The Neomedieval Diminished Seventh Sonority: In homage to Scott Dakota and his Monarda-12 tuning Recently Scott Dakota has been focusing on his Monarda-12 tuning as one manifestation of his Tannic temperament family, and his discussion of the diminished seventh chord in a tonal context has led me to this brief study of the diminished seventh in a neomedieval context. Many thanks, Scott, for your inspiration, creativity, and many theoretical insights! The diminished seventh sonority, or more specifically the "fully diminished" seventh sonority, may be defined in a neomedieval or more specifically Parapyth context as having three adjacent minor thirds. In the most typical Parapyth case, these thirds will have sizes ranging from around 7:6 (266.871 cents) to 13:11 (289.210 cents). In this typical form of the Parapyth diminished seventh, the three minor thirds will produce an outer interval of a diminished seventh equal to some form of middle sixth. A few examples in the MET-24 system (2/1, 703.711, 57.422) may illustrate some of the different possibilities: ...13;11 + 13;11 + 13;11 = 104;63...|....13;11 + 168;143 + 13;11 = 18;11.....| Bb4.. 866.6 ..104;63................|..B4.....853.7...18;11..................| ......................13;11 288.9...|.........................13:11. 288.9...| G4... 577.7 ...88;63................|..G#4... 564.8...18;13..................| ......................13;11 288.9...|........................168:143 276.0...| E4... 288.9 ...13;11................|..F^4... 288.9...13;11..................| ......................13;11 288.9...|.........................13;11. 288.9...| C#4 ....0.0.....1/1.................|..D^4......0.0....1/1...................| . . ...13;11 + 7;6 + 13;11 = 13;8.......|......7;6 + 13;11 + 168;143 = 21;13.......| Bb^4...842.6....13;8................|..C#^4...829.7....21;13...................| ......................13;11 288.9...|.......................168;143 276.0......| G^4....553.7....11;8................|..Bb^3...553.7....11;8....................| .......................7;6. 264.8...|........................13;11. 288.9......| F4.....288.9 ...13;11...............|..G^3....264.8.....7;6....................| ......................13;11 288.9...|.........................7;6.. 264.8......| D4 ......0.0.... 1/1................|..F3.......0.0.....1/1....................| Three usual sizes of minor thirds are involved in these diminished seventh sonorities: the regular minor third from three fourths at 496.3 cents each, at 288.9 cents (e.g. D4-F4), or a regular tone at 207.4 cents plus a regular minor second at 81.4 cents; a septimal minor third at 264.8 cents, equal to the regular tone at 207.4 cents plus the spacing generator at 57.4 cents (e.g. F3-G^3); and a "supraseptimal" minor third at 276.0 cents, equal to a regular tone at 207.4 cents plus a "middle thirdtone" formed from a chromatic semitone or augmented prime less a semitone, i.e. 126.0 - 57.4 = 68.6 cents (e.g. G^4-G#). This last interval may also be thought of as an augmented second less spacing (333.4 - 57.4 = 276.0 cents). A diminished seventh sonority invites many alternative resolutions, with wolf fifths or fourths as the main constraining factor. Each 12-note chain of fifths in a 24-note Parapyth system such as MET-24 has no 4;3 fourth at Eb or Eb^, and no 3;2 fifth at G# or G#^. Fortunately, the diversity of resolutions leaves open a number of options even in positions where these "wolf" intervals become relevant. -------------------------------------------------------- 1. Direct resolutions of the diminished seventh sonority --------------------------------------------------------- Let's consider the simplest case for building a diminished seventh sonority: C#-E4-G4-Bb4, where all the notes belong to a single chain of fifths, and all three minor thirds have the regular near-just 13:11 size, with the outer interval a large middle or submajor sixth at 866.6 cents, around 104:63 (867.8 cents) or 33:20 (867.0 cents). The first two options are "direct" resolutions, where the diminished seventh is a "penultimate" sonority resolving immediately to a stable one. In the first of these options, the outer diminished seventh or large middle sixth expands to the octave of a complete 2:3:4 sonority: Bb4----- +126.0 ----- B4 866.6...............1200.0 G4------ - 81.4 ----- F#4 577.7................703.7 E4------ +207.4 ----- F#4 288.9................703.7 C#4----- -207.4 ----- B3 The expansion from large middle or submajor sixth to octave in the outer parts helps convincingly release the tension, with the lowest part descending by a regular tone, and the highest part ascending by a near-14:13 or small middle second step of 126.0 cents which acts as a "supraminor second." Other details of counterpoint including a resolution between the middle parts from regular minor third to unison (E4-G4 to F#4), and a curious expansion between the two highest parts by stepwise contrary motion from minor third to fourth (G4-Bb4 to F#4-B4), with the lower voice of this pair descending by a regular near-22:21 semitone G4-F#4 (81.4 cents) and the higher voice ascending by near-14:13 supraminor second Bb4-B4 (126.0 cents). The other option for a standard direct resolution is for the outer diminished seventh or submajor sixth to contract to a stable fifth, with all voices again moving by step: Bb4----- - 81.4 ----- A4 866.6................703.7 G4------ +207.4 ----- A4 577.7................703.7 E4------ -207.4 ----- D4 288.9................ 0 C#4----- + 81.4 ----- D4 There's an interesting contrapuntal symmetry in this resolution: the outer parts ascending (lowest voice, C#4-D4) or descending (highest voice, Bb4-A4) by a regular diatonic semitone at 81.4 cents, to move from the diminished seventh or submajor sixth to a fifth; and the middle voices descending (E4-D4) or ascending (G4-A4) by a regular 207.4-cent tone to expand from regular minor third to fifth (E4-G4 to D4-A4). In these direct resolutions of a neomedieval diminished seventh sonority, we arrive at a stable sonority with the lowest voice either a tone lower (expansive resolution of dim7-8, C#4-B3) or a semitone higher (contractive resolution of dim7-5, C#4-D4). --------------------------------------------------------------------------------- 2. More intricate resolutions: The diminished seventh as antepenultimate sonority --------------------------------------------------------------------------------- When we consider more intricate resolutions, the possibilities multiply. In these more intricate resolutions, the diminished seventh serves as an "antepenultimate" sonority which leads to another unstable sonority, which in turn resolves to a stable sonority. ------------------------------------ 2.1. The antepenultimate contractive ------------------------------------ In one main variety of these resolutions, as in the second direct resolution, the outer diminished seventh contracts to a fifth -- but, this time, with the note at the diminished fifth becoming the perfect fourth of a 6:8:9 sonority treated as a suspension, with this fourth then resolving to a minor third, which resolves to the unison of a stable sonority completing the progression: (Contractive -- intensive manner) Bb4----- -207.4 ----- A4---------------------------------------------- + 81.4 --- Bb4 866.6................703.7.......................................................703.7 G4----------------------- -207.4 -- F4 -- -81.4 - E4 -- +81.4 -- F4 -- -207.4---- Eb4 577,7................496.3.........288.9.........207.4.........288.9.............. 0 E4------ -207.4 ----- D4---------------------------------------------- + 81.4 --- Eb4 288.9.................0 C#4----- + 81.4 ----- D4---------------------------------------------------------- r Here the 6:8:9 sonority (D4-G4-A4) resolves to D4-F4-A4 at 0-288.9-703.7 cents (~22:26:33), which after decoration by a momentary ~8:9:12 sonority (D4-E4-A4) and a return to D4-F4-A4 leads to a final cadence of D4-F4-A4 to Eb4-Bb4, an intensive manner of cadencing where the outer voices of 22:26:33 ascend by regular semitones (D4-Eb4, A4-Bb4) while the middle voice descends by a regular tone (F4-Eb4). While the diminished seventh sonority requires four voices, here the rest of the progression needs only three active voices, so that a rest (r) is shown at the end for the lowest written voice. I follow this convention in the following examples also. Another version of this cadence is identical, except for the last sonority: (Contractive -- remissive manner) Bb4----- -207.4 ----- A4---------------------------------------------- +207.4 --- B4 866.6................703.7.......................................................703.7 G4----------------------- -207.4 -- F4 -- -81.4 - E4 -- +81.4 -- F4 -- - 81.4---- E4 577,7................496.3.........288.9.........207.4.........288.9..............0 E4------ -207.4 ----- D4---------------------------------------------- +207.4 --- E4 288.9.................0 C#4----- + 81.4 ----- D4---------------------------------------------------------- r Here the final cadence is in the remissive manner, with the middle voice descending by a semitone (F4-E4), and the outer voices ascending by a tone (D4-E4 and A4-B4). Thus intensive cadences involve ascending semitonal motion, and remissive cadences descending semitonal motion, with the ascending semitonal motion often deemed in late medieval European theory more "intense" or decisive, while descending semitonal motion is more "relaxed," or, as one might say in current parlance, "laid back." To this point, all the notes needed for these diminished seventh sonorities and their resolutions have come from a single chain of fifths. For our third contractive resolution of the diminished seventh C#4-E4-G4-Bb4 as an antepenultimate sonority, however, we will need notes from both chains of fifths: Bb4----- -207.4 ----- A4---------------------------------------------- +138.9 --- Bb^4 866.6................703.7.......................................................703.7 G4----------------------- -207.4 -- F4 -- -81.4 - E4 -- +81.4 -- F4 -- -150.0---- Eb^4 577,7................496.3.........288.9.........207.4.........288.9..............0 E4------ -207.4 ----- D4---------------------------------------------- +138.9 --- Eb^4 288.9.................0 C#4----- + 81.4 ----- D4---------------------------------------------------------- r In this progression, D4-F4-A4 resolves in what is termed the equable manner to the fifth Eb^4-Bb4^, with the outer voices ascending by steps of 138.9 cents (D4-Eb^4, A4-Bb^4), very close to 13:12 (138.6 cents), while the middle voice descends by a step of 150.0 cents, very close to a just 12:11 (150.6 cents). The term "equable," originally suggested by George Secor, refers to the melodic division of a minor third into two middle second steps, here a 13:11 minor third (e.g. D4-F4) into the steps D4-Eb^4 (13;12) and Eb^-F4 (12:11). In a purely melodic context, we might play a tetrachord D4-Eb^4-F4-G4 with the lower minor third D4-F4 divided 13:12:11 (JI values 0-150.6-289.2 cents, or a tempered 0-150.0-288.9 cents). Here, the resolution of D3-F4 to a unison on Eb4 realizes this same division contrapuntally, with the lower voice ascending by 13;12 and the upper voice descending by 12;11. A variation on this last equable resolution involves a different tuning of the minor third in the penultimate sonority leading to the final cadence: Bb4----- -207.4 ----- A4------------------------------------------------ +138.9 --- Bb^4 866.6................703.7.........................................................703.7 G4----------------------- -207.4 -- E^4 -- -57.4 - E4 -- +57.4 -- E^4 -- -126.0---- Eb^4 577,7................496.3.........264.8.........207.4...........264.8..............0 E4------ -207.4 ----- D4------------------------------------------------ +138.9 --- Eb^4 288.9.................0 C#4----- + 81.4 ----- D4----------------------------------------------------------- r In this variation, the middle voice at the fourth of 6:8:9 resolves to E^4 at 264.8 cents above D4, a near-just 7:6 minor third (just size 266.9 cents), leading to another shading of an equable resolution where the 7:6 minor third cadences to a unison with the lower voice again ascending by a 13;12 step (D-Eb^4), and the middle voice descending by a step of 126.0 cents (E^4-Eb4), close to 14:13 (128.3 cents). Thus we have, in contrapuntal terms, a 12:13:14 division of the 7:6 minor third. We could also express this 12:13:14 division in purely melodic terms in a tetrachord such as D4-Eb^4-E^4-G4, or 1;1-13;12-7;6-4;3 (JI 0-138.6-266.9-498.0 cents; tempered 0-138.9-264.8-496.3 cents), or 24:26:28:32. ---------------------------------- 2.2. The antepenultimate expansive ---------------------------------- In an antepenultimate expansive progression, as in the first direct resolution of a diminished seventh sonority, the outer diminished seventh or middle sixth expands to an octave -- here, the octave of a 6:8:9:12 sonority, thus C#4-E4-G4-Bb4 to B3-E4-F#4-B4, with the minor third of the diminished seventh sonority, in this case E4, becoming the fourth of the 6:8:9:12. Then follows a resolution much like that of the 6:8:9 sonority in antepenultimate contractive progressions, with the fourth resolving to the minor third, and this sonority in turn resolving to a stable 2:3:4. (Expansive -- intensive manner) Bb4----- +126.0 ----- B4------------------------------------------------- + 81.4 --- C5 866.6...............1200.0.........................................................1200.0 G4--------------------F#4-------------------------------------------------+ 81.4 --- G4 577,7................703.7..........................................................703.7 E4------------------------ -207.4 -- D4 -- -81.4 -- C#4 -- +81.4 -- D4 -- -207.4 --- C4 288.9................496.3..........288.9..........207.4..........288.9...............0 C#4----- -207.4 ----- B3--------------------------------------------------+ 81.4 ----C4 Here the final cadence is B3-D4-F#4-B4 to C4-C4-G4-C5, or in approximate JI terms, 22:26;33:44 to 2:3:4. In this closing resolution, the second to lowest voice descends by a regular tone at 207.4 cents, while the other three voices ascend by regular diatonic semitones at 81.4 cents. The remissive version of this cadence differs only in the final sonority, and the concluding melodic motions: (Expansive -- remissive manner) Bb4----- +126.0 ----- B4------------------------------------------------- +207.4 --- C#5 866.6...............1200.0.........................................................1200.0 G4--------------------F#4-------------------------------------------------+207.4 --- G#4 577,7................703.7..........................................................703.7 E4------------------------ -207.4 -- D4 -- -81.4 -- C#4 -- +81.4 -- D4 --- -81.4 --- C#4 288.9................496.3..........288.9..........207.4..........288.9...............0 C#4----- -207.4 ----- B3--------------------------------------------------+207.4 ----C#4 Here the second to lowest voice concludes by descending a regular diatonic semitone, while the other three voices ascend by regular tones to arrive at C#4-G#4-C#4 (2:3:4). We might call this a "rebounding diminished seventh" progression, the lowest voice starts and ends on the same note, C#4. These two progressions are available using only the notes of a single 12-note chain of fifths, but from this location, C#4-E4-G4-Bb4, our third or equable manner of cadencing requires notes from both 12-note chains: (Expansive -- equable manner) Bb4----- +126.0 ----- B4------------------------------------------------- +138.9 --- C^5 866.6...............1200.0.........................................................1200.0 G4--------------------F#4-------------------------------------------------+138.9 --- G^4 577,7................703.7..........................................................703.7 E4------------------------ -207.4 -- D4 -- -81.4 -- C#4 -- +81.4 -- D4 -- -150.0 --- C^4 288.9................496.3..........288.9..........207.4..........288.9...............0 C#4----- -207.4 ----- B3--------------------------------------------------+138.9 ----C^4 This time the second to lowest voice descends 150 cents, D4-C^4, a near-just 12:11 step, while the other voices all rise by steps of 138.9 cents, a near-just 13:12, so that, for example, the minor second at B3-D4 (288.9 cents, a near-just 13:11) contracts to a unison on C#4; and the major third D4-F#4 between the two middle voices at 414.8 cents, which could be taken as 33:26 (412.7 cents) or 14:11 (417.5 cents), expands to a near-3:2 fifth at 703.7 cents. -------------------------------------------- 3. Many destinations -- and yet more nuances -------------------------------------------- We have now considered how a neomedieval diminished seventh sonority may invite no fewer than eight standard resolutions, as in our example of C#4-E4-G4-Bb4: Direct expansive: C#4-E4-G4-Bb4 to B3-F#4-B4 (lowest voice moves C#4-B4, -207.4 cents) Direct contractive: C#4-E4-G4-Bb4 to D4-A4 (lowest voice moves C#4-D4, +81.4 cents) Antepenultimate contractive resolutions: Intensive: C#4-E4-G4-Bb4 arriving at Eb4-Bb4 (lowest voice moves C#4-Eb4, +162.9 cents) Remissive: C#4-E4-G4-Bb4 arriving at E4-B4 (lowest voice moves C#4-E4, +288.9 cents) Equable: C#4-E4-G4-Bb4 arriving at Eb^4-Bb^4 (lowest voice moves C#4-Eb^4, +220.3 cents) Antepenultimate expansive resolutions: Intensive: C#4-E4-G4-Bb4 arriving at C4-G4-C5 (lowest voice moves C#4-C4, -126.0 cents) Remissive: C#4-E4-G4-Bb4 arriving at C#4-G#4-C#5 (lowest voice moves C#4-C#4, 0 cents) Equable: C#4-E4-G4-Bb4, arriving at C^4-G^4-C^5 (lowest voice moves C#4-C^4, -68.6 cents) If we take C#4, the lowest note of the diminished seventh, as a reference point or "1/1," we find that these eight resolutions have points of arrival as follows, using the abbreviations of DE and DC for the Direct Expansive and Direct Contractive; and for the more intricate resolutions AC (Antepenultimate Contractive) and AE (Antepenultimate Expansive) followed by a letter showing the manner of the progression: (I)ntensive; (R)emissive; or (E)quable. .DE.........AEI.........AEE........AER.......DC......ACI.....ACE......ACR .B4---------C4----------C^4--------C#4-------D4------Eb4-----Eb^4------E4.. 16/9.......13/7........52/27.......1/1.....22/21....11/10..143/126...13/11 992.6.....1074.0......1131.4........0......81.4.....162.9...220.3....288.9 In sum, a diminished seventh sonority can lead to lots of destinations, and these destinations tend to be fairly close to the starting point, no further than a minor third from it. This reflects the way that medieval European and neomedieval progressions mostly involve stepwise motion, or sometimes motion by a third, in all voices, including the lowest voice. Historically, this pattern holds during the 9th-14th centuries. In the 15th to early 17th centuries, there's a balance between stepwise or thirdwise motion in the lowest voice and motion by a fourth or fifth; in late 17th-19th century tonality, motions by a fifth or fourth have a special and central importance. Even more destinations can be available if we keep in mind the possibilities for resolutions involving ratios of 2-3-7, and more particularly the 6:7:9 sonority with a fifth divided into a lower 7:6 minor third and upper 9:7 major third (266.9-435.1 cents in JI, and 264.8-438.9 cents in MET-24). In two instances, there are septimal variations on resolutions we've already considered for C#4-E4-G4-Bb4 that open yet more points of arrival. The first is the Antepenultimate Contractive resolution (Section 2.1) in the intensive manner: (Contractive -- intensive manner) Bb4----- -207.4 ----- A4---------------------------------------------- + 57.4 --- A^4 866.6................703.7.......................................................703.7 G4----------------------- -231.4 -- E^4 -- -57.4 - E4 -- +57.4 --E^4 - -207.4 --- D^4 577,7................496.3.........264.8.........207.4..........264.8.............. 0 E4------ -207.4 ----- D4---------------------------------------------- + 57.4 --- D^4 288.9.................0 C#4----- + 81.4 ----- D4---------------------------------------------------------- r In place of F4, the note at 288.9 cents or a tempered 13:11 minor third above D4, we substitute E^4 at 264.8 cents or a tempered 7:6, a 24-cent comma lower in this temperament. This leads to an intensive cadence D4-E^4-A4 to D^4-A^4 with motions of a 207.4-cent tone down in the middle voice (E^4-D^4), and ascending small thirdtones of D4-D^4 and A4-A^4 in the outer voices at 57.4 cents, which could be taken as a narrow 28:27 step (63.0 cents). The destination of the lowest voice in this variation is thus D^4, a comma lower than Eb4 in the version in Section 2.2. The antepenultimate expansive resolution in the intensive manner in Section 2.2 is open to a similar variation: (Expansive -- intensive manner) Bb4----- +126.0 ----- B4------------------------------------------------- + 57.4 -- B^4 866.6...............1200.0.........................................................1200.0 G4--------------------F#4-------------------------------------------------+ 57.4 -- F#^3 577,7................703.7..........................................................703.7 E4------------------------ -231.4 -- C#^4 -- -57.4 - C#4 -- +57.4 -C#^4 - -207.4 --- B^3 288.9................496.3..........264.8..........207.4..........264.8...............0 C#4----- -207.4 ----- B3--------------------------------------------------+ 57.4 --- B^3 Here our destination for the lowest voice is B^3, a comma lower than C4 in the earlier version. At times, these resolutions with 6:7:9 sonorities may be of interest as a way to arrive at a destination on the upper chain of fifths, e.g. to access a mode of some kind centered on the step B^ where this transposition offers an attractive set of intervals, etc. Another motivation, more vertically oriented, is that after the tension of a diminished seventh, the special smoothness of sonorities such as 6:8:9 or 6:7:9 can be very pleasing. To borrow some words I recall from David Doty in reference to JI, here applied to a near-just tuning like MET-24, sometimes a rather consonant sonority such as 6:8:9 or 6:7:9 may very pleasantly resolve to a yet greater consonance such as a 2:3:4 sonority or a simple 3:2 fifth. If we add B^3 and D^4 to our catalogue of destinations for the lowest part in resolutions of C#4-E4-G4-Bb4, then we now have these options: .DE.....AEI*...AEI.........AEE........AER......DC....ACI*..ACI.....ACE......ACR .B4-----B^4----C4----------C^4--------C#4------D4----D^4---Eb4-----Eb^4------E4.. 16/9...11/6.,.13/7........52/27.......1/1.....22/21.13/12.11/10..143/126...13/11 992.6.1050.0.1074.0......1131.4........0......81.4..138.9.162.9...220.3....288.9 * Resolution with ~6:7:9 sonority As it happens, the C#4-E4-G4-Bb4 sonority permits access to all of these resolutions without any obstacles in the form of wolf fourths or fifths. At some other locations, these wolf intervals can constrain our freedom of choice -- or, in some instances, provide the opportunity to use a bold 16:21:24 sonority in place of 6:8:9, for example. Also, from some positions, we may find ourselves using 6:7:9 sonorities, for example, not out of free choice, but because they are the only minor third sonorities available (unless, for example, we extend our concept of a "minor third" to include smaller middle thirds, a fascinating variation). A larger study of the neomedieval diminished seventh would include various shadings and permutations, as very incompletely sketched out at the opening of this paper. But the flexibility of this sonority, and its options for many standard resolutions, are very noteworthy in even a brief survey like this. Margo Schulter 27 March 2018