Yahoo Tuning Groups Ultimate Backup tuning "best average" near-ET scale

Sometimes tuning (temperament) = scale, but here scale is a subset of tuning which is often the case. Maybe it would be good to make some order in this terminology and use it more consistently here on Tuning group. We should fight with entropy :-)

Also using alphabet for steps in the scale is confusing as they can remind names of standard notes (A to G, and in some countries H as well). Greek alphabet for steps? Or just Roman numbers?

Otherwise an interesting idea for tuning. Could you write please a comparison with 12ET? Also Cent table would be good for those here using it.

Daniel Forrro

On 31 Jan 2009, at 10:26 PM, djtrancendance wrote:

> This scale takes the square root of two, tries to estimate it best
> with a simple fraction (24/17).
>
> The idea is to
> 1) make the brain categorize the notes as two-tones, thus making the
> result seem more consonant than normal ET and thus allowing more clear
> chords, plus the usual transpositions you come to expect in ET.
> 2) focus on the idea of "near-perfect" ratios, rather than having many
> notes some out 99% perfect and others completely sour (IE the 3rd in
> 12ET).
>
> The notes: >> a,b,d,e,g,i,j,l,m <<
> comprise a 9-note scale (my intended equivalent to the 7-note scale in
> ET) that can be easily transposed.
>
> 18/17 = 1.05882 (A)
> 19/17 = 1.11764 (B)
> 20/17 = 1.17647
> 21/17 = 1.23529 (D)
> 22/17 = 1.2941176705 (E)
> 23/17 = 1.3529411
> 24/17 = 1.411764 (G, new "5th")
>
> 18/17 * 24/17 = 1.494809 (I)
> 19/17 * 24/17 = 1.57785 (J)
> 20/17 * 24/17 = 1.660899
> 21/17 * 24/17 = 1.743944 (L)
> 22/17 * 24/17 = 1.8269896 (M)
> 23/17 * 24/17 = 1.910034602
> 24/17 * 24/17= 1.99307968477
>
> -Michael
>

🔗djtrancendance@...

Again, the scale in question I created is (note the roman numerals are the notes which comprise a SCALE in the tuning which include notes intended to be used as chords and notes that can be easily transposed.

(I.) = root
18/17 = 1.05882 (II.)

19/17 = 1.11764 (III.)

20/17 = 1.17647

21/17 = 1.23529 (V.)

22/17 = 1.2941176705 (VI)

23/17 = 1.3529411

24/17 = 1.411764 (VIII, new "5th")

18/17 * 24/17 = 1.494809 (IX)

19/17 * 24/17 = 1.57785 (X)

20/17 * 24/17 = 1.660899

21/17 * 24/17 = 1.743944 (XII)

22/17 * 24/17 = 1.8269896 (XIII)

23/17 * 24/17 = 1.910034602

24/17 * 24/17= 1.99307968477

--Also using alphabet for steps in the scale is confusing as they can
---remind names of standard notes (A to G, and in some countries H as
---well)
Agreed, I think your idea of Roman numbers is a good one. I avoided using things like C, C-flat, C-double flat...because this scale is not made to be built relative to just-intonation diatonic and/or be treated in the same fashion as 12TET-like scales far as chords. I just don't want someone trying to make a C,D#,G triad in my scale thinking it's supposed to sound alike, for example (and then whining when it doesn't).

---Otherwise an interesting idea for tuning. Could you write please a
---comparison with 12ET? Also Cent table would be good for those here
---using it.
Hahaha, oh man and last time I got
grilled for using "only cents", argh, you can't win. Sure, though, I can post it in both formats. :-D

---Sometimes tuning (temperament) = scale, but here scale is a subset of
---tuning which is often the case. Maybe it would be good to make some
---order in this terminology and use it more consistently here on Tuning
---group. We should fight with entropy :-)

Agreed!
It took me a while to figure out as well, that a scale is a subset of a tuning used to produce chords, that it can often be transposed within a tuning, etc. AKA of course, for example, 12TET is a tuning, C major is a scale, and C can be transposed to the key of C#,D,D#,E...within the 12TET tuning. There's enough madness going on when people don't know or state the difference when they introduce a new scale or tuning, or "resurrect" an old one.

-Michael

--- On Sat, 1/31/09,
Daniel Forro <dan.for@...> wrote:

From: Daniel Forro <dan.for@...>
Subject: Re: [tuning] "best average" near-ET scale
To: tuning@yahoogroups.com
Date: Saturday, January 31, 2009, 6:00 AM

Sometimes tuning (temperament) = scale, but here scale is a subset of

tuning which is often the case. Maybe it would be good to make some

order in this terminology and use it more consistently here on Tuning

group. We should fight with entropy :-)

Also using alphabet for steps in the scale is confusing as they can

remind names of standard notes (A to G, and in some countries H as

well). Greek alphabet for steps? Or just Roman numbers?

Otherwise an interesting idea for tuning. Could you write please a

comparison with 12ET? Also Cent table would be good for those here

using it.

Daniel Forrro

On 31 Jan 2009, at 10:26 PM, djtrancendance wrote:

> This scale takes the square root of two, tries to estimate it best

> with a simple fraction (24/17).

> The idea is to

> 1) make the brain categorize the notes as two-tones, thus making the

> result seem more consonant than normal ET and thus allowing more clear

> chords, plus the usual transpositions you come to expect in ET.

> 2) focus on the idea of "near-perfect" ratios, rather than having many

> notes some out 99% perfect and others completely sour (IE the 3rd in

> 12ET).

> The notes: >> a,b,d,e,g,i, j,l,m <<

> comprise a 9-note scale (my intended equivalent to the 7-note scale in

> ET) that can be easily transposed.

> 18/17 = 1.05882 (A)

> 19/17 = 1.11764 (B)

> 20/17 = 1.17647

> 21/17 = 1.23529 (D)

> 22/17 = 1.2941176705 (E)

> 23/17 = 1.3529411

> 24/17 = 1.411764 (G, new "5th")

> 18/17 * 24/17 = 1.494809 (I)

> 19/17 * 24/17 = 1.57785 (J)

> 20/17 * 24/17 = 1.660899

> 21/17 * 24/17 = 1.743944 (L)

> 22/17 * 24/17 = 1.8269896 (M)

> 23/17 * 24/17 = 1.910034602

> 24/17 * 24/17= 1.99307968477

> -Michael

🔗Daniel Forró <dan.for@...>

On 31 Jan 2009, at 11:22 PM, djtrancendance@... wrote:

>
> Again, the scale in question I created is (note the roman > numerals are the notes which comprise a SCALE in the tuning which > include notes intended to be used as chords and notes that can be > easily transposed.
>
> (I.) = root
> 18/17 = 1.05882 (II.)
> 19/17 = 1.11764 (III.)
> 20/17 = 1.17647
> 21/17 = 1.23529 (V.)
> 22/17 = 1.2941176705 (VI)
> 23/17 = 1.3529411
> 24/17 = 1.411764 (VIII, new "5th")
>
> 18/17 * 24/17 = 1.494809 (IX)
> 19/17 * 24/17 = 1.57785 (X)
> 20/17 * 24/17 = 1.660899
> 21/17 * 24/17 = 1.743944 (XII)
> 22/17 * 24/17 = 1.8269896 (XIII)
> 23/17 * 24/17 = 1.910034602
> 24/17 * 24/17= 1.99307968477

Sorry, Michael, I meant just to number steps of the scale, not steps in tuning. So if you used 9 steps, the highest number will be IX :-)
>
> --Also using alphabet for steps in the scale is confusing as they can
> ---remind names of standard notes (A to G, and in some countries H as
> ---well)
> Agreed, I think your idea of Roman numbers is a good one. I > avoided using things like C, C-flat, C-double flat...because this > scale is not made to be built relative to just-intonation diatonic > and/or be treated in the same fashion as 12TET-like scales far as > chords. I just don't want someone trying to make a C,D#,G triad in > my scale thinking it's supposed to sound alike, for example (and > then whining when it doesn't).

For sure not my idea, this is used in theoretical modal or functional harmony for steps of scale used as roots for chords, analysis can be independent from the key.

> ---Otherwise an interesting idea for tuning. Could you write please a
> ---comparison with 12ET? Also Cent table would be good for those here
> ---using it.
> Hahaha, oh man and last time I got grilled for using "only > cents", argh, you can't win. Sure, though, I can post it in both > formats. :-D

Life would be more easy if everybody do so :-)

> ---Sometimes tuning (temperament) = scale, but here scale is a > subset of
> ---tuning which is often the case. Maybe it would be good to make some
> ---order in this terminology and use it more consistently here on > Tuning
> ---group. We should fight with entropy :-)
>
> Agreed!
> It took me a while to figure out as well, that a scale is a > subset of a tuning used to produce chords, that it can often be > transposed within a tuning, etc. AKA of course, for example, 12TET > is a tuning, C major is a scale, and C can be transposed to the key > of C#,D,D#,E...within the 12TET tuning. There's enough madness > going on when people don't know or state the difference when they > introduce a new scale or tuning, or "resurrect" an old one.
>
> -Michael

There's even more madness as the same scale can have different MODES, like Lydian is a 4th mode of major (Ionian), or minor anhemitonic pentatonics is a 5th mode of major anhemitonic pentatonics. Number of modes is equal to number of notes in the scale.

Daniel Forro

🔗Carl Lumma <carl@...>

--- In tuning@yahoogroups.com, Daniel Forro <dan.for@...> wrote:
>
> Sometimes tuning (temperament) = scale, but here scale is a
> subset of tuning which is often the case. Maybe it would be
> good to make some order in this terminology and use it more
> consistently here on Tuning group.

It's been tried. I took this as a question for the FAQ I've
been working on. The answer I wrote is pasted below.

> Otherwise an interesting idea for tuning. Could you write
> please a comparison with 12ET? Also Cent table would be good
> for those here using it.

If Michael would just use Scala format, there wouldn't be
any problem.

-Carl

Q: I see the terms "tuning", "temperament", and "scale" used
interchangeably. Is there any distinction between them?

A: There is no broad consensus distinction between these terms.
In traditional Western keyboard practice, a "temperament" is
usually what defines how the keys are tuned, and a "scale" is a
more abstract thing, often thought of in terms of patterns of
2nds (e.g. LLsLLLs for the diatonic scale) or subsets of whatever
temperament is used.

Microtonal music theory needs much more general terms, since the
scales and consonances being targeted are not taken as given.
Gene Ward Smith has suggested distinct meanings for each of these
three terms in order to make discourse about microtonal music
theory more precise. This usage has seen wide adoption on the
tuning, tuning-math, and to some extent the makemicromusic lists,
but a broad consensus has not emerged. Here are Gene's
definitions (paraphrased):

scale- An ordered list of intervals, which may be applied to a
given concert pitch to generate an ordered list of pitches that
can be used to tune an instrument. Scala's scale file format
(.scl extension) is an embodiment of this definition.

temperament- A mapping from just intonation to an abstract group
of smaller rank. This algebraic definition basically means that
for every interval in just intonation, which is composed of some
combination of primes, we express it instead as a combination of
generators (called the generators of the temperament) such that
there are fewer generators than there were primes. For example,
in 5-limit JI, every interval can be expressed as a combination
of 2s, 3s, and 5s. In meantone temperament, every interval can
be expressed as a combination of octaves and fifths. 5-limit JI
is therefore a rank-3 intonation, and meantone a rank-2
temperament. In 12-ET, every interval can be expressed as a
combination of semitones. It is therefore a rank-1 temperament.

tuning- A choice of intervals for the generators of a
temperament. For example, 1200 cents / 697 cents is a decent
tuning for meantone. So is 1200 cents / 698 cents. They lead to
different scales, but they are both good meantone tunings because
they allow meantone temperament to produce good approximations
of 5-limit JI. Sometimes tunings are named after the
optimization they solve. For example, "RMS-optimal meantone"
refers to the tuning that minimizes the RMS error from JI under
the constraint of the meantone mapping.

🔗Michael Sheiman <djtrancendance@...>

--If Michael would just use Scala format, there wouldn't be
---any problem.
Don't quite get it...when I save a .SCL file from scala it usually just creates a list of cents, not both fractions AND cents. And...it seems if do cents people who want fractions are pissed, and vice-versa if I do only fractions. Is there a way to make scala produce a list with BOTH?

--- On Sat, 1/31/09, Carl Lumma <carl@...> wrote:

From: Carl Lumma <carl@...>
Subject: [tuning] Re: "best average" near-ET scale
To: tuning@yahoogroups.com
Date: Saturday, January 31, 2009, 4:05 PM

--- In tuning@yahoogroups. com, Daniel Forro <dan.for@... > wrote:

> Sometimes tuning (temperament) = scale, but here scale is a

> subset of tuning which is often the case. Maybe it would be

> good to make some order in this terminology and use it more

> consistently here on Tuning group.

It's been tried. I took this as a question for the FAQ I've

been working on. The answer I wrote is pasted below.

> Otherwise an interesting idea for tuning. Could you write

> please a comparison with 12ET? Also Cent table would be good

> for those here using it.

If Michael would just use Scala format, there wouldn't be

any problem.

-Carl

Q: I see the terms "tuning", "temperament" , and "scale" used

interchangeably. Is there any distinction between them?

A: There is no broad consensus distinction between these terms.

In traditional Western keyboard practice, a "temperament" is

usually what defines how the keys are tuned, and a "scale" is a

more abstract thing, often thought of in terms of patterns of

2nds (e.g. LLsLLLs for the diatonic scale) or subsets of whatever

temperament is used.

Microtonal music theory needs much more general terms, since the

scales and consonances being targeted are not taken as given.

Gene Ward Smith has suggested distinct meanings for each of these

three terms in order to make discourse about microtonal music

theory more precise. This usage has seen wide adoption on the

tuning, tuning-math, and to some extent the makemicromusic lists,

but a broad consensus has not emerged. Here are Gene's

definitions (paraphrased) :

scale- An ordered list of intervals, which may be applied to a

given concert pitch to generate an ordered list of pitches that

can be used to tune an instrument. Scala's scale file format

(.scl extension) is an embodiment of this definition.

temperament- A mapping from just intonation to an abstract group

of smaller rank. This algebraic definition basically means that

for every interval in just intonation, which is composed of some

combination of primes, we express it instead as a combination of

generators (called the generators of the temperament) such that

there are fewer generators than there were primes. For example,

in 5-limit JI, every interval can be expressed as a combination

of 2s, 3s, and 5s. In meantone temperament, every interval can

be expressed as a combination of octaves and fifths. 5-limit JI

is therefore a rank-3 intonation, and meantone a rank-2

temperament. In 12-ET, every interval can be expressed as a

combination of semitones. It is therefore a rank-1 temperament.

tuning- A choice of intervals for the generators of a

temperament. For example, 1200 cents / 697 cents is a decent

tuning for meantone. So is 1200 cents / 698 cents. They lead to

different scales, but they are both good meantone tunings because

they allow meantone temperament to produce good approximations

of 5-limit JI. Sometimes tunings are named after the

optimization they solve. For example, "RMS-optimal meantone"

refers to the tuning that minimizes the RMS error from JI under

the constraint of the meantone mapping.

🔗Chris Vaisvil <chrisvaisvil@...>

Save two files?

A pain but it would work.

On Sat, Jan 31, 2009 at 9:50 PM, Michael Sheiman
<djtrancendance@...>wrote:

> --If Michael would just use Scala format, there wouldn't be
> ---any problem.
> Don't quite get it...when I save a .SCL file from scala it usually just
> creates a list of cents, not both fractions AND cents. And...it seems if do
> cents people who want fractions are pissed, and vice-versa if I do only
> fractions. Is there a way to make scala produce a list with BOTH?
>
>
> --- On *Sat, 1/31/09, Carl Lumma <carl@...>* wrote:
>
>
> From: Carl Lumma <carl@...>
> Subject: [tuning] Re: "best average" near-ET scale
> To: tuning@yahoogroups.com
> Date: Saturday, January 31, 2009, 4:05 PM
>
>
> --- In tuning@yahoogroups. com<http://mc/compose?to=tuning%40yahoogroups.com>,
> Daniel Forro <dan.for@... > wrote:
> >
> > Sometimes tuning (temperament) = scale, but here scale is a
> > subset of tuning which is often the case. Maybe it would be
> > good to make some order in this terminology and use it more
> > consistently here on Tuning group.
>
> It's been tried. I took this as a question for the FAQ I've
> been working on. The answer I wrote is pasted below.
>
> > Otherwise an interesting idea for tuning. Could you write
> > please a comparison with 12ET? Also Cent table would be good
> > for those here using it.
>
> If Michael would just use Scala format, there wouldn't be
> any problem.
>
> -Carl
>
> Q: I see the terms "tuning", "temperament" , and "scale" used
> interchangeably. Is there any distinction between them?
>
> A: There is no broad consensus distinction between these terms.
> In traditional Western keyboard practice, a "temperament" is
> usually what defines how the keys are tuned, and a "scale" is a
> more abstract thing, often thought of in terms of patterns of
> 2nds (e.g. LLsLLLs for the diatonic scale) or subsets of whatever
> temperament is used.
>
> Microtonal music theory needs much more general terms, since the
> scales and consonances being targeted are not taken as given.
> Gene Ward Smith has suggested distinct meanings for each of these
> three terms in order to make discourse about microtonal music
> theory more precise. This usage has seen wide adoption on the
> tuning, tuning-math, and to some extent the makemicromusic lists,
> but a broad consensus has not emerged. Here are Gene's
> definitions (paraphrased) :
>
> scale- An ordered list of intervals, which may be applied to a
> given concert pitch to generate an ordered list of pitches that
> can be used to tune an instrument. Scala's scale file format
> (.scl extension) is an embodiment of this definition.
>
> temperament- A mapping from just intonation to an abstract group
> of smaller rank. This algebraic definition basically means that
> for every interval in just intonation, which is composed of some
> combination of primes, we express it instead as a combination of
> generators (called the generators of the temperament) such that
> there are fewer generators than there were primes. For example,
> in 5-limit JI, every interval can be expressed as a combination
> of 2s, 3s, and 5s. In meantone temperament, every interval can
> be expressed as a combination of octaves and fifths. 5-limit JI
> is therefore a rank-3 intonation, and meantone a rank-2
> temperament. In 12-ET, every interval can be expressed as a
> combination of semitones. It is therefore a rank-1 temperament.
>
> tuning- A choice of intervals for the generators of a
> temperament. For example, 1200 cents / 697 cents is a decent
> tuning for meantone. So is 1200 cents / 698 cents. They lead to
> different scales, but they are both good meantone tunings because
> they allow meantone temperament to produce good approximations
> of 5-limit JI. Sometimes tunings are named after the
> optimization they solve. For example, "RMS-optimal meantone"
> refers to the tuning that minimizes the RMS error from JI under
> the constraint of the meantone mapping.
>
>
>

🔗Daniel Forro <dan.for@...>

On 1 Feb 2009, at 9:05 AM, Carl Lumma wrote:
> It's been tried. I took this as a question for the FAQ I've
> been working on. The answer I wrote is pasted below.
>
Great.
> -Carl
>
> Q: I see the terms "tuning", "temperament", and "scale" used
> interchangeably. Is there any distinction between them?
>
> A: There is no broad consensus distinction between these terms.
> In traditional Western keyboard practice, a "temperament" is
> usually what defines how the keys are tuned, and a "scale" is a
> more abstract thing, often thought of in terms of patterns of
> 2nds (e.g. LLsLLLs for the diatonic scale) or subsets of whatever
> temperament is used.
>
>
Can be said that temperament is historical term for tuning, as "to temper" means generally "to set, to modify", and "to tune" is more specific for frequency, pitch, musical use.

Both temperament and tuning are "physical" setting of the instrument.

"Scale" is musical abstract term, so it's on higher level than temperament or tuning. It's part of musical language and it doesn't have fixed connection with tuning (in the sense that even when we use a scale C-D-E-F..... in the score, we still don't know how these notes are tuned, there's always some difference between notes on paper and sounding tones). I would say it's subset of tuning, ordered list of steps, not intervals, as composer can work with almost any interval inside the possibilities defined by tuning and scale, scales tones can be used freely, not always in order up or down, so intervals are "hidden" in the scale, like another subset.

Scale can be created from steps of different size, in traditional music theory minimal size of step is small second, maximal size augmented second.

In 12ET and other historical tunings only chromatic scale is a special case when scale = tuning.

Scales don't need to have octave repetition even when tuning itself has octaves inside.

In traditional music theory abstract scales can be transposed (shifted) in the frame of tuning system, and can have more modes depending on number of possible rotations (different root notes). First condition is not perfectly applicable in practice on many scales with steps tuned microtonally as when we transpose, size of steps will change. Depending on scale and tuning such transposition will be possible only in limited way (but differently from Messiaen's term of "limited transpositions") or not possible at all. Second condition is applicable, even microtonal scales can have modes.
> Microtonal music theory needs much more general terms, since the
> scales and consonances being targeted are not taken as given.
> Gene Ward Smith has suggested distinct meanings for each of these
> three terms in order to make discourse about microtonal music
> theory more precise. This usage has seen wide adoption on the
> tuning, tuning-math, and to some extent the makemicromusic lists,
> but a broad consensus has not emerged. Here are Gene's
> definitions (paraphrased):
>
> scale- An ordered list of intervals, which may be applied to a
> given concert pitch to generate an ordered list of pitches that
> can be used to tune an instrument. Scala's scale file format
> (.scl extension) is an embodiment of this definition.
>
See above, I wouldn't use "interval" but step, and then in my opinion this could be good definition of tuning.
In Scala scale = tuning, which is not totally exact.
> temperament- A mapping from just intonation to an abstract group
> of smaller rank. This algebraic definition basically means that
> for every interval in just intonation, which is composed of some
> combination of primes, we express it instead as a combination of
> generators (called the generators of the temperament) such that
> there are fewer generators than there were primes. For example,
> in 5-limit JI, every interval can be expressed as a combination
> of 2s, 3s, and 5s. In meantone temperament, every interval can
> be expressed as a combination of octaves and fifths. 5-limit JI
> is therefore a rank-3 intonation, and meantone a rank-2
> temperament. In 12-ET, every interval can be expressed as a
> combination of semitones. It is therefore a rank-1 temperament.
>
Hm. If we don't use temperament = tuning, or reserve it just as a historical term, it can be said simply that temperament is a modification of some ideal abstract tuning system. Then the question is what can be considered ideal system. JI? 12ET?
> tuning- A choice of intervals for the generators of a
> temperament. For example, 1200 cents / 697 cents is a decent
> tuning for meantone. So is 1200 cents / 698 cents. They lead to
> different scales, but they are both good meantone tunings because
> they allow meantone temperament to produce good approximations
> of 5-limit JI. Sometimes tunings are named after the
> optimization they solve. For example, "RMS-optimal meantone"
> refers to the tuning that minimizes the RMS error from JI under
> the constraint of the meantone mapping.
>
Nice mix of tuning, temperament and scale... See above.

Tuning can use same size of steps, or similar to scales, can have steps of different size. They can be again some higher order in the relation of the different sizes and the way how these different steps are organized.

Just few ideas...

Daniel Forro

🔗Carl Lumma <carl@...>

Daniel wrote:
> "Scale" is musical abstract term, so it's on higher level than
> temperament or tuning.

Not in the microtonal realm. Because you can't get a diatonic
scale out of the father temperament
/tuning/database?method=reportRows&tbl=10

> > temperament- A mapping from just intonation to an abstract group
> > of smaller rank. This algebraic definition basically means that
> > for every interval in just intonation, which is composed of some
> > combination of primes, we express it instead as a combination of
> > generators (called the generators of the temperament) such that
> > there are fewer generators than there were primes. For example,
> > in 5-limit JI, every interval can be expressed as a combination
> > of 2s, 3s, and 5s. In meantone temperament, every interval can
> > be expressed as a combination of octaves and fifths. 5-limit JI
> > is therefore a rank-3 intonation, and meantone a rank-2
> > temperament. In 12-ET, every interval can be expressed as a
> > combination of semitones. It is therefore a rank-1 temperament.
>
> Hm. If we don't use temperament = tuning, or reserve it just as
> a historical term, it can be said simply that temperament is a
> modification of some ideal abstract tuning system. Then the
> question is what can be considered ideal system. JI? 12ET?

There are tools to answer this question, which have uncovered
a whole universe of temperaments, which Petr can tell you about
perhaps more easily than I. You see some in the table I linked
to above.

-Carl

🔗Daniel Forro <dan.for@...>

Thanks for a link.

Daniel Forro

On 1 Feb 2009, at 2:05 PM, Carl Lumma wrote:

> Daniel wrote:
> > "Scale" is musical abstract term, so it's on higher level than
> > temperament or tuning.
>
> Not in the microtonal realm. Because you can't get a diatonic
> scale out of the father temperament
> /tuning/database?method=reportRows&tbl=10
>
> > > temperament- A mapping from just intonation to an abstract group
> > > of smaller rank. This algebraic definition basically means that
> > > for every interval in just intonation, which is composed of some
> > > combination of primes, we express it instead as a combination of
> > > generators (called the generators of the temperament) such that
> > > there are fewer generators than there were primes. For example,
> > > in 5-limit JI, every interval can be expressed as a combination
> > > of 2s, 3s, and 5s. In meantone temperament, every interval can
> > > be expressed as a combination of octaves and fifths. 5-limit JI
> > > is therefore a rank-3 intonation, and meantone a rank-2
> > > temperament. In 12-ET, every interval can be expressed as a
> > > combination of semitones. It is therefore a rank-1 temperament.
> >
> > Hm. If we don't use temperament = tuning, or reserve it just as
> > a historical term, it can be said simply that temperament is a
> > modification of some ideal abstract tuning system. Then the
> > question is what can be considered ideal system. JI? 12ET?
>
> There are tools to answer this question, which have uncovered
> a whole universe of temperaments, which Petr can tell you about
> perhaps more easily than I. You see some in the table I linked
> to above.
>
> -Carl
>

🔗Mark Rankin <markrankin95511@...>

Carlos,

Your Saturday, January 31, 2009 9:05 PM e-mail to: tuning@yahoogroups.com
included a link to /tuningdatabase?method=reportRows@tbl=10

which yielded a nice table 10 columns wide and 5 pages long. When attempting to print out the table, however, I found that while it delivered the 5-page *length* of the table all right, in *width* it delivered only the first five and a half columns of the 10-column-wide-table.

It's obvious that the table is too wide to handle.

Do you or any of the tuning folks know how to make this link print out all 10 columns?

Thanks,

-- Mark Rankin

--- On Sat, 1/31/09, Carl Lumma <carl@...> wrote:

From: Carl Lumma <carl@...>
Subject: [tuning] Re: terminology
To: tuning@yahoogroups.com
Date: Saturday, January 31, 2009, 9:05 PM

Daniel wrote:
> "Scale" is musical abstract term, so it's on higher level than
> temperament or tuning.

Not in the microtonal realm. Because you can't get a diatonic
scale out of the father temperament
http://groups. yahoo.com/ group/tuning/ database? method=reportRow s&tbl=10

There are tools to answer this question, which have uncovered
a whole universe of temperaments, which Petr can tell you about
perhaps more easily than I. You see some in the table I linked
to above.

-Carl

"best average" near-ET scale

🔗djtrancendance <djtrancendance@...>

🔗Daniel Forro <dan.for@...>

🔗djtrancendance@...

🔗Daniel Forró <dan.for@...>

🔗Carl Lumma <carl@...>

🔗Michael Sheiman <djtrancendance@...>

🔗Chris Vaisvil <chrisvaisvil@...>

🔗Daniel Forro <dan.for@...>

🔗Carl Lumma <carl@...>

🔗Carl Lumma <carl@...>

🔗Carl Lumma <carl@...>

🔗Daniel Forro <dan.for@...>

🔗Mark Rankin <markrankin95511@...>

🔗Carl Lumma <carl@...>