** copyright © 2002 by Larry J Solomon**

**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.

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:

- With the dry-erase marker, mark the pitch-classes (pc) onto the inner, rotating disc, with each next to its pitch name or number on the outer circle. One can use an X to mark each pc.
- Rotate the disc until the largest gap between the marks lies to the left (counterclockwise) of the twelve o'clock (zero or C) position. Unless there is another gap just as wide, this is the Solomon prime, and may be read off the circle just as the numbers of a clock. Remember that 12 o'clock is read as zero in set-theoretic terms.
- If there is another gap just as large as the one found in step 1b, then notate the configuration you see as a possible prime-form and then rotate the clock to place the other large gap before the 12:00 position. Compare this configuration with the one previously notated. The one with the most compaction clockwise from zero and nearest to the zero is the prime-form.
- Example: Determine the set-name of A B C D# E. Mark the positions of these on the inner disc. Rotate the disc until the largest gap is to the left of 12:00 (this moves the A-mark to the C position). The prime form is read as 02367.

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:

- Find and notate the prime-form from the operations under number 1 above, turn the clock over to read the opposite side. This side displays the set's "inverse".
- Find the prime-form of the inverse by using the same procedure as in number 1 above. Compare this with the notated, original prime you found. The most compact of the two is Forte's "best normal order", or Forte prime, which can then be located and identified in his list (see Appendix of SAM). If the prime and inverse forms are identical, the set is a mirror.

3. Determining the set-name and set properties:

- Using the
*Table of Set Classes*, locate the prime-form in the list under the cardinality of the set; i.e., if the set has five pcs, then its name will appear in the list under the prefix "5-". This list contains much other information on the set's properties, including the interval-vector, mirror properties (an asterisk [*] in the set-name indicates the set is a mirror), complement identification (complement's normally have the same ordinal number-- see #7 below), and a descriptive or common name. - Example: The above set example, 02367, is looked up on the set list as a five note set, and therefore has "5-" at the beginning of its name. We find it listed as 5-Z18B<, the "Gypsy pentachord.2" with an interval-vector of 212221.

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

- Mark the smaller set on the calculator as in step number 1.
- Mark the larger set outside of the movable disc, i.e., near the pc names on the stationary part of the calculator.
- Circle one of the marks of the larger set and one from the smaller set to be used as references.
- Rotate the inner disc, placing each mark on it opposite the referenced pc on the stationary disc. If any of these positions show that all marks on the inner disc correspond with those on the stationary disc, the sets have the subset relation.
- Example: Mark an incomplete dominant seventh chord on the inner disc, such as F# A# E. Circle the X mark at F#. Mark a complete dominant seventh chord on the stationary part of the disc; e.g, G B D F. Circle the X marked at G. Rotate the disc so that the circled mark on the inner circle is next to the G, marked on the outer circle. This position shows that every mark on the inner disc has a correspondent on the outer circle. Therefore, the two sets have the subset, or "inclusion" relation.

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

- Mark the set on the inner disc. Circle one pc as a reference.
- Starting with the circled pc, place each marked pc in the 12:00 position and read off the other marks on the circle as intervals up to, but not including, the circled pc.
- Convert the intervals to interval-classes (ic) and tally them as you read them. All intervals up to number 6 (tritones) are equivalent to an ic, so that they are simply counted as they are. Any interval larger than 6 is read as the complement of twelve on the circle by reading it counterclockwise from the 12:00 position; e.g., 7 is read as a 5, counterclockwise.

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.