Irrational ratio using and other "exception to the rule" tunings

Virtually all tunings here seem to be based on the concept of
maintaining JI intervals and general low-numbered fractional ratios
(IE 3/2, 5/6, 12/11, etc.
In general this produces the most consonance and allows tunings to
be made which can use musical theory and chords relatively similar to
those used in 12TET.
**********************************************************
However, for example, the Golden Ratio Tuning, does not follow this
convention. I am also starting to wonder to what degree MOS scales
follow the convention.
And 10TET, for example, has a circle of thirds as opposed to a
circle of fifths. And, as I understand it, Lucy Tuning uses a circle
of 5ths in one direction and a circle of 4ths in the other.
************************************************************

So, what tunings and scales would you recommend to research...to
study the concept of using "far-from-JI" ratios to create a relatively
good degree of consonance?

Virtually all tunings here seem to be based on the concept of
maintaining JI intervals and general low-numbered fractional ratios
(IE 3/2, 5/6, 12/11, etc.
In general this produces the most consonance and allows tunings to
be made which can use musical theory and chords relatively similar to
those used in 12TET.
**********************************************************
However, for example, the Golden Ratio Tuning, does not follow this
convention. I am also starting to wonder to what degree MOS scales
follow the convention.
And 10TET, for example, has a circle of thirds as opposed to a
circle of fifths. And, as I understand it, Lucy Tuning uses a circle
of 5ths in one direction and a circle of 4ths in the other.
************************************************************

So, what tunings and scales would you recommend to research...to
study the concept of using "far-from-JI" ratios to create a relatively
good degree of consonance?

--- In tuning@yahoogroups.com, "djtrancendance" <djtrancendance@...>
wrote:
>.... fractional ratios
> (IE 3/2, 5/6, 12/11, etc....

> However, for example, the Golden Ratio Tuning, does not follow this
> convention. I am also starting to wonder to what degree MOS scales
> follow the convention...
http://en.wikipedia.org/wiki/Golden_ratio
http://mathworld.wolfram.com/GoldenRatio.html
http://www.research.att.com/~njas/sequences/A001622
"
(x+1)^n - x^(2n). (x+1)^n - x^2n = 0
has only two real roots x1 = -(sqrt(5)-1)/2 =
-.618033988749894848204586834...

x2 = (sqrt(5)+1)/2 = 1.618033988749894848204586834...
"

> So, what tunings and scales would you recommend to research...to
> study the concept of using "far-from-JI" ratios to create a
> relatively good degree of consonance?
>
From the mathematically point of view:
"well-approximated" in terms of
http://en.wikipedia.org/wiki/Dense_set
the theory makes no difference:
"# The real numbers with the usual topology have the rational numbers
and the irrational numbers as dense subsets."

Because that both sets are:
http://en.wikipedia.org/wiki/Dense-in-itself

In the psycho-acustically view there is also no difference due to the:
http://en.wikipedia.org/wiki/Just_noticeable_difference
of pitch-perception and
http://en.wikipedia.org/wiki/Absolute_threshold_of_hearing
and
http://hyperphysics.phy-astr.gsu.edu/hbase/sound/earsens.html
"The normal human ear can detect the difference between 440 Hz and 441
Hz. It is hard to believe it could attain such resolution from
selective peaking of the membrane vibrations. Some pitch sharpening
mechanism must be operating...."

bye
A.S.

Just to keep the record straight;

LucyTuning is an endless spiral (not circle) of fourths in the "flat" direction and fifths in the "sharp" direction.

Each new step arrives at a new and unique interval from the starting point.

So you can "map" any interval in terms of (multiple) steps of fourths and fifths to whatever level of resolution suits your purposes.

Should you wish to approximate LucyTuning with an equal interval system, you could use 88edo.

(L) Large interval = 14 steps (II)
(s) small interval = 9 steps (bII)

(14*5) + (9*2) = 88 steps per octave

1420 edo provides a more precise approximation

(L) Large interval = 226 steps
(s) small interval = 145 steps

On 9 Feb 2009, at 17:13, djtrancendance wrote:

> Virtually all tunings here seem to be based on the concept of
> maintaining JI intervals and general low-numbered fractional ratios
> (IE 3/2, 5/6, 12/11, etc.
> In general this produces the most consonance and allows tunings to
> be made which can use musical theory and chords relatively similar to
> those used in 12TET.
> **********************************************************
> However, for example, the Golden Ratio Tuning, does not follow this
> convention. I am also starting to wonder to what degree MOS scales
> follow the convention.
> And 10TET, for example, has a circle of thirds as opposed to a
> circle of fifths. And, as I understand it, Lucy Tuning uses a circle
> of 5ths in one direction and a circle of 4ths in the other.
> ************************************************************
>
> So, what tunings and scales would you recommend to research...to
> study the concept of using "far-from-JI" ratios to create a relatively
> good degree of consonance?
>
>
>
Charles Lucy
lucy@...

- Promoting global harmony through LucyTuning -

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

For LucyTuned Lullabies go to:
http://www.lullabies.co.uk

--LucyTuning is an endless spiral (not circle) of fourths in the "flat" --direction and fifths in the "sharp" direction.

   Sorry about that...I didn't mean to say that either "spiral" terminates or loops around to its starting value the way a "circle" does ("circle" was a bad word choice).

   Oddly enough, if I have it right, it seems neither the "spirals" of 1.618033^x or the fourths (flat, decending) and fifths (sharp,ascending) in Lucytuning terminate against themselves or at an exact 2/1 octave.  So, if I have it right, the Golden Ratio and Lucy Tuning are, when defined with perfect accuracy, "non-octave" tunings.

--- On Mon, 2/9/09, Charles Lucy <lucy@...> wrote:

From: Charles Lucy <lucy@...>
Subject: Re: [tuning] Irrational ratio using and other "exception to the rule" tunings
To: tuning@yahoogroups.com
Date: Monday, February 9, 2009, 2:55 PM

Just to keep the record straight; 
LucyTuning is an endless spiral (not circle) of fourths in the "flat" direction and fifths in the "sharp" direction.
Each new step arrives at a new and unique interval from the starting point.
So you can "map" any interval in terms of (multiple) steps of fourths and fifths to whatever level of resolution suits your purposes.
Should you wish to approximate LucyTuning with an equal interval system, you could use 88edo.
(L) Large interval = 14 steps  (II)(s) small interval = 9 steps  (bII)
(14*5) + (9*2) = 88 steps per octave
1420 edo provides a more precise approximation
(L) Large interval = 226 steps(s) small interval = 145 steps

On 9 Feb 2009, at 17:13, djtrancendance wrote:
Virtually all tunings here seem to be based on the concept of
maintaining JI intervals and general low-numbered fractional ratios
(IE 3/2, 5/6, 12/11, etc.
In general this produces the most consonance and allows tunings to
be made which can use musical theory and chords relatively similar to
those used in 12TET.
************ ********* ********* ********* ********* ********* *
However, for example, the Golden Ratio Tuning, does not follow this
convention. I am also starting to wonder to what degree MOS scales
follow the convention.
And 10TET, for example, has a circle of thirds as opposed to a
circle of fifths. And, as I understand it, Lucy Tuning uses a circle
of 5ths in one direction and a circle of 4ths in the other.
************ ********* ********* ********* ********* ********* ***

So, what tunings and scales would you recommend to research...to
study the concept of using "far-from-JI" ratios to create a relatively
good degree of consonance?

Charles Lucylucy@lucytune. com
- Promoting global harmony through LucyTuning -
for information on LucyTuning go to:http://www.lucytune .com
For LucyTuned Lullabies go to:http://www.lullabie s.co.uk

Hi Michael;

It's not that they are non-octave, as the octave ratio is
2.0000000000000.

I know that LucyTuning will always land on new intervals as Pi (π) is
both irrational, (its infinitely long string of decimal places never
repeats),
and transcendental, (it is not a solution of any polynomial equation
with whole number coefficients).

I don't know if this is also true of phi.

This page explains more about the way various meantone tunings are
calculated:

http://www.lucytune.com/tuning/mean_tone.html

On 9 Feb 2009, at 23:03, Michael Sheiman wrote:

>
> --LucyTuning is an endless spiral (not circle) of fourths in the
> "flat" --direction and fifths in the "sharp" direction.
>
> Sorry about that...I didn't mean to say that either "spiral"
> terminates or loops around to its starting value the way a "circle"
> does ("circle" was a bad word choice).
>
> Oddly enough, if I have it right, it seems neither the "spirals"
> of 1.618033^x or the fourths (flat, decending) and fifths
> (sharp,ascending) in Lucytuning terminate against themselves or at
> an exact 2/1 octave. So, if I have it right, the Golden Ratio and
> Lucy Tuning are, when defined with perfect accuracy, "non-octave"
> tunings.
>
>
> --- On Mon, 2/9/09, Charles Lucy <lucy@...> wrote:
>
> From: Charles Lucy <lucy@harmonics.com>
> Subject: Re: [tuning] Irrational ratio using and other "exception to
> the rule" tunings
> To: tuning@yahoogroups.com
> Date: Monday, February 9, 2009, 2:55 PM
>
> Just to keep the record straight;
>
>
> LucyTuning is an endless spiral (not circle) of fourths in the
> "flat" direction and fifths in the "sharp" direction.
>
> Each new step arrives at a new and unique interval from the starting
> point.
>
> So you can "map" any interval in terms of (multiple) steps of
> fourths and fifths to whatever level of resolution suits your
> purposes.
>
> Should you wish to approximate LucyTuning with an equal interval
> system, you could use 88edo.
>
> (L) Large interval = 14 steps (II)
> (s) small interval = 9 steps (bII)
>
> (14*5) + (9*2) = 88 steps per octave
>
> 1420 edo provides a more precise approximation
>
> (L) Large interval = 226 steps
> (s) small interval = 145 steps
>
>
>
>
> On 9 Feb 2009, at 17:13, djtrancendance wrote:
>
>> Virtually all tunings here seem to be based on the concept of
>> maintaining JI intervals and general low-numbered fractional ratios
>> (IE 3/2, 5/6, 12/11, etc.
>> In general this produces the most consonance and allows tunings to
>> be made which can use musical theory and chords relatively similar to
>> those used in 12TET.
>> ************ ********* ********* ********* ********* ********* *
>> However, for example, the Golden Ratio Tuning, does not follow this
>> convention. I am also starting to wonder to what degree MOS scales
>> follow the convention.
>> And 10TET, for example, has a circle of thirds as opposed to a
>> circle of fifths. And, as I understand it, Lucy Tuning uses a circle
>> of 5ths in one direction and a circle of 4ths in the other.
>> ************ ********* ********* ********* ********* ********* ***
>>
>> So, what tunings and scales would you recommend to research...to
>> study the concept of using "far-from-JI" ratios to create a>> relatively
>> good degree of consonance?
>>
>>
>
> Charles Lucy
> lucy@lucytune. com
>
> - Promoting global harmony through LucyTuning -
>
> for information on LucyTuning go to:
> http://www.lucytune .com
>
> For LucyTuned Lullabies go to:
> http://www.lullabie s.co.uk
>
>
>
>
>

Charles Lucy
lucy@...

- Promoting global harmony through LucyTuning -

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

For LucyTuned Lullabies go to:
http://www.lullabies.co.uk

Charles Lucy wrote:

> I know that LucyTuning will always land on new intervals as Pi (π) is both > irrational, (its infinitely long string of decimal places never repeats), > and *transcendental*, (it is not a solution of any polynomial equation with > whole number coefficients).
> > I don't know if this is also true of phi.

Phi is irrational but not transcendental. Which means it'll always give new intervals.

Graham

http://www.lucytune.com/midi/bb7.mid

You must be using non-standard chord-names, as this is clearly not a
standard Bb7 (dominant-7th) chord,

but on the other hand, many of the chords are close to standard, so
this seems confusing.

?

caleb

On Feb 9, 2009, at 6:33 PM, Charles Lucy wrote:

> Hi Michael;
>
>
> It's not that they are non-octave, as the octave ratio is
> 2.0000000000000.
>
> I know that LucyTuning will always land on new intervals as Pi (π)
> is both irrational, (its infinitely long string of decimal places
> never repeats),
> and transcendental, (it is not a solution of any polynomial equation
> with whole number coefficients).
>
> I don't know if this is also true of phi.
>
> This page explains more about the way various meantone tunings are
> calculated:
>
> http://www.lucytune.com/tuning/mean_tone.html
>
>
>
>
> On 9 Feb 2009, at 23:03, Michael Sheiman wrote:
>
>>
>> --LucyTuning is an endless spiral (not circle) of fourths in the
>> "flat" --direction and fifths in the "sharp" direction.
>>
>> Sorry about that...I didn't mean to say that either "spiral"
>> terminates or loops around to its starting value the way a "circle"
>> does ("circle" was a bad word choice).
>>
>> Oddly enough, if I have it right, it seems neither the "spirals"
>> of 1.618033^x or the fourths (flat, decending) and fifths
>> (sharp,ascending) in Lucytuning terminate against themselves or at
>> an exact 2/1 octave. So, if I have it right, the Golden Ratio and
>> Lucy Tuning are, when defined with perfect accuracy, "non-octave"
>> tunings.
>>
>>
>> --- On Mon, 2/9/09, Charles Lucy <lucy@...> wrote:
>>
>> From: Charles Lucy <lucy@...>
>> Subject: Re: [tuning] Irrational ratio using and other "exception
>> to the rule" tunings
>> To: tuning@yahoogroups.com
>> Date: Monday, February 9, 2009, 2:55 PM
>>
>> Just to keep the record straight;
>>
>>
>> LucyTuning is an endless spiral (not circle) of fourths in the
>> "flat" direction and fifths in the "sharp" direction.
>>
>> Each new step arrives at a new and unique interval from the
>> starting point.
>>
>> So you can "map" any interval in terms of (multiple) steps of
>> fourths and fifths to whatever level of resolution suits your
>> purposes.
>>
>> Should you wish to approximate LucyTuning with an equal interval
>> system, you could use 88edo.
>>
>> (L) Large interval = 14 steps (II)
>> (s) small interval = 9 steps (bII)
>>
>> (14*5) + (9*2) = 88 steps per octave
>>
>> 1420 edo provides a more precise approximation
>>
>> (L) Large interval = 226 steps
>> (s) small interval = 145 steps
>>
>>
>>
>>
>> On 9 Feb 2009, at 17:13, djtrancendance wrote:
>>
>>> Virtually all tunings here seem to be based on the concept of
>>> maintaining JI intervals and general low-numbered fractional ratios
>>> (IE 3/2, 5/6, 12/11, etc.
>>> In general this produces the most consonance and allows tunings to
>>> be made which can use musical theory and chords relatively similar
>>> to
>>> those used in 12TET.
>>> ************ ********* ********* ********* ********* ********* *
>>> However, for example, the Golden Ratio Tuning, does not follow this
>>> convention. I am also starting to wonder to what degree MOS scales
>>> follow the convention.
>>> And 10TET, for example, has a circle of thirds as opposed to a
>>> circle of fifths. And, as I understand it, Lucy Tuning uses a circle
>>> of 5ths in one direction and a circle of 4ths in the other.
>>> ************ ********* ********* ********* ********* ********* ***
>>>
>>> So, what tunings and scales would you recommend to research...to
>>> study the concept of using "far-from-JI" ratios to create a
>>> relatively
>>> good degree of consonance?
>>>
>>>
>>
>> Charles Lucy
>> lucy@lucytune. com
>>
>> - Promoting global harmony through LucyTuning -
>>
>> for information on LucyTuning go to:
>> http://www.lucytune .com
>>
>> For LucyTuned Lullabies go to:
>> http://www.lullabie s.co.uk
>>
>>
>>
>>
>
> Charles Lucy
> lucy@lucytune.com
>
> - Promoting global harmony through LucyTuning -
>
> for information on LucyTuning go to:
> http://www.lucytune.com
>
> For LucyTuned Lullabies go to:
> http://www.lullabies.co.uk
>
>
>
>
>

As far as I know there are more ways how to write chord signs, unfortunately not only one standard. The worse one is that one using "b" and "#" for alterations, not context independent "-" and "+".

Daniel Forro

On 10 Feb 2009, at 8:55 AM, caleb morgan wrote:

>
> http://www.lucytune.com/midi/bb7.mid
>
> You must be using non-standard chord-names, as this is clearly not > a standard Bb7 (dominant-7th) chord,
>
> but on the other hand, many of the chords are close to standard, so > this seems confusing.
>
> ?
>
> caleb

I think it's just a typo:

It should be B natural, which makes it a diminished chord, maybe?

On Feb 9, 2009, at 7:18 PM, Daniel Forro wrote:

> As far as I know there are more ways how to write chord signs,
> unfortunately not only one standard. The worse one is that one using
> "b" and "#" for alterations, not context independent "-" and "+".
>
> Daniel Forro
>
> On 10 Feb 2009, at 8:55 AM, caleb morgan wrote:
>
> >
> > http://www.lucytune.com/midi/bb7.mid
> >
> > You must be using non-standard chord-names, as this is clearly not
> > a standard Bb7 (dominant-7th) chord,
> >
> > but on the other hand, many of the chords are close to standard, so
> > this seems confusing.
> >
> > ?
> >
> > caleb
>
>

I must have done it about ten years ago, (copyright 1998)

I don't know how you triggered that midi file, nor where that chord is linked from, and unfortunately my online search facility gives no link to bb7.mid.

It looks as though you were right; it was a typo, yet nevertheless, as it is using midi pitchbend, it would only have 64th of a semitone resolution.

Since I no longer use pitchbend to microtune, maybe it's about time that I replaced the midi links with audio files;-)

Using the values at the bottom of this page:

http://www.lucytune.com/midi_and_keyboard/pitch_bend.html

And by reading the midi file as an event list I find that the notes played with pitchbend values based on 4096 are:

Channel 1 sounds D3 + 185 = LucyTuned D
Channel 2 sounds B2 + 923 = LucyTuned *error* = Bb *This seems to be where the typo occurred, as it should be sounding from midinote A#; instead of B natural.*
Channel 3 sounds G#3 + 1292 = LucyTuned Ab
Channel 4 sounds F3 + 738 = LucyTuned F

So the chord sounded is wrong!
and should be:

D
Bb
Ab
F

which would give us the chord of Bb7.

I have replaced the midifile so that it should sound right now for future visitors.

Thanks for the error report.

You tunaniks are really strict taskmasters;-)

but this one took you ten years to find!

On 10 Feb 2009, at 00:20, caleb morgan wrote:

>
> I think it's just a typo:
>
> It should be B natural, which makes it a diminished chord, maybe?
>
>
>
> On Feb 9, 2009, at 7:18 PM, Daniel Forro wrote:
>
>> As far as I know there are more ways how to write chord signs,
>> unfortunately not only one standard. The worse one is that one using
>> "b" and "#" for alterations, not context independent "-" and "+".
>>
>> Daniel Forro
>>
>> On 10 Feb 2009, at 8:55 AM, caleb morgan wrote:
>>
>> >
>> > http://www.lucytune.com/midi/bb7.mid
>> >
>> > You must be using non-standard chord-names, as this is clearly not
>> > a standard Bb7 (dominant-7th) chord,
>> >
>> > but on the other hand, many of the chords are close to standard, so
>> > this seems confusing.
>> >
>> > ?
>> >
>> > caleb
>>
>
>
>
Charles Lucy
lucy@...

- Promoting global harmony through LucyTuning -

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

For LucyTuned Lullabies go to:
http://www.lullabies.co.uk

The erroneous midifile was triggered from this page:

http://lucytune.com/listening_room/piano_play.html

Apologies for creating so much confusion.

The tunaniks who are discussing scales may find the files in this folder useful:

http://www.lucytune.com/scales/

finding the scales which have acquired names seems to indicate that they have received widespread use.

On 10 Feb 2009, at 01:20, Charles Lucy wrote:

> I must have done it about ten years ago, (copyright 1998)
>
>
> I don't know how you triggered that midi file, nor where that chord > is linked from, and unfortunately my online search facility gives no > link to bb7.mid.
>
> It looks as though you were right; it was a typo, yet nevertheless, > as it is using midi pitchbend, it would only have 64th of a semitone > resolution.
>
> Since I no longer use pitchbend to microtune, maybe it's about time > that I replaced the midi links with audio files;-)
>
> Using the values at the bottom of this page:
>
> http://www.lucytune.com/midi_and_keyboard/pitch_bend.html
>
>
> And by reading the midi file as an event list I find that the notes > played with pitchbend values based on 4096 are:
>
> Channel 1 sounds D3 + 185 = LucyTuned D
> Channel 2 sounds B2 + 923 = LucyTuned *error* = Bb *This seems to > be where the typo occurred, as it should be sounding from midinote > A#; instead of B natural.*
> Channel 3 sounds G#3 + 1292 = LucyTuned Ab
> Channel 4 sounds F3 + 738 = LucyTuned F
>
> So the chord sounded is wrong!
> and should be:
>
> D
> Bb
> Ab
> F
>
> which would give us the chord of Bb7.
>
> I have replaced the midifile so that it should sound right now for > future visitors.
>
> Thanks for the error report.
>
> You tunaniks are really strict taskmasters;-)
>
> but this one took you ten years to find!
>
>
>
>
>
>
>
> On 10 Feb 2009, at 00:20, caleb morgan wrote:
>
>>
>> I think it's just a typo:
>>
>> It should be B natural, which makes it a diminished chord, maybe?
>>
>>
>>
>> On Feb 9, 2009, at 7:18 PM, Daniel Forro wrote:
>>
>>> As far as I know there are more ways how to write chord signs,
>>> unfortunately not only one standard. The worse one is that one using
>>> "b" and "#" for alterations, not context independent "-" and "+".
>>>
>>> Daniel Forro
>>>
>>> On 10 Feb 2009, at 8:55 AM, caleb morgan wrote:
>>>
>>> >
>>> > http://www.lucytune.com/midi/bb7.mid
>>> >
>>> > You must be using non-standard chord-names, as this is clearly not
>>> > a standard Bb7 (dominant-7th) chord,
>>> >
>>> > but on the other hand, many of the chords are close to standard, >>> so
>>> > this seems confusing.
>>> >
>>> > ?
>>> >
>>> > caleb
>>>
>>
>>
>
> Charles Lucy
> lucy@...
>
> - Promoting global harmony through LucyTuning -
>
> for information on LucyTuning go to:
> http://www.lucytune.com
>
> For LucyTuned Lullabies go to:
> http://www.lullabies.co.uk
>
>
>
>
>
Charles Lucy
lucy@...

- Promoting global harmony through LucyTuning -

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

For LucyTuned Lullabies go to:
http://www.lullabies.co.uk

djtrancendance wrote:
> Virtually all tunings here seem to be based on the concept of
> maintaining JI intervals and general low-numbered fractional ratios
> (IE 3/2, 5/6, 12/11, etc.
> In general this produces the most consonance and allows tunings to
> be made which can use musical theory and chords relatively similar to
> those used in 12TET.
> **********************************************************
> However, for example, the Golden Ratio Tuning, does not follow this
> convention. I am also starting to wonder to what degree MOS scales
> follow the convention.
> And 10TET, for example, has a circle of thirds as opposed to a
> circle of fifths. And, as I understand it, Lucy Tuning uses a circle
> of 5ths in one direction and a circle of 4ths in the other.
> ************************************************************
> > So, what tunings and scales would you recommend to research...to
> study the concept of using "far-from-JI" ratios to create a relatively
> good degree of consonance?

There are a few of what Paul Erlich calls "exotemperaments", which are so wildly tempered that they're unlikely to be confused with any attempt to approximate JI. The two main ones that seem to have musical potential are father and bug.

Father temperament has a 5-note MOS with the opposite step size pattern of pentatonic: L-L-s-L-s instead of s-s-L-s-L. The generators can vary quite a bit, since precision isn't critical with exotemperaments, but the TOP tuning listed in Paul's "Middle Path" paper has generators of 1185.9 and 447.4 cents. Similar to the 5-note MOS in 13-ET that Margo Schulter has used.

Bug temperament seems to me a more musical tuning, with generators of 1200.0 and 260.3 cents. It has a 9-note MOS, a subset of which coincides very closely with a tuning described in the Scala archive as pelog_pb.scl

"Primitive" Pelog, step of blown semi-fourths, von Hornbostel, type b.

So the 9-note MOS of bug temperament allows you to take pelog-type melodies and harmonies, and transpose them to different keys. Quite a useful feature. It doesn't exactly match any pelog that I've heard, but no two pelog tunings are really the same in any case.