all regular meantones have rather low major sevenths, in comparison to our modern 12EDO or Pythagorean tuning, as a result of flattened fifths combined with low (closer to optimal 5/4) major thirds, or alternatively, because the flatness of the the fifth is multiplied 5 times, e.g. F-C-G-D-A-E. many people see this as a problem, a reason to choose a well-temperament or closer-to-pure-fifth tuning over meantone, so that those major sevenths can be closer to the octave, a smaller leading tone providing not only a bit of tension to keep things moving but a rather nice close to phrases, small melodic minor seconds (100c or less, say) contrasting possibly with a reasonably large whole step of perhaps 200 or 210 or even 220 cents.
small step resolutions are very inviting, heightening dissonance before a cadence, and bending the listener’s ear just a little before resolving. historically many sizes of minor second have been used, and even today with 12EDO as a standard there is some debate on how high leading tones should be to provide just the right feel.
my view is that it depends on context. most people have overlooked – or simply not had enough experience playing and listening to – proper low leading tones, being the default option in meantones. those of 1/5-comma meantone at a pure 15/8 (1088.3c), 1/4-comma at 1082.9c, or 2/7-comma at 1079.1c, all rather different from say 12EDO’s semitone. to my ears, these are just right as major sevenths in most music without too much movement, a lazy large minor second up to the octave is rather restful, warm and means a pretty well-tuned dominant chord if resolving a perfect cadence. a lot of musicians argue that dominant chords and especially dominant sevenths should be anything but restful, with a sharpened third and possibly even seventh heightening tension, an active dominant being more effective than a static one. but i myself love the sound of leading tones in the above meantone systems, with the 120c seconds of 50EDO being particularly effective, especially if the tonic is sounded against it or in close proximity (e.g. in a pre-emptive resolution ala Purcell, or in a trill)
but of course this isn’t just about harmony, the high leading-tone aesthetic has a lot to do with melodic intonation, where a large minor second (say, 110c or more) will supposedly ‘stick out’ and weigh the phrase down. larger minor seconds are perhaps heavier in that respect. but how small is a good small second? The 100c semitones of 12EDO? 90c Pythagorean minor seconds? 84c (21/20)? 71c (25/24, or roughly the minor second of 17EDO)? 63c (28/27, really beautiful septimal minor second, made much use of in that La Monte Young piece i posted the other day)? 56c (22EDO’s minor second)? surely we’ve already surpassed what we can call a minor second, right? not so. i’ve found even 48c makes a great small second, just less than a quarter tone with regards to 12EDO. where does this interval show up? in 50EDO of course, along with a great selection of other minor seconds.
50EDO’s intervals in the minor second range are the following:
48c (arto or diminished second), 72c (sub second), 96c (lesser second), 120c (minor second), and if you’re adventurous, 144c (lesser middle second)
one really cool thing about meantones is that while their regular minor second will be on the large side, they will always have a smaller interval available in the form of the augmented unison, which often also represents the septimal minor or sub second. so although 120c is considerably large, we can go for a more regular semitone (this time it is actually half of the mean tone) at 96c, or use the sub second to give that 28/27 feeling <3, or we can actually use the diesis at 48c as a leading tone, rather effectively.
which means if we are resolving some kind of dominant cadence, a 120c minor second will give us a very close to pure major chord on the V, the 96c lesser second will give us a Pythagorean or greater third (408c, very close to 12EDO at 400, and very active in comparison), the sub second at 72c will give us 432c (a super third representing 9/7, an altogether different mood, which some might call ‘high tension’, though it’s actually much more stable than the greater third), and finally the diminished second, when used as a second, will mean we get an augmented third on the V chord, looking rather like 10:13:15, a tendo triad rather than a usual major.
of course anther option is to change the position of the dominant, so that placing it on the greater fifth (720c) will provide a pure third when we have a melodic semitone, and a super third where we have the melodic diminished/arto second in the melody. one more degree up at 744c almost doesn’t look like a fifth (being about 1/2 way in between 12EDO’s fifth and minor sixth), but it represents the regular wolf in meantone, e.g. G# to Eb, or B to Gb, and in 50EDO, this interval is actually a lot more consonant (at least to my ears) than it is in usual varieties like 1/4 comma or 31EDO. a cadence of the kind Vicentino was very fond of (in, or very close to, quarter-comma) might mean the bass descending from the ‘wolf’ super fifth to the tonic, while the leading tone on the super major seventh provides a pure third to the bass, while at the same time keeping the voice leading smooth with only a small 72c sub second. a very nice melodic minor second indeed, 3/8 of a 50EDO mean tone. in my 50EDO notation, we might have (in Bb major):
Cantus: Bb—-A^-Bb Bass: Bb-G-F^-Bb
or in four parts, a little more elaborate possibly not quite Vicentino’s style any more):
Cantus: Bb-Bb—–A^———-Bb Alto: D–D–Eb-D#^-E^-F^-F Tenor: F–G——A^—C^—–D Bass: Bb-G——F^———–Bb
although this is just a I-vi-V(4-3)-I progression, it’s got a bit of attitude. the dominant chord is raised by the diesis which means when we tie the Eb (minor sixth above G) over to the dominant, we get a sub seventh instead of a minor seventh, which being enharmonically equivalent to an augmented sixth (respelled F^-D#^) we can resolve it upwards just for fun (historically incorrect voice-leading but this is just a [bad] experiment really), the very smooth passing major seventh to octave on F^ leading us not to Bb^ but to Bb, possibly a very unexpected jump, though back right where we started. good luck singing that alto part…
a better example might be a modernised medieval style cadence to an open fifth and octave, with ‘double leading tones’: sharpened fourth and seventh ascending to the fifth and octave, and a lower voice moving from second down to the tonic. on D we might have:
Cantus: D—–C#^(tr.)-D Alto: G-A–G#^——A Bass: G-Fv-E———D
the only problem with this kind of voicing is that the fifth at 696c is a little flat when left open (without a third in the chord), the ear notices it’s not quite a perfect fifth in certain timbres, so either choose your timbres carefully, or maybe just don’t use it as a closing sonority, unless you want that hint of instability. filling in the chord would give us a more stable sound, but it would kind of kill the style we’ve got going.
but back to minor seconds, if you want to do something like 12EDO, or want the ability to be a little ambiguous, use the (here 96c) semitone, which can stand in for a whole bunch of different ratios, just like in 12EDO, it can mean the ability to go in just about any direction, it’s a bit tense (because it’s not really close to simple consonant ratios). we still haven’t had an example with the semitone so why don’t we try a more harmonically adventurous one like this: F > Fsub7(b9) > Eb-m7 > D+M13 > D+sub7(b9) > Eb/G > F/G > G
F Gb- > Gb- >F#+ G > A > B A C Db- Ebv Eb- Db->C#+ D+ Eb > Eb F G C Ebv C Db- C Bb- B+ A+ Bb^ C- Bb C D F Eb- D+ G > G
ok that got way outta hand. but hopefully it at least helped demonstrate the ability of semitones to be ambiguous, and allow not only semitonal but greater tone modulations, as the semitone is a comma-inflected interval (either a comma-raised augmented unison or a comma-lowered minor second), comma-modulations are very easy, and I could just have easily moved to G+ instead of G (as the bass would suggest perhaps I should have, instead i held over the Eb, the b9 of D+ to give a minor sixth against G). there are just so many possibilities.
to be honest i’m usually much more concerned with harmonic events than melodic voice-leading, so i might use all of the different intervals in the ‘minor second’ range at different times, for different purposes, when i’m playing particularly involved passages. for me, the most obvious uses would be:
minor second: as regular diatonic step, whenever playing a modal line
lesser second (semitone): as a trill, usually above the target pitch but can work from below too, and the most obvious ‘flat nine’, closest to the 17th harmonic
sub second: whenever resolving sub and super intervals: e.g. a sub seventh to a major sixth, a sub sixth to a fifth, a super seventh to an octave, a super third to a fourth.
arto second: when resolving arto and tendo intervals, or when doing something whacky, like moving between sub and super intervals- super sixth to sub seventh, sub third to super second, or when moving between regular and sub/super- fourth to super fourth (as in 4/3-11/8), or minor third to sub third (as in 6/5 to 7/6).
hardly any of this is going to make much sense without hearing it. for those of you bearing with me, thank you. but do some experimenting, find a way to play these things for yourself (my favourite is the programs Scala + pianoteq). i’ll try and make some demos in the next wee while to show off different kinds of resolutions using these steps and other intervals, but my main point in this post was that there are always options, and having many intonational possibilities even with just one chord sequence makes the world of harmony that much more exciting.
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