New book on tuning and scales by Bruce R. Gilson

Sorry to cross-post this, I had intended it for this list, but
accidentally sent it to Make Micro Music first. I had thought it went
out yesterday, but apparently something happened and it didn't make it
to or through Yahoo.

Back in 1993, Bruce R. Gilson published an article in Xenharmonikon 15
titled "A Numerical Theory of Scale Invention." Recently Bruce
contacted me to let me know that he had expanded this article into a
book called "Construction of Musical Scales, A Mathematical Approach"
which is available from Createspace, a publish-on-demand site for
books, music and video, that is affiliated with Amazon Books. It may
be visited at the URL below: https://www.createspace.com/3353001.

I've been far too busy to write a formal review, but the Table of
Contents looks comprehensive, to say the least. A novel aspect of this
book is that it uses millioctaves rather than cents and contains a
very large table of intervals expressed in Monzos, ratios, decimals
and MO's as an appendix. As I recall, there was a recent discussion on
the advantages of MO's on this or the tuning math list recently. The
book also has a comprehensive index, making it quite easy to locate
topics quickly.

The mathematical level is rather elementary, being limited to ratios
and logarithms, and advanced techniques such as the predicate
calculus, matrices, and Grassman algebra that are to be found on the
Tuning Math list are avoided, so it should be readily accessible to
anyone who wishes to learn about tuning and scales.

I've appended the TOC below. Alas, it appears that his careful
formatting seems to have been mostly lost in going from PDF to email.

--John

Table of Contents
Chapter 1. Introduction: What this book is
about...................................................................................1
Chapter 2. What is a musical tone?
........................................................................................4
Chapter 3. Octaves and
fifths..................................................................................7
Chapter 4. Measuring
intervals...............................................................................9
Chapter 5. The circle of fifths; the Pythagorean
scale...................................................................................14
Chapter 6. Modes and
keys....................................................................................19
Chapter 7. Naming of
intervals...............................................................................21
Chapter 8. The (Pythagorean) chromatic
scale...................................................................................24
Chapter 9. Thirds and sixths; just
intonation..............................................................................28
Chapter 10. Properties of just-intonation
scales..................................................................................35
Chapter 11. Temperaments, part 1. Meantone
temperament.............................................................................37
Chapter 12. Temperaments, part 2. Twelve-tone equal
temperament.............................................................................41
Chapter 13. Some properties of regular
temperaments............................................................................44
Chapter 14. General properties of diatonic
scales..................................................................................46
Chapter 15. Generalizations of the just
scale...................................................................................48
Chapter 16. Other equal
temperaments............................................................................51
Chapter 17. Other scales extracted from chromatic
scales..................................................................................62
Chapter 18. General comments on regular
temperaments............................................................................65
Chapter 19. Non-octave-based
scales..................................................................................66
Appendix A. Binary logarithms for computation of millioctave
intervals...............................................................................67
Appendix B. Named musical
intervals...............................................................................68
Appendix C. Frequency ratios for various fractions, and millioctave
equivalents.............................................................................105
Appendix D.
Bibliography............................................................................140
Index...................................................................................141

correct link:
https://www.createspace.com/3353001

Thanks for that John.

To pick a nit:

IMHO, a more appropriate abbreviation or symbol for "millioctave"
would be _lowercase_ "mo", as an uppercase "M" suggests "Mega" and an
uppercase "O" suggests the unit is named after a Mr or Mrs Octave?
Also, in the plural case an "s" should not be added to the symbol, to
avoid the possibility of confusion with seconds (of time).

My pedantic best.
-- Dave Keenan

It's nice to see a decimal octave being applied, which to me makes
more sense than cents of a 12-edo semitone. I would personally
prefer duodecimal division of the octave, but since the principle's
the same either way, decimal is fine.

--- In tuning@yahoogroups.com, "John H. Chalmers" <JHCHALMERS@...>
wrote:
>
> Sorry to cross-post this, I had intended it for this list, but
> accidentally sent it to Make Micro Music first. I had thought it
went
> out yesterday, but apparently something happened and it didn't
make it
> to or through Yahoo.
>
> Back in 1993, Bruce R. Gilson published an article in
Xenharmonikon 15
> titled "A Numerical Theory of Scale Invention." Recently Bruce
> contacted me to let me know that he had expanded this article into
a
> book called "Construction of Musical Scales, A Mathematical
Approach"
> which is available from Createspace, a publish-on-demand site for
> books, music and video, that is affiliated with Amazon Books. It
may
> be visited at the URL below: https://www.createspace.com/3353001.
>
> I've been far too busy to write a formal review, but the Table of
> Contents looks comprehensive, to say the least. A novel aspect of
this
> book is that it uses millioctaves rather than cents and contains a
> very large table of intervals expressed in Monzos, ratios, decimals
> and MO's as an appendix. As I recall, there was a recent
discussion on
> the advantages of MO's on this or the tuning math list recently.
The
> book also has a comprehensive index, making it quite easy to locate
> topics quickly.
>
> The mathematical level is rather elementary, being limited to
ratios
> and logarithms, and advanced techniques such as the predicate
> calculus, matrices, and Grassman algebra that are to be found on
the
> Tuning Math list are avoided, so it should be readily accessible to
> anyone who wishes to learn about tuning and scales.
>
> I've appended the TOC below. Alas, it appears that his careful
> formatting seems to have been mostly lost in going from PDF to
email.
>
> --John
>
>
>
> Table of Contents
> Chapter 1. Introduction: What this book is
>
about................................................................
...................1
> Chapter 2. What is a musical tone?
> ...................................................................
.....................4
> Chapter 3. Octaves and
>
fifths...............................................................
...................7
> Chapter 4. Measuring
>
intervals............................................................
...................9
> Chapter 5. The circle of fifths; the Pythagorean
>
scale................................................................
...................14
> Chapter 6. Modes and
>
keys.................................................................
...................19
> Chapter 7. Naming of
>
intervals............................................................
...................21
> Chapter 8. The (Pythagorean) chromatic
>
scale................................................................
...................24
> Chapter 9. Thirds and sixths; just
>
intonation...........................................................
...................28
> Chapter 10. Properties of just-intonation
>
scales...............................................................
...................35
> Chapter 11. Temperaments, part 1. Meantone
>
temperament..........................................................
...................37
> Chapter 12. Temperaments, part 2. Twelve-tone equal
>
temperament..........................................................
...................41
> Chapter 13. Some properties of regular
>
temperaments.........................................................
...................44
> Chapter 14. General properties of diatonic
>
scales...............................................................
...................46
> Chapter 15. Generalizations of the just
>
scale................................................................
...................48
> Chapter 16. Other equal
>
temperaments.........................................................
...................51
> Chapter 17. Other scales extracted from chromatic
>
scales...............................................................
...................62
> Chapter 18. General comments on regular
>
temperaments.........................................................
...................65
> Chapter 19. Non-octave-based
>
scales...............................................................
...................66
> Appendix A. Binary logarithms for computation of millioctave
>
intervals............................................................
...................67
> Appendix B. Named musical
>
intervals............................................................
...................68
> Appendix C. Frequency ratios for various fractions, and millioctave
>
equivalents..........................................................
...................105
> Appendix D.
>
Bibliography.........................................................
...................140
>
Index................................................................
...................141
>

Tony wrote:

> I would personally
> prefer duodecimal division of the octave, but since the principle's
> the same either way, decimal is fine.

What do you mean by "duodecimal division"?

Petr

Using duodecimal numbering instead of decimal, where the octave still
equals one unit.

Decimal...

One octave equals:
10 decioctaves
100 centioctaves
1000 millioctaves

Duodecimal...

One octave equals:
10' (12) dezioctaves
100' (144) zentioctaves
1000' (1728) millioctaves

That way, with only 0'1 octave precision, you already have your basic
twelve-edo.

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
>
> Tony wrote:
>
> > I would personally
> > prefer duodecimal division of the octave, but since the principle's
> > the same either way, decimal is fine.
>
> What do you mean by "duodecimal division"?
>
> Petr
>

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@...> wrote:
>
> Thanks for that John.
>
> To pick a nit:
>
> IMHO, a more appropriate abbreviation or symbol for "millioctave"
> would be _lowercase_ "mo", as an uppercase "M" suggests "Mega" and an
> uppercase "O" suggests the unit is named after a Mr or Mrs Octave?
> Also, in the plural case an "s" should not be added to the symbol, to
> avoid the possibility of confusion with seconds (of time).
>
Actually, in the book I use "moc." -- which conforms to your desire to
use lower case, even if it doesn't agree exactly with your abbreviation.

--- In tuning@yahoogroups.com, "Tony" <leopold_plumtree@...> wrote:
>
> It's nice to see a decimal octave being applied, which to me makes
> more sense than cents of a 12-edo semitone. I would personally
> prefer duodecimal division of the octave, but since the principle's
> the same either way, decimal is fine.
>

I've been fighting this fight for at least 15 years, so I'm glad to see
that someone appreciates this. I _do_ mention duodecimal division in
the book in the chapter that discusses the various units that have been
proposed for measuring intervals.

--- In tuning@yahoogroups.com, "Bruce R. Gilson" <brgster@...> wrote:
>
> --- In tuning@yahoogroups.com, "Tony" <leopold_plumtree@> wrote:
> >
> > It's nice to see a decimal octave being applied, which to me makes
> > more sense than cents of a 12-edo semitone. I would personally
> > prefer duodecimal division of the octave, but since the principle's
> > the same either way, decimal is fine.
> >
>
> I've been fighting this fight for at least 15 years, so I'm glad to
see
> that someone appreciates this. I _do_ mention duodecimal division in
> the book in the chapter that discusses the various units that have
been
> proposed for measuring intervals.
>
From Robert. I am quite happy to promote 100 and 200tet but I am
unwilling to promote 1000tet. I am quite happy with 1200tet and I
am sure that all the keyboard companies agree with me.

--- In tuning@yahoogroups.com, "Bruce R. Gilson" <brgster@...> wrote:
>
> --- In tuning@yahoogroups.com, "Tony" <leopold_plumtree@> wrote:
> >
> > It's nice to see a decimal octave being applied, which to me makes
> > more sense than cents of a 12-edo semitone. I would personally
> > prefer duodecimal division of the octave, but since the principle's
> > the same either way, decimal is fine.
> >
>
> I've been fighting this fight for at least 15 years, so I'm glad to
see
> that someone appreciates this. I _do_ mention duodecimal division in
> the book in the chapter that discusses the various units that have
been
> proposed for measuring intervals.
>

I'm tempted to get a copy. I can understand why you'd have to use
decimal numeration for such a book, due to the limited use/familiarity
of duodecimal among most people.

For most 5-limit JI intervals, the nearest decimal millioctave actually
has less error than the nearest duodecimal millioctave, not that that
makes much of a difference, just saying. I believe duodecimal is
closer in the 7-limit (I only tested a couple intervals, though).

I can also appreciate that the decimal octave lends intself to an
interesting meantone with a 580 moc fifth (same as 50-edo), but the
duodecimal division better supports 12 (10') and 72 (60')-edo.

They both work.

--- In tuning@yahoogroups.com, "robert thomas martin"
<robertthomasmartin@...> wrote:
>
> --- In tuning@yahoogroups.com, "Bruce R. Gilson" <brgster@> wrote:
> >
> > --- In tuning@yahoogroups.com, "Tony" <leopold_plumtree@> wrote:
> > >
> > > It's nice to see a decimal octave being applied, which to me
makes
> > > more sense than cents of a 12-edo semitone. I would personally
> > > prefer duodecimal division of the octave, but since the
principle's
> > > the same either way, decimal is fine.
> > >
> >
> > I've been fighting this fight for at least 15 years, so I'm glad
to
> see
> > that someone appreciates this. I _do_ mention duodecimal division
in
> > the book in the chapter that discusses the various units that
have
> been
> > proposed for measuring intervals.
> >
> From Robert. I am quite happy to promote 100 and 200tet but I am
> unwilling to promote 1000tet. I am quite happy with 1200tet and I
> am sure that all the keyboard companies agree with me.
>

Multiple-radix systems are unnecessary. We don't need 6 tones of 200
cents when we can just divide a single fundamental unit (octave) with
prefixed powers of ten (or twelve, but the radix used isn't that
important). It makes for fewer keystrokes on the calculator. ;)

--- In tuning@yahoogroups.com, "Bruce R. Gilson" <brgster@...> wrote:
>
> --- In tuning@yahoogroups.com, "Tony" <leopold_plumtree@> wrote:
> >
> > It's nice to see a decimal octave being applied, which to me
> > makes more sense than cents of a 12-edo semitone. I would
> > personally prefer duodecimal division of the octave, but since
> > the principle's the same either way, decimal is fine.
>
> I've been fighting this fight for at least 15 years, so I'm glad
> to see that someone appreciates this. I _do_ mention duodecimal
> division in the book in the chapter that discusses the various
> units that have been proposed for measuring intervals.

I don't get it. Cents divide the octave into 2, 3, 4, 5, 6,
10, 12, etc. parts. What advantage could a millioctave confer?

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "Bruce R. Gilson" <brgster@> wrote:
> >
> > --- In tuning@yahoogroups.com, "Tony" <leopold_plumtree@> wrote:
> > >
> > > It's nice to see a decimal octave being applied, which to me
> > > makes more sense than cents of a 12-edo semitone. I would
> > > personally prefer duodecimal division of the octave, but since
> > > the principle's the same either way, decimal is fine.
> >
> > I've been fighting this fight for at least 15 years, so I'm glad
> > to see that someone appreciates this. I _do_ mention duodecimal
> > division in the book in the chapter that discusses the various
> > units that have been proposed for measuring intervals.
>
> I don't get it. Cents divide the octave into 2, 3, 4, 5, 6,
> 10, 12, etc. parts. What advantage could a millioctave confer?
>
> -Carl
>

The millioctave provides the simplicity of single number base. The
binary log of the ratio of frequencies gives you a decimal octave
without any need to perform additional operations. It makes number
crunching somewhat easier and faster.

Factorability is very desirable, though. You're certainly right about
that. The problem is that 1200-edo comes at the cost of losing single-
radix utility. 1200 isn't the integral exponent of any number, and
it's too large to be a practical base by itself (and if you were to
go that large, you'd be better off with 1260 to gain a factor of 7).

That's why duodecimal would be advantageous; we could use a single
unit with a single, reasonably-sized radix and still have greater
factorability than decimal.

> > I don't get it. Cents divide the octave into 2, 3, 4, 5, 6,
> > 10, 12, etc. parts. What advantage could a millioctave confer?
>
> The millioctave provides the simplicity of single number base. The
> binary log of the ratio of frequencies gives you a decimal octave
> without any need to perform additional operations. It makes number
> crunching somewhat easier and faster.

I think I can afford a single * 1200.

> The problem is that 1200-edo comes at the cost of losing single-
> radix utility.

I'm still not sure what that utility would be.

> That's why duodecimal would be advantageous; we could use a single
> unit with a single, reasonably-sized radix and still have greater
> factorability than decimal.

What are you calling duodecimal? Base-12 log?

-Carl

--- In tuning@yahoogroups.com, "Bruce R. Gilson" <brgster@...> wrote:
> Actually, in the book I use "moc." -- which conforms to your desire to
> use lower case, even if it doesn't agree exactly with your abbreviation.
>

Hi Bruce,

I find that acceptable (without the trailing period) beacuse "moc"
also obeys the SI symbol rules (which are now applied even to non-SI
units), as would "moct", which I like even better.

And I assume you (correctly) do not use "mocs" for the plural case,
but continue to write say "5 moc", just as with money we do write "5
c" not "5 cs" or "5 c's"?

http://physics.nist.gov/cuu/Units/rules.html

We'd just have to agree on what is the standard symbol for the octave,
"o", "oc", or "oct", considering past usage. And in fact when I do a
little googling I find that "oct" is already in wide usage,
particularly in the combination "dB/oct" which means decibels per
octave (for the slopes of audio filter response curves).

So it appears "moct" would be standard.

I don't really think it's that important. I was mostly just having a
bit of fun with John Chlamers who has since agreed, in private email,
that I'm right. ;-)

Regards,
-- Dave

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> > > I don't get it. Cents divide the octave into 2, 3, 4, 5, 6,
> > > 10, 12, etc. parts. What advantage could a millioctave confer?
> >
> > The millioctave provides the simplicity of single number base. The
> > binary log of the ratio of frequencies gives you a decimal octave
> > without any need to perform additional operations. It makes number
> > crunching somewhat easier and faster.
>
> I think I can afford a single * 1200.

It doesn't seem like a big deal, but in doing calculation after
calculation, it can become quite an annoyance.

> > The problem is that 1200-edo comes at the cost of losing single-
> > radix utility.
>
> I'm still not sure what that utility would be.

Ease of use with our decimal counting system.

> > That's why duodecimal would be advantageous; we could use a single
> > unit with a single, reasonably-sized radix and still have greater
> > factorability than decimal.
>
> What are you calling duodecimal? Base-12 log?
>
> -Carl
>

Still based on the octave (base-2 logs), but stated in duodecimal
numeration.

+'1 is a twelve-edo chromatic scale step (2^0'1:1).

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@...> wrote:
>
> I don't really think it's that important.

Shuuuure you don't. -Carl

> > I think I can afford a single * 1200.
>
> It doesn't seem like a big deal, but in doing calculation after
> calculation, it can become quite an annoyance.

What are you using - an abacus?

> Still based on the octave (base-2 logs), but stated in duodecimal
> numeration.
>
> +'1 is a twelve-edo chromatic scale step (2^0'1:1).

Hmm...

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> > > I think I can afford a single * 1200.
> >
> > It doesn't seem like a big deal, but in doing calculation after
> > calculation, it can become quite an annoyance.
>
> What are you using - an abacus?
>
> > Still based on the octave (base-2 logs), but stated in duodecimal
> > numeration.
> >
> > +'1 is a twelve-edo chromatic scale step (2^0'1:1).
>
> Hmm...
>
> -Carl
>

Well, if you're finding the difference between two intervals, that's
an extra eight keystrokes. Even more if you store one cent value
before calculating the next one and recall it after.

> Well, if you're finding the difference between two intervals, that's
> an extra eight keystrokes.

Not with the power of software!

> Even more if you store one cent value
> before calculating the next one and recall it after.

You're using a hand calculator I take it.

-Carl

Someone should probably mention that Erv Wilson typically
uses octaves rather than cents. I suppose one advantage is,
the number of octaves is obvious -- one does not have to
divide by 1200. On the other hand, with cents, the number
of 12-ET degrees is obvious. In the end I guess it doesn't
matter that much.

-Carl

To be absolutely honest, during a more ethnomusicologically radical phase, I was concerned that the use of cents was too eurocentric, reifying a particular division of the octave as a universal standard. Moreover, the whole discussion had precisely the same flavor as the eternal discussions over the best equal division representation of this or than just intonation. But with some of the moderation that comes with age and more practical experience, I've come 'round to the conclusion that any measurement unit is going to be based upon a quantity of some arbitrary local significance, for example the semitone or the octave. Using cents, a familiar and de facto standard, has the charm that any examples in a text can be translated immediately into the practice of making real musical sounds. Tuning devices and software are built with cent measurements and players in the western tradition are often trained to adjust intonation in cents. (In my opinion western "classical" musicians should be trained in performing cent deviations from 12tet, tuning JI intervals through the 11th partial when singing or playing instruments with regular harmonic spectra, as well as to have the experience of performing in at least one historical european and one non-western tuning system). Inasmuch as cents or saveurs or millioctaves or their alternatives can be translated from one another by the multiplication by a constant, then then the argument in favor of any single measure is trivial and unconvincing in the face of the more important argument that using cents can save one intellectual step and its requisite time in getting to the most urgent matter, which is making interesting music well.

Daniel Wolf

--- In tuning@yahoogroups.com, "Daniel Wolf" <djwolf@...> wrote:
>
> Using cents,
> a familiar and de facto standard, has the charm that any examples in a
> text can be translated immediately into the practice of making real
> musical sounds.

I don't agree, if by 'real' you mean 'non-electronic'.
Try translating the following immediately into 'real musical sounds',
either melodically or harmonically:
164 cents
436 cents
742 cents
928 cents

- 'Unfair' you might say, but why?

> players in the western tradition are often trained to
> adjust intonation in cents.

I haven't come across this in practice, except for oboists checking
their 'A' on a meter. Moreover I don't believe it can be done with any
reasonable amount of training, in the sense of someone asking
specifically for (say) 6 cents higher or lower, and the player
achieving it (to within, say a couple of cents) *and* reliably
remembering the interval for next time.
Of course it 'can be done' possibly with an immense amount of
microscopic ear training, but that isn't 'western' musicianship.
Perhaps Daniel means players sticking tuning meters on their music
stands and trying to hit the 'right' spot. I call that machinery, not
music-making.
Perhaps some musicians do say 'a few cents higher' and then someone
plays a bit higher and it sounds better and they are happy, but one
might as well say 'a few boojums'...

> western "classical" musicians
> should be trained in performing cent deviations from 12tet

Why, and how? If one is playing with an ET keyboard it is useful to
practice ET (but then, how much stretch?) - but even more useful to
practice what sounds good with an ET keyboard. If not, there is no
point in taking ET as a basis anyway, deviations from just intervals
are much more musically useful and practical (ie audible) to consider.

As to 'how', is there any way to achieve proficiency in 'performing
cent deviations from 12et' apart from continually staring at an
electronic meter?

> the most urgent matter, which is making interesting music well.
>

People seem to have managed that for some time without any scientific
units or electronic gadgets at all.

If you're going to write a book about intervals you can use any units
you like, but to imagine that such scholarly / scientific apparatus
has a practical connection with performing music on non-electronic
instruments is unrealistic.
~~~T~~~

--- In tuning@yahoogroups.com, "Tony" <leopold_plumtree@...> wrote:
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
> >
> > > > I don't get it. Cents divide the octave into 2, 3, 4, 5, 6,
> > > > 10, 12, etc. parts. What advantage could a millioctave confer?
> > >
> > > The millioctave provides the simplicity of single number base. The
> > > binary log of the ratio of frequencies gives you a decimal octave
> > > without any need to perform additional operations. It makes number
> > > crunching somewhat easier and faster.
> >
> > I think I can afford a single * 1200.
>
> It doesn't seem like a big deal, but in doing calculation after
> calculation, it can become quite an annoyance.
>
>
> > > The problem is that 1200-edo comes at the cost of losing single-
> > > radix utility.
> >
> > I'm still not sure what that utility would be.
>
> Ease of use with our decimal counting system.
>
>
> > > That's why duodecimal would be advantageous; we could use a single
> > > unit with a single, reasonably-sized radix and still have greater
> > > factorability than decimal.
> >
> > What are you calling duodecimal? Base-12 log?
> >
> > -Carl
> >
>
> Still based on the octave (base-2 logs), but stated in duodecimal
> numeration.
>
> +'1 is a twelve-edo chromatic scale step (2^0'1:1).
>

For easy hand calculation, the savart (actually invented by Sauveur,
but not a 'sa[u]veur') is absolutely the best: log-10 of the interval,
times 1000. In practice you just move the decimal point 3 places by
eye. Pure intervals up to 7-limit are also pretty near whole numbers
of savart, or to put it another way whole savarts are a surprisingly
good approximation to many just intervals.
~~~T~~~

--- In tuning@yahoogroups.com, "Daniel Wolf" <djwolf@...> wrote:

> But with some of the moderation that comes with age and more
> practical experience, I've come 'round to the conclusion that any
> measurement unit is going to be based upon a quantity of some
> arbitrary local significance, for example the semitone or the
> octave.

The least arbitrary unit might be neper:

http://en.wikipedia.org/wiki/Neper

It is also the most politically correct choice in not
being "octavecentric". I'm not serious. :D

Kalle

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
'Unfair' you might say, but why?
>
> > players in the western tradition are often trained to
> > adjust intonation in cents.
>
> I haven't come across this in practice, except for oboists checking
> their 'A' on a meter. Moreover I don't believe it can be done with
any
> reasonable amount of training, in the sense of someone asking
> specifically for (say) 6 cents higher or lower, and the player
> achieving it (to within, say a couple of cents) *and* reliably
> remembering the interval for next time.

Tom,

This _is_ a standard part of training for non-fixed pitch
instrumentalists. In my case, as a trombonist, back in the dark ages
of the mid-70s, even our suburban Californian High School marching
band instructor had each member of the band spend hours tuning to the
needle of a strobotuner, and to this day my estimation of cent
deviations is pretty accurate as a result of that practice. There is
obviously a lot of ear AND muscle memory to reliably acquiring such
skills, but I'm convinced the best players share this. (I've watched
rehearsal conducted by Pierre Boulez, Hans Zender, and Paul Zukofsky
in which the conductors corrected intonation by specifying cent
changes for individual players).

In new music contexts, notating cent deviations, with or without the
use of an electronic tuner, is frequently the most efficient way to
do complex tunings. A good example is that of the Arditti Quartet, by
some measures the best such ensemble of our time. Alvin Lucier and
Walter Zimmermann happen to have written quartets for the Arditti
about the same time featuring beating. Lucier asked for cent
deviations from 12tet, and Zimmermann notated just ratios. In both
live concerts and in recordings, the Lucier quartet always works,
with the progressive reduction in beat rate completely vivid. The
Zimmermann, on the other hand, is so dependent on a large number of
local variables, chief among which is the performing environment
which determines, to a large extent, whether and how well the
instruments can hear each other, that it has only been successful in
studio recordings using headsets with recorded reference pitches. If
Zimmermann had noted cents deviations in addition to the ratios, I
believe that the concert performances would have been much more
successful.

Johnny Reinhard, famously around these parts, seems to be very
successful with singers, string and wind players using cents for more
complex tunings, and James Tenney's well-known _Critical Band__ is
wisely notated, AFAIC, with both ratios and cents.

That said, for a less complex just intonation, i.e. with a limited
number of total pitches used, relatively slow introduction of
pitches, using a 7 limit or perhaps a few intervals from an 11 limit,
and in which absolute pitch height is not fixed (as was in the
Zimmermann for the fixed beating rates) then a just intonation
notation of ratios or Helmholtz-Ellis style is perfectly adequate.

djw

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Someone should probably mention that Erv Wilson typically
> uses octaves rather than cents. I suppose one advantage is,
> the number of octaves is obvious -- one does not have to
> divide by 1200. On the other hand, with cents, the number
> of 12-ET degrees is obvious. In the end I guess it doesn't
> matter that much.
>
> -Carl
>

That's why I like duodecimal for this - octaves and 12-edo degrees are
obvious.

And yes, I'm using a hand calculator.

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@...> wrote:
>
> --- In tuning@yahoogroups.com, "Bruce R. Gilson" <brgster@> wrote:
> > Actually, in the book I use "moc." -- which conforms to your desire
> > to use lower case, even if it doesn't agree exactly with your
> > abbreviation.
>
> Hi Bruce,
>
> I find that acceptable (without the trailing period) because "moc"
> also obeys the SI symbol rules (which are now applied even to non-SI
> units), as would "moct", which I like even better.
>
> And I assume you (correctly) do not use "mocs" for the plural case,
> but continue to write say "5 moc", just as with money we do write "5
> c" not "5 cs" or "5 c's"?
>

One agreement with you and one disagrement. I keep the trailing
period, but I don't use the "s" in the plural -- after all, we don't use "5
secs" or "5 kms."

--- In tuning@yahoogroups.com, "Tony" <leopold_plumtree@...> wrote:
>
>
>
> I'm tempted to get a copy.

I hope you do. I'm interested in feedback from people on this list, though
it's clearly at a level which most of you exceed. I think you might like to use
the book as a reference in introducing people to non-standard tunings,
though.

> ... I can understand why you'd have to use
> decimal numeration for such a book, due to the limited use/familiarity
> of duodecimal among most people.

We write our numbers in decimal, not duodecimal. so that's another point.

--- In tuning@yahoogroups.com, "Tony" <leopold_plumtree@...> wrote:
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
> >
> > > > I don't get it. Cents divide the octave into 2, 3, 4, 5, 6,
> > > > 10, 12, etc. parts. What advantage could a millioctave confer?
> > >
> > > The millioctave provides the simplicity of single number base. The
> > > binary log of the ratio of frequencies gives you a decimal octave
> > > without any need to perform additional operations. It makes
> > > number crunching somewhat easier and faster.

One of my reasons for using it.

> >
> > I think I can afford a single * 1200.
>
> It doesn't seem like a big deal, but in doing calculation after
> calculation, it can become quite an annoyance.
>
>
> > > The problem is that 1200-edo comes at the cost of losing single-
> > > radix utility.
> >
> > I'm still not sure what that utility would be.
>
> Ease of use with our decimal counting system.

The principal reason for my using it. Among other things, we know since
grade school that .25 is a quarter and .5 is a half; some of us have
memorized other decimal equivalents of fractions. All this information is
easy to bring to hand when using millioctaves, not cents, flus, or 730-to-
the-octave Woolhouse units.

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Someone should probably mention that Erv Wilson typically
> uses octaves rather than cents. I suppose one advantage is,
> the number of octaves is obvious -- one does not have to
> divide by 1200. On the other hand, with cents, the number
> of 12-ET degrees is obvious.

That's one of the reasons I do NOT use cents. I think it has the effect of
raising 12-ET to the standard by which everything else is judged.

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
>
> [...]
>
> For easy hand calculation, the savart (actually invented by Sauveur,
> but not a 'sa[u]veur') is absolutely the best: log-10 of the interval,
> times 1000. In practice you just move the decimal point 3 places by
> eye. Pure intervals up to 7-limit are also pretty near whole numbers
> of savart, or to put it another way whole savarts are a surprisingly
> good approximation to many just intervals.

Just a minor point: Sauveur did invent the unit that is now called the
savart, but in fact it was not named as a misspelling of his name, as your
post might suggest. About 2 centuries after Sauveur, there was another
man whose name was Savart, who advocated the same unit.

Bruce R. Gilson wrote:

> That's one of the reasons I do NOT use cents. I think it has the effect of
> raising 12-ET to the standard by which everything else is judged.

If you think the main point of using cents is the fact that 12-EDO is a
"standard", then I have to strongly disagree. The reason why I DO use cents,
for example, is the fact that I have an immediate imagination of how large
the interval "in question" is. If you said to me: "Sing the second tone 670
cents higher than the first one", I think I should be no more "out of tune"
than about +/-10 cents to each side before I have a chance to hear it -- of
course, if I trained it a lot, maybe I could get within +/-5 cents, I don't
know, but that's another matter. OTOH, if you asked me to sing 585 MO, first
of all, I would probably say to myself: "Goodness, how large is that?" And
after quite some time I would "maybe" figure out that it's actually a
perfect fifth. If I should go for MO, then I would often probably be very
slow at working with logarithmic interval measuring, because knowing only
that 1000 MO is an octave and 500 MO is the "tritone" (and nothing else)
seems not enough to me. And I really don't need to force myself to
mechanically learn "by heart" what MO quantity gives what interval size, if
I know about another unit for whose usage this skill is not required. And if
I eventually DID decide to use MO, then I would immediately find an EDO to
compare with anyway, which is, of course, 50. But because 50 is so much more
than 12, comparing is 50 is also more difficult than comparing with 12. Had
we grown up with music made in 7-EDO instead of 12, I think there would be a
lot of people defending something like 70 or 1400-EDO for measuring
intervals. The need for comparing (to be able to quickly know the interval
size) is so strong that forcing people to get rid of it, I think, is
probably not one of the "most fruitful paths". And to paraphrase you, that's
one of the reasons why I don't use MO or Woolhouse units or whatever.

Petr

--- In tuning@yahoogroups.com, "Bruce R. Gilson" <brgster@...> wrote:
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
> >
> > Someone should probably mention that Erv Wilson typically
> > uses octaves rather than cents. I suppose one advantage is,
> > the number of octaves is obvious -- one does not have to
> > divide by 1200. On the other hand, with cents, the number
> > of 12-ET degrees is obvious.
>
> That's one of the reasons I do NOT use cents. I think it has
> the effect of raising 12-ET to the standard by which everything
> else is judged.

The lot of this thread raises *units* to a level of importance
they do not have.

-Carl

Dave Keenan wrote:

> We'd just have to agree on what is the standard symbol for the octave,
> "o", "oc", or "oct", considering past usage. And in fact when I do a
> little googling I find that "oct" is already in wide usage,
> particularly in the combination "dB/oct" which means decibels per
> octave (for the slopes of audio filter response curves).
> > So it appears "moct" would be standard.
> > I don't really think it's that important. I was mostly just having a
> bit of fun with John Chlamers who has since agreed, in private email,
> that I'm right. ;-)
> > Regards,
> -- Dave Also, "o" is an abbreviation for "octet", which is French for "byte". Not that there's much chance for confusion, since a 1000th of a byte is not a meaningful unit, and a "megaoctave" would also be pretty useless, but it would be another argument for using "oc" or "oct" instead of "o".

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
>
> Bruce R. Gilson wrote:
>
> > That's one of the reasons I do NOT use cents. I think it has the
> > effect of raising 12-ET to the standard by which everything else
> > is judged.
>
> If you think the main point of using cents is the fact that 12-EDO
> is a "standard", then I have to strongly disagree. The reason why I
> DO use cents, for example, is the fact that I have an immediate
> imagination of how large the interval "in question" is. If you said
> to me: "Sing the second tone 670 cents higher than the first one",
> I think I should be no more "out of tune" than about +/-10 cents to
> each side before I have a chance to hear it -- of course, if I
> trained it a lot, maybe I could get within +/-5 cents, I don't
> know, but that's another matter. OTOH, if you asked me to sing 585
> MO, first of all, I would probably say to myself: "Goodness, how
> large is that?" And after quite some time I would "maybe" figure
> out that it's actually a perfect fifth.

All that you are saying is that you have _learned_ what an interval
of 670 cents sounds like. And, for you, that is actually a good
reason you may want to use cents. But the fact is that cents are so
strongly related to 12-EDO (or 12-tET, or however you want to
abbreviate it; there seem to be numerous ways of writing it!) that I
see a very strong bias there. That a "true" fifth is 702 cents and
a "true" major third is whatever 322x1.2 cents comes to (I've learned
it by now in moc.!) while an ET fifth is the round number of 700
cents and an ET major third is the round number of 400 cents shows
this.

Just like the fact that I can understand feet more intuitively than
meters, because I've been using feet & inches all my life and only
met with meters in my lab classes in school, you feel more
comfortable with cents. But for the purposes of the things I discuss
in my book, it's easier to compute (in my head!) that 91 moc. is 1/11
of an octave or that 143 moc. is 1/7 of an octave than to compute the
corresponding numbers in cents. And if you'd memorized that a perfect
fifth was 585 moc. your "how large is that?" moment would never occur.
I know already that 322 moc. is a just major third and 585 moc. is a
Pythagorean fifth, just by working with them a bit. You would too, if
you were interested enough to work with them for just a short while.

And I must admit, I don't have perfect pitch. So if I wanted to
generate a note 400 moc. above middle C, I'd probably have to program
it on a computer or a synthesizer. And if the machine were calibrated
in cents, that would be a little bit of work. On the other hand, on
the spreadsheets I used to compute the entries in my book,
multiplying by 1000 or by 1200 is about equally easy. And when I do
in-my-head calculations, which I did in many places, multiplying by
1000 is a LOT easier!

--- In tuning@yahoogroups.com, "djwolf_frankfurt" <djwolf@...> wrote:
>
> --- In tuning@yahoogroups.com, "Tom Dent" <stringph@> wrote:
>
> > > players in the western tradition are often trained to
> > > adjust intonation in cents.
> >
> > I haven't come across this in practice, except for oboists checking
> > their 'A' on a meter. Moreover I don't believe it can be done with
> any
> > reasonable amount of training, in the sense of someone asking
> > specifically for (say) 6 cents higher or lower, and the player
> > achieving it (to within, say a couple of cents) *and* reliably
> > remembering the interval for next time.
>
> Tom,
>
> This _is_ a standard part of training for non-fixed pitch
> instrumentalists.

Then we disagree about what is 'standard'. I thought we're pretty
clear that there isn't one standard classical training, let alone if
we consider earlier in the 20th century.
When could such strobotraining have started to become standard anyway?
Way, way after the first atonal music ... what business would we have
to apply it to earlier repertoire?

> In my case, as a trombonist, back in the dark ages
> of the mid-70s, even our suburban Californian High School marching
> band instructor had each member of the band spend hours tuning to the
> needle of a strobotuner

Since when had this been 'standard' practice, I wonder. I've never
heard of anything like it, but maybe if you have an large number of
strobotuners (why buy them in the first place though?) it could make
sense...

If you want to achieve (say) a pure fifth between two brass
instruments, is it necessary, or efficient, to train one of them to
play at deviation 0 and the other at deviation +2?

> obviously a lot of ear AND muscle memory to reliably acquiring such
> skills, but I'm convinced the best players share this.

But did you ask them (- I mean, the best players)?

> (I've watched
> rehearsal conducted by Pierre Boulez, Hans Zender, and Paul Zukofsky
> in which the conductors corrected intonation by specifying cent
> changes for individual players).

You can talk of cents, but that doesn't prove you can hear them. Were
the correct cent changes in fact executed? More to the point, did you
see if the players had electronic gadgets in front of them?

> (...) Lucier and
> Walter Zimmermann happen to have written quartets for the Arditti
> about the same time featuring beating. Lucier asked for cent
> deviations from 12tet, and Zimmermann notated just ratios. In both
> live concerts and in recordings, the Lucier quartet always works,
> with the progressive reduction in beat rate completely vivid.

Now is this because the players memorize the *pitches*, or the
*intervals*, or the *hand positions* ... or, actually, the beat rates?
Is the cent notation just a crutch to be thrown away once the desired
acoustic effect emerges?

> Johnny Reinhard, famously around these parts, seems to be very
> successful with singers, string and wind players using cents for more
> complex tunings

I know what Johnny claims. You listen to his 2nd Brandenburg slow
movement and tell me whether the recorder part is in any definable
tuning whatever... let alone Werckmeister. You can advertise a set of
cent-values all you want, but that doesn't mean the actual pitches
emitted by the performers conform to it.

Anyway, I think the point has been conceded that cent-deviations only
have immediate practical value for 1) people who have already spent
many hours over many months staring at electronic meters and 2) music
with a basis of intervals that are not very far from 12tet.
I would strongly resist the idea that the 'best musicians' (of this or
any other era) need fall into class 1).

On the other hand I would strongly endorse the idea that the 'best
musicians' of any era hear and play harmonically just intervals
(whichever necessary) or deviations from them without any electronic
aid at all, simply by judging the musical results. That's certainly
how good harpsichord tuners work. You can't tune a good meantone via
an electronic gadget preprogrammed in cents.

However, I will still be impressed by the cent-practice argument if
any trombonist plays my 'random'/'unfair' intervals
164 cents
436 cents
742 cents
928 cents
with reasonable precision purely by ear. Let alone any violinist!
Obviously, from traditional 'Western' perspective, fairly nasty and
pointless intervals, but if cent-practice really helps for
non-12tet-centred things it should be effective for them too.
~~~T~~~

I don't have time for an argument Tom, but the issue here is cents or another division as a reference point, and the fact is that there are real musicians out there who have trained with tuners that use cents (not to mention those with absolute pitch in 12tet) for whom it is a perfectly reasonable task to be given a notation with cent deviations.

As I made it clear in my initial post in this thread, and many other threads over the years, I believe that musicians should also learn to tune a repertoire of just intervals, when they are required or optimal, but the reality of performance under real conditions and in real spaces is that tuning by reduction of beats is not always possible as it presumes that (a) the instruments at hand all have regular harmonic spectra, (b) the players can physically hear one another, and (c) the music unfolds at a slow enough pace to allow for correcting intonation in real time. (As important as tuning is to me, it has been my experience that _sometimes_ intonation is more "convincing" if one sticks to a wrong pitch rather than over-audibly corrects it, when sliding around would disturb the immediate musical context).

Obviously live performance, which is the only form of music-making in which I have much interest, is an intonationally difficult environment, but given the fact that gathering musicians together is often very difficult and rehearsal time is consequently so precious, having tool sets that are practical and redundant is a far better proposition than wasting time and energy on the impractical and over-specialized. I, for one, have no problem in notating simultaneously in Helmholz/Ellis, ratios, and cents if that will get the job done, and I believe that some combination will probably reach a greater number of musicians. What is not helpful, AFAIC, are notations in which intervals are spelled inconsistantly or arguments over altervatives to cents which, for non-tuning specialists, are the equivalent over arguments over the numbers of angels on pinheads. Intonation is a central concern for all music-making, not just a preserve of a microtonal niche, and the urgent issue is improving and expanding intonational practice in general.

Daniel Wolf

for measurement one could just use the log2 and not 'translate' it further. I am afraid we are probably stuck with cents. on a practical level if you ask a performer to play a specific interval , they are probably going to want to know the difference in cents. Even if you can convince them to learn a new system. the tuners they will use will be marked in sense. i think there are more important issues that trying to undo a history that is so entrenched.
--

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',