> But where does he get the name "OnlyPure"? There are fewer pure
>> intervals in 19-edt than in 12-edo, aren't there?
>> >> ~D.
> > I hate to state the obvious, but anyone who brands a tuning
> as obvious as this is an idiot. I'm almost as bad with
> the Alaska tunings, but geez.
> > -Carl

He also said he wanted to patent it. Can you patent a mathematical formula?

~D.

Carl wrote:

> I hate to state the obvious, but anyone who brands a tuning as obvious as this is an idiot. I'm almost as bad with the Alaska tunings, but geez.

though I'm sure you have mentioned Alaska tunings some time ago, I'm not sure what they are.

Petr

> though I'm sure you have mentioned Alaska tunings some time ago,
> I'm not sure what they are.

Circulating temperaments with flat octaves in the 1997-cent
range.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> > though I'm sure you have mentioned Alaska tunings some time ago,
> > I'm not sure what they are.
>
> Circulating temperaments with flat octaves in the 1997-cent
> range.
>
> -Carl
>

Love the typo.

Anyway, the piano tech mailing list agrees with Carl about the
patenting of a piano tuning method using pure twelfths - namely, that
it's probably both bogus (in the sense of not patentable) and useless
(ie the patent is unenforceable). Search Google for 'Onlypure'.

The promulgator of it, Mr. Stopper, is now promoting the existence of
a 'Supersymmetry' between beats and frequencies. But you can't get an
explanation of what this could mean until you buy an Onlypure licence.

As a theoretical physicist who actually deals with genuine
supersymmetry (graded Lie algebra, for those in the know) I find this
half amusing and half pathetic.

~~~T~~~

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:

> As a theoretical physicist who actually deals with genuine
> supersymmetry (graded Lie algebra, for those in the know) I find this
> half amusing and half pathetic.

Have you ever taken a look at tuning-math? Not much in the way of Lie
algebras, but root lattice terminology is used anyway, for what that is
worth. And multilinear algebra galore...

> Love the typo.

Which typo is that?

-Carl

Carl Lumma schrieb:
>> Love the typo. > > Which typo is that?

You hid it well - I spent some time looking for it, too.

Think about what an unflat octave is.

klaus

> >> Love the typo.
> >
> > Which typo is that?
>
> You hid it well - I spent some time looking for it, too.
>
> Think about what an unflat octave is.

A pure octave?

-Carl

Carl Lumma schrieb:
>>>> Love the typo. >>> Which typo is that?
>> You hid it well - I spent some time looking for it, too.
>>
>> Think about what an unflat octave is.
> > A pure octave?

Undeniably.

And in quantitative terms?

> >> Think about what an unflat octave is.
> >
> > A pure octave?
>
> Undeniably.
>
> And in quantitative terms?

I give up.

-Carl

Carl Lumma schrieb:
>>>> Think about what an unflat octave is.
>>> A pure octave?
>> Undeniably.
>>
>> And in quantitative terms?
> > I give up.

No you don't �=

Blunt hint: the 2000 cent octave.

> >>>> Think about what an unflat octave is.
> >>> A pure octave?
> >>
> >> Undeniably.
> >>
> >> And in quantitative terms?
> >
> > I give up.
>
> No you don't ¿=
>
> Blunt hint: the 2000 cent octave.

I have no idea what you're talking about, but if anyone wants
to divulge the secret my e-mail address is carl@lumma.org.

-Carl

When Stanford patented FM synthesis, Allen Strange said this was akin
to "patenting water".

Some of us pianotech people were fortunate enough to meet Bernard
Stopper in Kansas City last month. He demonstrated his tuning on a
Yamaha upright and a Fazioli grand. He was touting his new pocket pc
tuning software based on his 3^(1/19) pure P12 tuning system.

The 19-tone P12 tuning sounded very nice- some couldn't tell the
difference, others like myself reveled in the way it masked
dissonance, especially in bitonal/pan-diatonic chord structures.

What I found intriguing was the use of any combination of 5ths and
Octaves in setting the tuning aurally. Apparently he has found the
formula by which the beat found in the P4-P5-P8 sonority is cancelled
by stretching the P12 just.

Has anyone else worked this out, mathematically?

-Richard Barber
Morgan Hill, CA

--- In tuning@yahoogroups.com, "Danny Wier" <dawiertx@...> wrote:
>
> From: "Carl Lumma" <clumma@...>
> To: <tuning@yahoogroups.com>
> Sent: Friday, March 23, 2007 6:11 PM
> Subject: [tuning] Re: Equally dividing the pure fifth into 7
>
>
> >> I was thinking someone would've come up with the idea before me.
> >> But where does he get the name "OnlyPure"? There are fewer pure
> >> intervals in 19-edt than in 12-edo, aren't there?
> >>
> >> ~D.
> >
> > I hate to state the obvious, but anyone who brands a tuning
> > as obvious as this is an idiot. I'm almost as bad with
> > the Alaska tunings, but geez.
> >
> > -Carl
>
> He also said he wanted to patent it. Can you patent a mathematical
formula?
>
> ~D.
>

> Some of us pianotech people were fortunate enough to meet Bernard
> Stopper in Kansas City last month. He demonstrated his tuning on
> a Yamaha upright and a Fazioli grand. He was touting his new
> pocket pc tuning software based on his 3^(1/19) pure P12 tuning
> system.
>
> The 19-tone P12 tuning sounded very nice- some couldn't tell the
> difference, others like myself reveled in the way it masked
> dissonance, especially in bitonal/pan-diatonic chord structures.
>
> What I found intriguing was the use of any combination of 5ths and
> Octaves in setting the tuning aurally. Apparently he has found the
> formula by which the beat found in the P4-P5-P8 sonority is
> cancelled by stretching the P12 just.
>
> Has anyone else worked this out, mathematically?
>
> -Richard Barber
> Morgan Hill, CA

Hi Richard,

There's only one equal step tuning with the P12 (what I would
prefer to call 3:1) just. By P4-P5-P8 sonority, what chord
or chords do you mean? And in what manner were beats cancelled?
If you can explain, maybe the list will answer.

-Carl

By P4-P5-P8 I mean, for instance, C F G c. According to Stopper, any
combination of fifths and octaves sound beatless due to a symmetry in
the tuning. My guess is they are either phase-reverse cancelled, or
masked by sidebands? Any thoughts?
Richard

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
> Hi Richard,
>
> There's only one equal step tuning with the P12 (what I would
> prefer to call 3:1) just. By P4-P5-P8 sonority, what chord
> or chords do you mean? And in what manner were beats cancelled?
> If you can explain, maybe the list will answer.
>
> -Carl
>

>
> > Some of us pianotech people were fortunate enough to meet Bernard
> > Stopper in Kansas City last month. He demonstrated his tuning on
> > a Yamaha upright and a Fazioli grand. He was touting his new
> > pocket pc tuning software based on his 3^(1/19) pure P12 tuning
> > system.
> >
> > The 19-tone P12 tuning sounded very nice- some couldn't tell the
> > difference, others like myself reveled in the way it masked
> > dissonance, especially in bitonal/pan-diatonic chord structures.
> >
> > What I found intriguing was the use of any combination of 5ths and
> > Octaves in setting the tuning aurally. Apparently he has found the
> > formula by which the beat found in the P4-P5-P8 sonority is
> > cancelled by stretching the P12 just.
> >
> > Has anyone else worked this out, mathematically?
> >
> > -Richard Barber
> > Morgan Hill, CA
>

I doubt anything like this can be made to occur in all
keys and in all registers. If beating were masked by
sidebands, I should think the sidebands would themselves
be quite unpleasant. And I'm not aware of any way to
control the relative phases of beating intervals in a
tuning.

Is there a recording anywhere of this stuff, or a paper
by Stopper anywhere?

By the way, I'm in Los Gatos, and tangentially thinking
of starting a piano tuning business in the area. If
you have any interest in getting together for tea or
something, feel free to write me off-list at carl at
lumma dot org.

-Carl

--- In tuning@yahoogroups.com, "Richard Eldon Barber"
<bassooner42@...> wrote:
>
> By P4-P5-P8 I mean, for instance, C F G c. According to
> Stopper, any combination of fifths and octaves sound beatless
> due to a symmetry in the tuning. My guess is they are either
> phase-reverse cancelled, or masked by sidebands? Any thoughts?
> Richard
>
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <clumma@> wrote:
> > Hi Richard,
> >
> > There's only one equal step tuning with the P12 (what I would
> > prefer to call 3:1) just. By P4-P5-P8 sonority, what chord
> > or chords do you mean? And in what manner were beats cancelled?
> > If you can explain, maybe the list will answer.
> >
> > -Carl
> >