Mavila -- preferences for tempering?

Hi everyone.

I've been experimenting with various stretched-octave versions of mavila but so far I don't have a chance to hear my results, I can only compare the numbers. Anyway, I would be interested in views of other people than myself as to if you have any particular preferences regarding these highly mistuned temperaments.

For unfamiliar readers, I'll quickly say that mavila is the temperament which more or less swaps major and minor intervals and where the limma of 135/128 is tempered out -- i.e. it can make a 7-tone "quasi-diatonic" scale of 5 + 2 intervals to the octave but the "notated" chord C-E-G approximates 10:12:15 while the notated C-Eb-G approximates 4:5:6 and therefore the notated major second is actually smaller than the notated minor second. For this reason, I'm going to use the term "primary step" for the notated major second and "secondary step" for the minor second in the following paragraphs.

One possibility with stretched octaves is, for example, to use a primary step of ~154.2753 cents and a secondary step of ~217.8571 cents. In this version, the 5/4 approximation is 2/13-limma narrower and the 6/5 approximation is 1/13-limma narrower, and the triad of 1:5:12 is untempered.

Another interesting example can be ~152.6996 cents and ~223.3720 cents. This time, both the 5/4 and 6/5 approximations are narrowed by 1/9-limma, and the triad of 2:5:12 is untempered. Although this may not be so obvious at the first glance, an important difference here is in the 3/1 approximation -- 1/9-limma narrower compared to 2/13-limma. The downside is that fourths are 1/3-limma wider, which I'm already finding a bit too wide for my ears, should I really consider it an approximation of 4/3.

Another nice compromise is ~152.0595 cents and ~222.7319 cents. In this example, both the 5/4 and 6/5 approximations are narrowed by 1/8-limma, and the triad of 1:5:24 is untempered. Understandably, among these particular examples, this one has the worst fifths. OTOH, the fourths are a bit smaller than in the second case (although still larger than in the first one). What's more, this version has the least mistuning in the 8/5 approximation, while the first one has the least mistuning in the 5/3 approximation.

Taking all of this into consideration, I'm finally getting to a conclusion that a very good "mean" of these three temperaments is the "simple" 2/7-limma mavila with pure octaves, which I've already used in one composition back in 2008. From another view, however, I'm still finding its fifths farily narrow, which is actually the primary reason why I got interested into the wider-octave variants in the first place. Of course, if my main aim is to get melodies with lots of unusual intervals, I could happily use fourths as large as, let's say, 4/9-octave. But if I want, above all, to get acceptable major/minor triads, I just have to change my way of thinking -- which is what I've probably just done.

As I've said earlier, I sadly can't try out my example temperaments now but I hope I'll eventually find a way to have a listen.

I appreciate any comments or suggestions from your side about this topic.

Petr

I wrote:

> But if I want, above
> all, to get acceptable major/minor triads, I just have to change my way of
> thinking -- which is what I've probably just done.

Maybe I should have said "recognizable triads". If an 8/5 of ~814 cents is approximated by ~835 cents, the recognizability is questionable.

Petr

Petr Pařízek wrote:
> Hi everyone.
> > I've been experimenting with various stretched-octave versions of mavila but > so far I don't have a chance to hear my results, I can only compare the > numbers. Anyway, I would be interested in views of other people than myself > as to if you have any particular preferences regarding these highly mistuned > temperaments.
> > For unfamiliar readers, I'll quickly say that mavila is the temperament > which more or less swaps major and minor intervals and where the limma of > 135/128 is tempered out -- i.e. it can make a 7-tone "quasi-diatonic" scale > of 5 + 2 intervals to the octave but the "notated" chord C-E-G approximates > 10:12:15 while the notated C-Eb-G approximates 4:5:6 and therefore the > notated major second is actually smaller than the notated minor second. For > this reason, I'm going to use the term "primary step" for the notated major > second and "secondary step" for the minor second in the following > paragraphs.
> > One possibility with stretched octaves is, for example, to use a primary > step of ~154.2753 cents and a secondary step of ~217.8571 cents. In this > version, the 5/4 approximation is 2/13-limma narrower and the 6/5 > approximation is 1/13-limma narrower, and the triad of 1:5:12 is untempered.
> > Another interesting example can be ~152.6996 cents and ~223.3720 cents. This > time, both the 5/4 and 6/5 approximations are narrowed by 1/9-limma, and the > triad of 2:5:12 is untempered. Although this may not be so obvious at the > first glance, an important difference here is in the 3/1 approximation -- > 1/9-limma narrower compared to 2/13-limma. The downside is that fourths are > 1/3-limma wider, which I'm already finding a bit too wide for my ears, > should I really consider it an approximation of 4/3.
> > Another nice compromise is ~152.0595 cents and ~222.7319 cents. In this > example, both the 5/4 and 6/5 approximations are narrowed by 1/8-limma, and > the triad of 1:5:24 is untempered. Understandably, among these particular > examples, this one has the worst fifths. OTOH, the fourths are a bit smaller > than in the second case (although still larger than in the first one). > What's more, this version has the least mistuning in the 8/5 approximation, > while the first one has the least mistuning in the 5/3 approximation.
> > Taking all of this into consideration, I'm finally getting to a conclusion > that a very good "mean" of these three temperaments is the "simple" > 2/7-limma mavila with pure octaves, which I've already used in one > composition back in 2008. From another view, however, I'm still finding its > fifths farily narrow, which is actually the primary reason why I got > interested into the wider-octave variants in the first place. Of course, if > my main aim is to get melodies with lots of unusual intervals, I could > happily use fourths as large as, let's say, 4/9-octave. But if I want, above > all, to get acceptable major/minor triads, I just have to change my way of > thinking -- which is what I've probably just done.
> > As I've said earlier, I sadly can't try out my example temperaments now but > I hope I'll eventually find a way to have a listen.
> > I appreciate any comments or suggestions from your side about this topic.
> > Petr

One of the things I look for in a tuning of a temperament is consistency: how far you can go in a chain of the temperament without running into better approximations of particular intervals. I did some charts a while back and found that temperaments often have more than one region of better consistency within broader areas of lower consistency. (See /tuning-math/files/Rank%202%20Consistency/ for a few of these charts, in particular mavila5.png)

For mavila, the most consistent tunings have fourths wider than 522 cents, which is very wide for a fourth, but within the same range as your suggested tunings. The large pink area at the top of the chart is a range where mavila is not consistent enough to be useful. Stretching the octaves does give you better consistency in some cases, as well as improving the fifths as you've noted. Which one sounds better will depend to a large extent on the timbres used.

Petr,

Mavila is a linear temperament? To give the properties that you mentioned,
is the generator somewhere in the vicinity of the 4th root of 24/5, or
something like that?

-Mike

2010/3/30 Petr Pařízek <p.parizek@...>

>
>
> Hi everyone.
>
> I've been experimenting with various stretched-octave versions of mavila
> but
> so far I don't have a chance to hear my results, I can only compare the
> numbers. Anyway, I would be interested in views of other people than myself
>
> as to if you have any particular preferences regarding these highly
> mistuned
> temperaments.
>
> For unfamiliar readers, I'll quickly say that mavila is the temperament
> which more or less swaps major and minor intervals and where the limma of
> 135/128 is tempered out -- i.e. it can make a 7-tone "quasi-diatonic" scale
>
> of 5 + 2 intervals to the octave but the "notated" chord C-E-G approximates
>
> 10:12:15 while the notated C-Eb-G approximates 4:5:6 and therefore the
> notated major second is actually smaller than the notated minor second. For
>
> this reason, I'm going to use the term "primary step" for the notated major
>
> second and "secondary step" for the minor second in the following
> paragraphs.
>
> One possibility with stretched octaves is, for example, to use a primary
> step of ~154.2753 cents and a secondary step of ~217.8571 cents. In this
> version, the 5/4 approximation is 2/13-limma narrower and the 6/5
> approximation is 1/13-limma narrower, and the triad of 1:5:12 is
> untempered.
>
> Another interesting example can be ~152.6996 cents and ~223.3720 cents.
> This
> time, both the 5/4 and 6/5 approximations are narrowed by 1/9-limma, and
> the
> triad of 2:5:12 is untempered. Although this may not be so obvious at the
> first glance, an important difference here is in the 3/1 approximation --
> 1/9-limma narrower compared to 2/13-limma. The downside is that fourths are
>
> 1/3-limma wider, which I'm already finding a bit too wide for my ears,
> should I really consider it an approximation of 4/3.
>
> Another nice compromise is ~152.0595 cents and ~222.7319 cents. In this
> example, both the 5/4 and 6/5 approximations are narrowed by 1/8-limma, and
>
> the triad of 1:5:24 is untempered. Understandably, among these particular
> examples, this one has the worst fifths. OTOH, the fourths are a bit
> smaller
> than in the second case (although still larger than in the first one).
> What's more, this version has the least mistuning in the 8/5 approximation,
>
> while the first one has the least mistuning in the 5/3 approximation.
>
> Taking all of this into consideration, I'm finally getting to a conclusion
> that a very good "mean" of these three temperaments is the "simple"
> 2/7-limma mavila with pure octaves, which I've already used in one
> composition back in 2008. From another view, however, I'm still finding its
>
> fifths farily narrow, which is actually the primary reason why I got
> interested into the wider-octave variants in the first place. Of course, if
>
> my main aim is to get melodies with lots of unusual intervals, I could
> happily use fourths as large as, let's say, 4/9-octave. But if I want,
> above
> all, to get acceptable major/minor triads, I just have to change my way of
> thinking -- which is what I've probably just done.
>
> As I've said earlier, I sadly can't try out my example temperaments now but
>
> I hope I'll eventually find a way to have a listen.
>
> I appreciate any comments or suggestions from your side about this topic.
>
> Petr
>
>
>

--
-Mike

Hi Mike.

Of course, this is one possible generator -- actually, the third of my example versions uses this.
For the pure-octave version, I prefer using the 7th root of 25/3 in place of a fourth -- i.e. both thirds (major and minor) are less mistuned than fifths and fourths.

Petr