Some comments on dispersed harmony, arranged in chronological order:
"If the intervals of a four-part harmony lie as near as possible together in the three highest voices, it is called condensed harmony. If they lie at at greater distance from each other, it is called dispersed harmony." -- From The Pianist's Handbook, by Carl Engel. London: Hope and Co., Published 1853. p. 21
"When the four-part harmony is written so that the three upper parts are contained in an octave, and can be played with one hand, it is called CLOSE HARMONY. If the interval between the soprano and base is about equally divided by the tenor and alto, it is called DISPERSED HARMONY." -- From Palmer's Theory of Music: Being a Practical Guide to the Study Thorough-Bass, Harmony, Musical Composition, and Form, by Horatio Richmond Palmer, Cincinnati, OH: John Church & Co., 1876 p. 85
"Open, or dispersed, harmony requires the voices to be so separated that by transposing the soprano one octave lower it would come between the alto and tenor; and also by transposing the tenor one octave higher it would come between the alto and soprano, as at a (illustration, rlv). If only one of such transpositions is possible, the harmony is partially open, as a b (illustration, rlv). If neither is possible, or in other words, if the upper three voices are as near each other as they can be under a certain soprano, the harmony is close as at c (illustration, rlv). Any two voices lying next each other may often sing the same note; but, in elementary harmony, a lower voice should not sing above a higher, or the reverse." -- From Elements of Harmony, by Stephen Albert Emery, Boston, MA: Arthur P. Schmidt Co., 1890, pp. 13-14
"If the three upper voices of a chord -- soprano, alto, and tenor -- lie outside the compass of one octave, the harmony is usually said to be dispersed, or in open position." -- From Harmony: A Course of Study, by George Whitefield Chadwick, 1897, Boston, MA: B. F. Wood Music Co., p. 61
"Extended or scattered harmony: dispersed or extended harmony is that in which the notes composing a chord are so far apart that the upper three parts, treble, alto, and tenor, exceed an octave in compass; and between any two of the parts of a chord in dispersed harmony there is space for insertion of some one of the notes belonging to that chord." -- From Musical Dictionary, by W. L. Hubbard, New York, NY: Irving Squire, 1908. (2006 reprint) p. 164. This book also (p.576) calls it wide harmony, extended harmony, and open harmony.
"Dispersed harmony (Music), harmony in which the tones composing the chord are widely separated, as by an octave or more." -- From Webster's Revised Unabridged Dictionary (1913)
"25. When the alto is given the next member of the chord below the soprano, it forms what is called Close Harmony. When the alto skips the next member of the chord below the soprano and the tenor is given the one skipped, it forms what is called Dispersed Harmony.
"26. A low soprano should be written in close harmony and a high soprano should be written in dispersed harmony while a soprano on the third line or adjoining degrees, may be written in either close or dispersed harmony according to the surroundings." -- From Modern Harmony and Voice Leading, by Adger M. Pace, Cleveland, TN: James D. Vaughan Co., 1916, p. 9
J. S. James defines it as harmony in which “the notes forming the various chords [are] separated from each other by wide intervals.” -- From An Explanation of The Sacred Harp, 1920
"Harmony, close - A harmony whose tones are compact, the upper three voices lying within the compass of an octave.
"Harmony, dispersed - A harmony in which the notes forming the different chords are separated by wide intervals." -- From Music Theory Dictionary: the Language of the Mechanics of Music, W. F. Lee, editor, New York: Charles Hansen Educational Music and Books, 1965, p. 29
Dispersed harmony "occurs whenever a chord exceeds two octaves or the alto goes above the soprano." Two essential characteristics of Sacred Harp dispersed harmony: "the upper members of a chord are often dispersed rather than closely grouped...[and]...the crossing of voices." -- The Sacred Harp: a Tradition and Its Music, Buell Cobb, 1978, p. 36
"I think dispersed harmony is created throughout a piece of music where it is composed as most Sacred Harp composers do -- as horizontal strands rather than vertical chords. Each part is written so as to make it a singable tune of its own and this leads to 'open chords'. (Open chords being defined as chords in which another note can be inserted between the upper voices.) Sacred Harp composers move from 'open' to 'closed' chords without regard to where the change takes place in the phrase. Add to this the frequent crossing of voices throughout the composition, especially between treble and tenor, and we have two essential points in the style known as 'dispersed harmony'." -- Raymond Hamrick (in a letter to Hugh McGraw, November 1981)
"In this style of composition (dispersed harmony)...each vocal part--treble, alto, tenor, and bass--contributes a sort of tune, occupying its own separate staff, with the parts freely crossing one another and the tenor, or third line, carrying the chief melody." -- From The Tradition, by Jim Carnes, January 1989
Joel Cohen's notes on the The Boston Camerata's An American Christmas speaks of "dispersed harmony, with both soprano and tenor lines doubled..." (1993)
"Each part (treble, alto, tenor or lead, and bass) retains a degree of melodic independence ("dispersed harmony")..." -- From Sacred Harp Singing: History & Tradition, by Steven Sabol, 2005
Dispersed Harmony "...the parts cross over each other rather than running parallel." -- From Stormy Banks and Sweet Rivers: a Sacred Harp Geography, by James B. Wallace, 2007
DISPERSED HARMONY on Sympathetic Vibratory Physics
Note: in some Sacred Harp comments it seems the thought is that music written on four separate staffs is dispersed harmony. For example, "The parts in shape-note singing are so distinct that traditional tune books like The Sacred Harp print them on separate staves, displaying what is called dispersed harmony." -- New Georgia Encyclopedia, 2005