Charles Ives’s Approach to Intonation
© 2005 Johnny Reinhard
Published by NewMusicBox
It is generally thought that, except for a few pieces with specifically notated quartertones, such as the Three Quarter-tone Pieces for Two Pianos and the Symphony No. 4, the remainder of Charles Ives’s music was conceived for conventional twelve-tone equal temperament. But after reviewing Ives’s writings on the subject in letters, essays, and marginalia, I think differently. I believe there is a general tuning for Ives’s music that can best be described as Extended Pythagorean tuning.
Pythagorean tuning is the most widely-used term used for deriving a twelve-note scale from a spiral of pure perfect fifths (the relationship of the third harmonic to the second harmonic: 3/2). Music in Mediaeval Europe was replete with explanations of this tuning which was common practice for most instrumental music until the rise of keyboard temperaments. It is named after the ancient Greek mathematician and musician Pythagoras though its usage in other parts of the world predates him. The Babylonians left tablets to explain how spiraling fifths were used for making musical scales.
Spiraling untempered fifths, one upon another through 12 fifths, leaves an intervallic remainder. There is an excess of nearly an eighth-tone. Conventional equal temperament essentially chops this eighth-tone “comma” into 12 equal parts, and then subtracts each 1/12th of this comma from each of the spiraled perfect fifths in order to close the circle.
(If we use cents to measure the differences between musical intervals, with 1200 cents to the octave, the piano’s “not quite perfect” equal-tempered fifth measures 700 cents. The ideal perfect fifth—which is the basis of Pythagorean tuning—rounds out in whole numbers to 702 cents. Within Pythagorean tuning, the interval of a major third is interpreted as the result of a chain of four 3/2 perfect fifths A-E-B-F#-C# or 81/64 which rounds out to 408 cents. It is somewhat sharper than the equal tempered major third of 400 cents and also nearly an eighth tone sharper than the pure major third of just intonation, 5/4 [80/64], which is 386 cents.)
Extended Pythagorean tuning derives scales with a greater number of total pitches by spiraling beyond 12 perfect fifths. Extended Pythagorean tuning could allow for a different pitch frequency for each distinct spelling of a note: e.g. there would be no enharmonic equivalency of D# and Eb as there is on a conventional piano keyboard, rather they would be two distinct pitches. The chromaticism of Extended Pythagorean tuning offers benefits for both greater consonance and richer dissonance in comparison to 12-tone equal temperament. By spiraling 20 pure perfect fifths the following chromatic scale is formed:
Such a scale would not require the creation of any additional accidental symbols beyond those already in use in standard musical notation and thus could be easily written. Such a scale would also cover all the accidentals that Ives notated in his compositions and explain his often bizarre spellings of pitches which in standard 12-tone equal temperament would be enharmonically identical, e.g. a Db going up to an F# rather than an Gb, which occurs in a solo cello line in an Ives manuscript, etc.
Since Ives rarely experienced performances of his compositions, and heard practically none of his mature works, it was impracticable for him to demand much in the way of intonational subtlety from musicians of his time period. However, Ives had good reason to anticipate that musicians would eventually make sense of his intentions regarding intonation.
Consonance is a relative thing (just a nice name for a nice habit). It is a natural enough part of music, but not the whole, or the only one. The simplest ratios, often called perfect consonances, have been used so long and so constantly that not only music, but musicians and audiences, have become more or less soft. If they hear anything but doh-me-soh or a near-cousin, they have to be carried out on a stretcher (Ives, Memos, p. 42).
Charles’s father, George Ives, who imparted to his son a lifelong quest for new sounds, was experimenting in the 19th century with new tuning possibilities, easily earning himself the highest rank among American microtonalists of that era. Among his inventions were a slide cornet, filling glasses with differing degrees of liquid to get microtonal intervals (a microtonal version of a glass harmonica), and a machine involving violins stretched across a clothes press and let down with weights. While the elder Ives frequently spoke about quartertones, he also frequently explored subtler intervallic gradations and at one point tuned a piano to the overtone-based relationships of just intonation. This must have been a major effort for him, as there is great planning necessary to achieve this. Charles described being impressed that his father could master the transference of the overtone series to the piano without the aid of any contemporaneous outside tool or device such as the Acousticon, a relative of the oscilloscope.
George Ives was quite fond of Hermann Helmholtz’s book On the Sensations of Tone in its English translation by Alexander J. Ellis which had been published in 1875. Significantly, Helmholtz outlined Extended Pythagorean notation in detail. Although Helmholtz admitted that he thought it difficult to hear all the interval relations in Extended Pythagorean, he was able to depict it with clarity. Helmholtz argued for Extended Pythagorean interpretation. “[T]here is no perceptible reason in the series of fifths why they should not be carried further, after the gaps in the diatonic scale have been supplied.” In an appendix to his translation, Ellis had taken Extended Pythagorean tuning through 26 perfect fifths. Ellis also successfully made the case that musical staff notation was invented for both just-biased temperaments (like meantone), and Pythagorean interpretations, “with a distinct difference of meaning between sharps and flats, although that difference was different in each of the two cases.” These notational speculations by Ellis might have been the institutional justification, and invitation, for young Charles Ives to employ such an alternative notation for his original music.
After George Ives died, Charles turned to another mentor, Dr. John Cornelius Griggs (1865-1932), a sought-after baritone and doctoral candidate from Leipzig University nine years Charles’s senior whom Ives scholar Howard Boatwright has described as “next to Ives’s father, the earliest supporter” of Ives’s music (Boatwright Essays Before a Sonata, p. 81, footnote z). Early on in their lifelong friendship, Griggs notably wrote Ives that: “The tempered system at its best is not conducive to correct and vigorous musical thinking, as has been the violin and voice training of earlier centuries.” On a marginalia on a photostat copy of the First Piano Sonata, Ives scribbled out a little polemic. “I was asked once by Dr. G., ‘Was this E# written instead of F natural because E# and F natural are not (not always) the same note?’ ‘Yes…only in the piano machine.’ ‘But they can be [different] if they are in the thought, and in a certain imaginary way in the ears as such” (Memos, p. 255).
Ives left crucial evidence for a unified “acoustical plan” to be applied to the majority of his later music:
“[W]hen a movement, perhaps only a section or passage, is not fundamentally based on the diatonic (and chromatic) tonality system, the marked notes (natural, # or b) should not be taken as literally representing those implied resolutions, because in this case they do not exist” (Memos, p. 190).
In response to a certain “Prof. $5000,” a.k.a. “Grandma Prof.,” Ives castigated the “g—d—sap!” for objecting to “a B# and a B natural in the same chord.” In addition, Ives thought it significant to point out that B natural and B# have a harmonic relationship in a full chord. As if to personally respond to potential accusations against his use of two different Bs together, Ives explained:
Now when both the two Bs are used in chord, there is a practical, physical, acoustical difference (overtonal, vibrational beats) which make it a slightly different chord than the Bs of an exact octave—and [even] on the piano the player sees that and feels that, it goes into the general spirit of the music—though on the piano this is missed by the imaginative (Memos, p. 189).
In other writings, Ives asserted that a B# sounds an eighth-tone higher in pitch than its nearest C. An Extended Pythagorean interpretation of this notation satisfies this condition perfectly. By continuing the spiral of fifths from C from B#, after 12 perfect fifths have been stacked and octave displaced, B# is indeed nearly eighthtone higher (23.4 cents) than its nearest C. Once again, Helmholtz foretold exactly what Charles Ives would later claim for his own music in Memos. “Hence the tone B# is higher than the octave of C by the small interval 74/73″ (Helmholtz, On the Sensations of Tone, p. 312).
Ives speculated about the difficulties a composer faced in attempting to circumvent the tyranny of the perfect fifth. “But this is doubtful; the octave and fifth are such unrelenting masters in the realm of the physical nature of sounds” (“Some ‘Quarter-tone’ Impressions,” p. 112-114). His acoustical plan was based entirely on these two intervals, the perfect fifth and the octave. Typically, the ear picks out the dominant perfect fifth—”inexorably,” said Ives—as the dominant harmonic. “The fifth seems to say, ‘You can’t get away from the fact that I am boss of the overtones—the first real partial.’” But the fifth is the same in both Pythagorean (including Extended Pythagorean) and just intonation tunings.
They want to have Ralph Waldo Emerson or Henry Thoreau sing Do-Me-Soh—but those men were men—they didn’t sing Doh-Me-Soh—they knew the Doh-Me-Soh, but they didn’t sell it to the ladies all the time, they used it as one of the windows, not the whole parlor, etc. etc. (Memos, p. 188).
Based on Ives’s musical notation, it could be argued that Ives eschewed the simple consonances derived from the major and minor thirds of just intonation (e.g. 5/4, 6/5), favoring the significantly larger di-tone 81/64 major third of 408 cents as his tonal norm, which offers strong melodic counterpoint. By using Extended Pythagorean tuning, Ives could elicit a just quality major chord through use of what his father had termed a First spelling, instead of the usual Second spelling, e.g. A-Db-E rather than A-C#-E. Helmholtz had stipulated that by stacking eight pure fifths, the bottom pitch would create an interval with the top pitch that was only two cents shy of the 386 cents just major third, at 384 cents. This two cents difference is as small as the difference between a pure perfect fifth and the equal-tempered fifth, and it could be effectively masked by the use of even minimal vibrato. It would need to be spelled A-Db, with Db being an eighthtone flatter than C# (C# forming a Pythagorean third of 408 cents above A) satisfying Ives’s insistence that C# is higher than Db. Ives was careful in his writings to distinguish between the overtones and his own preferred acoustical plan. Ives properly recognized the overtones as “the vibrations of all the partials as a sounding unit” (“Some ‘Quarter-tone’ Impressions,” pp. 112-114).
Ives struggled with many editors of his music expecting to change Ives’s “mismatched” notation spellings. In one instance, copyist George F. Roberts declared Ives a stickler on notation accidentals and related to oral historian Vivian Perlis how the composer had cautioned Roberts’s colleague, George Price, not to alter any note spellings. Most emphatically, Ives wrote a message to his editor on the front page of the score to The Fourth of July. “Mr. Price, Please don’t try to make things nice. All the wrong notes are right. Just copy as I have. I want it that way.” (Perlis, p. 188)
It obviously exacted quite a bit from the infirm Ives to spar with respected editors. He was surely grateful to have people accept his music for publishing. Ives apparently had to resort to rationalizing the situation in Memos, in an attempt to quell his internal inferno:
“Price never made a mistake! What mistakes he made were yours. If he thought you had put down the wrong note, he would put it right (right or wrong), and blame you when you had the audacity to say that, not every time there’s a C natural, all C sharps that happen on the same beat in the chord should be scrapped. Then he would get mad and want to charge for correcting your right notes into mistakes. In business and in politics, and in almost every department of life, I know many Prices. But his penmanship was as beautiful as a Michelangelo—to look at a page of those ‘statuettes’ made it hard to jump on him.” (Memos, p. 65)
A 1944 letter written in his wife’s hand, addressed to music editors Sol Babitz and Ingolf Dahl, was in response to potential altering of Ives’s notation in a planned publication of Violin Sonata No. 3. By 1944, Ives was unable to draw a simple straight line as a result of health frailties. It also seems that he felt rather resigned to only a minimal influence on the situation at best (Burkholder, ed., Charles Ives and His World, p. 250), as illustrated by the letter below:
Babitz / Dahl
…He is rather sorry that some flats and sharps have been changed into each other. Mr. Ives usually had a reason technically, acoustically or otherwise, for using sharps and flats. If a D-flat is in one part and C-sharp in another on the same time beat, it was mainly due to some acoustical plan—which he had in mind or was working out or trying to in those days.
But after the first page, whatever changes there are in accidentals (which he hopes are not many, especially in the 3rd movement) do not bother to put them back as the old copy—Either way won’t ‘make or break’ the listener’s ear.
With our kindest wishes to you both
[signed Harmony T. Ives]
Like so many editors before him, John Kirkpatrick had falsely reasoned that “wherever Ives’s non-conformist spellings offer unreasonable hindrances to memorizing, they are changed to what is hoped will be helpful” (Kirkpatrick, Editor’s Notes for Ives’s 3-Page Sonata, 2nd edition, p. 14). Kirkpatrick made it crystal clear that his purpose in limiting the use of accidentals was to facilitate memorization for the performer. By 1974 Kirkpatrick had enough time to reflect on Ives’s tuning and had reported these personal revelations at the Ives Centennial symposia. Reminiscing about his turbulent experiences with the composer regarding the acceptable notation for Ives’s song, “Maple Leaves”:
At the time, I thought it was sheer nonconformism, but then, the more I got into this music generally, the more it seemed to me that he had unexpected tunings in mind, that actually the core of the passage was probably a real A-sharp reaching up toward B and a slightly low F-natural reaching down toward E—what used to be called a fourth and a comma. From then on, I had great reverence for these things (Kirkpatrick, An Ives Celebration, p. 139).
Partly as a result of increasing respect for Ives’s notation decisions, the Ives Society is reportedly restoring Ives’s original spellings in recent editions.
Scholars have certainly rallied around the importance of keeping a composer’s choice of notes as sacrosanct, even if there are no clear reasons for the original choices. Carol Baron was the first scholar to make the case for Ives in this regard. In her 1987 dissertation, Baron sheds light for the first time upon the implications of Ives’s notation choices by taking an errant Kirkpatrick to task. “[I]f standards of conformity are applied to Ives’s spelling, then distortions will result. For example, Ives’s innovative pitch organization was not based on the diatonic melodic and harmonic directions that Kirpatrick apparently assumes to be operating” (Ph.D. Dissertation, City University of New York, 1987, p. 115). Thanks to this kind of attention, to what must have seemed a petty detail to the unilluminated at the time, greater focus on intonational issues is made in the performance of Ives’s music today.
Ives’s elaborate verbal defenses—there are several in Memos—articulate his sensitivity and extraordinary concern with the direction, as well as the stasis and degrees of motion, of individual tones in complex harmonic and also microtonal contexts as they relate to the resultant overtones; rational decisions were made. The integrity of Ives’s choices, clearly tied to his compositional process, must be respected in editions of his music (Baron, Ph.D. Dissertation, City University of New York, 1987, p. 118).
By reading Ives’s notation as reflective of an idealized—and organic—Extended Pythagorean tuning, musical passages gain added significance. One example is the final chord of Section B of Ives’s Universe Symphony, which was written by the composer by the stacking of letter names for the intended notes:
Bb B C C# Eb D# E F F# G G# A C#
Philip Lambert, not recognizing the intended intonational consequences of Ives’s manuscript, wrote that the final chord of Section B should have all 12 tones in the chord, except that there is a D# and an Eb. Based on conventional interpretation, Lambert concluded that the D# was an error made by Ives, and that he meant to leave a plain D, but somehow added a “#” marking to a D because he was distracted:
And at the very end of the page—after a double bar—he makes a vertical list of twelve notes that are probably supposed to form an aggregate. (all pitch classes are represented except pc 2, and pc 3 occurs as both D# and Eb—perhaps Ives erroneously placed a sharp sign after the D) (Lambert, A Universe in Tones, p. 197).
I believe that Lambert is mistaken in his interpretation. In my estimation, no composer would make such a mistake on a final chord of such importance in a symphonic movement, especially when it is written out using letters rather than note heads.
The previous measure has half note Ds played by a solo flute and bassoon under a fermata, before resolving into this tutti no-D chord. When performed in Extended Pythagorean tuning, the fantastic coloring of twelve notes from a total of 21 different possibilities is formidable.
Even larger clusters of pitches occur elsewhere in the Universe Symphony. There are fully 13 distinguishable notes spelled in OU measure 106, Prelude #2 – Birth of the Oceans. Here we have the elusive D natural. Ah, but there is no A natural this time among the 13 notes given. There is no Bb, B#, Db, E#, Fb, Gb and Ab as well. The notes that gradually swell up into this 13-note chord are: B, C, C#, D, D#, E, Eb, E, F, F#, G, G#, A#. Another prominent chord appears at OU measure 27. The Earth Orchestra plays 11 differently written notes which are tied over a barline: A, A#, B, C, D, D#, Eb, E, F#, G, G#. Group I of the Heavens Orchestra adds another two pitches, Bb and F natural, on the downbeat of OU measure 27, making for 13 different notes. And Group II of the Heavens Orchestra plays on the downbeat as well, with an added Db in its top voice, making a grand total of 14 different and distinct notes heard at once.
If interpreted in Extended Pythagorean tuning, there are many other examples of clear microtonalisms in the Universe Symphony. OU measure 78 has the celli playing a Db along with an F# above it, and a B in the double bass. The outlined pure perfect fifth B-F# is averaged up to 702 cents (rather than 701.955…etc). The Db divides the perfect fifth into intervals of 180 cents and 522 cents: quite exotic intervals to hold in harmony for half an OU in duration. Even for modern ears, 522 cents is still quite a fresh interval.
If there was no semantic meaning for distinguishing the note E from a written Fb, then Ives wouldn’t choose both to be played simultaneously in a harmonic lock, like he did in OU measure 112. Similarly, if B natural and Cb were meant to be the same note, why feature a melody alternating between the two which Ives does in the final solo at the end of Section C? In an Extended Pythagorean interpretation of both examples, the interval is a Pythagorean comma, which is only one cent away from an equal-tempered eighth-tone (24 cents).
In an interesting pencil addition to the cover of a copy of the Concord Sonata, Ives rationalized that the “mind, ear, and thought don’t have to be always limited by the ‘twelve’—for a B# and a C are not the same—a B# may help the ear-mind get higher up the mountain than a C natural always” (Memos, p. 189). Here we have a non-tuning system explanation for his insistence on retaining specific choices for notation spellings. But whether reflected as a genuine tuning difference with audible distinctions to be heard, or only the mere psychological trappings of meaning, there can be no doubt that Ives was fully aware of the difference between them.
It is as if Ives walked a fine line between actually hearing the Pythagorean tuning that his writings indicate and a more psychological heightening of musical intent. Ives has made it clear he could use the symbols for notation in different ways. Sometimes the notation corresponded to equal temperament in the usual way, but more often, the signposts had a distinctive meaning to the imaginative sign maker.
On a conventionally tuned piano, where there can only be a psychological factor for an interpretation of Ives’s notation, there is no actual change of pitch to be heard. A “psychological factor” might have some impact on a pianist who imagines B# as higher in pitch than C, according to the composer. However, the piano ultimately tells a different story for it has no residual effect on listeners. We will all hear 12-tone equal temperament when Ives’s music is heard on a 12-tone equal tempered piano. It is only through the elevation of Ives’s ideas in tuning, following his acoustical plan, that the desired effect can be accomplished. If Ives could have had his way, he would have purged the piano of all its tuning limitations.
Based on George Ives’s mentoring (which included a thorough understanding of Helmholtz’s On the Sensation of Tone), Charles Ives’s own comments, and the corroborating evidence of “misspellings” found throughout the manuscripts of his musical compositions, Extended Pythagorean tuning is ideal for interpreting most of Charles Ives’s later music without keyboard. However, Extended Pythagorean tuning is only a part of the polymicrotonalism that Ives included in the Universe Symphony, which was Ives’s musical interpretation of the multiple possibilities found in nature.
Contained in the sketches are seven listed pitch orientations designed for inclusion in the Universe Symphony. The first listing is for “perfectly tuned correct scales.” Extended Pythagorean would fit the description for “perfectly tuned correct scales” like a glove. On one occasion Ives introduced his notational ideas to six violinists in a rehearsal of The St. Gaudens. The violinists were informed of the distinctions of Pythagorean tuning with the emphasis placed on the notated Db being lower than the notated C#. “After the players had sensed this difference in playing the passage—say B-B#-C, D-Db-C (to remember the B# and C, and the D and Db etc.)—to me they usually sounded nearer to each other than a quarter-tone, though in the upper and the lower movements I noticed very little difference. Then [I] would try to have the player think and so play the Db as it had been played in going up to D, and then play with the others in a chord—and this had its own way [of being] different to the usual” (Memos, p. 191).
The next tuning Ives refers to is “well-tempered little scales” which would best be represented by equal temperament, rather than irregular keyboard temperaments associated with the likes of Andreas Werckmeister. Third listed is “a scale of overtones with the divisions as near as determinable by Acousticon.” Just Intonation tuning fits impeccably with “a scale of overtones.” Fourth listed is a scale of smaller division than a semitone. “Scales of smaller intervals” would include the quartertones. Fifth listed are scales of uneven division greater than a whole tone, and further non-octave tuning divisions offset by eighth-tones. Listed sixth are “scales with no octave, some of them with no octave for several octaves. Finally seventh, “scales of uneven division greater than a whole tone” which are uneven compound intervals.
The Universe Symphony sketches feature specifically notated quartertones. Sometimes these quartertones appear within the context of the dominant acoustical plan of Extended Pythagorean tuning throughout the piece, at other times whole sections are quartertonal in conception, as with Prelude #3 titled And Lo, Now It Is Night. There even seems to be a special allowance for the vagary a quartertonal orchestra might generate (at least back in 1916). On sketch page Neg. = q3039/Copyflow = 1846, Ives wrote that quartertones need not be exactly pitched. “A kind of 24 scales, varying intervals and overtone vibrations, each its own tonal plan.” This may indicate some added flexibility to the exactly specific quartertones.
But, if Ives intended the music we assume to be in 12-tone equal temperament to be tuned to an Extended Pythagorean scale with a total of 21 possible pitches, what are we to make of the music in which he actually specified quartertones, which performers and musicologists have been presenting as music with a total of 24 equidistant pitches?
The common interpretation of quartertones stems from a 12-tone equal temperament model which is then perfectly divided in half, with the size of a quarter-step having the value of 50 cents in every instance. But Ives’s flexibility about the size of quartertones—”A kind of 24 scales, varying intervals and overtone vibrations, each its own tonal plan”—implies a different conception entirely.
Once, young Charles got his curiosity piqued to return to a church to play two pianos that were further apart than a quartertone from each other. When he found they were not available, his disappointment was so great that he determined to describe his experience of lost epiphany in his diary:
In the Sunday-School room of the Central Presbyterian Church, New York, there were, for a while, two pianos which happened to be just about a quarter tone apart, and I tried out a few chords then.
In this connection, and also referring to Father’s glasses tuned in different intervals larger and less than quarter tones, after hearing the two pianos out of tune in Central Church (but as near as I could tell by listening and with tuning forks, [they] were about a quarter tone apart)—a scale (to knock the octaves and fifths out by wider intervals, stretching [the] whole and half tones a little, but keeping the proportions of the scale)—it was started or suggested by these two pianos, and glasses between [the quarter tones]. But one piano was moved before I could get it well grasped in my ears (Memos, p. 108-9).
Ives is expressing something special about the sonority of two pianos tuned well enough each to itself, but somewhat more than a quartertone apart. The sound of quartertones as they sound on pianos is rather fixed in the imagination of contemporary composers. An exact quartertone tuning effectively invokes the 11th harmonic (the interval of 11/8 which is approximately 551 cents, a cent away from the interval between C and F quartertone sharp which would be 550 cents). If, however, two pianos were tuned 60 cents apart instead—as they were for duo-pianists Joshua Pierce and Dorothy Jonas performances and recording of Ives’s Three Quarter-tone Pieces for Two Pianos in the 1990s—the harmonically dreamy properties of the 13th harmonic could be effectively invoked.
Charles Ives envisioned a myriad of tuning approaches and the rewards of hearing the intonational distinctions that he specified make a profound difference in the resulting sound and its reception. The intonational clarity is subtle, though fine musicians are sensitive to a perception of improvement in the difference. It is a similar sensation to hearing early composers in their respective intonation-preferred models (e.g., Dietrich Buxtehude’s music in his favored Werckmeister III well-temperament, or John Dowland performed in his own personal irregular tuning, published by his son Robert). Listening to music that matches the sensibilities of the various tuning arrangements reveals a new dimension of meaning, allowing for greater intimacy with the composers through their music creations.
That is, if one can learn to like and use a consonance (so called), why not a dissonance (so called)? If the piano can be tuned out of tune to make it more practicable (that is, imperfect intervals), why can’t the ear learn a hundred other intervals if it wants to try?—and why shouldn’t it want to try? (Memos, p. 140).
Johnny Reinhard is the director and founder of the American Festival of Microtonal Music (established 1981). He has been responsible for premieres of major works by Lou Harrison, Harry Partch, Edgard Varèse, and Charles Ives, among many others, as well as the first modern-day performances of works by Johann Sebastian Bach, John Dowland, Gesualdo, Telemann, Beethoven, and Mendelssohn in their original tunings. As an internationally-travelled conductor, composer, publisher, and bassoon soloist, Reinhard has explored a myriad of tunings. This past year, Reinhard has also become a record producer of a new line of compact discs featuring a wide variety of microtonal music on the PITCH label.