--------------------------------------------- Neo-Gothic tunings and temperaments: Meantone through a looking glass (Part 1 of 2) --------------------------------------------- One artistic source for tunings and temperaments both old and new is the Western European musical tradition of the Gothic era, and especially the complex polyphony of the 13th and 14th centuries. This music, and "neo-Gothic" styles drawing inspiration from it, invite approaches to interval aesthetics quite distinct from those of European music from the Renaissance to the Romantic era. The historical Gothic tradition itself offers a consummate tuning system: Pythagorean tuning, or 3-limit just intonation (JI), which results in a subtle "balance of power" between the stable and unstable intervals and sonorities of Gothic polyphony. This system deserves attention both for its intrinsic beauty and for its historical role as the one JI system to win widespread acceptance on standard keyboard instruments. Neo-Gothic tunings and temperaments strive to develop this historical tradition further in one of at least two directions: either the further extension of Pythagorean tuning beyond the 17 notes recognized by medieval theorists (Gb-A#); or the use for Gothic and neo-Gothic music of "reverse meantone" temperaments with fifths somewhat _wider_ than a pure 3:2. Both approaches maintain a Pythagorean flavor while offering new types of intervals or artfully altered interval sizes, blending tradition with innovation. Starting with Pythagorean tuning, or the almost identical 53-tone equal temperament (53-tet), the neo-Gothic spectrum moves through a universe of "reverse meantone" temperaments with the fifths becoming increasingly larger than pure: for example 41-tet, 29-tet, exponential meantone (see Section 2), and 17-tet. Thirds and sixths become even more active than in Pythagorean, and their resolutions to stable 3-limit intervals even more economic and efficient, involving diatonic semitones increasingly smaller than the already compact Pythagorean limma at 256:243, or about 90.22 cents. The region from around Pythagorean or 53-tet to 17-tet (with fifths about 3.93 cents wider than pure) might be considered the central or "quintessential" neo-Gothic zone, where (as the latter term may suggest) fifths and fourths are pure or reasonably close to pure. These 3-limit concords coexist with and provide resolutions for a kaleidoscopic variety of unstable intervals and sonorities, some approximating higher-prime ratios such as 13:11 or 17:14. While the progressions retain a Gothic logic, these sonorities add a color and flavor radically distinct from that of historical 3-limit or 5-limit European practice. Beyond 17-tet, we enter a "far neo-Gothic" zone ranging out to around 22-tet, where fifths and fourths are less smooth but still acceptable primary concords, especially with timbral adjustments (Darregization or Sethareanization). At 22-tet, where fifths are about 7.14 cents wide, the diatonic semitone is reduced to only 1/4-tone (~54.55 cents), and we encounter a Wonderland of interval spellings and alterations -- yet standard Gothic progressions and some modern offshoots still succeed musically, and indeed delightfully. From one viewpoint, the neo-Gothic spectrum ranging from around Pythagorean to 22-tet is a kind of mirror-reversed image of the historical European meantone spectrum from Pythagorean to 1/3-comma meantone or 19-tet (with fifths tempered in the _narrow_ direction by about 7.17 cents and 7.22 cents respectively). Thus the subtitle of this article, "Meantone through a looking-glass." ------------------------------------------------------------------- 2. The neo-Gothic spectrum, Blackwood's R, and exponential meantone ------------------------------------------------------------------- Neo-Gothic temperaments with fifths larger than pure may be seen (and heard) as artful variations and distortions of classic medieval Pythagorean tuning, the amount of distortion increasing as we move from 53-tet or Pythagorean to 17-tet, and from there to 22-tet. At the outset of this exploration of the "reverse meantone" continuum, it would be well to distinguish between the scales themselves and the specific Gothic or neo-Gothic applications which are the focus of this article. Musicians who use 53-tet as a system of 5-limit or 7-limit rather than Pythagorean JI, or who use 22-tet for Paul Erlich's tetradic 7-limit system developed by analogy to 5-limit major/minor tonality[1], will be well aware of this distinction. Not only our aesthetic appreciation of a scale, but our reckoning of its vital statistics, can change with musical viewpoint. From our Pythagorean perspective, 53-tet has whole-tones of 9 steps and diatonic semitones of 4 steps; in 22-tet, these intervals are 4 steps and 1 step. For a musician using 53-tet to approximate 5-limit JI, or using Erlich's tetradic 22-tet system, these basic intonational metrics can and will vary.[2] To describe the aesthetics of neo-Gothic tunings and temperaments in a nutshell, we might focus on three main themes: (1) Pure or near-pure fifths and fourths, the prime concords; (2) Active and dynamic thirds and sixths inviting efficient resolutions to stable intervals; and (3) Large whole-tones and small diatonic semitones, facilitating expressive melody and incisive cadential action. In medieval Pythagorean intonation, we have a classic balance between these elements; neo-Gothic temperaments with fifths wider than pure compromise the first element in order to accentuate the second and third. Unstable thirds and sixths become yet more active, and can yet more efficiently resolve to stable intervals; the contrast between large whole-tones and small semitones becomes yet greater. Looking more closely at some specific tunings in the context of Gothic or neo-Gothic parameters of musical style may help in understanding what happens as we move along the reverse meantone continuum from Pythagorean to 22-tet. For a more thorough discussion of medieval Pythagorean tuning in the context of Gothic musical style, and of medieval sonorities and cadences, see [32]http://www.medieval.org/emfaq/harmony/pyth.html [33]http://www.medieval.org/emfaq/harmony/13c.html ------------------------------------------------ 2.1. Artistic parameters and Blackwood's R (T/S) ------------------------------------------------ The following table surveys a few tunings and temperaments at various points along the neo-Gothic continuum, and may become more meaningful as we relate interval sizes to traits and constraints of musical style: ---------------------------------------------------------------------- tuning/ fifth M2 M3 m3 m2 R=T/S temperament (+/-3:2) (+/-9:8) ====================================================================== Central or quintessential neo-Gothic 53-tet 701.89 203.77 407.55 294.33 90.57 2.25 (-0.07) (-0.14) (~81:64) (~32:27) 9/4 ---------------------------------------------------------------------- Pythagorean 701.96 203.91 407.82 294.13 90.22 ~2.26 (0.00) (0.00) (81:64) (32:27) ---------------------------------------------------------------------- 41-tet 702.44 204.87 409.76 292.68 87.80 2.33... (+0.48) (+0.96) (~19:15) 7/3 ---------------------------------------------------------------------- 29-tet 703.45 206.90 413.79 289.66 82.76 2.50 (+1.49) (+2.99) (~13:11) 5/2 ---------------------------------------------------------------------- exponential 704.61 209.21 418.43 286.18 76.97 ~2.71828 meantone (+2.65) (+5.30) (~14:11) (e) ---------------------------------------------------------------------- 17-tet 705.88 211.76 423.53 282.35 70.59 3.00 (+3.93) (+7.85) 3/1 ====================================================================== Far neo-Gothic 39-tet 707.69 215.38 430.77 276.92 61.54 3.50 (+5.73) (+11.47) 7/2 ---------------------------------------------------------------------- 22-tet 709.09 218.18 436.36 272.72 54.55 4.00 (+7.14) (+14.27) (~9:7) 4/1 ====================================================================== As even this small sampling shows, neo-Gothic tuning systems represent a variety of approaches. Pythagorean intonation is a "tuning" in the strict sense, a JI system based on integer ratios only; 17-tet, 22-tet, 39-tet, 41-tet, and 53-tet all belong to the family of equal temperaments or "n-tet's." Exponential meantone, like more familiar historical meantone temperaments not based on an equal division of the octave, would be classified in some schemes as "non-just, non-equal." "Exponential meantone" is defined as having a ratio between its whole-tone and diatonic semitone equal to Euler's exponential _e_, ~2.71828.[3] The result is a temperament with qualities somewhere between those of 29-tet and 17-tet on our chart, with the versatile 46-tet providing a yet closer approximation, and 109-tet a nearly exact one.[4] More generally, as we shall see, the ratio between whole-tone and diatonic semitone, termed "R" by tuning theorist and composer Easley Blackwood[5], provides one measure of the extent of the Neo-Gothic continuum and its place in the larger intonational universe. We may also express this ratio as T/S, using medieval initials for "tone" and "semitone." For neo-Gothic tunings and temperaments, as the last column of our chart shows, R or T/S varies from around 2.25 to 4. Taking the columns of our table from left to right, let us consider how the intonational qualities of these tunings interact with the artistic parameters of medieval or neo-medieval styles. -------------------------------- 2.1.1. Smooth fifths and fourths -------------------------------- In Pythagorean tuning, fifths and fourths, the choice medieval concords rightfully having pride of place on the first column of our chart, have pure ratios of 3:2 and 4:3; in the almost identical 53-tet, fifths are very slightly narrow (~0.07 cents). As we move out along the central neo-Gothic zone through 41-tet and 29-tet and exponential meantone, this ideal is rather mildly compromised; at 17-tet, fifths are about 3.93 cents wide. As we move into the far neo-Gothic zone, this compromise becomes more pronounced, with fifths at 22-tet about 7.14 cents wide. One approach might be to compare these temperaments with historical meantones where the fifths are narrowed. With 41-tet and 29-tet, the tempering is less than in 12-tet (~1.95 cents), whose fifths are often considered "near-pure"; exponential meantone is comparable to 1/8-comma meantone (~2.69 cents), and 17-tet to 2/11-comma meantone (~3.91 cents). Further out, 39-tet compares to something between 1/4-comma (~5.38 cents) and 1/7-comma (~6.14 cents)[6], and 22-tet to 1/3-comma meantone (~7.17 cents). With neo-Gothic or reverse meantones, as with historical meantones, a bit more than 7 cents of tempering seems to mark the limit of tenable compromise for the fifths, and this constraint provides one motivation for placing the far end of our spectrum around 22-tet.[7] To avoid confusing stylistic norms with universal values, we might add that other world musics such as as Balinese or Javanese gamelan quite pleasingly use fifths and fourths much further from 3:2 or 4:3, while musics based on 11-tet or 13-tet get along without any intervals resembling these ratios. -------------------------------------------------- 2.1.2. Compatible major seconds and minor sevenths -------------------------------------------------- In addition to pure fifths and fourths, Pythagorean features pure ratios for major seconds or ninths and minor sevenths (9:8, 9:4, 16:9) with ideal ratios to form relatively concordant sonorities in combination with fifths or fourths: e.g. 4:6:9, 6:8:9, 8:9:12, 9:12:16.[8] In a medieval setting, I term these combinations "mildly unstable quintal/quartal sonorities"; in certain neo-medieval styles, they might be treated not merely as relatively blending but as stable. As the second column of our table shows, the variance of major seconds (and likewise of major ninths and minor sevenths) from their ideal Pythagorean ratios is equal to twice the tempering of the fifths (a relationship sometimes slightly obscured by rounding adjustments). At 17-tet, this variance is around 7.85 cents; by 22-tet, it is around 14.27 cents. Although the just intonation of relatively concordant quintal/quartal sonorities is a special charm of medieval Pythagorean tuning, and these combinations are better within the quintessential neo-Gothic zone from Pythagorean to 17-tet, I agree with Paul Erlich that they remain acceptable in 22-tet[9] -- especially, I would add, with some Darregian/Setharean timbre adjustments. ------------------------------- 2.1.3. Active thirds and sixths ------------------------------- In medieval Pythagorean tuning, major thirds at ~407.82 cents (81:64) and minor thirds at ~294.13 cents (32:27) have an active and unstable but relatively blending or "imperfectly concordant" quality; 53-tet (interpreted in a Pythagorean manner) offers almost identical ratios. Major sixths at ~905.87 cents (27:16) are regarded in the 13th century as somewhat more tense, and minor sixths at ~792.18 cents (128:81) as yet more tense, but play a vital role in cadential progressions, often expanding by contrary motion to octaves. In the 14th century, major and minor sixths gain a status as "imperfect concords" on par with the thirds. To borrow the term of modern composer and theorist Ludmila Ulehla[10], these intervals act as "dual-purpose" sonorities (as distinguished in her terminology from stable "concords" or urgent "discords"), at once inviting directed resolutions to stable intervals and serving as moments of diverting vertical color. As the third and fourth columns of our table show, major and minor thirds (and sixths also, their octave complements) become even more active and dynamic as we move from Pythagorean into the realm of neo-Gothic temperaments with fifths wider than pure. By 17-tet, major thirds have expanded to ~423.53 cents while minor thirds have contracted to ~282.35 cents. In 22-tet, these intervals have sizes of ~436.36 cents (~9:7) and 272.72 cents (not far from 7:6). As we move beyond Pythagorean, Darregian/Setharian timbre adjustments can help in keeping a _relatively_ blending quality for these intervals while enjoying the superefficient cadential action featured by these reverse meantone temperaments (see Section 2.1.5). The expansion of major thirds (and sixths), and contraction of minor thirds (and sixths), may also serve as a possible constraint placing the far end of the neo-Gothic spectrum not too far from 22-tet. At this point, major and minor thirds are near 9:7 and 7:6 respectively, still quite distinct (to my ears) from narrow fourths or wide major seconds. Likewise, major sixths are near 12:7, and minor sixths near 14:9. Going well beyond 22-tet, at a fifth size of around 712 cents we would find major thirds expanding into the region near 450 cents, and minor thirds yet later contracting into the region near 250 cents, etc., where questions of categorical ambiguity could become more important. This is by no means to suggest that such temperaments are undesirable, only to suggest that they may belong to a somewhat different realm than neo-Gothic from Pythagorean to 22-tet.[11] -------------------------------------------------------------------- 2.1.4. Large whole-tones and small diatonic semitones: Blackwood's R -------------------------------------------------------------------- In Pythagorean tuning, whole-tones are a generous 9:8 (~203.91 cents), and diatonic semitones a compact 256:243 (~90.22 cents). This contrast facilitates expressive melody and efficient cadential action. A useful measure of this contrast is the ratio between the sizes of these two intervals, T/S or Blackwood's R, about 2.26 for Pythagorean and precisely 2.25 or 9/4 for the almost identical 53-tet, where a whole-tone is equal to 9 steps and a diatonic semitone to 4 steps. As we move into the spectrum of neo-Gothic temperaments where fifths are tempered increasingly wide of pure, the second column of our table (already met in Section 2.1.2) shows how major seconds increase in size; column 5 shows how diatonic semitones shrink even more rapidly. Column 6 follows the consequent accentuation of the contrast between these intervals as measured by T/S or R. Note that for equal temperaments, R is given both as a decimal and as a fraction showing the number of steps for each interval, e.g. 9/4 for 53-tet and 5/2 for 29-tet. Moving from Pythagorean to 29-tet, we find that whole-tones have expanded rather moderately to ~206.90 cents while diatonic semitones have contracted to ~82.96 cents, with R increasing from ~2.26 to 2.5. In exponential meantone, these intervals are ~209.21 cents and ~76.97 cents, with the defining ratio R of Euler's e, ~2.71828. At 17-tet, whole-tones and diatonic semitones are at ~211.76 cents and ~70.59 cents -- 3 steps and 1 step respectively -- so that R is 3. Travelling into the far neo-Gothic zone, we find that at 39-tet, these intervals are ~215.38 cents and ~61.54 cents, with R at 3.5; the diatonic semitone has become slightly smaller than the _chromatic_ semitone of 19-tet (~63.16 cents, very close to 28:27). At 22-tet, the whole-tone has grown to ~218.18 cents and the diatonic semitone has contracted to ~54.54 cents, with R at 4. While this semitone -- literally a "diatonic quartertone" -- may look very small on paper, I find that my ears can routinely accept it as a regular semitone. Thus the contrast between large whole-tones and concise diatonic semitones, already a notable attraction of Pythagorean, becomes yet more accentuated as we progress along the reverse meantone spectrum: the "minor semitone" of 53-tet or Pythagorean (at or around 4/9-tone) shrinks to the thirdtone of 17-tet and the literal quartertone of 22-tet. This contrast, in its continuum of Neo-Gothic shades, can lend an expressive air to melodic lines and vertical progressions alike, and leads to our fifth artistic theme of efficient cadences. --------------------------------- 2.1.5. Efficient cadential action --------------------------------- In Gothic music, cadential progressions are typically guided by directed resolutions from unstable intervals to stable ones by stepwise contrary motion (e.g. 2-4, 3-1, 3-5, 6-8, 7-5). In the 14th century, such resolutions where one voice moves by a whole-tone and the other by a semitone are especially favored (e.g. m3-1, M3-5, M6-8, m7-5, M2-4). Late medieval theorists tell us that the unstable interval should "approach" its stable goal as closely as possible, resolving with an ideally efficient motion. This cadential aesthetic nicely fits both the melodic and vertical parameters of Pythagorean tuning. A major third at 81:64 (~407.82 cents), for example, has a large size which at once lends it a degree of dynamic tension because of its acoustical complexity, and permits it to expand more economically to a stable fifth, as one voice moves by a whole-tone and the other by an incisive diatonic semitone.[12] In neo-Gothic temperaments with fifths wider than pure, both aspects of this musical equation are further accentuated. As major thirds and sixths grow larger and larger (Section 2.1.3), they take on an even more active and dynamic quality, with this tension released by yet more efficient expansion to fifths and octaves (M3-5, M6-8) involving yet more narrow and incisive diatonic semitones (Section 2.1.4). One measure of cadential efficiency or incisiveness is the total distance an unstable interval must expand (M3-5, M6-8, M2-4) or contract (m3-1, m7-5) in order to reach its stable goal. Since in these "closest approach" progressions one voice moves by a whole-tone and the other by a diatonic semitone -- whose sum is a minor third -- this distance will be equal to a minor third. Thus column 4 of our table, showing the size of a minor third, can also serve as an index of cadential efficiency; as this size gets smaller, cadences become more efficient, involving smaller and more incisive semitonal motions (column 5). In Pythagorean tuning, our "closest approach" progressions are already admirably efficient, involving only ~294.13 cents of expansion or contraction (the size of a 32:27 minor third). By 17-tet, it has decreased to ~282.35 cents, and by 22-tet to ~272.72 cents. This trend correlates intimately with the shrinking of the diatonic semitone from ~90.22 cents in Pythagorean to ~54.55 cents in 22-tet.[13] The "closest approach" aesthetic, as realized by medieval Pythagorean tuning, may combine the satisfying contrast between a tense interval and its stable resolution; the release of this tension through economically directed motion; and the melodic as well as vertical appeal of concise cadential semitones. Neo-Gothic temperaments with fifths larger than pure offer accentuated variations on these themes in assorted shades of intonational Mannerism. --------------- Notes to Part 1 --------------- 1. Paul Erlich, "Tuning, Tonality, and Twenty-Two-Tone Temperament," _Xenharmonikon_ 17 (Spring 1988), pp. 12-40. 2. From a 5-limit JI perspective, 53-tet would mix large whole-tones of 9 steps (~203.77 cents, ~9:8) and small-whole tones of 8 steps (~181.13 cents, ~10:9), with usual diatonic semitones of 5 steps (~113.21 cents), quite close to 16:15 -- an interval which in Pythagorean terms closely approximates the _chromatic_ semitone or apotome at 2187:2048 (~113.69 cents). In Erlich's 22-tet system, see n. 1 above, pp. 22-25, a "large" interval (L) is equal to 3 steps (~163.64 cents), and a "small" interval (s) to 2 steps (~109.09 cents); from a Pythagorean viewpoint, 3 steps is a chromatic semitone, and 2 steps a curious "intermediate semitone" between this and the diatonic semitone of 1 step. 3. This temperament, with fifths about 2.65 cents wider than pure, might be taken as a kind of neo-Gothic counterpart to the "Golden Meantone" developed by Thorvald Kornerup and advocated by Jacques Dudon, where this same ratio is equal to the golden mean, ~1.61834 (and fifths are about 5.74 cents narrow). The latter temperament is about midway between 1/4-comma meantone (~5.38 cents, major thirds pure) and Zarlino's 2/7-comma meantone (~6.14 cents, major and minor thirds equally impure). 4. Emphasizing that the table represents an arbitrary sample of tunings and temperaments at a few points on the neo-Gothic spectrum, I might out of psychological curiosity note my selection of 17-tet, 29-tet, 41-tet, and 53-tet -- possibly because Pythagorean tunings of 17, 29, 41, and 53 notes represent "Moments of Symmetry" as described by theorist Ervin Wilson. As for 39-tet, it is the one equal temperament with 53 or fewer notes having a ratio of whole-tone to diatonic semitone (T/S, or Blackwood's R, see text below and n. 5) greater than 3 but less than 4. The purpose for including exponential meantone is, of course, unabashed promotion. Equally meritorious temperaments such as 46-tet might just as well have been included. 5. Easley Blackwood, _The Structure of Recognizable Diatonic Tunings_ (Princeton: Princeton University Press, 1985). 6. As it happens, 39-tet involves almost exactly the same amount of tempering in the wide direction as Golden Meantone (see n. 3) in the narrow direction. 7. While "mirror-image" comparisons can be engaging, there is an important artistic asymmetry. In historical meantones, fifths are compromised (in the narrow direction) in order to optimize thirds and sixths, the primary Renaissance-Romantic concords. In Gothic or neo-Gothic music, where fifths and fourths are the primary concords and pure Pythagorean intonation provides a superb solution (except for special "neo-medieval" styles where circularity might be sought in less the 53 notes), temperament is more of an artistic liberty, and indeed an artful Manneristic distortion. To tune 19-tet for 5-limit music may be motivated in good part by a desire to optimize overall consonance; to tune 22-tet for Neo-Gothic music is likely more an expression of calculated xenharmonic zest. 8. Examples of these medieval sonorities, using a MIDI-style notation where C4 indicates middle C and higher note numbers show higher octaves, would be C3-G3-D4 (M9 + 5 + 5); C3-F3-G3 (5 + 4 + M2); C3-D3-G3 (5 + 4 + M2); and C3-F3-Bb3 (m7 + 4 + 4). 9. Erlich, see n. 1 above, p. 26 and n. 31. Erlich's 22-tet sonorities forming "the decatonic equivalent of 'quartal' or 'quintal' harmony," ibid. n. 31, involve four notes, e.g., in a conventional Pythagorean spelling, C3-F3-G3-Bb3 or C3-D3-G3-A3. However, I find typical Gothic quintal/quartal sonorities of the kind we have been discussing with three notes and intervals (see n. 8 for examples) to be also satisfactory, at least in apt timbres. 10. Ludmila Ulehla, _Contemporary Harmony: Romanticism through the Twelve-Tone Row_ (New York, 1966), p. 428. 11. In practice, the limit of acceptable temperament for fifths and fourths (especially in styles where they are the main concords) may take priority as a constraining factor, and this limit is arguably reached around 22-tet. With gamelan-like timbres, however, some experiments exploring and possibiy circumventing such constraints might be very interesting. See also n. 17 below on the sizes of major thirds and diatonic semitones in tunings with fifths ranging from 710 to 715 cents. 12. For classic statements of this Pythagorean cadential aesthetic with its "incisive" melodic semitones, see Mark Lindley, "Pythagorean Intonation and the Rise of the Triad," _Royal Musical Association Research Chronicle_ 16:4-61 (1980), ISSN 0080-4460; and "Pythagorean Intonation," _New Grove Dictionary of Music and Musicians_ 15:485-487, ed. Stanley Sadie, Washington, DC: Grove's Dictionaries of Music (1980), ISBN 0333231112. 13. Since "closest approach" progressions involve motion of a whole-tone in one voice and a semitone in the other, the expansion of the major second from Pythagorean to 22-tet by ~14.27 cents (Section 2.1.2, column 2 of table) partially offsets the shrinking of the diatonic semitone by ~35.68 cents, resulting in a net gain in efficiency of ~21.41 cents. Increasing the size of the fifth produces a twofold expansion of the major second (formed from two fifths up minus an octave) but a fivefold reduction of the diatonic semitone (formed from five fifths down). Most respectfully, Margo Schulter --------------------------------------------- Neo-Gothic tunings and temperaments: Meantone through a looking glass (Part 2 of 2) --------------------------------------------- -------------------------------- 2.2. The neo-Gothic region and R -------------------------------- Our survey suggests one possible definition for the neo-Gothic spectrum as the portion of the continuum of regular tunings ("meantone" in the most generic sense) where the value of T/S, or Blackwood's R, ranges from 2.25 (53-tet) to 4 (22-tet). We can subdivide this Neo-Gothic region into a "central" neo-Gothic zone from 53-tet or Pythagorean through 17-tet, where R ranges from 2.25 through 3; and a "far" neo-Gothic zone beyond 17-tet through 22-tet, with R ranging from 3 through 4. The following chart may illustrate this possible mapping; equal temperaments are identified by the number of steps per octave, Pythagorean tuning as "Py," and exponential meantone as "Ex": Py 53 41 29 Ex 17 39 22 |--------|--------|--------|--------|--------|---------|--------| R 2.25 2.5 2.75 3 3.25 3.5 3.75 4 |--------------------------|------------------------------------| Central neo-Gothic Far neo-Gothic (2.25 <= R <= 3) (3 < R <=4) As is the case with many such mappings ranging from music history periodizations to stratigraphic boundaries in paleobiology, different schemes for "drawing lines" can bring out interesting concepts, whatever scheme we choose to adopt at a given moment or for a given purpose. It might seem natural to adopt Pythagorean (or the almost identical 53-tet) as the lower boundary of the neo-Gothic region, given our focus on "reverse meantone temperaments" with fifths wider than pure. Pythagorean tuning, or "zero-comma meantone" with pure fifths is the lower limit of this zone, just as it is the upper limit of meantone temperaments with fifths narrower than pure.[14] If we focus on Blackwood's R, then 53-tet has the attraction of placing the lower boundary at the neat integer ratio of 9/4, or 2.25. Since 53-tet so closely resembles Pythagorean (when its steps are used to define intervals in a Pythagorean manner, of course), and has such a great potential for neo-Gothic music, both tunings seem to belong in the same category.[15] Interestingly, Blackwood suggests a musical basis for setting a lower boundary slightly _below_ Pythagorean, but not much lower than 53-tet. In his view, 406 cents is about the maximize size at which major thirds can acceptably serve as stable concords; beyond this point, they become too acoustically complex and active to form stable 5-limit triads.[16] Since the use of thirds as unstable although _relatively_ blending intervals is a cardinal feature both of historical Gothic polyphony and of its neo-Gothic offshoots, Blackwood's observation suggests a lower boundary at a fifth size of around 701.5 cents (~0.46 cents narrower than pure), producing a major third of 406 cents. Such a tuning would have a whole-tone of 203 cents and a diatonic semitone of 92.5 cents, yielding an R of ~2.195. Taking 53-tet as our lower limit provides a bit of artistic margin: in 53-tet or Pythagorean, thirds have a distinctly active and dynamic quality fitting Gothic and neo-Gothic styles. Setting the high end of the neo-Gothic spectrum at around 22-tet may reflect mainly the problem of the increasing temperament of the fifths (~7.14 cents in 22-tet) in styles where fifths and fourths are the primary concords. Disregarding this constraint, we might engage in interesting dialogues regarding the point at which major thirds growing into narrow fourths, or diatonic semitones shrinking into comma-like intervals difficult to recognize as "half-steps," might clearly place us in a new musical terrain.[17] If we do take the quality of fifths as a governing constraint, and draw an upper boundary at 22-tet, then these other factors may become more academic. In a neo-Gothic spectrum running from 53-tet and Pythagorean to 22-tet, diatonic semitones remain larger than 50 cents throughout the region, and major thirds range from a Pythagorean 81:64 (or minutely smaller in 53-tet) to just larger than 9:7. Minor thirds range from around a Pythagorean 32:27 to slightly larger than 7:6, and major seconds from around a Pythagorean 9:8 to a kind of "mean-tone" in 22-tet about midway between a 9:8 and an 8:7.[18] Following this approach, tunings and temperaments along a neo-Gothic spectrum of R=2.25-4 share a common family resemblance to medieval Pythagorean, giving the region a somewhat unified quality, and yet featuring an impressive range of variation. One purpose for proposing such boundaries, of course, should be not to discourage but to provoke experimentation beyond recognized metes and bounds. If this discussion leads to more exploration of the world beyond R=4, it will have served its purpose. ----------------------------------------- 3. Alternative thirds and 17-tet symmetry ----------------------------------------- If we provisionally accept R=2.25-4 as a range for distinctively neo-Gothic tunings -- as opposed to _meritorious_ tunings! -- there remains the question of why 17-tet (R=3) should serve as a line of demarcation between the central and far neo-Gothic zones. One might reply that 17-tet is a well-known "exaggerated Pythagorean" temperament roughly in the middle of our spectrum[19], that it is the point where a diatonic semitone is equal to precisely 1/3-tone, or roughly the point beyond which the tempering of the fifth becomes a more substantial issue, or simply that R=3 is a nice round number. There is, however, another basis for regarding 17-tet as a point of symmetry in relation to the neo-Gothic spectrum, a basis providing a connection between musical developments of the early 15th century and a possible "reenvisioning of history" through the use of new intonational variations on familiar 14th-century progressions. In the decades around 1400, as Mark Lindley[20] has documented through theoretical sources and actual music, musicians became intrigued with a variation of Pythagorean intonation on keyboards where sharps were tuned at the flat end of the chain of fifths. In this tuning, thirds involving written sharps -- in such sonorities as D-F#-A, A-C#-E, E-G#-B -- had a distinctively smooth quality differing from the active flavor of regular Pythagorean thirds. These alluring major and minor thirds were actually Pythagorean diminished fourths at 8192:6561 (~384.36 cents) and augmented seconds at 19683:16384 (~317.60 cents) -- e.g. D-Gb-A, A-Db-E, E-Ab-B -- intervals removed only by a schisma of 32805:32768 from pure ratios of 5:4 (~386.31 cents) and 6:5 (~315.64 cents). Thus they are often known today as "schisma thirds." By around 1450, the appetite of musicians for these "alternative" thirds evidently led to meantone temperaments seeking them in as many places as possible. This intonational shift may be seen as one aspect of the transition from Gothic to Renaissance musical style, often placed somewhere in the early to middle 15th century. Reflecting on the 15th-century role of Pythagorean augmented seconds and diminished fourths as schisma thirds leads us to a new world of possibilities: the varied flavors these "alternative thirds" take on at different points along the neo-Gothic spectrum. In Pythagorean tuning, as we have just seen, augmented seconds are very close to 6:5 and diminished fifths to 5:4; and this situation is almost identical in 53-tet.[21] More generally, the augmented second is smaller than the diminished fourth. Let us see what happens to these alternative thirds as we move along the neo-Gothic spectrum. In the following table, the last column shows for equal temperaments the number of steps in these intervals, along with the arithmetic of their derivation: an augmented second from a whole-tone plus a chromatic semitone (equal to the difference between T and S); and a diminished fourth from a fourth minus a chromatic semitone. ---------------------------------------------------------------------- tuning/ fifth aug2 dim4 T/S=R aug2/dim4 temperament (+/-3:2) steps ====================================================================== 53-tet 701.89 316.98 384.91 2.25 14 17 (-0.07) (~6:5) (~5:4) 9/4 9+5 22-5 ---------------------------------------------------------------------- Pythagorean 701.96 317.60 384.86 ~2.26 (0.00) (~6:5) (~5:4) ---------------------------------------------------------------------- 41-tet 702.44 321.95 380.49 2.33... 11 13 (+0.48) 7/3 7+4 17-4 ---------------------------------------------------------------------- 29-tet 703.45 331.03 372.41 2.50 8 9 (+1.49) (~17:14) 5/2 5+3 12-3 ---------------------------------------------------------------------- exponential 704.61 341.46 363.14 ~2.71828 meantone (+2.65) (~17:14) (~21:17) (e) ---------------------------------------------------------------------- 17-tet 705.88 352.94 352.94 3.00 5 5 (+3.93) (~11:9) (~11:9) 3/1 3+2 7-2 ====================================================================== 39-tet 707.69 369.23 338.46 3.50 12 11 (+5.73) (~21:17) (~17:14) 7/2 7+5 16-5 ---------------------------------------------------------------------- 22-tet 709.09 381.82 327.27 4.00 7 6 (+7.14) (~5:4) 4/1 4+3 9-3 ====================================================================== As we move outward from 53-tet or Pythagorean, our near-6:5 augmented seconds grow larger while our near-5:4 diminished fifths grow smaller. At 41-tet, this process is not too pronounced, so that we can still speak of "schisma thirds" in a quasi-Pythagorean sense at ~6.31 cents wider than 6:5 and ~5.83 cents narrower than 5:4. By 29-tet or exponential meantone, however, we have moved into a different region where augmented seconds and diminished fourths take on a flavor of "alternative thirds" quite different from 5-limit intervals. There remains a certain polarity between these thirds, ~341.46 cents and ~363.17 cents in exponential meantone: we might term them "superminor/submajor," or in a more medieval Pythagorean fashion "suprasemiditonal/subditonal." Such thirds may suggest various integer ratios such as 17:14 and 21:17, for example. At 17-tet, our expanding augmented second and shrinking diminished fourth converge into a single "neutral" interval at ~352.94 cents, rather close to 11:9 and even closer to the 4-step interval of Gary Morrison's 88-cent equal temperament (88-cet). Thus 17-tet (R=3) represents a special "moment of convergence" on our spectrum.[22] Beyond 17-tet, the augmented second represents the _larger_ alternative third, and the diminished fourth the _smaller_ one. Thus at 39-tet, we have a situation somewhat comparable to exponential meantone, but with the roles of these two intervals reversed. At 22-tet (R=4), our augmented second has expanded to an interval almost equivalent to the Pythagorean diminished fourth, a major third at ~381.82 cents (~4.49 cents narrower than 5:4); our diminished fourth has shrunk to a minor third of ~327.27 cents (~11.73 cents wider than 6:5), somewhat akin to a Pythagorean augmented second. From a composer's perspective, the diverse alternative thirds of the neo-Gothic spectrum present a resource permitting us to "reenvision history" by experimenting with familiar Gothic progressions in new shades of intonation. The "suprasemiditonal/subditonal" thirds found in the portion of the central neo-Gothic zone around exponential meantone, and again in the far zone around 39-tet (with the roles of diminished fourth and augmented second reversed), offer special possibilities in this direction. For example, it is quite possible to take standard 14th-century progressions where minor thirds contract to unisons and major thirds expand to fifths, and substitute these alternative thirds. Like Pythagorean thirds, they seem to have an active and passionate quality lending intensity to cadential action; yet their color is very different. Also, such progressions involve melodic motions by strikingly large _chromatic_ semitones, giving then a definite contrast in flavor to usual Gothic or neo-Gothic progressions with compact diatonic semitones.[23] From the perspective of theory, these alternative thirds also provide a basis for distinguishing between a central neo-Gothic zone where augmented seconds are smaller than diminished fourths (R=2.25-3), and a far neo-Gothic zone where augmented seconds are the larger intervals (R=3-4). At 17-tet, R=3, these two thirds converge into one, a moment of symmetry (in a general as opposed to Wilsonian sense) serving as a kind of "continental divide" for the neo-Gothic spectrum. ---------------------------------------- 4. Through a looking glass enigmatically ---------------------------------------- If we view the neo-Gothic meantone spectrum from Pythagorean or 53-tet to 22-tet (R=2.25-4) together with the historical European meantone spectrum from 19-tet to Pythagorean (R=1.5-~2.26), we may see both regions in better perspective. Here the main number line shows the tempering of the fifth in cents in the narrow (negative) or wide (positive) direction. As with the chart in Section 3, numbers immediately above this line identify equal temperaments, "Py" shows Pythagorean, and "Ex" exponential meantone; fractions represent various historical meantone tunings: 1/3 2/7 1/4 1/5 1/6 1/8 Py Ex 19 31 12 53 41 29 17 39 22 -|----|----|----|---|---|---|---|----|---|---|---|---|---|---|- 5th -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 R 1.5 1.75 2 2.25 2.5 2.75 3 3.5 4 |--------------------------------|------------------------------| Historical meantone Neo-Gothic meantone (1.5 <= R <=~2.26) (2.25 <= R <=4) |------------------| |---------------|--------------| Characteristic Central Far (1.5 <= R <= ~1.82) (2.25 <= R <=3) (3 < R <= 4) Proceeding from left to right, we travel first through the zone of "characteristic" meantone or meantone in the usual sense of temperaments from around 1/3-comma to 1/6-comma where fifths are narrowed in order to obtain pure or near-pure thirds, including 19-tet and 31-tet. Here R is appreciably less than 2, with diatonic semitones larger than chromatic semitones -- an especially colorful contrast in the area of 1/3-1/4 comma. We next move through the intermediate region between 1/6-comma meantone and 53-tet or Pythagorean, including 12-tet (R=2). At about R=2.2, we reach Blackwood's limit of acceptability for 5-limit music, from another perspective our beckoning portal to the Gothic or neo-Gothic world of active and dynamic thirds and sixths efficiently resolving to stable 3-limit intervals. At R=2.25 (53-tet) or R=~2.26 (Pythagorean) we are at the point of symmetry on our chart where fifths are virtually or precisely pure. As we continue through the central neo-Gothic zone, thirds and sixths become even more active while diatonic semitones shrink from around 4/9-tone to 1/3-tone in 17-tet (R=3). Much beyond Pythagorean, at least when seeking to maintain a 13th-14th century balance of concord/discord, we may "Darregize/Sethareanize" our timbres to keep our thirds _relatively_ blending while enjoying the "superefficient" resolutions they invite and the other special qualities of these tunings. With the right timbre for 17-tet, a half-cadence in Machaut on a major third can have the expected quality of a charming but pregnant pause rather than an acute clash! Beyond 17-tet, we move through the far neo-Gothic zone where timbre adjustments can also mitigate the increasingly pronounced temperament of our fifths. We arrive at 22-tet (R=4), a xenharmonic outpost or resort where many features of the classic Pythagorean world are dramatically and intriguingly distorted, and yet the musical terrain remains recognizable. Just as Ivor Darreg courageously asserted that _every_ equal temperament has its own potential for beautiful music[24], so each point and region on our meantone continuum has its own musical virtues and attractions. Some of these beautiful possibilities may more fully reveal themselves if we are familiar with the whole spectrum, including the region of Pythagorean and beyond, a region too often represented by the cartographic legend: "And here there be Wolves." For example, a quite familiar and conventional temperament such as 12-tet can take on new qualities when we appreciate its intriguing ambivalence: is it a compromised 5-limit meantone, or a somewhat subdued "semi-Gothic" 3-limit tuning? One might creatively play on this ambiguity, with the 20th-century use of this scale for both tertian and quartal/quintal harmony (e.g. Bartok, Hindemith) a possible precedent. Once 12-tet is viewed as one member of a vast society of scales, not a substitute for all the others, this temperament and its surrounding "middle country" between Renaissance meantone and Pythagorean may flower in ways not yet imagined. Returning to our main focus on the vibrant world of Gothic and neo-Gothic music and tunings, temperaments beyond Pythagorean could lead in many directions. One of the most intriguing is what I might term direct chromaticism. While Gothic music of the 13th and 14th centuries has many routine and unconventional uses of accidentalism, chromaticism in the proper sense of melodic motion by a chromatic semitone is rather less common -- in contrast to the many examples of such chromaticism in the 16th-century ambience of tunings such as 1/4-comma meantone. In these tunings the difference in size between the large diatonic and small chromatic semitones is dramatic, equal to a diesis of ~41.06 cents in 1/4-comma meantone and ~63.16 cents (1/19 octave) in 19-tet, lending a special air to 16th-century chromatic progressions alternating between the two semitones. While the contrast between the small _diatonic_ and large chromatic semitones is less dramatic in Pythagorean (~23.46 cents), as fifths get wider than pure the disparity rapidly increases to ~41.38 cents in 29-tet (about the same as in 1/4-comma meantone) and ~55.28 cents in exponential meantone (not too far from 19-tet). A millennial era of neo-Gothic chromaticism may be at hand.[25] Microtonalist and tuning theorist Graham Breed[26] has offered some interesting observations raising the question of what I might term "direct commaticism": the use of the Pythagorean comma (e.g. Ab-G#), or its equivalents in various neo-Gothic temperaments as direct melodic intervals. This comma is equal to the difference between the diatonic and chromatic semitones discussed just above. While I have used the direct melodic Pythagorean comma, e.g. Ab-G#, in "neo-medieval" interpretations of certain early 15th-century cadential progressions, such idioms may take on new flavors as we move through the neo-Gothic spectrum. By 29-tet or exponential meantone, as we have seen, this "comma" has grown to the size of a Renaissance meantone diesis, used as a direct melodic interval in the "enharmonic" style of Nicola Vicentino (1511-1576). Might one emulate Vicentino by using the diesis-like commas of these neo-Gothic tunings in amazing shifts and variations on standard medieval progressions?[27] Then again, in 22-tet, the normal diatonic semitone of ~54.55 cents is around the size of a largish diesis, making the above categories rather problematic.[28] This temperament, like a good science fiction novel or physics book, is a feast for the intellect and imagination, as well as actually working musically -- and beautifully -- for Gothic or neo-Gothic music. Whether I compare the experience to travelling near the speed of light, or observing the behavior of photons near an extreme gravity well, 22-tet as a neo-medieval tuning deserves its own theory of relativity. There is also the question of the world beyond 22-tet, possibly a region where neo-Gothic and gamelan may meet. To consider such musical possibilities of the realm beginning with rather than ending at Pythagorean, a realm where the high art of the Gothic era may hold up "a distant mirror" not only to the present but to the future, is to gaze through a looking glass enigmatically, but the enigma is pleasant, enticing one to new music. --------------- Notes to Part 2 --------------- 14. This boundary would accord with the classification of equal temperaments as "positive or negative (that is, fifths that fall short of 701.955-cent third harmonic or fifths that exceed the third harmonic)." See Brian McLaren, "A Brief History of Microtonality in the Twentieth Century," _Xenharmonikon_ 17:57-110 (Spring 1998), at p. 78, describing the work of M. Joel Mandelbaum. 15. Thus the Neo-Gothic spectrum begins with a tuning which has fifths very slightly narrower than pure; in technical terms, 53-tet might be described as ~1/315-comma meantone. 16. Blackwood, n. 5 above, pp. 202-203. Here Blackwood is suggesting a fifth size of 701.5 cents or major third size of 406 cents (R=~2.2) as an upper limit of acceptability for tertian music of the European Renaissance-Romantic repertory. Note that tunings somewhat below this limit, for example 12-tet (R=2), may be quite acceptable for Gothic or neo-Gothic as well as tertian music, but do not have the _distinctive_ "Gothic/neo-Gothic" quality of tunings in the region of R=2.25-4 (from 53-tet and Pythagorean to 22-tet). 17. At a fifth size of 710 cents (~8.04 cents wider than pure), we would have a major third of 440 cents and a minor second of 50 cents; at 712 cents (~10.04 cents wider than pure), 448 cents and 40 cents; at 714 cents (~12.04 cents wider than pure), 456 cents and 30 cents; at 715 cents (~13.04 cents wider than pure), 460 cents and 25 cents. At 710 cents, the value of R is 4.4; at 712 cents, 5.6; at 714 cents, 7.6; at 715 cents, 9.2. 18. Thus two such 22-tet whole-tones form a near-9:7 major third, just as two whole-tones in 1/4-comma meantone (midway between 9:8 and 10:9) form a pure 5:4. 19. My warm thanks to John Chalmers for introducing me to this tuning; the "exaggerated Pythagorean" description may come from Ivor Darreg. 20. For Lindley's impressive thesis, see the articles cited in n. 12 above. 21. From a Pythagorean point of view, people who treat 53-tet as a 5-limit JI system are actually redefining major and minor schisma thirds (17 and 14 steps respectively) as regular thirds. 22. Here I am inspired by Ervin Wilson's "Moment of Symmetry." Whether there is any connection between Pythagorean 17 being a moment of symmetry and 17-tet being a moment of convergence for augmented seconds and diminished fourths I leave as an intriguing question. 23. In neo-Gothic temperaments with fifths wider than pure, the greater-than-Pythagorean difference between these semitones can heighten such a contrast. In exponential meantone, for example, the diatonic semitone is ~76.97 cents and the chromatic semitone ~132.25 cents, a situation somewhat resembling that of Renaissance meantones (where the diatonic semitone is the larger interval). 24. See McLaren, n. 14 above, pp. 80-81. 25. Interestingly, one medieval theorist who does demonstrate and advocate use of the direct chromatic semitone is Marchettus of Padua in his _Lucidarium_ (1318), a treatise also advocating that singers use cadential semitones or "dieses" equal to only "one of the five parts of a tone." Whether or not Marchettus should be read as describing cadences with semitones considerably narrower than in Pythagorean and extra-wide major thirds and sixths -- and he is addressing vocal intonation rather than keyboard tunings -- his treatise has played a central role in my musical odyssey leading to this paper, which I therefore warmly dedicate to him. 26. Graham Breed, Tuning List, Tuning Digest 700 (6 July 2000), Message 25; and Tuning Digest 703 (7 July 2000), Message 8. Breed finds that the comma in 53-tet (1 step, ~22.64 cents) "is too small to be clearly comprehended," and therefore has a "troubling quality," but that the 41-tet comma (1 step, ~29.27 cents) is satisfactorily "heard as a melodic elaboration." He finds a comma at a round 24 cents (in a synthesizer approximation of Pythagorean with fifths tuned to an even 702 cents, ~0.04 cents wide) to be "adequate," although "it takes a bit of getting used to." In addition to directing my attention to the theme of "commatic" progressions, these observations invite further investigation of listener thresholds for the "comprehension" of very small melodic inflections. 27. This line of development may already be a reality, and in historically informed performances of the medieval repertory as well as in new compositions or improvisations in allied styles, since I have seen reports that at least one noted group (Mala Punica) is using "micro-intervals" for 14th-century Italian music. 28. Technically speaking, a diatonic semitone (or quartertone, R=4) in 22-tet is equal to _half_ of the comma, the 2-step difference between the chromatic semitone of 3 steps and the diatonic semitone of 1 step. However, commalike distinctions in this temperament between regular and "schisma" thirds, for example (augmented second vs. major third, minor third vs. diminished fourth), involve a difference of 1 step, the same interval as the usual minor second. Most respectfully, Margo Schulter [