O Europae Middle Portion (mm. 22-32) in JI Mykhaylo Khramov has raised the fascinating question of how a piece in a tempered tuning such as Peppermint 24 (12-note chains of fifths at 704.096 cents, two chains at 58.680 cents apart) might be realized in JI. Here I will focus specifically on the middle portion of _O Europae_, where a JI rendition is possible with values generally within a few cents of the tempered ones, but with pure vertical concords and often also optimized melodic steps. Currently a difficulty I am having in reading Mykhaylo's notation for his MIDI arrangement of my piece may possibly not be so disadvantageous here, since it gives me an opportunity to propose a JI tuning for this middle section of _O Europae_ without already understanding his solution, then allowing us to compare notes on our approaches. To give an overview of the piece from my perspective as the composer, I should briefly explain that although the style is generally medieval or neomedieval European, I borrow a device common in the later style of the 16th-century madrigal. This is to have rather straightforward and cheerful passages, mostly diatonic, that contrast with slower sections featuring various kinds of unusual inflections, dissonances, and the like. As in many of these madrigals, so in _O Europae_, these contrasting styles often reflect the varying moods of a text. Thus opening (mm. 1-21) and closing (mm. 33-41) passages of _O Europae_ are celebratory and outgoing, saluting the European Union and especially honoring France as a voice for peace. The middle portion, however, has words which invite a neomedieval equivalent of the kind of chromaticism and/or dissonance found, for example, in the madrigals of Don Carlo Gesualdo or Claudio Monteverdi. Ex dumeto illaqueante belli ducite nos Out of the entangling thicket of war lead us Translating this section of the piece from Peppermint temperament to JI is fortunately not too difficult, because unlike some other tempered schemes such as meantone, the commas disregarded by Peppermint are generally not too large. Specifically, the 81:80 or Didymic comma (21.51 cents) does not apply, because simple ratios of 5 are not sought in this style -- in contrast to meantone, where they are, of course, the main point of the temperament. Also, the 64:63 comma of Archytas (27.26 cents) and some close variants are observed, so that translating from Peppermint to JI usually involves quite small shifts in the positions of notes. Given my own intonational formation and medieval orientation, it is perhaps not surprising that I find ratios as a natural language, with their equivalent in cents, savarts equal to 2^(1/301), or units of 1024-ed2 or sometimes 2048-ed2 supplementing the ratios. Thus I will show notes in two systems of alphabetical notation (Sagittal and simple keyboard notations); as JI ratios from F, the final of this piece in F Lydian which can be regarded as the 1/1; and by their differences in cents from the same note spellings in Peppermint. --------------------------------------- 1. Version with Sagittal note spellings --------------------------------------- In the notations for this portion of _O Europae_ as it appears in the PDF version, defining a few important symbols may suffice to clarify the notation. In Peppermint these symbols inflect notes by precisely defined amounts, but in a JI realization their sizes may be somewhat negotiable, although typically not varying more than a relatively few cents from the tempered values. The regular flat sign \||/ simply lowers a note by an apotome or chromatic semitone, equal in Peppermint to 128.7 cents, a near-just 14:13 (128.3 cents), with this ratio a general guide for JI. Thus B\||/ is identical to the familiar Bb. The half-flat or diesis sign \|/ lowers a note by the spacing between the keyboards, 58.680 cents, very close to 121:117 (58.2 cents) or 91:88 (58.0 cents) -- but also often representing the larger 28:27 (63.0 cents), or the smaller 33:32 (53.3 cents). We will see that in progressions where a 7:6 minor third contracts to a unison, or a 9:7 major third expands to a fifth, it is ideal that one voice should move by a 9:8 tone (203.9 cents), and other by the 28:27 thirdtone of Archytas (63.0 cents), which in the temperament is somewhat compressed. Another sign providing alternative spellings for some notes shown with the \|/ sign, e.g. Bb\|/ (the note a spacing interval below B\|/) is the |\ sign, as in A|\, which shows a note raised in Peppermint by the difference between a regular diatonic semitone at around 79.5 cents (e.g. A-Bb or A-B\||/) and the spacing interval at 58.7 cents (e.g. Bb\|/-Bb), a tempered comma of 20.8 cents. Thus A|\ or Bb\|/ defines the note a 20.8-cent comma higher A and a spacing interval of 58.7 cents lower than Bb or B\||/. While the sizes of the |\ comma and \|/ spacing interval are fixed in Peppermint, in JI they may be more fluid. For example, in the context of directed resolutions for such intervals as 7:6 to a unison, 9:7 to a fifth, 12:7 to an octave, or 7:4 to a fifth -- each with one voice moving by a 9:8 tone and the other by a 28:27 small semitone or thirdtone -- it may be natural to think of \|/ as showing the lowering of a note by 28:27. In other contexts it might represent 33:32, 91:88, 121:117, or some other ratio. The size of the |\ comma is also somewhat fluid or negotiable in JI. For example, if we take a regular semitone as around 22:21 (80.5 cents), and the spacing variable in a certain context as 28:27 (63.0 cents), this would imply a |\ comma at the rather small size of 99:98 (17.5 cents). We will see this kind of relationships in our example between steps at 14/11 above F (A) and 9/7 above F (Bb\|/ or A|\). In other JI contexts, the |\ comma might also represent 78:77 (e.g. 7/6 vis-a-vis 13/11) at 22.3 cents; or the Archytan comma at 64/63, e.g. 21/16 vis-a-vis 4/3 (27.3 cents). At the beginning of the middle Section at measure 22, if we take the opening note A in the lowest voice as 14/11 (417.5 cents) in relation to the final of F, then this tuning might result, with notes shown as alphabetical spellings, JI ratios, JI values in cents, and differences from the tuning of the same note in Peppermint, and ratios shown above each voice for each melodic interval (e.g. 9:8 shows an ascending 9:8 tone, while 8:9 shows a descending tone of the same size): 22 23 24 25 | 1 2 3 4 5 6 | 1 2 3 4 5 6 | 1 2 3 4 5 6 | 1 2 3 4 5 6 | 27:28 28:27 22:21 21:22 -62.961 62.961 80.537 -00.537 E4 E4 E\|/4 E4 F4 F4 F4 21/11 21/11 81/44 21/11 2/1 2/1 2/1 1119.463 1119.463 1056.502 1119.463 1200 same same +1.015 +1.105 -5.296 +1.105 0 same same 9:8 8:9 9:7 203.910 -203.910 435.084 A3 A3 B3 A3 D\|/4 D\|/4 D\|/4 14/11 14/11 63/44 14/11 18/11 18/11 18/11 417.508 417.500 621.418 417.508 852.592 same same +1.126 +1.126 -3.156 +1.126 -1.015 same same 27:28 28:27 -62.961 62.961 A3 A3 A\|/3 A3 A3 A3 A3 14/11 14/11 27/22 14/11 14/11 14/11 14/11 417.508 417.500 354.547 417.508 417.508 same same +1.126 +1.126 -3.156 +1.126 +1.126 same same 3:2 3:2 6:7:9 3:2 7:9:11 7:9:11 7:9:11 Ex du- me- to Il- la- que- 26 27 28 29 30 | 1 2 3 4 5 6 | 1 2 3 4 5 6 | 1 2 3 4 5 6 | 1 2 3 4 5 6 | 1 2 3 4 5 6| 22:21 27:28 28:27 80.537 -62.961 62.961 E4 F4 E|\4 E|\4 E|\4 F4 21/11 2/1 27/14 27/14 27/14 2/1 1119.463 1200 1137.039 1137.039 1137.039 1200 -1.015 0 -4.281 -4.281 -4.28 0 33:32 8:9 27:28 28:27 8:9 14:11 53.273 -203.910 -62.961 62.961 -203.910 417.508 D\|/4 D\|/4 D4 C4 B|\4 C4 B\||/3 18/11 18/11 27/16 3/2 81/56 3/2 4/3 852.592 852.59 905.865 701.955 638.994 701.955 498.045 -1.015 -1.015 -6.422 -2.141 -6.422 -2.141 +2.141 99:98 28:27 17.576 62.961 A3 A3 A|\3 A|\3 A|\3 B\||/3 14/11 14/11 9/7 9/7 9/7 4/3 417.508 417.508 435.084 435.084 435.084 498.045 -1.126 -1.126 -2.141 -2.141 -2.141 +2.141 14:18:21 7:9:11 16:21:24 6:7:9 8:9:12 6:7:9 3:2 an- te bel - - - - li 31 32 | 1 2 3 4 5 6 | 1 2 3 4 5 6 |.... 112:99 9:8 213.598 203.910 F4 F4 G4 A4 2/1 2/1 224/99 28/11 1200 1200 1413.598 1617.508 0 0 +5.407 +1.126 9:8 203.910 D4 D4 D4 E4 56/33 56/33 56/33 21/11 915.553 same same 1119.463 +3.266 same same -1.105 21:22 80.537 B\||/3 B\||/3 B\||/3 A3 4/3 4/3 4/3 14/11 498.045 same same 417.508 +2.141 same same +1.126 22:28:33 22:28:33 33:42:56 2:3:4 du- ci- te nos First, a bit of overall musical context may place these intonational details in better perspective. In the opening and closing mostly diatonic sections, we have more conclusive intensive cadences (ascending semitonal motion, descending whole-tone motion) on the modal final of F, contrasting with internal remissive cadences (descending semitonal motion, ascending whole-tone motion) on A. There are also in these sections intensive cadences on G, the step above the final, to provide a bit of variety and color. This contrast between intensive cadences on F as a modal final, and remissive cadences on A, is a standard pattern of 14th-century Western European vertical organization in polyphony. In the middle chromatic section, however, this usual vertical order is reversed with an intensive cadence on A. In conventional notation, this would be G#3-B3-D#4, here tuned at a just 6:7:9, resolving to the stable fifth A3-E4, with the 7:6 minor third G#3-B3 contracting to a unison and the 9:7 major third B3-D#4 expanding to a fifth. Here the small semitones G#3-A3 and D#4-E4 are, in JI, melodic steps of 28:27 (62.961 cents); while the descending tone B3-A3 in the middle voice is tuned at 9:8 (203.910 cents). In the Sagittal notation of the PDF file, based largely on the system of George Secor and David Keenan (with contributions by others), the unstable 6:7:9 sonority is written A\|/3-B3-E\|/4, with the \|/ sign lowering a note by an amount represented in the Peppermint temperament by a spacing interval of 58.680 cents. In a JI rendition, as here, the precise size of this interval may be defined according to the musical context, which here suggests 28:27. We will note that all vertical intervals of the 6:7:9 sonority are pure, as is the resolving fifth of this intensive cadence, A3-E4. Differences from Peppermint steps are generally quite small, but illustrate some distinctions between JI and tempered tuning. Here I place the opening fifth A3-E4, in relation to the final of F3, at 14/11-21/11. Since Peppermint is designed closely to approximate these ratios, only very small differences are required in JI: 1.126 cents higher for a just 14/11 step at A3; and 1.015 cents lower for a just 21/11 step at E4. This fifth is followed by an intensive cadence in which it is the goal, a kind of very small-scale departure and return: thus A3-E4 to A\|/3-B3-E4 to A3-E4. Taking A3 in the lowest voice at 14/11 as a reference pitch uniting the departure and the return, this implies that A\|/3 should be a 28:27 lower, or at 27/22 (354.547 cents). This is lower by 3.156 cents than the Peppermint position for this note at 357.703 cents. The tempered 357.703 cents is indeed a compromise. In a Near Eastern context, for example, it provides a reasonable representation of 27/22 as the wusta Zalzal or middle finger Zalzalian third fret in al-Farabi's Mode of Zalzal; and, at the same time, a somewhat better approximation of 16/13 (359.472 cents), an interval whose octave complement or inversion is 13/8 (840.528 cents), with various uses in Near Eastern and neomedieval contexts. Thus the object of Peppermint as a fixed temperament is to set A\|/ so as to be reasonably accurate for either 27/22 (as the step an approximate 28:27 below A, or al-Farabi's wusta Zalzal) or 16/13. In JI, however, it is possible to make slight melodic adjustments so as to choose 27/22, as here, or 16/13, or possibly some intermediate value such as 59/48 (357.217 cents), which occurs in a tetrachord of Safi al-Din al-Urmawi. After this intensive cadence on A, the lowest voice stays at A3 or 14/11 while the upper voices shift to form the unstable sonority A3-D\|/4-F4 or 7:9:11 (0-435.084-782.492 cents). Tuning a just 7:9:11 results in 14/11-18/11-2/1. Note that the outer minor sixth at 11:7, A3-F4, is a routine part of the diatonic structure; but 7:9:11 has a chromatic atmosphere, something like an augmented or diminished sonority in a 16th-century European setting. The highest voice momentarily descends from the minor sixth to the fifth and then ascends back (F4-E4-F4), thus 2/1-21/11-2/1, with semitone motions of 22:21 (80.537 cents). In terms of the overall ensemble or concentus, thus produces a momentary shift from the prevailing 7:9:11 to 14:18:21 (a lower 9:7 and upper 7:6 third, 0-435.084-701.955 cents), and then back to 7:9:11. The 7:9:11 is also a fine example of what is termed in modern theory an isoharmonic sonority, since the adjacent terms 7:9 and 9:11 have identical differences of 2, George Secor introduced me to these combinations, and they play a very important role in neomedieval verticality. In a usual neomedieval idiom, we might expect this 7:9:11 to move to a sonority above the same lowest note with a 4:3 fourth and a 3:2 fifth, here A3-D4-E4, emulating a Renaissance European suspension with a fourth above the bass and a major second between the fourth and another voice at the fifth. In a neomedieval context, A3-D4-E4 would be considerably more concordant than in a Renaissance idiom, especially when tuned as a just or near-just 6:8:9 (0-498.045-701.955 cents), a rather simple ratio, although the 9:8 tone adds a degree of tension (being in itself, as a simple two-voice interval, relatively but not acutely tense). However, in place of the expected A3-D4-E4, we have (measure 27) A|\3-D4-E|\4, with two surprising elements. The first is the melodic shift of the lowest voice by a direct comma step, A3-A|\3, here 14/11-9/7, a shift of 99:98 or 17.576 cents. The second is a vertical sonority to which George Secor initiated me in 2001: A|\3-D4-E|\4 at 16:21:24, with a narrow 21:16 fourth (470.781 cents) at a 64:63 comma smaller than a just 4:3, and a just 3:2 fifth, and therefore a large 8:7 tone (231.174 cents) between the upper voices. As built above the 9/7 step for A|\3, this 16:21:24 sonority at A|\3-D4-E|\4 in its just form calls for 9/7-27/16-27/14 (435.084-905.865-1137.039 cents) in reference to F3. In Peppermint, the tunings for these same steps would be 437.225-912.287-1141.320 cents). Here the greatest difference involves D4, which in the usual diatonic order would placed above F3 at around 22/13 (910.790 cents) or 56/33 (915.553 cents). In order here to obtain a just 21:16 narrow fourth above 9/7, however, this must be lowered from a tempered 912.287 cents to a Pythagorean 27/16 -- a lowering of 6.422 cents. The considerable dissonance of the 21:16 narrow fourth now resolves to produce two relatively concordant or mildly unstable sonorities. The outer voices hold the fifth A|\3-E|\4 or 9/7-27/14, while the middle voice moves down by a 9:8 tone from D4, the 21:16 fourth above A|\3, to C4 at a smooth 7:6 minor third above it, forming a directed 6:7:9 sonority of A|\3-C4-E|\4 (0-266.871-701.955 cents), with the notes positioned at 9/7-3/2-27/14. In a Renaissance-like figure, the middle voice momentarily descends by a 28:27 step to 81/56, forming A|\3-B|\4-E|\4 as a just 8:9:12 sonority (0-203.910-701.955 cents) with a lower 9:8 tone, an upper 4:3 fourth, and an outer 3:2 fifth, another relatively concordant or mildly unstable sonority. Then it returns to 3/2, the 7:6 third of our 6:7:9 at 9/7-3/2-27/14. A notable difference here is that in Peppermint, B|\4 is placed at 645.416 cents, often for example approximating a 16/11 above F (648.683 cents); but to obtain an 81/56 step, we must lower this tempered position by 6.422 cents. This 6:7:9 sonority then resolves to the fifth Bb3-F4, at 4/3-2/1, with melodic motions of an ascending 28:27 in the outer voices of A|\3-Bb3 (9/7-4/3) and E|\4-F4 (27/14-2/1), and a descending 9:8 tone in the middle voice, C4-Bb3 (3/2-4/3). We now return to the usual diatonic order as the fifth Bb3-F4 leads to a routine _quinta fissa_ or "split fifth" sonority of Bb3-D4-F4 at 22:28:33 with a lower 14:11 major third and an upper 33:28 minor third (0-417.508-701.955 cents), here placed at 4/3-56/33-2/1, with the highest voice then moving from the fifth to the major sixth of a directed Bb3-D4-G4 or 33:42:56 sonority (14:11 major third, 56:33 major sixth, 0-417.508-915.553 cents) placed at 4/3-56/33-224/99. Note that D4 at 56/33 above F is in a considerably higher position than it was for the striking 16:21:24 suspension (measure 27), where it was at 27/16 -- a difference of 896:891 or 9.688 cents. In Peppermint, small compromises including a slight stretching of fifths and compressing of fourths permit a single tempered value of 912.287 cents for F3-D4 with reasonable results; but in JI, some fluidity in the placement of notes permits pure vertical intervals often combined with superparticular or other desired ratios for melodic steps. Even in JI, some more more complex melodic steps may result, as is true here at measure 31, where the highest voice moves from a 3:2 fifth to a 56:33 major sixth above Bb3, F4-G4, a step of 112:99 or 213.590 cents, greater than 9:8 by 896:891. Such melodic steps of a tone somewhat greater than 9:8 seem characteristic of Byzantine music, for example, and also some idioms of Persian music. In Peppermint, all regular tones are 208.191 cents, a reasonable compromise; but in JI, some variation may be not merely tolerable but desirable. The Bb3-D4-G4 sonority now resolves in a regular remissive cadence to A3-E4-A4 at a just 2:3:4, a sonority which as Johannes de Grocheio declares around 1300 represents _trina harmoniae perfectio_, the "threefold perfection of harmony" with its outer 2:1 octave, lower 3:2 fifth, and upper 4:3 fourth. In this neomedieval JI tuning, Bb3-D4-G4 at 4/3-56/33-224/99 resolves to A3-E4-A4 at 14/11-21/11-28/11. As expected in a progression using _musica recta_, the regular gamut of the diatonic notes plus Bb (with B/Bb, as in _O Europae_, often a fluid step), we have a descending semitonal motion of 22:21 (80.537 cents), here in the lowest voice (Bb3-A3, 4/3-14/11), combined with ascending 9:8 motions in the upper voices (D4-E4, 56/33-21/11; G4-A4, 224/99-28/11). Both the Peppermint temperament and the JI realization for this section of _O Europae_ are based mainly on primes 2-3-7-11-13. Within the JI version, we find a variability for the step D4 as great as 896:891 or 9.688 cents: 27/16 (905.865 cents) at measure 27, versus 56/33 (915.553 cents) at measure 31. In terms of the Peppermint tuning of D4 at 912.287 cents, the 27/16 is 6.422 cents lower; and the 56/33, 3.266 cents higher. An advantage here in the process of translating from a tempered tuning to JI is that Peppermint aims rather closely to emulate JI, with 896:891 about the largest comma dispersed rather than observed within the usual diatonic system (for example with four or five fifths up producing a near-just 14/11 major third and 21/11 major seventh, larger by 896:891 than the Pythagorean 81/64 and 243/128 respectively). In meantone temperaments which disperse the larger syntonic or Didymic comma of 81:80 (21.506 cents), so that three or four fifths up approximate a 5/3 major sixth and 5/4 major third, translating to JI may be a more complicated process; and likewise with Archytan temperaments which disperse the septimal or Archytan comma of 64:63 (27.264 cents), so that three or four fifths up represent a 12/7 major sixth and 9/7 major third. Thus a neomedieval style may be especially congenial both to near-just temperaments like Peppermint, and to JI realizations. There are related topics concerning the consonance/dissonance continuum and the like that I would like to address elsewhere; but the purpose here is to address the tuning of vertical sonorities and melodic steps when translating neomedieval music from a temperament to a JI realization. ----------------------------------------- 2. Version with simple keyboard spellings ----------------------------------------- While the PDF file for _O Europae_ uses Sagittal notation, it is possible to notate the steps of Peppermint, or their variations in a JI realization, using a simple keyboard notation. For this notation, when the upper 12-note chain of fifths is chosen as the reference, then notes on the lower chain (lowered by a spacing of 58.680 cents) are shown by a "v" sign -- thus Dv-D where Sagittal notation might have D\|/-D. Although not relevant here, the "^" sign similarly serves, when a note on the lower chain of fifths is chosen as the reference, to show a note on the upper chain of fifth, raised by the spacing of 58.680 cents. The v (or ^) sign if applicable, plus the usual note spellings for the 12 notes on either chain (Eb-G#), suffice to define all 24 notes of the tuning -- whose positions become somewhat variable in JI. Apart from this simplified notation, the following is identical to the Sagittal version presented above: 22 23 24 25 | 1 2 3 4 5 6 | 1 2 3 4 5 6 | 1 2 3 4 5 6 | 1 2 3 4 5 6 | 27:28 28:27 22:21 21:22 -62.961 62.961 80.537 -00.537 E4 E4 Ev4 E4 F4 F4 F4 21/11 21/11 81/44 21/11 2/1 2/1 2/1 1119.463 1119.463 1056.502 1119.463 1200 same same -1.015 -1.015 -5.296 -1.015 0 same same 9:8 8:9 9:7 203.910 -203.910 435.084 A3 A3 B3 A3 Dv4 Dv4 Dv4 14/11 14/11 63/44 14/11 18/11 18/11 18/11 417.508 417.500 621.418 417.508 852.592 same same +1.126 +1.126 -3.156 +1.126 -1.015 same same 27:28 28:27 -62.961 62.961 A3 A3 Av3 A3 A3 A3 A3 14/11 14/11 27/22 14/11 14/11 14/11 14/11 417.508 417.500 354.547 417.508 417.508 same same +1.126 +1.126 -3.156 +1.126 +1.126 same same 3:2 3:2 6:7:9 3:2 7:9:11 7:9:11 7:9:11 Ex du- me- to Il- la- que- 26 27 28 29 30 | 1 2 3 4 5 6 | 1 2 3 4 5 6 | 1 2 3 4 5 6 | 1 2 3 4 5 6 | 1 2 3 4 5 6| 22:21 27:28 28:27 80.537 -62.961 62.961 E4 F4 Fv4 Fv4 Fv4 F4 21/11 2/1 27/14 27/14 27/14 2/1 1119.463 1200 1137.039 1137.039 1137.039 1200 -1.015 0 -4.281 -4.281 -4.28 0 33:32 8:9 27:28 28:27 8:9 14:11 53.273 -203.910 -62.961 62.961 -203.910 417.508 Dv4 Dv4 D4 C4 Cv4 C4 Bb3 18/11 18/11 27/16 3/2 81/56 3/2 4/3 852.592 852.59 905.865 701.955 638.994 701.955 498.045 -1.015 -1.015 -6.422 -2.141 -6.422 -2.141 +2.141 99:98 28:27 17.576 62.961 A3 A3 Bbv3 Bbv3 Bbv3 Bb3 14/11 14/11 9/7 9/7 9/7 4/3 417.508 417.508 435.084 435.084 435.084 498.045 -1.126 -1.126 -2.141 -2.141 -2.141 +2.141 14:18:21 7:9:11 16:21:24 6:7:9 8:9:12 6:7:9 3:2 an- te bel - - - - li 31 32 | 1 2 3 4 5 6 | 1 2 3 4 5 6 |.... 112:99 9:8 213.598 203.910 F4 F4 G4 A4 2/1 2/1 224/99 28/11 1200 1200 1413.598 1617.508 0 0 +5.407 +1.126 9:8 203.910 D4 D4 D4 E4 56/33 56/33 56/33 21/11 915.553 same same 1119.463 +3.266 same same -1.015 21:22 80.537 Bb3 Bb3 Bb3 A3 4/3 4/3 4/3 14/11 498.045 same same 417.508 +2.141 same same +1.126 22:28:33 22:28:33 33:42:56 2:3:4 du- ci- te nos Margo Schulter 7 June 2017