re-tuning methods (was Re: odeion1-003)

On Sat, 13 Dec 2003 21:37:24 -0000 Robert wrote

>I see that really you have two notes to set there
>- the bridge note and the new tonic.

>So that is more of a challenge to the User Interface.
>I suppose you could do it by saying that
>the bridge note is to be the current note played
>in a melody line, or a best fit for the notes currently in play
>for a chord,and user just changes the tonic. Come to think
>of it that is probably what Peter is describing.

>That sounds like a far more natural system
>to play with. User doesn't need to bother
>about the bridge notes, so the music always
>dovetails about at least one note, and just set the
>tonic to change to.

>So if piece moves to a perceived new key,
>user can simply set the
>new tonic to that key. Or to the roots
>of the chords perhaps in j.i. where you can't
>even keep triads nicely tuned within a single
>diatonic key.

Yes, that is exactly what I was talking about,
( and have implemented ).

>Maybe one wants to do a bridge somewhere else
>other than the tonic.

>E.g. set E = 5/4 to the 3/2 of the new scale.
>I.e. change the key to A major, with E as the
>bridge note.

That would give more flexibility but I couldn't
see a convenient way of doing it.

Peter
www.midicode.com

On Sat, 13 Dec 2003 19:53:30 -0800 Kurt wrote

>>> But just to pose the question to you again, with a specific example. Lets
>>> say you have a 12-tone scale based on the harmonic series. Carl gave me
>>> this one, which is quite useful, perhaps in some sense optimal:
>>
>>> 16:17:18:19:20:21:22:24:26:27:28:30
>>
>>> And suppose you have a 3:4:5 (12:16:20) chord at G-C-E and you want the E to
>>> remain at fixed pitch as you hold the same 3 keys and you want the chord to
>>> become an 11:15:19, with the bottom 2 notes retuning to create this. (I
>>> hope I got that right.)
>>
>> I have tried to analyse this and can not see what your new tuning would be.
>> Can you give a full set of ratios for your scale after modulation or tell
>> me the
>> new key note, please?

>Yes, I gave the absolute minimum information there.

>The scale before and after modulation are the same scale with the 16..30
>ratios shown above. The 16 corresponds to the "tonic" and the tonic is
>shifted to a new position. In this case the tonic is shifted one note down
>from C to B. Lets measure frequencies in units contrived to make the
>frequency of C equal to 1 in the first chord.

>Then the G-C-E 3:4:5 (or equally 12:16:20) chord is initially:

> 3/4 : 1 : 5/4

>and after the modulation the G-C-E chord is, because I specified that it
>have the ratios 11:15:19 and that the top note remain at constant pitch:

> (5/4)*(11/19) : (5/4)*(15/19) : 5/4

>which makes the calculations explicit but can be simplified to

> 55/76 : 75/76 : 5/4

OK, I think I got it, but it looks to me like you have gone up
to C# not down to B.

So you could achieve the re-tuning you require using the dynamic
re-tuning of Midicode Synth and hitting a new key note of C#,
in this particular example.

( But it took me half an hour with an Excel spread sheet to
figure that out ! )

Interesting scale, I have an equivalent in Midicode Synth as
Wendy Carlos Harmonic.

Peter
www.midicode.com

On Sun, 14 Dec 2003 07:30:13 +0100 Werner wrote

>
>So in this example the notes of your pivot chord do not move but other
>scale
> degrees are now tuned to the new tonic of G.

> But what are you doing at the following chord sequence in C major:
> C-E-G, F-A-C, D-F-A, G-B-D-(F), C-E-G..

> At the point D-F-A the fifth D-A shows a "Wolf". If yo accept this, the
>idea of "just intonation" has a break.
> I f you change the scale at this point to a D Scale, you will climb 22
>Cents deeper and if you repeat
> this sequence 4 times, you will end in B major.
> Changing the key without any modulation...

> Werner Mohrlok

Yes, that is the basic flaw of dynamic re-tuning which I alluded to
in my original post when I said :

Paul Erlich and others have pointed out that a problem with this
approach is that it leads to a shift of absolute pitch with each
modulation which is difficult to reverse.

Peter
www.midicode.com

On Sat, 13 Dec 2003 20:11:16 -0800 Kurt wrote

>>> There's a problem in that too. The modulation bridge note
>>> (or 'pivot') need not be the tonic of the new key. So if you do not
>>> retune relative to the bridge note then you introduce an unwanted
>>> dissonance into the melody immediately following the pivot.
>>
>> For Just Intonation I think that the relation of the new key note to
>> the bridge note would ( usually ) be such that dissonance does not
>> occur. Can you give me an example in which dissonance would
>> occur, please?

>The example scale you considered as a 12-tone "Just Intonation" scale is
>exactly the kind of scale that will minimize some problems, especially by
>making it possible to retain tunings of more notes when cycle-of-5th
>modulations are done.

>However, what I have been recently interested in is using the harmonic
>scale:

> 16:17:18:19:20:21:22:24:26:27:28:30

>and doing things like modulating by major and minor thirds. From XMW (with
>some conceptual modification) came the the idea that the current state of
>the tuning system includes a scale, a reference midi note #, and a reference
>pitch.

Yes, that is what I have in Midicode Synth for an individual tuning, also
octave tuning in case you want to stretch or squash it. Plus an overall
note offset ( to provide transposition ) and a tuning adjust slider for
overall pitch.

>In my current way of doing things I think of a modulation as consisting of
>two aspects mathematically:

> adding an offset (positive or negative) to the reference midi note #

> this represents the movement of the tonic in MIDI note # space

> multiplying a reference pitch (frequency actually) by a ratio

> this represents the movement of the tonic in frequency space

That is what Midicode Synth dynamic re-tuning does.

>(Personally I prefer to call this pitch but represent it in Hz. Others
>object to that.)

>Currently I am working with the assumption that all playing notes are
>retuned to the new scale. XMW also allow the option that playing notes are
>not retuned, but retain their previous pitch.

This is an issue which Rick McGowan raised with me off-list.
Based on Rick's suggestion the current version of my software
provides an option for whether sounding notes are re-tuned or
just subsequent notes.

>Without returning,
>dissonances can occur. But even with returning, if you are not careful what
>you are doing musically, dissonances can occur. There are plenty of
>dissonant-sounding triads you can pull out of the harmonic scale above. In
>fact the very example I gave for the resultant chord 11:15:19 might not
>sound that consonant.

It certainly doesn't sound that consonant to me! But it is a very
good example numerically which is what you raised it for.

>Another rather rushed attempt at clarifying. I hope it helps.

>-Kurt

Indeed it does.

Peter
www.midicode.com

On Sat, 13 Dec 2003 17:45:01 -0800 Kurt wrote

>> Yes, my program accepts midi streams. I will consider providing
>> programmatic access next time I update it but addressing by
>> frequency rather than midi note number is probably too big a
>> change.

>There are also other problems with such a change in model. Not necessarily
>problems, but possibly. The MIDI note number also serves as a unique
>identifier for a playing note. This is important for keyboard playing
>purposes (that is when an actual keyboard isused for input) when retuning
>can happen due to some protocol "on the side" which controls modulation.
>Playing notes may or may not be retuned dynamically during modulations. In
>any case the frequency does not uniquely identify the playing note in this
>case, in a way that is helpful when keeping track of notes on and off. For
>example if a modulation occurs and you don't keep track of the note number
>*somewhere*, you will be unable to turn off the note playing at the previous
>frequency.

>In my current implementation of XMW I keep track of both note numbers and
>frequency. Note numbers flow fairly transparently through the system and
>frequencies change rather dynamically. But the note numbers keep everything
>from going awry.

>-Kurt

Absolutely. There are several structures in my software keyed by midi note
number. It is fundamental. That is why I will probably not make the change.

Peter.
www.midicode.com

-----Urspr�ngliche Nachricht-----
Von: Peter Frazer [mailto:paf@easynet.co.uk]
Gesendet: Dienstag, 16. Dezember 2003 16:35
An: tuning@yahoogroups.com
Betreff: [tuning] re-tuning methods (was Re: odeion1-003)

On Sun, 14 Dec 2003 07:30:13 +0100 Werner wrote

>
>So in this example the notes of your pivot chord do not move but other
>scale
> degrees are now tuned to the new tonic of G.

> But what are you doing at the following chord sequence in C major:
> C-E-G, F-A-C, D-F-A, G-B-D-(F), C-E-G..

> At the point D-F-A the fifth D-A shows a "Wolf". If yo accept this, the
>idea of "just intonation" has a break.
> I f you change the scale at this point to a D Scale, you will climb 22
>Cents deeper and if you repeat
> this sequence 4 times, you will end in B major.
> Changing the key without any modulation...

> Werner Mohrlok

Yes, that is the basic flaw of dynamic re-tuning which I alluded to
in my original post when I said :

Paul Erlich and others have pointed out that a problem with this
approach is that it leads to a shift of absolute pitch with each
modulation which is difficult to reverse.

Peter
www.midicode.com

Yes - and this shows that this approach to just intonation is not a very
practical one. Indeed,
the problem shown above, caused by the syntonic comma and some others,
caused
by other commas, show that it is impossible to create a closen system of
just intonation
without any break.
If you feel interest to know how different ideas of just intonation handle
this problem
you will find examples on our websites www.hermode.com at the end of the
"historical"
chapter, at "software driven tunings".
We present four different ideas in a just intonation model by a musical
example.
The third one (I called it "pulk idea") is the same than with the model
described above,
the forth one is mine (Hermode Tuning), a more compromise model.

Kind regards
Werner

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>

Hello Peter and Paul!
The d-f-a chord is a great and very musical useful chord. Trhe objection to it ereminds me of
the dreaded tritone 400 years ago. As music ventured out, it began to realized how important the
tritone is chords was in helping to 'define' where one is in a scale this wolf can and does this
very thing. In fact by the time of Wagner chords with tritone practally out numbered those
without. As Carl jung pointed outm there is not life without tension. the absence is death

>
> From: Peter Frazer <paf@easynet.co.uk>
> Subject:
>
>
> > At the point D-F-A the fifth D-A shows a "Wolf". If yo accept this, the
> >idea of "just intonation" has a break.
> > I f you change the scale at this point to a D Scale, you will climb 22
> >Cents deeper and if you repeat
> > this sequence 4 times, you will end in B major.
> > Changing the key without any modulation...
>
> > Werner Mohrlok
>
> Yes, that is the basic flaw of dynamic re-tuning which I alluded to
> in my original post when I said :
>
> Paul Erlich and others have pointed out that a problem with this
> approach is that it leads to a shift of absolute pitch with each
> modulation which is difficult to reverse.
>
>

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

--- In tuning@yahoogroups.com, "Werner Mohrlok" <wmohrlok@h...> wrote:
>
> -----Ursprüngliche Nachricht-----
> Von: Peter Frazer [mailto:paf@e...]
> Gesendet: Dienstag, 16. Dezember 2003 16:35
> An: tuning@yahoogroups.com
> Betreff: [tuning] re-tuning methods (was Re: odeion1-003)
>
>
> On Sun, 14 Dec 2003 07:30:13 +0100 Werner wrote
>
> >
> >So in this example the notes of your pivot chord do not move but
other
> >scale
> > degrees are now tuned to the new tonic of G.
>
> > But what are you doing at the following chord sequence in C
major:
> > C-E-G, F-A-C, D-F-A, G-B-D-(F), C-E-G..
>
> > At the point D-F-A the fifth D-A shows a "Wolf". If yo accept
this, the
> >idea of "just intonation" has a break.
> > I f you change the scale at this point to a D Scale, you will
climb 22
> >Cents deeper and if you repeat
> > this sequence 4 times, you will end in B major.
> > Changing the key without any modulation...
>
>
> > Werner Mohrlok
>
> Yes, that is the basic flaw of dynamic re-tuning which I alluded
to
> in my original post when I said :
>
> Paul Erlich and others have pointed out that a problem with this
> approach is that it leads to a shift of absolute pitch with each
> modulation which is difficult to reverse.
>
>
> Peter
> www.midicode.com
>
> Yes - and this shows that this approach to just intonation is not
a very
> practical one. Indeed,
> the problem shown above, caused by the syntonic comma and some
others,
> caused
> by other commas, show that it is impossible to create a closen
system of
> just intonation
> without any break.

Not impossible at all. I invented a technique for doing precisely
this some 15 years ago. I will post some samples - just as soon as I
have my old 486 DOS computer set up again.

-Peter

> If you feel interest to know how different ideas of just
intonation handle
> this problem
> you will find examples on our websites www.hermode.com at the end
of the
> "historical"
> chapter, at "software driven tunings".
> We present four different ideas in a just intonation model by a
musical
> example.
> The third one (I called it "pulk idea") is the same than with the
model
> described above,
> the forth one is mine (Hermode Tuning), a more compromise model.
>
> Kind regards
> Werner

-----Urspr�ngliche Nachricht-----
Von: sault@cyberware.co.uk [mailto:sault@cyberware.co.uk]
Gesendet: Mittwoch, 17. Dezember 2003 13:48
An: tuning@yahoogroups.com
Betreff: [tuning] re-tuning methods (was Re: odeion1-003)

--- In tuning@yahoogroups.com, "Werner Mohrlok" <wmohrlok@h...> wrote:
>
> -----Urspr�ngliche Nachricht-----
> Von: Peter Frazer [mailto:paf@e...]
> Gesendet: Dienstag, 16. Dezember 2003 16:35
> An: tuning@yahoogroups.com
> Betreff: [tuning] re-tuning methods (was Re: odeion1-003)
>
>
> On Sun, 14 Dec 2003 07:30:13 +0100 Werner wrote
>
> >
> >So in this example the notes of your pivot chord do not move but
other
> >scale
> > degrees are now tuned to the new tonic of G.
>
> > But what are you doing at the following chord sequence in C
major:
> > C-E-G, F-A-C, D-F-A, G-B-D-(F), C-E-G..
>
> > At the point D-F-A the fifth D-A shows a "Wolf". If yo accept
this, the
> >idea of "just intonation" has a break.
> > I f you change the scale at this point to a D Scale, you will
climb 22
> >Cents deeper and if you repeat
> > this sequence 4 times, you will end in B major.
> > Changing the key without any modulation...
>
>
> > Werner Mohrlok
>
> Yes, that is the basic flaw of dynamic re-tuning which I alluded
to
> in my original post when I said :
>
> Paul Erlich and others have pointed out that a problem with this
> approach is that it leads to a shift of absolute pitch with each
> modulation which is difficult to reverse.
>
>
> Peter
> www.midicode.com
>
> Yes - and this shows that this approach to just intonation is not
a very
> practical one. Indeed,
> the problem shown above, caused by the syntonic comma and some
others,
> caused
> by other commas, show that it is impossible to create a closen
system of
> just intonation
> without any break.

Not impossible at all. I invented a technique for doing precisely
this some 15 years ago. I will post some samples - just as soon as I
have my old 486 DOS computer set up again.

-Peter

Peter:

It is impossible. And it is not a computer question, it is a
mathematic question.

I modify the example shown above. To clarify things we assume that
the complete sequence will be played legato:

C-E-G, F-A-C, D-F-A, G-B-D-F (the position of the F to the G
looks already dangerous...),
F-C-G: Crash!!!
You cannot tune the C to the G and simultaneous to the F as a
pure fifth.

Another example? Also legato:

E-G#, C-E-G#, C-Eb-Ab

best:
Werner

Hi, Peter,

on 12/16/03 6:44 AM, Peter Frazer <paf@easynet.co.uk> wrote:

> On Sat, 13 Dec 2003 19:53:30 -0800 Kurt wrote
>
>>>> But just to pose the question to you again, with a specific example. Lets
>>>> say you have a 12-tone scale based on the harmonic series. Carl gave me
>>>> this one, which is quite useful, perhaps in some sense optimal:
>>>
>>>> 16:17:18:19:20:21:22:24:26:27:28:30
>>>
>>>> And suppose you have a 3:4:5 (12:16:20) chord at G-C-E and you want
> the E to
>>>> remain at fixed pitch as you hold the same 3 keys and you want the
> chord to
>>>> become an 11:15:19, with the bottom 2 notes retuning to create this. (I
>>>> hope I got that right.)
>>>
>>> I have tried to analyse this and can not see what your new tuning would be.
>>> Can you give a full set of ratios for your scale after modulation or tell
>>> me the
>>> new key note, please?
>
>> Yes, I gave the absolute minimum information there.
>
>> The scale before and after modulation are the same scale with the 16..30
>> ratios shown above. The 16 corresponds to the "tonic" and the tonic is
>> shifted to a new position. In this case the tonic is shifted one note down
>> from C to B. Lets measure frequencies in units contrived to make the
>> frequency of C equal to 1 in the first chord.
>
>> Then the G-C-E 3:4:5 (or equally 12:16:20) chord is initially:
>
>> 3/4 : 1 : 5/4
>
>> and after the modulation the G-C-E chord is, because I specified that it
>> have the ratios 11:15:19 and that the top note remain at constant pitch:
>
>> (5/4)*(11/19) : (5/4)*(15/19) : 5/4
>
>> which makes the calculations explicit but can be simplified to
>
>> 55/76 : 75/76 : 5/4
>
> OK, I think I got it, but it looks to me like you have gone up
> to C# not down to B.

Yes, pretty much, though I might rather say that I brought B up to C
(because the 11:15:19 chord appeared at Gb-B-Eb prior to modulation). In
any case it was that thinking that was the source of my confusion. In the
truest description I brough Eb up to E.

> So you could achieve the re-tuning you require using the dynamic
> re-tuning of Midicode Synth and hitting a new key note of C#,
> in this particular example.

Really? I doubt that it is the E pitch that would remain constant then,
based on what you have written elsewhere. I understood your retuining
method to agree with what Robert Walker was doing, which in "xmw" terms is
equivalen to "crossfree" mode, in which the current note played on the
modulation channel determines the new tonic, and specifically that the new
tonic retains the pitch that that note had prior to modulation. That is,
the pitch of C# would remain constant rather than the pitch of E remaining
constant. As a result none of the pitches in the G-C-E chord would remain
constant. The result would be "close" but not at the same absolute pitch.
I gave up on trying to calculate the absolute result because of the
complexities of the fractions involved, but I think you can get the idea.
Otherwise let me know.

> ( But it took me half an hour with an Excel spread sheet to
> figure that out ! )

Yikes, you worked harder than I did.

-Kurt

> Interesting scale, I have an equivalent in Midicode Synth as
> Wendy Carlos Harmonic.
>
> Peter
> www.midicode.com

Peter,

on 12/16/03 7:15 AM, Peter Frazer <paf@easynet.co.uk> wrote:

> On Sat, 13 Dec 2003 20:11:16 -0800 Kurt wrote
>
>>>> There's a problem in that too. The modulation bridge note
>>>> (or 'pivot') need not be the tonic of the new key. So if you do not
>>>> retune relative to the bridge note then you introduce an unwanted
>>>> dissonance into the melody immediately following the pivot.
>>>
>>> For Just Intonation I think that the relation of the new key note to
>>> the bridge note would ( usually ) be such that dissonance does not
>>> occur. Can you give me an example in which dissonance would
>>> occur, please?
>
>> The example scale you considered as a 12-tone "Just Intonation" scale is
>> exactly the kind of scale that will minimize some problems, especially by
>> making it possible to retain tunings of more notes when cycle-of-5th
>> modulations are done.
>
>> However, what I have been recently interested in is using the harmonic
>> scale:
>
>> 16:17:18:19:20:21:22:24:26:27:28:30
>
>> and doing things like modulating by major and minor thirds. From XMW (with
>> some conceptual modification) came the the idea that the current state of
>> the tuning system includes a scale, a reference midi note #, and a reference
>> pitch.
>
> Yes, that is what I have in Midicode Synth for an individual tuning, also
> octave tuning in case you want to stretch or squash it. Plus an overall
> note offset ( to provide transposition ) and a tuning adjust slider for
> overall pitch.
>
>> In my current way of doing things I think of a modulation as consisting of
>> two aspects mathematically:
>
>> adding an offset (positive or negative) to the reference midi note #
>
>> this represents the movement of the tonic in MIDI note # space
>
>> multiplying a reference pitch (frequency actually) by a ratio
>
>> this represents the movement of the tonic in frequency space
>
> That is what Midicode Synth dynamic re-tuning does.
>
>> (Personally I prefer to call this pitch but represent it in Hz. Others
>> object to that.)
>
>> Currently I am working with the assumption that all playing notes are
>> retuned to the new scale. XMW also allow the option that playing notes are
>> not retuned, but retain their previous pitch.
>
> This is an issue which Rick McGowan raised with me off-list.
> Based on Rick's suggestion the current version of my software
> provides an option for whether sounding notes are re-tuned or
> just subsequent notes.

Yes, it also occurred to me that having an option to do the same for
*octaves* of the sounding notes might be desirable, in case the player adds
an octave. Of course, you could take the same argument in many other
directions. It is a question of what protocols end up being useful.

-Kurt

Peter, and Robert Walker also if you are reading, and Carl also,

on 12/16/03 1:54 AM, Peter Frazer <paf@easynet.co.uk> wrote:

> On Sat, 13 Dec 2003 21:37:24 -0000 Robert wrote
>
>> I see that really you have two notes to set there
>> - the bridge note and the new tonic.
>
>> So that is more of a challenge to the User Interface.
>> I suppose you could do it by saying that
>> the bridge note is to be the current note played
>> in a melody line, or a best fit for the notes currently in play
>> for a chord,and user just changes the tonic. Come to think
>> of it that is probably what Peter is describing.
>
>> That sounds like a far more natural system
>> to play with. User doesn't need to bother
>> about the bridge notes, so the music always
>> dovetails about at least one note, and just set the
>> tonic to change to.
>
>> So if piece moves to a perceived new key,
>> user can simply set the
>> new tonic to that key. Or to the roots
>> of the chords perhaps in j.i. where you can't
>> even keep triads nicely tuned within a single
>> diatonic key.
>
> Yes, that is exactly what I was talking about,
> ( and have implemented ).
>
>> Maybe one wants to do a bridge somewhere else
>> other than the tonic.
>
>> E.g. set E = 5/4 to the 3/2 of the new scale.
>> I.e. change the key to A major, with E as the
>> bridge note.
>
> That would give more flexibility but I couldn't
> see a convenient way of doing it.

Convenience will always be relative to a given set of musical needs, also
given the limitations of what 2 hands and 2 feet can do. At the moment it
is very interesting to me to look for protocols that offer more retuning
choices in the context of an instrument with 2-octave or greater organ
pedalboard, beause this is what I have. On a one octave pedalboard I could
not differentiate upward movements from downward movements, for example.
With 2.5 octaves that becomes a possibility, which remains useful in most
circumstances given the always-available option to substitute octaves in
advance of a modulation such that the desired upward/downward movement
direction is available.

The current questions for me are how to make best use of:

differentiating upward/downward movements on the pedalboard

detection of overlapping versus non-overlapping depression of
consecutive pedals

possible use of "tapping" (stacatto) pedal presses
(obviously such would rule out actual use of stacatto!)

support for modes which impose other constraints or models
that are in keeping with a particularly compositional purposes
For example, one way to keep pitches from going astray
would be to map each modulation onto the shortest sequence
of circle-of-5th modulations required to get to that note.
This might work well for certain scales, within limits,
but I have not yet tried it.
Nonetheless it is an example of a model that could be imposed.

-Kurt

>
> Peter
> www.midicode.com

-----Urspr�ngliche Nachricht-----
Von: Kurt Bigler [mailto:kkb@breathsense.com]
Gesendet: Donnerstag, 18. Dezember 2003 03:10
An: tuning@yahoogroups.com
Betreff: Re: [tuning] re-tuning methods (was Re: odeion1-003)

Hi, Peter,

on 12/16/03 6:44 AM, Peter Frazer <paf@easynet.co.uk> wrote:

> On Sat, 13 Dec 2003 19:53:30 -0800 Kurt wrote
>
>>>> But just to pose the question to you again, with a specific example.
Lets
>>>> say you have a 12-tone scale based on the harmonic series. Carl gave
me
>>>> this one, which is quite useful, perhaps in some sense optimal:
>>>
>>>> 16:17:18:19:20:21:22:24:26:27:28:30
>>>
>>>> And suppose you have a 3:4:5 (12:16:20) chord at G-C-E and you want
> the E to
>>>> remain at fixed pitch as you hold the same 3 keys and you want the
> chord to
>>>> become an 11:15:19, with the bottom 2 notes retuning to create this.
(I
>>>> hope I got that right.)
>>>
>>> I have tried to analyse this and can not see what your new tuning
would be.
>>> Can you give a full set of ratios for your scale after modulation or
tell
>>> me the
>>> new key note, please?
>
>> Yes, I gave the absolute minimum information there.
>
>> The scale before and after modulation are the same scale with the
16..30
>> ratios shown above. The 16 corresponds to the "tonic" and the tonic is
>> shifted to a new position. In this case the tonic is shifted one note
down
>> from C to B. Lets measure frequencies in units contrived to make the
>> frequency of C equal to 1 in the first chord.
>
>> Then the G-C-E 3:4:5 (or equally 12:16:20) chord is initially:
>
>> 3/4 : 1 : 5/4
>
>> and after the modulation the G-C-E chord is, because I specified that
it
>> have the ratios 11:15:19 and that the top note remain at constant
pitch:
>
>> (5/4)*(11/19) : (5/4)*(15/19) : 5/4
>
>> which makes the calculations explicit but can be simplified to
>
>> 55/76 : 75/76 : 5/4
>
> OK, I think I got it, but it looks to me like you have gone up
> to C# not down to B.

Yes, pretty much, though I might rather say that I brought B up to C
(because the 11:15:19 chord appeared at Gb-B-Eb prior to modulation). In
any case it was that thinking that was the source of my confusion. In the
truest description I brough Eb up to E.

> So you could achieve the re-tuning you require using the dynamic
> re-tuning of Midicode Synth and hitting a new key note of C#,
> in this particular example.

Really? I doubt that it is the E pitch that would remain constant then,
based on what you have written elsewhere. I understood your retuining
method to agree with what Robert Walker was doing, which in "xmw" terms is
equivalen to "crossfree" mode, in which the current note played on the
modulation channel determines the new tonic, and specifically that the new
tonic retains the pitch that that note had prior to modulation. That is,
the pitch of C# would remain constant rather than the pitch of E remaining
constant. As a result none of the pitches in the G-C-E chord would remain
constant. The result would be "close" but not at the same absolute pitch.
I gave up on trying to calculate the absolute result because of the
complexities of the fractions involved, but I think you can get the idea.
Otherwise let me know.

> ( But it took me half an hour with an Excel spread sheet to
> figure that out ! )

Yikes, you worked harder than I did.

-Kurt

> Interesting scale, I have an equivalent in Midicode Synth as
> Wendy Carlos Harmonic.
>
> Peter
> www.midicode.com

It seems that a lot of us have such tools. I also - and one of the group
of scales is exactly the same as shown above. I also can call the changes
"gliding" as it is interesting to follow the shift of combination tones.
I wrote these scales as I assumed that by changing the scale the "root"
(this means the note with the frequency "16") could become recognized
(even when it actually would not sound) as a result of the combination
tones.
Indeed, one can. I know some persons who are perfect in such
hearing.
But unfortunately, I cannot. Or at least not in every case.

Werner