Yahoo Tuning Groups Ultimate Backup tuning A guitar for Justin White's scale (was: re:microtemperament)

Perhaps Paul Erlich would take the omnitetrachordal MOS that he suggested
in schismic temperament and turn it back into a rational tuning (a
periodicity chunk, if not a periodicity block, as in block =
parallelipiped) that differs from Justin's original scale in as few notes
as possible. Then if that scale is acceptable to Justin, I could see if
that one comes out any better in 9-limit kleismic (or whatever).

I gave the open string tuning and fretting in cents in
/tuning/topicId_27317.html#27329

Here's how the note-names map to the fingerboard.
approx position of
7/4 9/8 45/32 7/4 9/8 45/32 nut 12-tET frets

16/9 16/9 1fr
9/5 9/5 2fr
7/6 35/24 7/6 35/24 3fr
189/128 189/128 4fr
32/27 32/27 5fr 1fr
15/8 6/5 3/2 15/8 6/5 3/2 6fr
243/128 243/128 7fr

5/4 25/16 5/4 25/16 8fr
63/32 81/64 405/256 63/32 81/64 405/256 9fr 2fr

1/1 1/1 10fr

21/16 105/64 21/16 105/64 11fr

25/24 4/3 5/3 25/24 4/3 5/3 12fr 3fr
135/128 27/20 27/16 135/128 27/20 27/16 13fr

35/32 45/32 7/4 35/32 45/32 7/4 14fr
4fr
10/9 16/9 10/9 16/9 15fr
9/8 9/5 9/8 9/5 16fr

7/6 3/2 15/8 7/6 3/2 15/8 17fr 5fr
243/128 243/128 18fr

405/256 63/32 405/256 63/32 19fr
6fr
5/4 1/1 5/4 1/1 20fr

5/3 25/24 5/3 25/24 21fr
21/16 27/16 135/128 21/16 27/16 135/128 22fr 7fr

7/4 35/32 7/4 35/32 23fr

16/9 10/9 16/9 10/9 24fr 8fr
45/32 9/5 9/8 45/32 9/5 9/8 25fr

35/24 15/8 7/6 35/24 15/8 7/6 26fr
9fr
32/27 32/27 27fr
3/2 6/5 3/2 6/5 28fr

25/16 1/1 5/4 25/16 1/1 5/4 29fr 10fr
405/256 81/64 405/256 81/64 30fr

105/64 135/128 21/16 105/64 135/128 21/16 31fr
11fr
5/3 4/3 5/3 4/3 32fr
27/16 27/20 27/16 27/20 33fr

7/4 9/8 45/32 7/4 9/8 45/32 34fr 12fr 8ve

-- Dave Keenan
Brisbane, Australia
http://dkeenan.com

🔗Dave Keenan <D.KEENAN@UQ.NET.AU>

🔗David C Keenan <D.KEENAN@UQ.NET.AU>

Justin,

Instead of the rationalisation that Paul posted for the true MOS scale, I
suggest you consider one where:
35/32 is replaced by 16/15
35/24 is replaced by 64/45
105/64 is replaced by 8/5
and the choice between 25/18 and 27/20 still remains.

This is the same scale that Paul posted, just a rotation so you don't lose
note names 6/5 and 9/5.

A schismic guitar could retain standard open string tuning. We'd only need
to find the optimal scale rotation for fretting.

Here are the cents for your original tuning when it is 9-limit schismic
tempered (4.7c errors). The modifications required to make it MOS are in
parenthesis:

0.0
63.8
89.4
153.2 (114.9)
178.7
204.3
268.1
293.6
319.1
383.0
408.5
472.3
497.9
523.4 or 561.7
587.2
651.1 (612.8)
676.6
702.1
766.0
791.5
855.3 (817.0)
880.9
906.4
970.2
995.7
1021.3
1085.1
1110.6
1174.5
1200.0

Here are the cents for your original scale when 9-limit kleismic tempered
(3.0c errors) (as per the guitar design I posted). The cents I posted
earlier were for the guitar _fretting_, not the scale (different rotation
and 5 extra frets).

0.0
67.0
85.1
152.1 (115.8)
182.8
200.9
267.9
298.6
316.7
383.7
401.9
468.8
499.5
517.7 or 566.5
584.6
651.6 (615.4)
669.8
700.5
767.4
785.6
852.6 (816.3)
883.3
901.4
968.4
999.1
1017.2
1084.2
1102.3
1169.3
1200.0

So you've got 4 tunings to choose from there. We could also do a Miracle
tempered version which would have errors between the two above, and would
give you some usable ratios of 11.

-- Dave Keenan
Brisbane, Australia
http://dkeenan.com

🔗Paul Erlich <paul@stretch-music.com>

🔗Dave Keenan <D.KEENAN@UQ.NET.AU>

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., David C Keenan <D.KEENAN@U...> wrote:
> > Then if that scale is acceptable to Justin, I could see if
> > that one comes out any better in 9-limit kleismic (or whatever).
>
> ??? I still think you're missing the point of periodicity blocks and
> their relationship to temperament.

No. I think I understood that quite well, some time ago. And I am very
grateful to you for that. But it simply isn't the point at issue here.

Justin asked for a fingerboard design for his _just_ scale. He didn't
particularly ask for it to be tempered. Nor did he ask for it to be
modified to become MOS or omnitetrachordal.

But if a practical fingerboard design can be made for it, by using a
microtemperament (i.e. one whose errors are small enough that it still
sounds just in most circumstances) then he may be willing to wear it.

At least that's my take.

Justin?

-- Dave Keenan

🔗Justin White <justin.white@davidjones.com.au>

--- In tuning@y..., David C Keenan <D.KEENAN@U...> wrote:

<No. I think I understood that quite well, some time ago. And I am very
grateful to you for that. But it simply isn't the point at issue here.

Justin asked for a fingerboard design for his _just_ scale. He didn't
particularly ask for it to be tempered. Nor did he ask for it to be
modified to become MOS or omnitetrachordal.

At least that's my take.

Justin?>

Well that was originally what I wanted but I am still going to make a strictly
just version of my scale so it would seem to be doubling up. I am now not going
to use the fourth interchangeable fretboard for the tempered scale but I will
intead use all the spare parts I have to make a new guitar. With this guitar I
have will have a tremelo arm so I am happy to go into tempering more. But I
would like it to be as close as possible to just without throwing away
thebenefits tempering can provide.

Now the main purpose of this guitar is for live playing and for trying out
parallel modulations not possible on the strictly just fretboards,

I still would like to use the scale I presented as a basic template, but I
welcome improvements that will make it MOS. I dont know what omnitetrachordality
is. But I like tetrachords as a basic building block for scales and I would like
to include them.

I would be nice to have no missing notes and frets that go stright across with
no gaps.

Sorry for any misunderstandings.
Justin White

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@y..., "Justin White" <justin.white@d...> wrote:
>
> I still would like to use the scale I presented as a basic
template, but I
> welcome improvements that will make it MOS.

I'll try to work with both versions in parallel.

> I dont know what omnitetrachordality
> is.

It means that every octave species of the scale contains two
identical 4:3 spans. I showed that your scale was tetrachordal in
many octave species but not all of them. I believe the MOS version is
omnitetrachordal.

> But I like tetrachords as a basic building block for scales and I
would like
> to include them.
>
> I would be nice to have no missing notes and frets that go stright
across with
> no gaps.

I'm suggesting a 41-fret-per-octave guitar, with no frets closer than
26 cents. Your original scale, as well as the MOS scale, will be
available on every string. You will be able to transpose your scale
to many different keys, and also the fact that there are only two
step sizes between frets, and that the pattern is very repetitive,
will make it very easy to find any chord "shape" you discover,
anywhere on the fingerboard. All fingerings will be fairly familiar,
as the open strings will be tuned roughly in standard tuning. You'll
really be able to get around on this guitar.

None of the other proposals by Dave Keenan will have these properties.

🔗Justin White <justin.white@davidjones.com.au>

--- In tuning@y..., "Justin White" <justin.white@d...> wrote:
>>
>> I still would like to use the scale I presented as a basic
template, but I
>> welcome improvements that will make it MOS.

>I'll try to work with both versions in parallel.

>> I dont know what omnitetrachordality
>> is.

--- In tuning@y..., "Paul Erlich" wrote:

>It means that every octave species of the scale contains two
>identical 4:3 spans. I showed that your scale was tetrachordal in
>many octave species but not all of them. I believe the MOS version is
>omnitetrachordal.

>> But I like tetrachords as a basic building block for scales and I
would like
>> to include them.
>>
>> I would be nice to have no missing notes and frets that go stright
across with
>> no gaps.

>I'm suggesting a 41-fret-per-octave guitar, with no frets closer than
>26 cents. Your original scale, as well as the MOS scale, will be
>available on every string. You will be able to transpose your scale
>to many different keys, and also the fact that there are only two
>step sizes between frets, and that the pattern is very repetitive,
>will make it very easy to find any chord "shape" you discover,
>anywhere on the fingerboard. All fingerings will be fairly familiar,
>as the open strings will be tuned roughly in standard tuning. You'll
>really be able to get around on this guitar.

This sounds exciting.

>None of the other proposals by Dave Keenan will have these properties.

The verdict of the first round of listening tests is that all the proposed
scales have almost identical basic melodic gestalts. My listening tests were
somewhat limited by the 12 note span of the pads on my soundmodule/drum machine
[Ensoniq ASR-X]. The results so far were that out of the three scales I tried
[my own, Daves proposal and Pauls proposal]. Dave's one had most variety with
lot's of near just intervals and slow beating. I quite liked this effect. Pauls
scale seemed good for modulation and very even which I liked also. My own scale
seemed very unasuming almost not there. Anyway it was not the greatest test.

But it has influenced my own theories on scales a bit. I think that every melody
whether just or tempered or whatever seems to fit best to a certain genetic blue
print and that the whole can be deduced somewhat from the part. Erv Wilson has
talked about this sort of idea and also I think Yasser had a similar theory re:
the expansion of tonal systems.

I am interested in reading his book now !

I'll use my keyboard and do it properly on the weekend.

Justin White

🔗Dave Keenan <D.KEENAN@UQ.NET.AU>

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> None of the other proposals by Dave Keenan will have these
properties.

And none of the other proposals (so far) by Dave Keenan have errors as
large as 4.3 cents, or as many as 41 frets for a 29 note scale.

Here's my own fretting proposal for the 29 note 9-limit schismic MOS
(with 4.3c errors). I gave the cents for the scale earlier, but not
the fretting for the guitar. It uses EADGBE tuning. It has 30 frets
and no missing notes. The full scale is available on the two E strings
and the A string. If you accept 27/20 ins place of of 25/18 then that
scale is available in full on the E strings and the B string. It would
only require 33 frets to have the full scale on every string. No frets
are closer than 26 cents.

fret
tuning step
(cents) (cents)
------- -------
0.0 26.3
26.3 62.7
89.0 26.3
115.3 62.7
178.1 26.3
204.4 26.3
230.7 62.7
293.4 26.3
319.7 62.7
382.5 26.3
408.8 26.3
435.1 36.4 *
471.5 26.3
497.8 26.3
524.1 62.7
586.8 26.3
613.2 62.7
675.9 26.3
702.2 26.3
728.5 62.7
791.2 26.3
817.5 62.7
880.3 26.3
906.6 26.3
932.9 62.7
995.6 26.3
1021.9 62.7
1084.7 26.3
1111.0 62.7
1173.7 26.3
1200

-- Dave Keenan

🔗Dave Keenan <D.KEENAN@UQ.NET.AU>

--- In tuning@y..., "Justin White" <justin.white@d...> wrote:
> --- In tuning@y..., David C Keenan <D.KEENAN@U...> wrote:
> > Justin asked for a fingerboard design for his _just_ scale.
...
> >
> > But if a practical fingerboard design can be made for it, by using
a
> > microtemperament (i.e. one whose errors are small enough that it
still
> > sounds just in most circumstances) then he may be willing to wear
it.
>
> Well that was originally what I wanted but I am still going to make
a strictly
> just version of my scale so it would seem to be doubling up.

Yes Justin. If you're still planning to do a fingerboard directly from
the rational definition of the scale, then I have failed. I'd really
like to know where my proposal fell down, so I can do better next
time. Is it the non-standard open string tuning, or the minimum fret
spacing, or the errors too large (beating too fast)? Or some
combination of these?

How many frets or fretlets will you have with the rational fretting?

There are some other options for microtemperament. One has only 1.9 c
errors but will probably need a non-standard open tuning if it is to
minimise the number of frets.

The other will use standard open tuning and only 0.4 c errors but it's
not likely to reduce the number of frets much below that of the
rational fretting (if at all).

-- Dave Keenan

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> > None of the other proposals by Dave Keenan will have these
> properties.
>
> And none of the other proposals (so far) by Dave Keenan have errors
as
> large as 4.3 cents,

Easily bent away.

> or as many as 41 frets for a 29 note scale.

When the pattern is so simple (there are always 12 large spaces),
then I don't see that the number of frets should be a complicating
factor, as long as the smallest step size remains the same.

> Here's my own fretting proposal for the 29 note 9-limit schismic
MOS
> (with 4.3c errors). I gave the cents for the scale earlier, but not
> the fretting for the guitar. It uses EADGBE tuning. It has 30 frets
> and no missing notes. The full scale is available on the two E
strings
> and the A string. If you accept 27/20 ins place of of 25/18 then
that
> scale is available in full on the E strings and the B string. It
would
> only require 33 frets to have the full scale on every string.

33 out of my 41, with the same smallest and second-smallest steps.

Why not put down all 41?

And why the keys of E and B? Justin, what key do you think you'd like
to have 1/1 on most of the time? With 41 frets per octave, you'll be
able to have 1/1 on 9 or 10 different root-notes in a chain of fifths.

🔗Paul Erlich <paul@stretch-music.com>

🔗Justin White <justin.white@davidjones.com.au>

In Gui tar Wars --- "Paul Erlich" <paul@s...> wrote:

>33 out of my 41, with the same smallest and second-smallest steps.

>Why not put down all 41?

>And why the keys of E and B? Justin, what key do you think you'd like
>to have 1/1 on most of the time? With 41 frets per octave, you'll be
>able to have 1/1 on 9 or 10 different root-notes in a chain of fifths.

I sing best in the keys of G and A and also D and C. These are the most
important keys for me.

Also I'm not that hung up on standard tuning if it makes designs easier. If
standard tuning is the optimum fine. But I only need the open strings to be
tuned in fourths or thirds for it to be easy to play.

Also you mentioned planar temperament. Could this be visualised as a horizontal
ribbon that meets a the ends while a normal linear temperament would look like a
doughnut ?

Justin White

🔗Justin White <justin.white@davidjones.com.au>

Dave wrote:

--- In tuning@y..., "Justin White" <justin.white@d...> wrote:
>> There was no problem with the scale. In a previous post I said that
I actually
>> enjoyed your scale most of all in my limited listening tests. My
main
>> consideration is that I will be using the strict JI guitar to play
along with
>> smaplers and synthesiser playing in just intonation. For this reason
I obviously
>> want have the same tuning [or as close as I can get] on my guitar.

>"or as close as I can get on my guitar". I'll come back to that later.

>>It was never
>> my intention to take your tuning over mine. I consider the extra
difficluty in
>> construction to be minor point. I was more interested in taking that
scale as a
>> spring board to create something new in a tempered paradigm with all
the
>> attendant benefits of modulation straight frets.

>Then one of us is a little confused. You started this thread by
>writing:

Probably me. I thought that your method of temperament as well as making guitars
simpler to build provided many more consonant intervals and modulations.

I realise now that temperaments based on periodicty blocks seem to be more
suited to this. I thought that you did use periodicity blocks in your method. I
only realised when you demonstrated how it was done that you could microtemper
any JI scale and that it had nothing to do with whether the scale was a
periodicty block or not.

>"I am in fact interested in how Daves developments could be applied to
>a CS [periodiciy block ?] that I came up with last weekend."

>At the time, the thread was called "microtemperament", and I
>understood, "Dave's development" to mean the minimising of frets for
>JI guitars by using microtemperaments. i.e. temperaments
>indistinguishable from just in normal use.

>If you're willing to temper it enough so you can modulate it by fifths
>and you don't care how many frets, then certainly use 41 of 7-limit
>schismic as Paul suggests. But this has very little to do with "Dave's
>developments".

This was a misnomer. I look at yourself, Paul and Graham as temperament gurus of
a sort and merely wished to find out what could be done with my scale if
tempered, not nescessarily with your latest developments but in any way that
seemed natural.

Sorry if I have not communicated clearly, and for wasting your time.

One of the major barriers to tempering scales etc for me at least is my
mathematical ineptitude. Thats mostly why I would present scales rationally
rather than in cents. For me ratios represent audible experiences. Cents do not
so much. I only know the cents values approximatly or exactly for the basic
intervals.

>> >How many frets or fretlets will you have with the rational
fretting?
>>
>> I haven't worked it out yet. There will be a few I guess. It doesn't
bother me
>> though.

>Will they all be continuous? Smallest spacing? Let me know when you
>work it out.

Yep I will certainly show it to you before I go ahead.

>> >There are some other options for microtemperament. One has only 1.9
c
>> >errors but will probably need a non-standard open tuning if it is
to
>> >minimise the number of frets.
>
>> >The other will use standard open tuning and only 0.4 c errors but
it's
>> >not likely to reduce the number of frets much below that of the
>> >rational fretting (if at all).
>
>> These would all be good if I was not trying to match up to very
accurate synths.
>> Metasynth is very accurate and my ensonq has resolution of 0.37
cents which is
>> pretty good too.

>But Justin, even if you fret for the rational scale it would be some
>kind of miracle if even half the notes you play are within 3 cents of
>the intended tuning! It is ridiculous to think you could get anywhere
>near to matching the accuracy of these synths with a guitar, and
>completely unnecessary.

Well, what would make the kleismic scale fret any nearer? I would probably get
your scale as heard on my sampler if I went for the rational fretting, and get
something more inaccurate if I went for the kleismic scale.

I guess I agree with Paul that the main benefits of temperament are modulation,
a simplified way of mastering the tonal universe. For this benefit I can accept
temperament to this degree. But to make the guitar easier to build ? I would
rather put in the extra work and get as close as I can.

Also your data re. the 3 cents vagaries of guitar intonation are based on
guitars fretted to equal temperament. Guitar companies fret to the given formula
rather than match the tuning by ear or with electronic tuners. I will be tuning
the guitar by ear by eliminating beats not by fractions of string length etc.

Justin White

🔗Paul Erlich <paul@stretch-music.com>

🔗Dave Keenan <D.KEENAN@UQ.NET.AU>

--- In tuning@y..., "Justin White" <justin.white@d...> wrote:
> --- In tuning@y..., "Justin White" <justin.white@d...> wrote:
> I thought that your method of temperament as well as
making guitars
> simpler to build provided many more consonant intervals and
modulations.

Yes, these functions do blur into one another. If you accept larger
errors extra consonances and modulations become available.

> I look at yourself, Paul and Graham as temperament gurus of
> a sort and merely wished to find out what could be done with my
scale if
> tempered, not nescessarily with your latest developments but in any
way that
> seemed natural.

Ok. Paul is right, it's easy to just see the Paul/Dave/Graham
temperament powerhouse as a single entity. It has been a privelege to
be part of that.

> Sorry if I have not communicated clearly, and for wasting your time.

Ah. Don't worry about it. I think I was just being grumpy because you
liked Paul's better than mine. :-)

> One of the major barriers to tempering scales etc for me at least is
my
> mathematical ineptitude. Thats mostly why I would present scales
rationally
> rather than in cents. For me ratios represent audible experiences.
Cents do not
> so much. I only know the cents values approximatly or exactly for
the basic
> intervals.

Yes. I think that's why a lot of people stick to ratios, and that's
fair enough. But then too many "mistake the map for the territory", as
I feel you are still doing (explained more below).

> Yep I will certainly show it to you before I go ahead.

Please email me. I'm going off list for some months.

> >But Justin, even if you fret for the rational scale it would be
some
> >kind of miracle if even half the notes you play are within 3 cents
of
> >the intended tuning! It is ridiculous to think you could get
anywhere
> >near to matching the accuracy of these synths with a guitar, and
> >completely unnecessary.
>
> Well, what would make the kleismic scale fret any nearer?

Of course it wouldn't be any nearer, but the additional errors would
be insignificant. The kleismic (or other microtemperament) errors are
as likely to improve on the guitar errors as they are to make them
worse.

> I would probably get
> your scale as heard on my sampler if I went for the rational
fretting, and get
> something more inaccurate if I went for the kleismic scale.

The typical errors don't simply add. From statistical theory, crudely
used, if the guitar has typical errors (rms or standard deviation) of
3 cents and the tuning has typical errors (rms or standard deviation)
of 1 cent then the result will be a typical error of sqrt(3^2 + 1^2) =
3.2 cents.

So a microtemperament with 1c rms (prob 3c max) errors would only add
about 0.2c to the total error for the guitar.

> I guess I agree with Paul that the main benefits of temperament are
modulation,
> a simplified way of mastering the tonal universe. For this benefit I
can accept
> temperament to this degree. But to make the guitar easier to build ?
I would
> rather put in the extra work and get as close as I can.

Ok. But you might also consider the fact that using the same
microtemperament on the synthesizers would be a way of "humanising"
them.

> Also your data re. the 3 cents vagaries of guitar intonation are
based on
> guitars fretted to equal temperament. Guitar companies fret to the
given formula
> rather than match the tuning by ear or with electronic tuners. I
will be tuning
> the guitar by ear by eliminating beats not by fractions of string
length etc.

Do you mean that you will place fretlets individually for each string
in each position, and never change string guages or manufacturers? I
have assumed you will have a bridge that is individually adjustable
for each string.

That data on 3 cents vagaries, does not include those due to
accidental variations in finger pressure on the part of the player. I
understand that these are easily +-3 cents too, possibly +-6 cents.

Paul agreed that you can tune out microtemperament errors as well as
you can tune out guitar errors, by _deliberate_ variations in finger
pressure. But that takes skill, and even assuming the skill is
aquired, it takes _time_. i.e. the notes must be sustained long enough
to hear at least a cycle of the slowest beat you intend to cancel.

And Carl, this applies as much to FFT's as it does to human hearing.
The uncertainty principle says that your frequency resolution is the
inverse of the time over which you sample. You want 0.1 Hz resolution,
you need a 10 second sample. A 0.1 Hz change represents a 1 cent
change at 173 Hz, or a 0.1 c change at 1730 Hz, etc.

1730 Hz is kind of a magic number that can be worth remembering. It's
where a _small_ change in cents (say < 20) corresponds to about the
same number of hertz. Also, a 17.3 cent change is a 1% change in
frequency.

Regards,
-- Dave Keenan