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So what's the deal with the shapes?
The shapes are a teaching tool. Since the Middle Ages and Guido of Arezzo, solmization syllables have been used to help singers train their voices and ears. Most people will at least recognize Do, Re, Mi, Fa, Sol, La, Ti, Do, the traditional Western solfege pattern for the major scale. These syllables are used to learn intervals and patterns in music. Trained musicians can apply these syllables without the aid of shapes, but around the late 1700's, shaped notes were invented so that singers would not have to understand music theory to know which syllable to assign to a given note. Each syllable is associated with a shape, and the solfege is built right in to the notation. The Sacred Harp uses the same idea, but with a 4-syllable system. In order to further simplify the process, a four-shape notation was adopted. A major scale would be sung Fa, Sol, La, Fa, Sol, La, Mi, Fa. The above figure (from George Pullen Jackson's White Spirituals in the Southern Uplands, p. 14) illustrates this system, introduced by William Little and William Smith in The Easy Instructor, published in 1801. As you can see, a triangle corresponds to "Fa", a circle to "Sol", rectangle to "La", and diamond to "Mi". The interval from a shape to its next occurrence in the key (Fa to Fa, Sol to Sol, etc.) is always a perfect fourth, and there is always a half-step leading to Fa (La-Fa and Mi-Fa). It may seem confusing at first, because of the repetition of shapes, but most singers find it very helpful once they grow accustomed to the system.
The History and development of shape notation
The early colonists had interesting congregational singing practices. In the eighteenth century, they would sing from "tuneless" books, such as the Bay Psalm Book from 1640. The words found in these books would be sung to familiar folk tunes, usually originating from the native lands of the colonists. This amounted to singing all of their psalms to a handful of folk tunes. Boring singing practice often leads to poor singing, as it seems to have done in New England. Some preachers attempted "lining out" tunes, but this quickly became cumbersome. The first logical step towards improving musical ability and widening the range of musical literature is to teach the people how to read music. Most non-musicians shudder at the thought of learning to read music, so simplified instruction methods were needed.
The first American music instruction book that provided a systematic approach to teaching music was written around 1714 by Reverend John Tufts. His book, A Very Plain and Easy Introduction to the Art of Singing Psalm-tunes, featured a system of four solmization syllables: fa, sol, la, and mi. He employed the five-line staff, but with the letters F, S, L, and M substituting noteheads. Thirty-seven three-part psalm tunes were included in his instruction book.