I wrote some more of my Nursery Tunes for Weird Children that I recently mentioned here. They're silly little pieces, but they serve a serious purpose for me. I've been doing a lot of sketches for large works in what I call my 8x8 tuning, which contains potentialities that I need to explore and learn to hear before I can commit myself to basing entire pieces on them. In particular I've been trying out, in the second and fourth pieces here, more exotic triads a little higher in the harmonic series, such as 7:9:11, 10:13:15, and 8:11:13, which resolve in both pieces to conventional major (4:5:6) and minor (10:12:15) triads. "Tiger, Tiger" climbs through the 33-pitch scale step by step to control the harmony, just as so much late Romantic music harmonized the charomatic scale. And the final piece moves between major triads on the 5th and 13th harmonics in kind of a super-Neo-Riemannian voice-leading. Anyone capable of the requisite math can quickly see what I mean.

Down to the End of the Town
Up the Hill and Up Again
Tiger, Tiger Turning Right
The Cracked Bells of St. Swithun's
Jack Ate a Blackbird

I just love that microtonality enables me to write music with which I can confuse my own ear. And I think it's fun to occasionally hear conventional musical textures reworked microtonally, as a way of imagining what we might have been doing all these decades had Europe decided not to cut us off above the 5th harmonic. I also realize occasionally that I could have been a passable neoclassicist had I decided to go that route, though that would have seemed like career suicide when I was first starting out. But as it turns out, almost everything interesting is career suicide, microtonality most of all.

I can use these pieces to illustrate a didactic point, though. My 1/1, the reference point of my tuning, is Eb, but that doesn't mean the pieces are all in that key. The first piece is in G, the last in C13-flat, and the penultimate begins every phrase on D7-flat; the others move around. I tend to avoid Eb, in fact, because that's where the least exotic intervals are. I say this to confute all those music professors who find it droll to smugly remind us that Harry Partch's music is all in the key of G. (Partch's 1/1 is G 392 Hz.) Actually, Partch's music employs many tonalities, and sometimes none. It would be exactly as accurate to claim that all orchestra music is in the key of A, since that's the pitch the orchestra tunes to before the performance. I could rename any of my 33 pitches 1/1, but Eb is the reference point that provides the simplest fractions. There's a precise analogy with a meantone keyboard: in the 17th century, writing in the key of Bb made more of the sharp side of the circle of fifths available, while the key of A major made more of the subdominant side possible, so you chose your key according to the mood you were aiming for. So you could as easily claim that all pre-1800 keyboard music is in the key of C. The attractiveness of having an implicit center is the subtle tension of leaning away from it. Resisting gravity is how artists create a feeling of lightness.

COMMENTS:

Michael Golzmane says: Really enjoying seeing your process here, Kyle. And listening to this music. What a new world of sound!

Ian Stewart says: Interesting approach Kyle, the pieces certainly have a aural character my ears can't quite define. I really liked the chromatic run in your tuning. At the Proms I heard a piece by Michael Gordon that produced microtones by a process I did not really understand.
I have never used microtones but frequently upset people when I say the problem with equal temperament is that the thirds are all wrong and the fifths sound strange. Ideally I would only like to use one of the Baroque tunings, but as most of the music I compose is for classical saxophone, frequently with piano or tuned percussion, I don't think this is likely to change.

mclaren says: Psychoacoustics will help you predict what these things will sound like. Since everything Harry Partch ever said about microtonality contradicts the audible reality, we must fall back on the observed evidence of how the human ear/brain system actually works in response to music. Fortunately such information abounds in published peer-reviewed scientific journals. Unfortunately most musicians remain utterly unaware of this information -- courtesy of an American university musical curriculum which exhibits the kind of insensate hostility toward these scientific results about human hearing normally seen nowadays only by Republicans frantically denying global warming.
The interval 15/13 has a width of 247.7 cents. At any range this will sound audibly and disturbingly rough when played loudly with sustained near-harmonic-series timbres. The interval 13/10 has a width of 454..2 cents. If the timbre gets matched to the tuning, this interval can sound smooth. Since it's impossible to simultaneously match a single timbre with the two intervals 15/13 and 13/10 to make 'em both sound smooth, the proximate solution is 1) either to accept the 10:13:15 triad as a momentary acoustically rough vertical complex that demands resolution to a more stable (smoother) vertical complex, or 2) to play the 10:13:15 triad softly or with inharmonic percussion short-lived timbres, like a xylophone or celesta. The interval twixt the root and highest rote is a 3/2, the familiar just perfect fifth, and will sound acoustically smooth with almost any near-harmonic-series timbre and thus is not an issue.
We know that the 15/13 will sound unacceptable acoustically rough because this interval is less than the critical bandwidth of ~270 cents. See "Tonal consonance and critical bandwidth," Plomp and Levelt, Journal of the Acoustical Society of America, Oct., 38 (4): pp. 548-60. The solution there involves short-lived percussion timbres or inharmonic timbres.
The interval 11/9 has a width of 347.4 cents and sounds like a neutral third in between a major and a minor third. It will sound acoustically smooth with almost any near-harmonic-series timbre. The interval 11/8 has a width of 551..3 cents and will sound acoustically smooth provided the timbre is approximately matched to the tuning. The 11/8 sounds alien to conventional Western music and by no means unpleasant -- it simply takes some getting used to. It can produce musically beautiful effects in the right context. The interval twixt the lowest and highest notes of the 8:11:13 triad is 13/8, with a width of 840.5 cents. This neutral sixth will sound acoustically rough with harmonic series timbres rich in 3rd and 6th harmonics, so best to avoid timbres of that kind.
The interval 9/7 has a width of 435 cents and if the timbre gets matched approximately to the tuning, will sound acoustically smooth since it exceeds the critical bandwidth of ~270 cents. The interval 11/7 has a width of 782.5 cents and if the timbres gets matched approximately tot he tuning will also sound acoustically smooth. The interval 11/9 is the familiar neutral third which sounds fine with any timbre.
As a practical matter, the individual partials of the bottom note in a 7:9:11 will interfere acoustically quite roughly with the individual partials of the top note in a 7:9:11 triad, since these partials fall about 50 cents (half a semitone) away from another, in the range of 1/4 of the critical bandwidth. This will produce extreme acoustic roughness if sustained loud harmonic series tones get used. So the solution is either to accept this extreme beating as a vertical complex requiring resolution to a more stable (smoother) vertical tone complex, or to use tones with either shorter-lived or less harmonic partials, such as a vibraphone or electric piano, or breathy noisy timbres without a lot of harmonic series overtones, like the flute or shakuhachi or quena.
Likewise, the partials of the bottom note in the 10:13:15 triad will tend to beat roughly with the partials of the middle note, falling roughly 47.7 cents away from one another, which is around 1/4 of the critical bandwidth -- the interval width Plomp & Levelt identified from listening tests as maximum acoustic roughness, and thus unstable and requiring resolution if loud sustained harmonic series timbres are used. The solution once again is either to use the middle note in the 10:13:15 as a passing tone which gets resolved (the 15:10 interval obviously requires no resolution since it's a just perfect fifth and sound entirely stable with sustained harmonic series timbres), or to once again use a shorter-lived timbre like a celesta or vibraphone, or an inharmonic timbre.
None of this proves mysterious or unexpected. The human ear/brain system operates in predictable ways, and ignoring the known operation of the human ear/brain system produces music of the kind generated at Darmstadt after WW II and at Princeton in the 1950s, now securely covered by the lid of history's trash can.
There's nothing magical or exotic about any specific integer or combination of integers used as musical ratios. In the end, every musical ratio boils down to an interval width measured and heard in cents, which has predictable perceptual qualities well defined by listening experiments over the last 50 years. We know that the human ear/brain system hears intervals in cents rather than integer ratios because MEG (magnetoencelalography) experiments on listeners have shown that the human acoustic cortex detects pitch in a semicircular wedge of cells whose response to pitch ranges logarithmically from low to high. Logarithmic response is, of course, the same as cents. (See Music, Cognition and Computerized Sound: An introduction to Psychoacoustics, ed. Perry R. Cook, 2000; also see "The Neurobiology of Harmony Perception," chapter 9 in The Cognitive Neuroscience of Music, ed. Isabelle Peretz and Robert J. Zatorre, Oxford University Press, 2009, pp. 127-151..)
We also know why the human ear/brain system hears pitch logarithmically rather than by integer ratios. From a Darwinian viewpoint, mammals that respond quickest and with the best approximate accuracy to sounds (Is a lion roar rising or falling in pitch? tells you whether the lion is running toward or away from you because of the Doppler effect) will leave the most offspring, so speed and rough general accuracy get genetically selected (since mammals that take too long to judge the pitch of sounds, or judge them too poorly, get eaten). It turns out that an infinite number of integer ratios correspond to any given audible interval (in cents), so obviously integer ratios are not useful for detecting pitch quickly -- it takes too long, because there are too many ratios to test (infinity turns out to be a large number).
None of this invalidates integer ratios as musical intervals. They have been used in music for far longer than the johnny-come-lately intervals of our "conventional" (read: recently confected for financial reasons involving standardized industrially mass-produced Western orchestral musical instruments) Western tuning of 12 logarithmically equally spaced intervals. As you imply, the human ear/brain system is capable of distinguishing far more pitches in the octave than 12, and there's no reason why we shouldn't extend that number to 33 or even more, for expressive melodic or harmonic purposes.
Left out of my precis? The emotional effect of various xenharmonic musical intervals. Acoustical roughness or smoothness may dictate the overall harmonic stability of a given just ratio, but in the end, no law graved on tablets of stone requires us to compose using triads or with conventional harmonic progressions. Moreover, acoustically rough vertical complexes can and often do add a delightful moeity of musical spice to the composition, in the same way that a few drops of tobasco sauce can make a bland meal delicious. The Darmstadt crew ran off the rails when they avoided smooth acoustic vertical complexes entirely, disallowing points of acoustical rest -- this produces music that grows unutterably wearisome, since the acoustical tension never gets resolved. Conventional Western music ran off the rails by unnecessarily limiting its harmonic and melodic resources to an overly small range of pitches, which impoverishes the emotional palette of music in the same way that forcing, say, Leonardo da Vinci to paint the Mona Lisa using only 12 distinct specific colors would have impoverished his art. This explains the hijinks of various extended technique characters like Helmut Lachenmann disassembling their saxophones and using the parts to blow bubbles in a glass of water, since once you've run out of pitches and the musical resources of 12 equal have been stripped mined to exhaustion by many generations of great composers, what else can you do, musically, but produce bizarre stunts for shock effect?
The exciting frontier for 21st century music of course lies in between these two extremes of all-acoustically-rough-vertical-intervals and jagged-haphazard-melodies all the time (the New Complexity/serialism/chance music, etc.) and the "free style" of Lou Harrison or Percy Grainger's free music machine, which present the composer with too many pitch choices for a practical extended composition...at least, until audiences and composers have become sufficiently familiar with the enormous range of music outside the conventional 12 tones per octave to navigate with equanimity such a limitless sea of potential intervals.
Short version: psychoacoustics lets us accurately predict the overall general musical properties of integer ratios (vertical stability or unstability, smoothness or roughness) but doesn't specify the detailed emotional effects of new intervals, particularly as they appear in context (no law requires that we use triads: we could just as easily employ JI tone clusters, JI quartal harmony, etc.). So psychoacoustics and the published peer-reviewed perceptual science of audition gives us an indispensable head start in dealing with JI, necessary but not by itself sufficient for composition.
Since these facts have been documented beyond dispute, it stands to reason that hordes of ill-informed musicians will rush forward to deny them. In the 2014 America of global warming denial, Darwinian evolution denial, vaccine denialism, and claims that "soup kitchens caused the Great Depression," we can expect nothing less. Bonus points for commenters who claim the earth is cubical, or that the sun rotates around it.
KG finally replies: Now that I've time to read all this, thanks for it. Very helpful.

Graham Clark says: "The exciting frontier for 21st century music of course lies in between these two extremes of all-acoustically-rough-vertical-intervals and jagged-haphazard-melodies all the time (the New Complexity/serialism/chance music, etc.) and the "free style" of Lou Harrison or Percy Grainger's free music machine, which present the composer with too many pitch choices for a practical extended composition..."
Sounds thrilling.

fm says: Kyle Gann demonstrating some of the limitless technical and expressive capabilities of the meruvina - bravo!
At first hearing, it would be musically delicious to have a carefully adjusted clarinet timbre play the main melodic voice of Tiger, Tiger Turning Bright, with those tunings illuminated ever so bright through the different registers, and, likewise, a perfectly contoured trumpet timbre for the main melodic voice of Jack Ate a Blackbird, the tunings enhanced by warm, metallic articulations and colorations. (By coincidence, I recently learned that an extremely sweet little dog I befriended hunts and eats birds...hard to believe.)
KG replies: Nice instrumental ideas. Maybe someday when the music world has realized what will be gained.

fm says: One central key is to realize that the congregation of sounds represented by traditional instrumental timbres, which comprise the history of Western music, including the orchestra, and absolutely including the equally splendid instrumental tone colors from myriad other lands, are part of the "found sounds" in the reservoir of aural experience we experience growing up. Composers cannot be limited in their aural responses to environment and the passing of time (both historical and personal), straitjacketed by how traditional sounds are utilized and combined in the past, as if musicians own these timbres exclusively. No one is threatening the traditional use of instrumental sounds by traditional musicians, which clearly forms a different type of music.

Seek those images
That constitute the wild;
The lion and the virgin,
The harlot and the child.

... or in a much more famous poem than "Those images", also by William Butler Yeats, "Among School Children":

O chestnut tree, great rooted blossomer,
Are you the leaf, the blossom, or the bole?
O body swayed to music, O brightening glance,
How can we know the dancer from the dance?

Susan Scheid says: fm's last comment is lovely, both in concept and as stated. This is very much a "lay listener" response to your post, Kyle, and without the requisite math in hand, but as to those "'found sounds' in the reservoir of aural experience we experience growing up": in my childhood home, we had a piano in the basement on which I, as a small child, propped a keyboard chart and tried to teach myself to read music. The piano, shall we say, wasn't "properly" tuned in the Western tradition, and every time I hear a microtonal work on keyboard (or perhaps MIDI here), like these small, demented delights, I'm transported to that piano (in a further confusion to the ear, I didn't realize that, in addition to the notes I saw, there were sharps and flats). Thanks, as always, for letting us in on your process, and I look forward to more.

mjy says: Forgive me if you've mentioned this previously, but have you ever considered writing a book that is sort of a primer on microtonality? Said book with accompanying songs on cd and scores would be a very, very worthwhile Kickstarter campaign should you ever write it. These songs and your few posts on the subject really make clear info that other writers make seem extremely murky.
KG replies: Thanks for asking. I am already under contract with U. of Illinois Press to publish my The Arithmetic of Listening, the text I wrote for use with my microtonality classes, in late 2016. It needs a lot of beefing up and musical examples, which in class I simply provide as I need them, and the audio examples of which I will indeed put on the internet. Until then, I have some basic information here:
http://www.kylegann.com/microtonality.html

Copyright 2015 by Kyle Gann

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