proper, propriety

A function defined over the interval matrix of a scale, revealing the perceptual properties of that scale.

Strictly Proper scales are those in which all intervallic size classes are distinct and non-overlapping.

Scales in which two or more pairs of interval classes contain identical members are Proper.

Scales with overlapping or contradictory interval classes are Improper and are less likely to be perceived as musical gestalts than proper or strictly proper scales.

This concept of Propriety was introduced by David Rothenberg (Rothenberg, 1969).

[from John Chalmers, Divisions of the Tetrachord]

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The following was submitted by Robert Walker in Yahoo Tuning Group, Message 21421:

From: "Robert Walker"
Date: Sun Apr 22, 2001 10:43 pm
Subject: Re: Propriety

Hi Monz,

I wonder if your propriety page may need some clarification for mathematicians.

"Strictly Proper scales are those in which all intervallic size classes are distinct and non-overlapping."

I took it in the set theoretic sense, that two classes overlap if they have members in common, translating "overlap" as "set intersection".

So, came to the preliminary, somewhat puzzling conclusion that you could have a scale that is strictly proper, and also improper.

However here it means that A and B overlap if B has a member smaller than one of the members of A.

Once I realised that, all became clear.

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From Dave Keenan:
Yahoo Tuning-Math List, message 918 (Tue Aug 28, 2001 9:24pm)

A scale is proper if all intervals spanning the same number of scale steps, have a range of sizes (in cents) that does not overlap but may meet, the range of sizes for any other number of scale steps. A scale is strictly-proper if all intervals spanning the same number of scale steps, have a range of sizes (in cents) that is disjoint from (does not meet or overlap), the range of sizes for any other number of scale steps.

Examples.

1. Improper
Number of steps in interval                               4
              1                2                  3 <---------->
Ranges   <-------->        <-------->        <---------->
|        |        |        |        |        |        |        |
0       100      200      300      400      500      600      700 etc.
Interval size (cents)

2. Proper
Number of steps in interval
              1                2                  3       4
Ranges   <-------->        <-------->        <--------x-------->
|        |        |        |        |        |        |        |
0       100      200      300      400      500      600      700 etc.
Interval size (cents)

3. Strictly proper
Number of steps in interval
              1                2                  3       4
Ranges   <-------->        <-------->        <-------> <------->
|        |        |        |        |        |        |        |
0       100      200      300      400      500      600      700 etc.
Interval size (cents)

. . . . . . . . . . . . . . . . . . . . . . . . .

See also:

Lumma impropriety
Lumma stability
Rothenberg efficiency
Rothenberg stability
Rothenberg redundancy
MOS (moment of symmetry)

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Updated: 2002.2.4

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