Constructing a Simple Clock Calculator for Music Set Theoretical Analysis

copyright © 2002 by Larry J Solomon

Table of Contents

Caution: Set calculators found on the web are convenient and yield good results most of the time, but their algorithms have been found to be defective. A good test is to input the set 0,2,4,7,8,11 (C,D,E,G,G#,B). The correct prime should read 013589. If the set calculator yields a different result, it is defective. The following calculator, constructed from simple everyday materials is always accurate when used correctly, and it can be carried conveniently anywhere, flat, in a notebook, and it requires no power.

1. How to Construct the Calculator

Materials: 3 pieces 8½ x 11" light-colored, heavy card-stock paper, a sharp knife, a small brass paper fastener, an 8½ x 11" vinyl sheet protector, Xacto knife, scissors, and permanent marker/s.

1. Use a permanent marker for the following marks: With a compass, mark a 6" circle in the center of one of the pieces of card stock, which is to become the clock. Also mark the center of the circle. Make hour-lines extending on both sides of the circle (see picture) with a permanent marker; these must be placed accurately. It is best to use a protractor with marks placed every 30 degrees around the circle. Mark the pitch names around the outside of the circle opposite the hour lines. Slip the clock into an 8½" by 11" vinyl sheet protector

2. Protect your working surface with a piece of card stock, cardboard, or newspaper placed underneath the whole assembly. Cut out a sixteenth-inch central circle through ALL layers of the vinyl and paper. Discard the small center pieces.

3. Place a piece of the card stock into the sheet protector under the clock. Using an Xacto razor-knife, or scissors, carefully and smoothly cut out the 6 inch disc, but only through the upper two layers; i.e., through the upper layer of vinyl and through the clock paper, but NOT through the lower layer of vinyl (the cardboard only serves to protect the lower layer). This circle must be cut as smoothly as possible through the vinyl layer.

4. Discard the cut-out paper disc and the protective stock beneath it. Keep the vinyl disc.

5. Place the clear vinyl disc back onto the clock, centering it. There should now be a large transparent disc in the center of the clock. (See above picture) Place a small brass paper fastener through the small central hole and through the back. Bend back the tails of the fastener to secure the front and back of the calculator. It is best to cut the tails of the fastener to a length of ¼" with scissors.

6. This completes the assembly of the set-theory clock calculator. The central transparent disc should rotate smoothly around the fastener. As the disc is rotated the hour-lines should line up in every position.

2. How to Use the Calculator

Tools needed: Clock Calculator, kleenex or handkerchief, a "dry-erase" marker, paper and pencil. The kleenex tissue is used to wipe off the marks that are made on the face of the calculator between analyses; this is done in order to freshly start each new set analysis. Don't leave marks on the face of the calculator, for if they are left for too long, they will be difficult to remove. Place a plain white sheet of paper or card stock behind the calculator when using it.

1. To obtain the Solomon prime:

2. To find the Forte Prime and set name (Appendix 1 of SAM), you must determine the "best normal order" (BNO) before you can determine his prime and set-name. To do this:

3. Determining the set-name and set properties:

4. Determining if one set is a subset of another; i.e., determining if one set is contained inside another.

5. Determining the interval-vector without a set list.

6. Similarity relations of sets of equal cardinality can be determined by comparing their interval-vectors. See the definitions of these relations in the Set Theory Glossary.

7. Complements may be determined by marking the clock with all the pitches that are not in a given set, and then finding its Prime form.

Complements can also be read from the set list as the set-name with the same ordinal number but whose cardinal number is the complement of 12; e.g., 4-27 has a complement that is set 8-27. If you are using the Table of Set Classes and there is no "B" at the end of the set-name, add a "B" at the end of the complement's name ; i.e., the complement of 4-27 is 8-27B. Non-Z Hexachords are their own complements, one with a "B" ending and the other without when the set is not a mirror; however, if the set has a "Z" in its set-name, the complement's ordinal designation is the number following two dots near the end of the set-name; e.g., 6-Z12..41B has a complement that is 6-Z41B. A set whose name ends with a "<" indicates that the complement name has the same ending; e.g., the example in step 2, above, 5-Z18B< has a complement that is 7-Z18B<, both with B< endings. Non-hexachordal sets that have the double dots near the end of the set-name designate their Z counterpart as the number following the dots; e.g., 4-Z15..29, means that set 4-Z29 has the same interval vector as 4-Z15.

8. Row structure can also be analyzed with this calculator by placing ordinal numbers (e.g., 1-12) instead of Xs on the inner disc. Analysis of such ordered-sets can be easily done; e.g., a transposition of any inversion of a row is read on the opposite side of the disc as numbers 1-12. The RI form can be read on the opposite side as 12-1. Other properties, such as the directed-interval structure and ordered subsets can be read with the calculator.