Bench Degree·MUSICchapter

Appendix A: The Bench

The whole volume’s kit in one place. Almost all of it is already in your pocket or on a shelf; the one thing you may need to build is a single string on a board.

One string carries the book. A guitar’s G string, or the monochord below, gives you the chord of Chapter 1, the harmonic ladder of Chapter 3, the ratios of Chapter 5, the Pythagorean scale of Chapter 6 and the bugle of Chapter 26. A phone carries most of the rest, on free apps. Everything else is a keyboard you can open in a browser, a notation program that costs nothing, recordings, and your own voice.

Nothing here is specified by brand. Every item and every app is described by what it does, so the list stays usable for as long as the book exists.

Section 1: The kit

Total: nothing, if you own a guitar and a phone. About $15 if you build the monochord instead.

Section 2: The free apps, by function

Volume. A pure sine tone in headphones is more harmful than it sounds, because it lacks the harshness that warns you to turn it down. Square and sawtooth waves sound much louder than a sine at the same setting. Start every experiment quiet and turn up.

Section 3: Building a monochord

A monochord is one string stretched between two fixed bridges over a board, with a movable bridge to divide it (Chapter 5). This one is built to the same 648 mm (25.5 in) vibrating length as a standard steel-string guitar, so every table in the book applies to it unchanged.

Steps:

1. Mark the two fixed-bridge positions 648 mm (25.5 in) apart, each about 50 mm (2 in) from an end of the board.
2. At the left end, drill a pilot hole and drive the anchor screw at a slight backward angle, leaving its head standing about 6 mm (1/4 in) proud. Drive it at least 25 mm (1 in) into solid wood. A screw in a shallow hole or near a split will pull out under load.
3. At the right end, fit the tuning machine with its own screws, or build the eye-bolt block and fit the bolt, washer and wing nut.
4. Glue the two dowels across the board at the marks, or simply let the string’s tension hold them in place.
5. Glue the scale beside the string, its zero exactly at the left bridge.
6. Hook the string’s ball end over the anchor, lay it over both bridges, and wind it onto the tuning post or thread it through the eye bolt.
7. Put on safety glasses. Tighten slowly, checking the tuner, until the open string reads G at 196 Hz, the guitar’s third string and the note of Chapter 1. Go no higher than A at 220 Hz. Tension rises with the square of the frequency, so that small step adds a quarter more pull.
8. Slide the pencil under the string. The two lengths either side of it now sound separately; the scale tells you the fraction.

Set the board on a table, or across an open cardboard box, and it will sound louder.

Hazards. A guitar string at pitch pulls with roughly 7 to 14 kg (15 to 30 lb), depending on the string. If it breaks or a screw lets go, the free end whips. Wear safety glasses whenever you tune it up, keep faces away from the string’s line, and do not let a child tighten it. The cut end of a steel string is sharp: bend it back on itself or cover it with tape. If the board visibly bows, it is too thin; the pitch will drift and the screws are under more strain than they should be. The pencil raises the string’s tension a little as it is pushed under, so notes on the monochord can read a few cents sharp of the tables. That is a property of the instrument, not a mistake in your measuring.

Section 4: Every experiment in the book

Section 5: A sample answer to the Chapter 17 exercise

The cantus firmus is D F E D G F E D in whole notes, starting on the D just above middle C. Here is one counterpoint above it that obeys every rule of Chapter 17. It is not the only right answer; there are dozens.

Upper voice: D C E F E D C sharp D, starting on the D an octave above the cantus firmus.

Check it against the rules, one at a time.

1. Start on a perfect consonance. It opens on an octave. End on the octave, approached by step. It closes on an octave, and the upper voice arrives from C sharp, a half step below.
2. Only consonances. Every interval in the table is an octave, a fifth, a tenth (a compound third) or a sixth. No seconds, fourths or sevenths.
3. No parallel fifths or octaves. There is only one fifth, at note 2, so no two fifths can follow each other. The octaves at notes 1, 3 and 8 are each reached by contrary motion, and none follows another directly. The only parallel motion is the run of sixths at notes 5 to 7, and parallel sixths are allowed: they are imperfect consonances, which do not fuse the voices.
4. Contrary motion preferred. Five of the seven moves are contrary. Steps preferred. The upper voice moves by step everywhere except one leap of a third, C up to E, and its single high point, F, comes once.

Why C sharp. The written mode is Dorian on D, which has C natural. With C natural at note 7, the interval over E would be a minor sixth, still a consonance, and the exercise would still pass rule 2. But a sixteenth-century singer would have raised it by musica ficta (Chapter 11), so that the upper voice leans a half step into the final, and the major sixth opens outward to the octave. That move, a major sixth expanding to an octave as one voice rises a half step and the other falls a step, is the standard two-voice cadence of Renaissance polyphony. A teacher would mark the C natural as a missed cadence, not an error.

Play the two lines together on the online keyboard or in MuseScore. If you wrote your own answer first, run it through the same four checks. If it passes all of them and sounds calm and singable, it is as right as this one.

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