Billbrpt write: <<What gets me is that no matter how many notes these people can imagine being in a scale, it still can only be ET. Their whole world would explode if anyone dared to think of something like a 31 Meantone.<< There is no need to suffer the meantone restrictions in a scale with 31 notes to the octave. Harmonically, it lets us have it both ways, it is just physically difficult to manipulate an instrument with that many keys. In fact, the meantone tunings are simply subsets of the higher ET's. i.e. a 54 TET has all the notes found in 12 TET as well as those we form by moving from ET to well-temperaments. >> The greater than 12 tone scale was being discussed and dismissed as foolish intellectual pondering in the early 19th Century too.<< I would like to post below a recent posting from Margo Schulter that address the question in more detail. I don't think it was considered foolish, and it certainly predates the early 19th century. -------------------------------------------------- 3. Adaptive structure: a comparison with Vicentino -------------------------------------------------- While realizing tetrachords and scales like those of Archytas and al-Farabi in a "classic JI" manner, our 24-note adaptive scheme also somewhat resembles the 38-note system of Vicentino for obtaining vertical concords with pure ratios. Here we focus on some similarities and differences between these two adaptive systems, from the viewpoint of the keyboard performer as well as the tuning theorist. Vicentino's system for obtaining both "perfect fifths" and "perfect thirds," as mentioned at the opening of Part I, involves two 19-note manuals evidently each tuned in 1/4-comma meantone (Gb-B#) with pure major thirds at a distance of 1/4 syntonic comma (~5.38 cents) apart. A likely arrangement for the keyboards of Vicentino's _archicembalo_ or _arciorgano_ is the following, with the usual five accidental keys of each manual split front-to-back (e.g. G#/Ab, Eb/D#), and extra small keys for E# and B#. Here Vicentino's comma sign (') -- for Vicentino, the term "comma" may refer to various small intervals -- serves to show the raising of the notes on the second manual by 1/4 syntonic comma: Db' D#' Gb' Ab' A#' C#' Eb' E#' F#' G#' Bb' B#' C' D' E' F' G' A' B' C' ----------------------------------------------- Db D# Gb Ab A# C# Eb E# F# G# Bb B# C D E F G A B C Using this instrument, the player can attain pure ratios for complete 5-limit sonorities of the 16th century extolled both by Vicentino and Gioseffo Zarlino (1558) as manifesting "perfect" harmony (Zarlino's _harmonia perfetta_). Here are just realizations on Vicentino's keyboard for four forms of these complete sonorities regarded as concordant in practice and theory, and also two forms discussed by Zarlino[6] combining a sixth with a fourth above the lowest part, commonly subject to many of the same restrictions as more unequivocal discords. Interval arrangements for each sonority are shown by the notation (outer|lower + upper), and the just tuning is shown using Zarlino's string ratios, more modern frequency ratios, and rounded cents with respect to the lowest voice: ----------------------------------------------------------------------- Example Intervals String ratio Freq ratio Cents ----------------------------------------------------------------------- C3-E3-G3' (5|M3 + m3) 15:12:10 4:5:6 0-386-702 D3-F3'-A3' (5|m3 + M3) 6:5:4 10:12:15 0-316-702 E3-G3'-C4 (m6|m3 + 4) 24:20:15 5:6:8 0-316-814 F3'-A3'-D4 (M6|M3 + 4) 5:4:3 12:15:20 0-386-884 ....................................................................... G3'-C4-E4 (M6|4 + M3) 20:15:12 3:4:5 0-498-884 A3'-D4-F4' (m6|4 + m3) 8:6:5 15:20:24 0-498-814 ----------------------------------------------------------------------- Pure major thirds and minor sixths are played with both notes on the same keyboard. Fifths and minor thirds, tempered narrow by Vicentino's "comma" of 5.38 cents on either keyboard, are restored to their just proportion by playing the lower note on the lower manual, and the upper note on the upper manual (e.g. D3-F3'-A3'). Fourths and major sixths, tempered wide by this same amount, are played in their just forms with the lower note on the upper keyboard and the upper note on the lower keyboard (e.g. F3'-A3'-D4). As Vicentino's notation shows, narrow intervals are thus enlarged by this "comma," and wide intervals reduced by it, achieving pure ratios. In his circular of 1561 advertising the arciorgano, this "perfect diatonic music" is described along with its marvellous effects[7]: First there are obtained perfect fifths above the white keys of the common organ, which make a wonderful sound; then two kinds of thirds, one major, one minor, and similarly, two kinds of sixths, in which case it happens that whenever perfect fifths are struck together with perfect thirds, they fill the ears with such harmony that no better can be heard on earth. Both Vicentino's adaptive tuning and the adaptive Xeno-Gothic scheme appear to have the following properties in common: (1) Both systems make it possible to realize a complete set of sonorities featuring pure ratios: Vicentino's stable sonorities based on 5-limit ratios of 2-3-5, or stable and unstable Gothic/neo-Gothic sonorities based on ratios of 2-3-7. (2) Each of the two manuals in itself represents a standard regular tuning: Vicentino's 19-note meantone (Gb-B#), or a Gothic/neo-Gothic 12-note Pythagorean (Eb-G#). (3) The notes of the upper keyboard are shifted by a slight interval to obtain pure ratios for vertical sonorities: by 1/4 syntonic comma (~5.38 cents) in Vicentino's scheme, or a septimal schisma (~3.80 cents) in the neo-Gothic scheme. While both schemes yield pure vertical sonorities combining multiple prime factors (2-3-5 or 2-3-7), Vicentino's is based on a regular temperament, evidently 1/4-comma meantone: melodic intervals, and also vertical intervals in unstable sonorities, typically have irrational meantone ratios. Indeed the beauty and elegance of this system lies in its synthesis between the flexibility and regularity of a Renaissance temperament and the purity of just vertical concords. In contrast, the adaptive neo-Gothic scheme is entirely based on integer ratios, a much easier solution in a medieval or neo-medieval setting than in a 5-limit Renaissance setting.[8] Such an "adaptive rational" scheme satisfies two additional properties: (1) The regular tuning on each keyboard is also a just tuning based on integer ratios, namely Pythagorean intonation; and (2) The small interval of adjustment between the two keyboards is itself rational, here the septimal schisma of 33554432:33480783 (~3.80 cents), so that all ratios in the scheme remain integer-based. In one obvious respect, Vicentino's system shares the melodic smoothness and regularity of meantone while our neo-Gothic adaptive system exemplifies the intricacy or unevenness of classic multi-prime JI systems: the matter of commas and comma shifts. ------------------------------------------- 3.1. Commas and melodic evenness/unevenness ------------------------------------------- Vicentino's system, like a usual meantone temperament, disperses the syntonic or 3-5 comma (81:80, ~21.51 cents) by narrowing each regular fifth by 1/4 of this amount, equal to the 5.38-cent adjustment between the two manuals. In progressing from one pure vertical sonority to the next, the usual meantone melodic intervals need vary by only this small amount, rather than by a full syntonic comma as often happens in classic Renaissance JI. In contrast, the adaptive neo-Gothic system features the classic septimal or 3-7 comma (64:63, ~27.26 cents); notably unequal whole-steps of 9:8 and 8:7, and sometimes direct melodic progressions by a septimal comma (Section 4), are characteristic, whether regarded as blemishes or adornments. As discussed in Part I, Section 1, this latter system has effectively exchanged one comma for another: the Pythagorean or 2-3 comma which would normally obtain between the manuals is enlarged by a septimal schisma to a septimal comma, making pure 7-based sonorities possible. Here we might add that Vicentino himself ardently advocates another kind of striking melodic shift: the enharmonic shift of a meantone diesis, equivalent to a 2-3 comma (in 1/4-comma tuning, typically 128:125 or ~41.06 cents). However, such shifts are an optional and expressive effect providing a prime motivation for his other archicembalo/arciorgano tuning dividing the octave into 31 essentially equal steps of 1/5-tone, rather than an integral and necessary feature of his adaptive JI system.[9] The question of the commas invites a comparison of the practical challenges and opportunities which keyboardists may encounter using these two systems in the lively musical settings for which they are designed. ----------------------------------------------------- 3.2. Commas and stable concords: A keyboardist's view ----------------------------------------------------- While the presence of an unexpurgated septimal comma in the neo-Gothic system makes it in some ways more intricate, Vicentino's system may actually offer a more arduous task for the performer who wishes to maintain pure concords wherever possible. This is true in part because of the distinctions of vertical stability/instability involved in each case. In Gothic or neo-Gothic music, the complete stable sonority is the three-voice trine (e.g. D3-A3-D4, 2:3:4). In our adaptive scheme, as in a usual Pythagorean tuning, all regularly fifths and fourths on each keyboard -- and therefore complete trines -- are pure. One need only play them in the usual manner.[10] In Vicentino's 16th-century system, however, playing any full 5-limit concord (Zarlino's _harmonia perfetta_) requires mixing notes from the two keyboards in order to obtain pure fifths (or fourths) and minor thirds (or major sixths). Of course, such mixing of notes from the two manuals is required in the neo-Gothic scheme to obtain pure 7-flavor versions of unstable sonorities. However, the typical vertical spacing of such medieval or neo-medieval sonorities can simplify this task. In many three-voice or four-voice progressions, the widest vertical interval negotiated by a single hand is characteristically a fifth or fourth: E^4 F4 F4 E^4 E^4 F4 D4 C4 F4 E^3 D^4 E^4 D4 E^4 F4 B^3 C4 B^3 C4 Bb3 A^3 Bb3 A^3 C4 B^3 C4 G3 F3 G3 F3 G^3 A^3 G^3 A^3 G3 F3 While keeping the right hand on the right manual at the right time remains a challenge, the keyboardist is at least assured of a friendly "landing" on a trine or fifth conveniently located on a single manual. In contrast, let us see what is involved in some routine Renaissance progressions at Vicentino's keyboard if we wish to maintain pure concords for any length of time: G4 F'4 B4 C'5 A4 B4 A'4 G#4 G4 F#4 D'4 D4 G4 A4 E'4 G4 D4 E4 D4' D4 B3 A'3 D'4 F4 C'4 D'4 A'3 B'3 Bb'3 A'3 G3 D3 G3 F3 A3 G3 F'3 E3 G2 D3 As these examples may illustrate, we can indeed progress between pure vertical sonorities with smooth and virtually regular melodic steps, but with a complication more generally noted for the archicembalo by Ercole Bottrigari in 1594[11]: "[T]he player must many times press and hold down down with one hand some keys of both keyboards at the same time, and occasionally do this with both hands at once." Possibly new and more ergonomic keyboard designs may alleviate the mechanical difficulties of needing to mix notes from both keyboards, frequently within a single hand, "whenever perfect fifths are struck with perfect thirds," to borrow Vicentino's phrase. Happily, both adaptive systems provide a "safety valve" for the intrepid keyboardist, who is free to revert to a regular and stylistically felicitous tuning system readily at hand as a subset of the full scheme: 19-note Renaissance meantone for the player of Vicentino's instrument, or 12-note medieval Pythagorean for the player of the neo-Gothic instrument. This flexibility might serve not only as a concession to the beleaguered performer, but as a musical virtue: the opportunity to mix and contrast pure sonorities with more complex ones. In a neo-Gothic setting, complex Pythagorean versions of unstable sonorities are valued and cultivated in their own right alongside pure 7-flavor versions. In Vicentino's system, for example, the playing of certain passages with a subtle contrast between pure major thirds and the tempered minor thirds in a usual meantone fashion could add a welcome note of musical variety[12], alongside other passages featuring pure fifths and thirds. In short, both adaptive systems may have the advantage of combining the musically routine with the extraordinary, giving the performer the creative scope for choice in seeking out effects both old and new. ----- Notes ----- 1. See, e.g., Margo Schulter and David Keenan, "The Golden Mediant: Complex ratios and metastable musical intervals," Tuning Digest 810:3, 18 September 2000; http://www.egroups.com/message/tuning/12915. Note that the term "Noble Mediant" might now be preferred in the title, as suggested by Dave Keenan. 2. For these standard forms of 7-flavor cadences with melodic steps of 9:8 and 28:27, the general rule is that intensive progressions (with ascending semitones) resolve to a trine or fifth on the lower manual, while remissive progressions (with descending semitones) resolve to the upper manual. Another way of stating this pattern: melodic whole-tones are played on the same keyboard (e.g. G3-F3, E^4-F#^4); ascending semitones move from the upper to the lower keyboard (e.g. B^3-C4, E^4-F4); and descending semitones from the lower to the upper keyboard (e.g. F3-E^3, C4-B^3). 3. In contrast, the 15th-century shift in Western Europe from active Pythagorean thirds and sixths to 5-limit "valley" tunings resulted in larger cadential semitones and less efficient resolutions. Major thirds at 5:4 (~386.31 cents) and major sixths at 5:3 (~884.36 cents) are a syntonic comma (81:80, ~21.51 cents) _narrower_ than their Pythagorean counterparts, while minor thirds at 6:5 (~315.64 cents) are _wider_ by this same amount. The m3-1, M3-5, and M6-8 resolutions thus require this extra amount of expansion or contraction, a total distance of a rounded 316 cents (the size of the 6:5 minor third). Cadential semitones in a pure 5-limit tuning are typically 16:15 (~111.73 cents), again a syntonic comma larger than Pythagorean. On early Renaissance tension between the musical values of smoother thirds and incisive cadential semitones, see also Mark Lindley, "Pythagorean Intonation and the Rise of the Triad," _Royal Musical Association Research Chronicle_ 16:4-61 at 45 (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. 4. While the musical style is derived from that of Gothic Europe, the mixture of melodic semitones and major thirds might somehow evoke for me also the Balinese or Javanese pelog and Japanese pentatonic scales featuring these intervals. 5. In the right setting, these narrow fourths (21:16, ~470.78 cents) and wide fifths (32:21, ~529.22 cents) might present an opportunity for special effects rather than a problem. Keenan Pepper, for example, has extolled the "crunchiness" of 21:16, and neo-Gothic styles favor many varieties of altered or "less usual" intervals. 6. Gioseffo Zarlino, _The Art of Counterpoint: Part Three of Le Istitutioni harmoniche, 1558_, trans. Guy A. Marco and Claude Palisca (W. W. Norton, 1976), ISBN 0-393-00833-9, Chapter 60, pp. 190-193. On Zarlino's approach to multivoice sonorities, see also my article, http://www.egroups.com/message/tuning/15397, and also emendations in http://www.egroups.com/message/tuning/15434 in response to valuable corrections by Paul Erlich also available as part of that thread, e.g. http://www.egroups.com/message/tuning/15400. 7. Henry W. Kaufman, "Vicentino's Arciorgano: An Annotated Translation," _Journal of Music Theory_ 5:32-53 (1961) at 34-35. 8. In a neo-Gothic JI system (factors of 2-3-7) based entirely on integer ratios, stable trines (2:3:4 or 3:4:6) are free from the complications of the septimal or 3-7 comma at 64:63. In contrast, in a Renaissance JI system (factors of 2-3-5), stable sonorities combining Vicentino's "perfect fifths and perfect thirds" (4:5:6 or 10:12:15) squarely confront the problem of the syntonic or 3-5 comma at 81:80. As we shall see in Section 3.2, even Vicentino's system, while it succeeds in dispersing the syntonic comma, bears some sign of this greater complication by requiring the performer to mix notes from both manuals whenever such a "perfect" Renaissance sonority is desired. 9. Vicentino's adaptive system offering sonorities where all fifths and thirds are "perfect," as well as his first or circulating 31-note scheme for his archicembalo and arciorgano, would nicely fit a base tuning of 1/4-comma meantone for the first 12 notes of the lower manual, which Vicentino simply directs should be tuned as on usual keyboard instruments, with the fifths slightly narrowed or "blunted." 10. In an "avant-garde" neo-Gothic style which acted on the view of Jacobus of Liege that 9:1 (a major 23rd, e.g. D3-E6) is a "perfect concord" by treating sonorities such as D3-A5-E6 (1:6:9) as stable, these 3-prime-limit sonorities would also be available in their usual locations on either keyboard. 11. Ercole Bottrigari, _Il Desiderio, 1594_, trans. Carol MacClintock, Musical Studies and Documents 9 (American Institute of Musicology, 1963), p. 51, where it is noted that tuners and organists may be rather intimidated by "the keys separated, as I have said, into two keyboards with the usual black semitones divided in two and others added." 12. On such a contrast in 1/4-comma meantone "between pure major 3rds and tempered minor 3rds," see Mark Lindley, "Temperaments," _New Grove Dictionary of Music and Musicians_ (n. 3 above), 18:660-674 at 663. Most respectfully, Margo Schulter mschulter@value.net You do not need web access to participate. You may subscribe through email. 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