Lipps–Meyer Law

Hi Members,

I was brushing up on some composition skills today and ran across a term I was not familiar with. Its called, "Interval Strength". Not quite sure what this is defined as. I'm sure this question can be applied to other tunings as well besides 12tEQ.

The links below try to explain the term "Interval Strength" but I don't understand. I searched the net for awhile but came up empty as far as a more detailed explanation. .

I clearly understand their mentioned terms "effect of indicated continuation" and "effect of finality" but it's the relationship of the note names that confuse me.

Both definitions (below) are referring to two different scenarios. Very confusing. I believe it would be easier to understand if they referred to actual frequencies ("A" = 440 etc.) versus letter names.

If anyone can please chime in and help clear up my confusion I would much appreciate the help.

(Definition #1)

Lipps–Meyer law

From Wikipedia, the free encyclopedia

The Lipps–Meyer law, named for Theodor Lipps (1851–1914) and Max F. Meyer (1873–1967), hypothesizes that the closure of melodic intervals is determined by "whether or not the end tone of the interval can be represented by the number two or a power of two",[1][verification needed] in the frequency ratio between notes.

"The 'Lipps-Meyer' Law predicts an 'effect of finality' for a melodic interval that ends on a tone which, in terms of an idealized frequency ratio, can be represented as a power of two."[2]

Thus the interval order matters — a perfect fifth, for instance (C,G), ordered <C,G>, 2:3, gives an "effect of indicated continuation", while <G,C>, 3:2, gives an "effect of finality".

This is a measure of interval strength or stability and finality. Notice that it is similar to the more common measure of interval strength, which is determined by its approximation to a lower, stronger, or higher, weaker, position in the harmonic series.

Source: http://en.wikipedia.org/wiki/Lipps%E2%80%93Meyer_law

(Definition #2)

Interval strength

David Cope (1997) suggests the concept of interval strength,[5] in which an interval's strength, consonance, or stability (see consonance and dissonance) is determined by its approximation to a lower and stronger, or higher and weaker, position in the harmonic series. See also: Lipps–Meyer law.

Thus, an equal tempered perfect fifth ( play (help·info)) is stronger than an equal tempered minor third ( play (help·info)), since they approximate a just perfect fifth ( play (help·info)) and just minor third ( play (help·info)), respectively. The just minor third appears between harmonics 5 and 6 while the just fifth appears lower, between harmonics 2 and 3.

Source: http://uk.ask.com/wiki/Harmonic_series_%28music%29#Interval_strength

Thank you
Wally

--- In tuning@yahoogroups.com, "Wally" <soundmaker@...> wrote:
>
> Hi Members,
>
> I was brushing up on some composition skills today and ran across a term I was not familiar with. Its called, "Interval Strength". Not quite sure what this is defined as. I'm sure this question can be applied to other tunings as well besides 12tEQ.
>
> The links below try to explain the term "Interval Strength" but I don't understand. I searched the net for awhile but came up empty as far as a more detailed explanation. .
>
> I clearly understand their mentioned terms "effect of indicated continuation" and "effect of finality" but it's the relationship of the note names that confuse me.

I'm not exactly sure what your confusion is, but I'll say some stuff that might help.

I'm going to use cents in my explanation. See http://en.wikipedia.org/wiki/Cent_%28music%29 if you're unfamiliar with it. Everybody uses cents and cents are your friend.

> Both definitions (below) are referring to two different scenarios. Very confusing. I believe it would be easier to understand if they referred to actual frequencies ("A" = 440 etc.) versus letter names.

There's nothing wrong with letter names. The interval C-G is 7 semitones, so it's 700 cents, and once you learn about cents it should be easy to convert that into a frequency ratio. The answer is 1.49830708. That means that whatever frequency C is, G is 1.49830708 times that. The specific base frequency you're using doesn't matter.

> If anyone can please chime in and help clear up my confusion I would much appreciate the help.
>
> (Definition #1)
>
> Lipps–Meyer law
>
> From Wikipedia, the free encyclopedia
>
> The Lipps–Meyer law, named for Theodor Lipps (1851–1914) and Max F. Meyer (1873–1967), hypothesizes that the closure of melodic intervals is determined by "whether or not the end tone of the interval can be represented by the number two or a power of two",[1][verification needed] in the frequency ratio between notes.
>
> "The 'Lipps-Meyer' Law predicts an 'effect of finality' for a melodic interval that ends on a tone which, in terms of an idealized frequency ratio, can be represented as a power of two."[2]
>
> Thus the interval order matters — a perfect fifth, for instance (C,G), ordered <C,G>, 2:3, gives an "effect of indicated continuation", while <G,C>, 3:2, gives an "effect of finality".

Here it says that C and G have the ratio 2:3, which might be confusing you because above I said that the ratio between C and G is 1.49830708, which is not the same as 2:3 (which is 1.5). This is because an equal tempered fifth is exactly 700 cents, but a just intonation fifth is 701.955001 cents.

However, this article is not incorrect to associate C-G with the ratio 2:3, because the human perception of intervals is fuzzy and any interval within about 10 or 20 cents of 3/2 will be heard as 3/2. That is to say, if I play for you a 3/2 interval and then a 700-cent interval, the 700-cent interval will give you the same musical impression as the 3/2 - it sounds like a perfect fifth. The only difference is that you might notice a slight beating of the 700-cent interval, because the 3rd harmonic of the lower note is beating with the 2nd harmonic of the upper note.

The main point is that 700 cents cannot be radically different in consonance than 701.955001 cents.

> This is a measure of interval strength or stability and finality. Notice that it is similar to the more common measure of interval strength, which is determined by its approximation to a lower, stronger, or higher, weaker, position in the harmonic series.
>
> Source: http://en.wikipedia.org/wiki/Lipps%E2%80%93Meyer_law
>
> (Definition #2)
>
> Interval strength
>
> David Cope (1997) suggests the concept of interval strength,[5] in which an interval's strength, consonance, or stability (see consonance and dissonance) is determined by its approximation to a lower and stronger, or higher and weaker, position in the harmonic series. See also: Lipps–Meyer law.
>
> Thus, an equal tempered perfect fifth ( play (help·info)) is stronger than an equal tempered minor third ( play (help·info)), since they approximate a just perfect fifth ( play (help·info)) and just minor third ( play (help·info)), respectively. The just minor third appears between harmonics 5 and 6 while the just fifth appears lower, between harmonics 2 and 3.

This seems like a clear explanation to me. An equal tempered fifth approximates 3/2 and an equal tempered minor third approximates 6/5.

Another thing that might be confusing you is how you tell which specific just interval a given equal tempered interval is "approximating". For example, the tritone, 600 cents, is close to 7/5 (582.512193 cents), but it's equally close to 10/7 (617.487807 cents). So which one is the tritone "approximating"?

The correct answer is, of course, both of them. Both 7/5 and 10/7 are less than 20 cents away from 600, and both of them are also simple enough fractions that they make an important contribution to the consonance of the 600 cent tritone (as opposed to, say, 41/29, which is too complex to matter). Thus we say that the difference between 7/5 and 10/7, i.e. 50/49, is a comma that is tempered out of 12-tone equal temperament.

Here's a table of all 12-equal intervals with their related just intervals and relative consonance:

P1 (0 cents): {1/1}, unison
m2 (100 cents): {15/14, 16/15, 17/16, 18/17, 19/18, 20/19, 21/20}, strong dissonance
M2 (200 cents): {10/9, 9/8, (8/7)}, mild (borderline) dissonance
m3 (300 cents): {(7/6), 6/5}, consonance
M3 (400 cents): {5/4, (14/11, 9/7)}, consonance
P4 (500 cents): {4/3}, strong consonance
A4/d5 (600 cents): {7/5, 10/7}, mild dissonance
P5 (700 cents): {3/2}, strong consonance
m6 (800 cents): {(14/9, 11/7), 8/5}, consonance
M6 (900 cents): {5/3, (12/7)}, consonance
m7 (1000 cents): {(7/4), 16/9, 9/5}, mild dissonance
M7 (1100 cents): {15/8, 17/9, 19/10}, strong dissonance
P8 (1200 cents): {2/1}, strong consonance

Ratios in parentheses may be too far away from the actual interval, or too complex, to contribute to its perception, but it's debatable.

For more about a specific mathematical model of this, read about "harmonic entropy", for example here: http://www.soundofindia.com/showarticle.asp?in_article_id=1905806937

My apologies if I'm telling you things you already know, or answering the wrong questions, but as I said, I'm not sure what your specific point of confusion is.

Keenan

On Sun, Oct 16, 2011 at 8:44 PM, Wally <soundmaker@...> wrote:
>
> Hi Members,
>
> I was brushing up on some composition skills today and ran across a term I was not familiar with. Its called, "Interval Strength". Not quite sure what this is defined as. I'm sure this question can be applied to other tunings as well besides 12tEQ.
>
> The links below try to explain the term "Interval Strength" but I don't understand. I searched the net for awhile but came up empty as far as a more detailed explanation. .

It's not a term with an explicit, well-defined, single common use as
far as I know. Just sounds like floaty-sounding jargon that people
might use to be colorful sometimes.

> (Definition #1)
>
> Lipps–Meyer law
>
> From Wikipedia, the free encyclopedia
>
> The Lipps–Meyer law, named for Theodor Lipps (1851–1914) and Max F. Meyer (1873–1967), hypothesizes that the closure of melodic intervals is determined by "whether or not the end tone of the interval can be represented by the number two or a power of two",[1][verification needed] in the frequency ratio between notes.
//snip

This must be relative to the tonic, or something, or else it doesn't
make sense. If we're saying that the tonic note is "1", then all
octave-equivalent notes to the tonic will be some power of 2. Assuming
that's what this law means, then it just states that melodic motions
that end on the tonic will be heard as the most "final," or something
like that.

This paper from 1926 (!) sounds incredibly interesting:

http://psycnet.apa.org/journals/xge/9/3/253/

It's not free on this site, but maybe you can hunt it down.

> David Cope (1997) suggests the concept of interval strength,[5] in which an interval's strength, consonance, or stability (see consonance and dissonance) is determined by its approximation to a lower and stronger, or higher and weaker, position in the harmonic series. See also: Lipps–Meyer law.
>
> Thus, an equal tempered perfect fifth ( play (help·info)) is stronger than an equal tempered minor third ( play (help·info)), since they approximate a just perfect fifth ( play (help·info)) and just minor third ( play (help·info)), respectively. The just minor third appears between harmonics 5 and 6 while the just fifth appears lower, between harmonics 2 and 3.
>
> Source: http://uk.ask.com/wiki/Harmonic_series_%28music%29#Interval_strength

This has to do with interval consonance, also called "concordance."
Harmonic Entropy models this pretty well, although it should be
understood that it's a very preliminary model that ignores any concept
of musical training or other learned factors affecting the perception
of the listener. Other than that, I think Keenan addressed this pretty
well, so I'll leave it there.

-Mike