Why are scales with a minimal number of different intervals more easy to sing?

Hi,

At various places I read that reducing the number of different intervals in a scale makes it more easy to sing. For example, this notion is one of the underlying principles in Paul Erlich's writing which led to his decatonic scales. Also, Carl mentions it in his too-condensed-tuning-math-outline. I wonder on what evidence this observation is based. Is this personal experience? Or have there been any empirical studies conducted?

I simply would like to better understand this idea, because it does not convince me spontaneously and I bet you guys gave this matter some though. For example, following this idea 5-limit music in 12 ET would be more easy to sing than in JI. Is this really so?

Thank you!

Best
Torsten

--
Torsten Anders
Interdisciplinary Centre for Computer Music Research
University of Plymouth
Office: +44-1752-586219
Private: +44-1752-558917
http://strasheela.sourceforge.net
http://www.torsten-anders.de

2008/8/2 Torsten Anders <torsten.anders@...>:
> Hi,
>
> At various places I read that reducing the number of different
> intervals in a scale makes it more easy to sing. For example, this
> notion is one of the underlying principles in Paul Erlich's writing
> which led to his decatonic scales. Also, Carl mentions it in his too-
> condensed-tuning-math-outline. I wonder on what evidence this
> observation is based. Is this personal experience? Or have there been
> any empirical studies conducted?

Carl's outline is here:

http://www.lumma.org/music/theory/tctmo/

He does say that. I don't agree with it. It may be that scales with
two step sizes are easier to hear in some sense. There haven't been
many empirical studies about any of this.

Having two step sizes can minimize the number of notes that change
with a modulation. With an octave-based MOS only one note changes.
It's been suggested in different places that this is important for
tonal music.

> I simply would like to better understand this idea, because it does
> not convince me spontaneously and I bet you guys gave this matter
> some though. For example, following this idea 5-limit music in 12 ET
> would be more easy to sing than in JI. Is this really so?

No, and I'd expect JI to be easier, but not with any evidence. At
some level that diatonic scale's going to be heard with two step
sizes, though, whatever the precise tuning.

Graham

I think it has more to do with the capacity of the human memory than
the functioning of the ear. The intervals of 12-tet are pretty easy to
memorize. Nonetheless, a vocal group that has rehearsed together for a
long time will intuitively learn to sing that major third 14 cents
flat to get a shining, beatless chord of justice. But it's easier to
think about a "whole step" rather than think in terms of the "major
and minor whole tones", if you know what I mean.

It's like what I was saying a while ago about the modes: All of them
are sort of a "blueprint" for a certain sound. But when played live,
with an orchestra, there will be little comma adjustments here and
there so that in D dorian, the top D in the chord D F A C E G B D is
different than the lowest D; it's boosted up by 81/08. But the
musicians are still likely going to treat it like it's 2 octaves above
the root, play it on their violins or sing it or what not, and then
boost that top note 20 cents just because it sounds better. They'd
probably think they're singing the octave flat or something.

-Mike

Hi Torsten,

I'll answer backwards.

> I simply would like to better understand this idea, because it
> does not convince me spontaneously and I bet you guys gave this
> matter some though. For example, following this idea 5-limit
> music in 12 ET would be more easy to sing than in JI. Is this
> really so?

As long as we're talking about unaccompanied melodies alone,
yes.

> At various places I read that reducing the number of different
> intervals in a scale makes it more easy to sing. For example,
> this notion is one of the underlying principles in Paul Erlich's
> writing which led to his decatonic scales. Also, Carl mentions
> it in his too-condensed-tuning-math-outline. I wonder on what
> evidence this observation is based. Is this personal experience?
> Or have there been any empirical studies conducted?

I'm not aware of any studies, but I can offer you anecdotal
evidence. Paul Erilch, myself, and Rothenberg seem to agree.
And according to Paul, "Even Mathieu admits that a 12-tone
chromatic scale is melodically smoother in 12tET than in an
unequal tuning."

Monz provides the following from a Ben Johnston article,
"Scalar Order as a Compositional Resource"
Perspectives of New Music vol.2 #2, Spring-Summer '64, p.59

"The two conflicting criteria which condition this are simplicity
and symmetry: that is, a preference for simplicity or consonance
of harmonic pitch ratios, and a preference for dividing melodic
intervals symmetrically, into "equal" smaller intervals.
...
"Harmonic listening is too easy and too basic to be ignored, even
in purely melodic music. Yet melodic preference for equal scale
intervals is also strong. If a scale is derived harmonically, it
must consist of intervals whose melodic sizes differ by what seems
a negligible amount. What seems negligible depends mostly upon
relative sizes but also upon cultural conditioning and upon "how
good an ear" an individual listener has."

-Carl

Torsten wrote:
> I simply would like to better understand this idea,

The phenomenon is very simply explained. Humans judge
melodic jumps that represent equal log-pitch differences
to be of equal size, and we are generally capable of
hearing transposition as a symmetry. Therefore, the
fewer different interval sizes in each interval class,
the more numerous will be subsections of a melody that
are identical under transposition.

In 12-ET, the phrase D-E-F-A-B-C consists of the congruent
subsections D-E-F and A-B-C (among others). In the
5-limit intense diatonic scale, the symmetry is slightly
imperfect (10/9 - 16/15 vs. 9/8 - 16/15) and this is
audible. One may prefer it or not, but one cannot deny
the difference. With adaptive JI, melodic regularity and
pure harmonies should be simultaneously realizable in
almost every case.

Tetrachordality ("omnitetrachordality" in Paul Erlich's
work) is explained the same way, except transpositions
are deemed more effective if they are at a 5th. This
might be refined by weighting each possible transposition
by the harmonic complexity of the associated interval.
But simply minimizing the variety of different interval
sizes in the scale should improve such symmetries across
the board.

That's a lot of words but a very simple idea. These
scales are easier to sing in because when you come to a
certain part, you're more likely to have sung it before!

-Carl