THE

AMERICAN

JOURNAL OF SCIENCE AND ARTS.

[SECOND SERIES.]

ART. I. - On Perfect Harmony in Music, the Double Diatonic Scale,
and an Enharmonic Key-board for Organs, Pianofortes, etc.;
by HENRY WARD POOLE, Of South Danvers, Mass.

  1. SEVENTEEN years ago I published in this Journal* a theory of Perfect Intonation in music, with description of an organ made to obtain this result, which had then just been completed. The organ was provided with pedals and mechanism by which the large number of pipes necessary for perfect tuning could be played by the common key-board. It was supposed that all music, for the moment, was in some key or scale. This scale the organist could prepare by putting down a single pedal, which had the effect of uniting the twelve finger-keys of each octave with twelve valves, and disconnecting all the others. As the music passed into other scales, by modulation, and other less marked transitions, the player by touching the pedal of the new scale, made the changes of sounds required. In the present paper, I shall describe a new key-board in which all the sounds contained in the organ are represented, and placed within control of the organist, without aid from pedals or any interior mechanism; and which is practicable for any extent of medulation, or number of notes in the octave. It is uniform in all keys, and the same succession of melodies or harmonies is fingered the same in every key signature. The pedal-base is also provided for by an appropriate key-board. I shall also treat of the scale heretofore unnoticed by theorists, to which I have given the name of Double Diatonic, [+] together with other matters bearing on the theory and practice of perfect harmony.

* vol. ix, Jan., Mar., 1850.
[+] See the Mathematical Monthly, ii, 16, 1859.

  2. In my former article in this Journal it was maintained that the Prime Seventh with the ratio 4:7 was harmonious, admissible and used in music, although this, so far as I have seen, was asserted for the first time. * It is clearly evident that this element enters into music of all kinds, and that the diatonic scale must contain it, or that there must be two diatonic scales: which latter supposition is most correct. If only fifths and thirds are admitted in forming a diatonic scale it will naturally be made of the common chords of three roots, namely the tonic or key-note, the dominant or fifth above, and the subdominant or fifth below. This I have distinguished as the triple diatonic scale, which has three intervals in the ratio of 8:9, 9:10, and 15:16. The notes are represented by the syllables Do, Re, Mi, etc., which always bear the same relation to the key-note and to each other. Thus, note Re is always as 8:9 or a major tone, Re to Mi as 9:10 or a minor tone, Mi to Fa as 15:16 or a diatonic semitone. No exception is admitted in this rule. Considering the key-note to make 48 vibrations in a given time, we have the

  3. But if the ear prefers - and it often does prefer - the subdominant harmony may be suppressed, and the fourth of the scale, Fa, and the sixth, la, may be replaced by the perfect seventh and ninth of the dominant harmony; so that if we still take Do as a starting point or first of the scale, we require a new Fa and La, for which formerly there have been no names. But the perfect seventh, or pur seventh as it is called, is already in solmization sung as Sib, taking the sound of Se - pronounced by the Italian rules as are all these syllables, and like the English Say - and no other name is needed. Below Si therefore we take Se, and take as first of a scale the Fa already given. Then we have the

DOUBLE DIATONIC SCALE. FA to FA.

Common chord on Fa, Chord Of 7 and 9 on Do.

Rel. vibrations, or

FA

SOL

LA

SI

DO

RE

MI

FA

32

36

40

52

48

54

60

64

48

54

60

63

72

81

90

96

First.

Second.

Third.

Fourth.

Fifth.

Sixth.

Seventh.

Octave.

Intervals

8:9

9:10

20:21

7:8

8:9

9:10

15:16

* The German "Jarbuch" of Liebig and Kopp, in a discriminating review of my article in this Journal in 1850, specified this declaration.

  4. The reasons for taking the key-note on Fa will appear on consideration, but for the present the reader will recollect that if the flat seventh of the natural key is taken - for example Bb with the common chord of C - the ear requires a resolution on the chord of F or Fa, which is the controlling note.

  5. To sing this scale is easy, provided the intervals of the triple scale are well fixed by their syllables; and it only remains to learn the intervals La to Se, 20:21, and Se to Do, 7:8, which are easily recognised on hearing the harmony which should accompany a lesson in singing. In the last scale are five different intervals in place of the three of the first scale, by which more variety is secured.

  6. These two scales contain all that belongs to the major ; the additional notes required to complete the minor keys will be considered afterwards, as well as those called "accidentals" which are borrowed from related scales The notation generally employed in music is practically correct, and, without changing the letters or sharps and flats, scales may be noted so that the exact sounds shall be indicated. In all times a singer most know or feel the pitch of each note, if he would sing it correctly. If he has learned the intervals by solmization, in the only rational way, or by always giving the same intervals to the same succession of syllables, and if he knows by the written music what intervals are called for, he will give them equally well in the key of C or in C#, or on the dozen different pitches which can be given between these two sounds. But when we are to deal with fixed sounds, as is necessary when constructing an instument, or when tow fixed instruments may have to play together, it is neccessary to know and express the exact sounds required. If the note be C, it will not do to use that which is the key-note of the natural scale for the third of four flats, which is a comma lower, nor for the perfect seventh of D, two sharps, which is lower still. I formerly indicated this distinction by a numerical index, but the following system presents advantages.

  7. Every key-note is marked as usual, but with a Roman capital; every major third to these key-notes with Roman lower case, and every perfect seventh with a Gothic capital. The second, fourth and fifth of the triple diatonic scale, being key-notes in other scales, and in the series of key-notes, each a fifth one from another, are accordingly in Roman capitals. So the sixth and seventh of the same scale are thirds of other keys, and marked in letters of the lowercase. The two diatonic series will then be represented thus in the natural key.

  8. Much that is curious and interesting concerning this double diatonic scale could be shown, did the character of this article admit, and were it practicable to give musical examples from the masters. It would be seen that the most beautiful, varied and ornate compositions are made from the elements it contains. It has the capacity in certain styles of music of using with much grace accidentals, or chromatics, as they are called; for example, the si, the regular leading note to Do, and the Sol#, diatonic semitone below la, or the leading note to the relative minor; these chromatics always ascending diatonic semitone (15:16) to the notes above. Especially is the si, or major seventh, used with Do, making the ratio of 8:15, if it is to be considered as claiming to be attended to as concordant, or as otherwise than as a passing note of a melody. But when perfectly tuned it is heard in harmony, especially with the mi and Sol with which it is sounded. An example will be given to illustrate this. First, it may be mentioned that besides the three series of notes - key-notes, thirds and sevenths - another series is used, that of the dominant's thirds in the minor scales, the leading notes to the relative minor's key note. This in each key is Sol#, and is tuned a major third above mi; and mi, Sol# and si form a major common chord (4:5:6). These notes are expressed in italic letters of the lowercase. The example being the double diatonic scale of G, in one sharp, I shall give this scale with the accidentals introduced in the following melody from Rossini's 'Il Barbiere di Seviglia.' The air will be remembered as appearing in the accompaniment to the song or recitative (for it is all on one note, D or Do, the part which this accompanies), in which Figaro describes his place of business, (Numero quindici, etc.), and afterward is the air sung by Almaviva when be has a prospect of seeing Rosina, while at the same time Figaro sings in joy at the sound of his patron's gold.

Almaviva sings

  9. Although it is convenient to consider a special strain of music as being in a definite key or scale, and to consider the notes which are sometimes prone to introduce themselves as "accidentals," and in a manner extraneous, yet the truth is that all such intruders have the excuse of being relatives, with the right of entrance, under certain rules, which the great masters understand. Among the related notes in the scale in G, last given, are C and e, or the fourth and sixth of the double diatonic on the same key-note, G. These are introduced in a passing manner, as in the cadence - a familiar example is found in the Oh! dolce concento* of Mozart where the subdominant harmony, not before heard, comes in just before the final dominant and tonic chords. In the example given from Rossini, the third note of the first measure may be C, as well as c-Fa of the triple scale as Se of the double. In the fifth measure the fourth note may be the same C, and the sixth note may be e, the sixth or la of the triple scale of G. But the following note on the same degree in the sixth measure is clearly and necesarily E, Re, or the ninth of the chord of the seventh on D. So the third note of the first measure may be e. The enharmonic change from e to E, a rise of a comma, is often required, is very beautiful, and I have proved that it can readily be made for my singers, who know this change of a comma as well as others know the tone or semitone, will give it, even without accompaniment, with perfect accuracy, as proved by the harmony afterward applied as a test. All this variety within the limits of musical laws - which only forbid what is disorderly or what the ear will not distinguish - adds to the pleasure of music,and it is the exact rendering of all the melodies and harmonies which gives the charm to a good singer. When acutely perceptive of such accuracy, I had the good fortune to listen to Alboni on all the occasions when it was possible to do so. I thought her then, and still am of the opinion, that she was the best singer I have ever heard. It is certain that she had a wonderful exactness in executing whatever she undertook. There was no "temperament" in her scales, and what the strictest theory requires in intonation she understood and gave. She sang music whose analysis would alarm a student with its apparent difficulties; but the delighted auditors perceived only a delicious and "easy" flow of melody.

  10. Fortunately, the greater part of the difficulties in the higher class of melodies ore overcome by the unconscious or instinctive talent of the singers. The accompaniment of such melodies is not difficult and the harmonies attending make clear what the melody must be. No instrument will ever compete with the voice in its peculiar department, but may surpass it in that which it is fitted for. Neither voices nor instruments separately can produce the highest effects in music; those will be attained by the combination of the two. Improvement in the instruments which accompany will be followed by better vocal music.

* Generally so called. It is the air in Mozart's Il Flauto Magico, "Oh. cara armonia." From this is taken the song "Away with melancholy."

  11. For understanding what is to follow, I would have it borne in mind, that I consider that all musical ratios derived from the primes 3, 5, and 7 are appreciable by the ear, and may be used in all their combinations and transpositions into different keys, which is already done in a series of perfect fifths. The next prime, the eleventh, does not present sufficient claims to be admitted to the musical canon, except under regulations which as yet would not undertake to define. I can tune it, and can perceive that it yields harmony so far as to give coincidences in its vibrations with the other prime chords, the fifth, the third and the seventh. It is not impossible, when a great refinement is made in music, and the sense highly cultivated, that this class of novel sounds may be introduced and appreciated. But except under such conditions, and without the most exact intonation, the eleventh would fail to give any effect other than incomprenhensible discord. It might be admissible in the harmonic stops of an organ - those called mixtures, sesquialtras, etc. - but only under a system of perfect harmony.

  12. For a practical instrument I would provide five series of sounds - a series signifying that each sound is a fifth from that which precedes, and that which follows it. These five series, arranged in the order of their importance, are as follows, the notes of each series being marked with the latter and sharp or flat in common use, but in a distinctive type for each series:

  13. This being understood, it will be known that the same letter in the same type is always the same sound, and its relation evident; that the same letter in Roman lower-case is a comma lower than the same in capitals, and a quarter (0.256) of a comma higher than the seventh in Gothic capitals. A letter of the III series, in Italics, is two commas below the same in the I series, or one comma below that of the II. The sevenths of the V are one comma below those of the III; the lower-case letter indicating this difference below the capital.

  14. Among the names which have to be remembered as advocates of perfect harmony and just ideas in music, perhaps the first in modern times is that of Gen. T. Perronet Thompson of London. My first knowledge of his valuable services came during the preparation of the second part of my article in this Journal (in March, 1860), by an allusion in the Westminster Review to an enharmonic organ which he had just brought out in London. This led to the reading of his spirited articles on music and other subjects in the Westminster Review, and to the seeking his acquaintance, which, through a considerable correspondence, I have had the good fortune to make. I have also received his "Theory and Practice of Just Intonation" and "Description and use of the Enharmonic Organ" of his invention, "built for the Exhibition of 1851, and an Appendix: tracing the identity of design with the Enharmonic of the Ancients. London, 1850." 8vo. I also obtained, after much search, and by the kind efforts of the author, what appears to be the initiative work of Gen. Thompson in musical doctrines. Although a work requiring thorough classical and mathematical knowledge, as well as information in several departments of literature and taste, it bears the modest title, "Instructions to my Daughter for Playing on the Enharmonic Guitar, being an attempt to effect the execution of correct harmony, on principles analogous to those of the ancient Enharmonic. By a Member of the University of Cambridge. London, 1829." In folio, with illustrations. This work seems to have been stimulated by a fine perception of the delicate harmonies of which the guitar strings are capable, and be falling the collection of the "Seven Ancient Greek authors, on Music," collected and publisbed by Meibomius and printed on the Elzevir press in 1652, a copy of which is in the library of Harvard College. The Euclid of geometry is one of these seven, and there is an eighth author who is not reckoned an ancient, as he lived as late as A.D. 470. It is clear that there was something which these called "enharmonic," which is declared to be the "most accurate." (Aristides Quintilianus, lib. i, p. 19, ed. Meib.) That "the name of enharmonic [or harmony] is given to the genus abounding in the smallest intervals; from the harmonizing." (Idem, i, 18.) "The enharmonic, so called from being taken in the perfect intervalling of whatever is subjected to harmony." (Id., ii, 111.) With much more to justify Gen. Thompson in adopting the title of Enharmonic: which name I also take as appropriate to a system of perfect harmony, and to the instruments which are constructed on its principles.

  15. The enharmonic organ of Gen. Thompson had key-boards in which, without any change in the interior of the organ, all sounds contained therein could be given. Every sound of the organ was represented in three key-boards, except some very rarely called for, which had exchangeable pipes. The organ of Mr. Alley and myself had a key-board like that of the common organs, and the fingering was the same: all changes were made by pedals, one for each key, which put the organ in tune for its own scales. It could be played without the player knowing what sounds he used - he only needed to keep the organ in the right key. Gen. Thompson justly remarks that his system would have merits over ours, in compelling a musician to know what he is doing. But in the dark days of enharmonic science it may be excusable not to demand too much of the organists.

  16. There are great difficulties which present themselves in admitting to a key-board the multitude of sounds required if several transpositons or signatures are to be played in. If the five series of sounds already described, �12, are carried into the keys from nine flats to nine sharps-nineteen signatures - just one hundred notes to the octave are required! But an octave is limited in width by the span of the fingers. Six and a half inches is about the convenient average measure, and this is adopted by organ and piano-forte makers. If the notes we want are divided equally into this space each will get the hundredth part of it, or sixty-five thousandths an inch. The pins of a barrel organ might play upon them, but with human fingers it is hardly possible.

  17. But there is a fortunate circumstance in the relation of the sounds which comes to our aid. All are not wanted at the same time; when we are near the key of nine sharps there is no possibility of our requiring the notes of nine flats. These we may arrange therefore at a distance front or back, and place near by the related notes which may be required in connection with those already in use.

  18. At least seven finger keys should be in convenient relation to each other, and of sufficient size and position to allow of their being touched, and for the changes of fingers necessary in running scales and taking chords in different positions. Such a key-board I have endeavored to devise, the result of which may be seen from the following description and figures.

  19. The first point I took in the resolution of the problem was, the convenience of the broad white ivory keys of the common key-board, and the elevated black keys - the white especially affording room for shifting the fingers, and the raised keys making it easy to touch a narrow key which it would be hard to do if all were in one level. The second: that the key-notes and the thirds, being of different classes, might be assigned to these two classes of finger-keys, naturally giving to the first class the more extended keys, or the white. So the octave ought to have its seven notes. Pieces of bristol-board were out to the width of the common white keys, or nearly an inch, and in length double that of the part in front of the black keys. With the same material I made elevated black keys of the width and height of those of the common key-board, and of the length of 2.7 inches which were arrangged in hopes of getting at least the diatonic scales, triple and double, which could be easily managed, and in a manner uniform for all keys.

* At the time of writing (April, 1867) I have hade application for a patent for this key-board.

  20. The provisional key-board is not figured here, but may be understood by reference to figs. 1, 2, and 4, which contain also the additional series of sounds, IV and V, �12. In fig. 1 let the black key, d, be moved to the left until its left edge concides with the right edge of C, its back end as now being in contact with the front end of the white key D; let e, f#, and the two keys (F⁷ and another not marked but really Eb⁷) marked with vertical lines indicating their color, red, be moved in the same direction and distance as d, till they are in contact, respectively, with D and E. There is no room near for d# and d⁷ and their two companions by f# but we have all the key�notes, thirds and perfect sevenths, and the advantage of greater width in the white keys, which are nearly an inch and a quarter wide (1.21 in., the black keys being 0.45)). This key-board, although in the minor mode, from the absence of the series IV and V, is still recommendable where economy is necessary, as all the music of the major key, including the beautiful ohord of the seventh, can be played; except in cases where certain accidentals are introduced from the minor mode, as illustrated in the example from Rossini in �8.

  21. A portion of the complete enharmonic key-board is shown in perspective in fig. 1, in the natural size -the length of the keys being reduced to one half by the perspective. The keys are of five different colors - represented here, the white and black by their natural colors, and the rest according to heraldic rules, viz., the red by vertical lines, the blue by horizontal, the yellow by white stippled with black - and of as many different elevations. The following table represents this.

  22. These keys all have vertical rectilinear motion so that a touch on any part of their surface produces always the same effect. This is attained-in one method - by attaching each to a pair of guiding rods, passing down through a couple of horizontal tables where they are secured to a piece which communicates in the usual manner with the valves or hammers. It will be understood that the mechanical construction of the instrument beyond the key-board may be the same as usual, except that its number of pipes or strings must be multiplied.

  23. As the assemblage of all the notes mav confuse the eye at first sight, I have drawn on half scale the plan of the finger-keys which in every signature - or commencing on any white key whatever-give the triple and double diatonic scales. The white keys are of the width of 0.993 in., the black keys having the width of 0.45.* The white keys on the common-key-board are but 0.93, or the seventh of 6 inches. The plan in fig. 2 will enable a player to judge whether the scale can be executed. It is immediately intelligible even to a child, who, having learned the order in one key, knows it in every other. The keys are considered by their relations to each other, that is, as Do, Re, etc., and (to repeat it again), Do may be taken on any white key. The fingering in the triple diatonic scale is the same as in that of the natural key, and of others, on the common key-board, and the fingers easily reach the keys and change on Fa and Do as in the latter case. In the following scales the usual signs represent the thumb and four fingers; see fig. 2.

TRIPLE DIATONIC SCALE - Its fingering.
DO	Re	me	FA	SOL	la	si	DO
+	1	2	+	1	2	3	4 (or +)
DOUBLE DIATONIC SCALE - Its fingering.
FA	SOL	la	Se	DO	RE	mi	FA
+	1	2	3	+	1	2	3

The player is recommended to complete the octave, Fa to Fa, by copying the four lower keys, Do, Re, mi, Fa, figs. 2, 3, and placing the lower Do upon the upper one, or by conceiving that this has been done. This will show the double scale in its regular order.

  24. I regret that the limits of these pages did not allow a larger portion of the key-board, and permit reference to it in the manner of taking the several clords in their various positions. But I think that those interested can extend the diagrams by the data given, and I shall, therefore, give the fingering for several chords, which being understood in one key will be the same in all transpositions.

* The widths are established thus: first determine the width of the octave, and that of the black keys. Representing these respectively by O, and b, the width of the white key (W), is obtained thus: (O + b) / 7 = W. For the key-board for major keys, only, �20, the formula is (O - b)/6 = W.

A Portion of the Enharmonic Key-board....p.12

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