Literary-friendly entertainment fiction: the “progressive rock” of books

 

 

I had an interesting discussion with the Beautiful but Evil Space Mistress about more literary sci-fi and its sales potential the other day, and was reminded of this when I heard the audio of a Coldplay concert last night.
My daughter had attended live and went gaga about how this was the best concert she had ever heard. Having listened to about 2 hours worth of Coldplay, I can hear why this band is popular. It’s definitely a cut above today’s pop music — which is, of course, damning with very faint praise. It also has a few clever harmonic touches — but they are added in, for my taste, homeopathic doses, and smothered in pop cheese and endlessly milked hooks. (I was actually surprised they had a live drummer: the drum parts are so monotonous to my ears that I was convinced they were using a drum machine with very good samples.)

But apparently, that is as much musical sophistication as you can add and keep a mainstream pop audience nowadays. Sgt. Pepper era Beatles, or ELO in their heyday, are far-out progressive rock in comparison — heck, so is even Pet Sounds by the Beach Boys! But here’s the point exactly: they know their market, and they produce exactly what the market wants. If they are at all disaffected musically, they are suffering all the way to the bank.

All this has a parallel in fiction. Allow me to explain.

In pure entertainment fiction, everything is secondary to “the meat and potatoes” of the genre: in military sci-fi or space opera, for example, those would be a strong, sweeping plot, protagonists you can root for, credible antagonists, lots of action, and the like.

And no, that does not mean “dumb” fiction: plots and characters can be very complex, and about certain special interests that are central to their chosen bailiwick, genre writers and their fans alike can be detailed and obsessive to the point of “weaponized autism”. So if, say, Larry Correia geeks out about guns, or David Weber about orbital mechanics, that is not just tolerated but expected. (Never mind the level of painstaking detail seen in police procedurals and the better techno thrillers.)

But for anybody who is well-read, the prose in many works of genre fiction can be irritatingly pedestrian, if not outright “dumbed-down”. This is especially true of the American market: a few recent contemporary romances I read for “market research” seemed to have been deliberately written at an 8th-grade reading level. British readers seem to expect somewhat more polished prose.

At the other extreme, in pure “literary fiction”, one can often be forgiven for assuming that entertaining the reader is an afterthought at best. Writers of such books are often academics or academic wannabes, and peer approval is more important to them than anybody actually reading their book for pleasure.

There are writers that are trying to steer a middle course: combine entertainment mainstays with prose that those who love the language for its own sake will find pleasant to read — and if some readers need to look up a word on their Kindles, well, so be it. Often such books have philosophical, sociological, and/or spiritual themes running through them. “Literary-friendly entertainment fiction”, if you like. Dave Freer and Brad Torgersen are examples in the sci-fi field. Yet discussions with, among others, our BbESM suggested that the market for “literary-friendly entertainment fiction” is limited.

The musical equivalent of “literary fiction” would be contemporary classical music — which for the most part has no mainstream audience, and is written by academic musicians for academic musicians.

The Freers and Torgersens (and up to a point, Lois McMaster Bujold) would be like progressive rock: music that tries to retain some of the accessibility and energy of rock and pop music while also seeking a broader musical horizon. Now the best bands in that genre — take Yes or ELP in their heyday, or Genesis before they turned pop, or Rush and Dream Theater in a somewhat harder vein — have rabid and faithful cult fan bases — but “cult” is the operative word. I am sure the three guys in Rush made a ton of money over the years by building up a like-minded audience and reliably delivering the goods to it — but aside from airplay for “Tom Sawyer” , “The Spirit of Radio”, or “Limelight”, their mainstream appeal is always going to be limited.

There are artists with “two-track” outputs: nice tuneful pop songs or rock anthems for the casual listener, and more profound work for the serious aficionado. There’s a couple Jimi Hendrix or Led Zeppelin songs that everybody knows, for instance, but they only scratch the surface of what these artists could do. Similarly, there are fiction writers who bankroll their more serious output with occasional crowd-pleaser offerings. Some people would call this “selling out”, but the “two-trackers” are basically following in the footsteps of Ludwig van Beethoven (!): he would write reams of bagatelles (short, inventive piano pieces playable by amateurs) and song arrangements to pay his bills and stave off the creditors, while he worked on the symphonies and piano sonatas that earned him his place in the pantheon of classical music.

Sarah Hoyt seeks a slightly different course (like her acknowledged literary guiding light, Robert A. Heinlein). Yes, there are philosophical elements and literary allusions in works like Darkship Revenge — and not always in small doses, mind you —  but they are blended in subtly, carefully mixed in with the “meat and potatoes”.  Similarly, Lois McMaster Bujold (possibly my favorite living writer in any genre) has deep psychological material in her novels, but makes sure not to skimp on the things that Baen’s target audience wants in a novel. In music, this would be a “progressive pop” approach, with Coldplay at the more timid edge and Steely Dan at the more daring edge.  (More experimental artists like Frank Zappa would actually derive quite a bit of their appeal from extra-musical factors, such as shockingly bawdy lyrics.)

What does all this mean for a writer? The answer will, of course, be different for those trying to make a partial or complete living off our pens, and for those of us for whom fiction writing is a labor of love rather than an income stream. Robert Heinlein used to quip that his favorite five words in the English language were “pay to the order of”, and until late in his career made sure to write books that are easy to read on the surface — but that reveal several layers when read in depth.

 

What the heck are stem mixes?

 

Suppose you’re learning your part (guitar, keyboards,…) in one of the more challenging rock or metal sounds, and you don’t just want to “play the song” — you want to hear exactly what your hero plays and reproduce it to the best of your ability.

There are a number of YouTube videos floating around of isolated vocal, guitar, bass, keyboard… parts of various classic rock and metal tracks. For some instruments, they can be extracted from the stereo mix by EQing and stereo manipulation. Cranking up bass in an EQ while rolling off midrange and treble, for example, will leave you with mostly bass guitar and kick drum, while doing the opposite may be handy if you’re trying to learn a complex bass part. However, if the instrument ranges cross, this doesn’t work — bass and guitar parts in Tool songs often cross, for instance, and Yes bassist Chris Squire (RIP) never stayed in his lane.

Also, as vocals tend to be placed in the center of the stereo image, mixing the left and right channels in counterphase will reduce the vocal a lot, allowing a “poor man’s karaoke”.

A “stem track”, on the other hand, is derived from the original multitrack recording: it is best defined as the submix from a single instrument group. A “guitar stem” would be all the guitar parts of a track (mixed, among them, like in the final master), a “drum stem” the final submix of all the drum channels, and the like.

 

 

In a way it’s an inversion of those old “Music Minus One” recordings for classical musicians: those were ensemble or orchestral recordings with one solo instrument omitted, sold together with sheet music for that solo part.

Here are a couple of examples:

It is very educational sometimes to hear how parts interlock, how minor imperfections in an individual part get buried in the overall mix, and how the whole of a song can be greater than the sum of the parts. Compare, for example, Steve Harris’s bass part on “Two Minutes To Midnight”  with the final complete song:

 

Enjoy!

 

 

 

 

Concert pitch, or how we came to tune (mostly) to A=440 Hz

Yesterday I stumbled onto another of these “A=432 Hz” advocacy pages: it got me thinking that “how did we get to A=440 Hz?” would be a good subject for a post. So here goes. TL;DR summary: there is neither conspiracy nor deep ‘harmony with the cosmos’: the standard came about for purely pragmatic reasons. Let me explain.

In antiquity and the Middle Ages, there were no absolute pitch standards. Sure, theoretical math about the construction of scales goes back all the way to the School of Pythagoras, but that concerns itself with relative pitch (intervals), not with an absolute reference pitch. Whichever fixed-pitch instrument was part of the ensemble would have dictated the reference pitch for the others — and since this was long before the era of mass manufacturing, those were all one-of-a-kind instruments.

The German composer and theoretician Michael Praetorius (1571-1621)
did mention that in his day, there was “chamber tuning” (Kammerton) and there was “choir tuning” (Chorton, which followed the church organ) and that those were a whole tone apart. So historical organs would give us a clue as to historical pitch, right?

Well… it was indeed so that the pipes for the lowest note of an organ “principal” stop were by convention made eight foot long. (Hence the practice of labeling organ stops, or later drawbars on a Hammond organ, by “foot”: 16’ will sound one octave below, 4’ one octave above, 5 1/3’ a fifth above,… the notes played on the keyboard.) So that would seem to impose standardization at least for church music, right?

Not quite. Whose foot are we talking? Each principality in those days had its own set of customary units. We do know that German Baroque organs that have been preserved are almost invariably sharp of modern concert pitch, typically by about a semitone, but sometimes as much as a whole tone. An even-tempered semitone, 2^(1/12)≈1.05946, up from A=440 Hz would be about 466 Hz, two semitones about A=494 Hz. A whole tone down from A=466 Hz would imply a chamber pitch somewhere around A=415 Hz, as favored today by many ‘historically informed performance’ ensembles — but this would not have been universal, and actual concert pitch may have been higher.

The first tuning fork wasn’t invented until 1711, by an English court musician (trumpeter) named John Shore. Tuning forks (or “pitchforks”, as Shore punningly called them) are small and portable, drift very little with temperature and over time, and yield a nearly pure sinusoidal sound (i.e. devoid of overtones).

One of Shore’s London customers was the great expat German composer Handel. Händel’s tuning fork has actually been preserved

handels-tuning-fork

and sounds at A=422.5 Hz. A number of other historical tuning forks have been preserved, e.g., those used by fortepiano (and later piano) manufacturers for initial setup and tuning. The record shows that pitch kept drifting up and up, as orchestras kept pursuing an ever brighter sound. (This is not mere psycho-suggestion, particularly for the string section: tuning string instruments higher means increasing the tension on the strings, leading to more overtones in the sound.) Two other developments took place in parallel: the opera genre became a mainstay of classical music throughout Europe, and as long-distance travel became more practical and affordable thanks to the Industrial Revolution, star opera singers would travel widely.

What this also meant, however, is that an opera diva could be traveling to a new city, and suddenly would be unable to hit the highest notes as the orchestra was tuning higher. The resulting protests led to a pushback against “pitch inflation”, and hence to efforts to arrive at a standard.

[[[sidebar: scientific tuning, a.k.a. Sauveur pitch, philosophical tuning, Verdi tuning.

The French courtier and physicist Joseph_Sauveur, who first coined the term “acoustics” for that subfield of physics, in 1713 proposed an absolute pitch standard based on the frequencies of all C’s being powers of two: middle C=256 Hz, C’=512 Hz, and so forth. In Pythagorean tuning, that implies A=256 x (27/16) = 432 Hz. [in 5-limit just intonation, that would be A=256 x (5/3) = 426.666… Hz; in 12-tone equal temperament A=256 x 2^(9/12)= 430.54 Hz.] This was considerably sharp of French Baroque practice and was hence not adopted by performers. A century and a half later, the composer Giuseppe Verdi tried to revive this proposal, at a time when orchestras routinely turned way sharp of this. In recent years, various mystical and numerological ideas have been attached to Sauveur pitch, which has led to some (usually nonclassical) musicians adopting it.]]]

In 1834, a German industrialist, inventor, and amateur acoustician named J. H. Scheibler  devised an array of tuning forks pitched at multiples of 4 Hz, permitting pitch measurements at 4 Hz resolution. (As Heinrich Hertz yet had to be born, the unit was of course not named “Hz.” but cycles per second, or Schwingungen pro Sekunde.) At an 1834 meeting in Stuttgart of the German Society of Natural Scientists and Physicians , Scheibler demonstrated the device, and a motion was adopted to use A=440 Hz as standard concert pitch (the first time that this proposal was made). As Scheibler had many contacts in the German-speaking music world, a number of musicians informally adopted his standard.

The trailblazer for standardization of measurement units in Europe was, of course, France, with its 1799 adoption of the metric system, which eventually became the standard for nearly the entire industrialized world as well as of the worldwide scientific community. (In 1875, seventeen countries would sign a metric system convention, which led to the creation of the International Bureau for Weights and Measures outside Paris.) In the same vein, the French government issued a ministerial decree in 1859 that mandated a “diapason normal” (standard tuning fork) throughout France at A=435 Hz: this compromise value had been recommended by an ad hoc commission advised by the likes of the composers Halévy, Meyerbeer, Auber, Ambroise Thomas and Rossini. A number of continental European countries adopted the French standard.

In Britain, on the other hand, attempts had been made to standardize to an A=452 Hz concert pitch (almost a quarter-tone sharp of modern concert pitch). Protests by singers led to the adoption of a modified French standard: concert hall organs were tuned at A=435 at about 15 degrees centigrade: assuming a thermal expansion coefficient of about 0.1% per degree Fahrenheit (actually, closer to 0.067%) of the air column in the  organ pipes, it was argued that pitch would drift up to about 439 Hz in a heated concert hall, and hence in 1896, A=439 Hz was adopted by the Royal Philharmonic as the “new philharmonic pitch”. The older standard was then being referred to as “old pitch” or “high pitch”.

Recording and broadcast technology gave a new impetus to international standardization. On June 11, 1925, the US recording industry adopted A=440 as a standard. and eventually, this revived “Stuttgart value” was agreed upon at the 1939 London meeting of the International Standards Association (the predecessor of today’s ISO). What tipped the scales for 440 Hz rather than 439 Hz was again a practical argument: the BBC’s engineers could generate a stable, invariant A=440 Hz from a 1 MHz quartz crystal oscillator through a combination of frequency division and multiplication circuits (divide by 1000, then by 25, then multiply by 11).  This was an impractical approach for 439, which is a prime number. Eventually, A=440±0.5 Hz would be enshrined as ISO standard 16.

Many symphony orchestras actually tune slightly higher, A=442 or 443 Hz: a list of reference pitches for orchestras worldwide can be found here (in German). The Berlin Philharmonic, in the halcyon days of Karajan, actually tuned to A=445, to revert to the more common 443 Hz under later conductors.

In the “historically informed performance” community, A=415 Hz is commonly used for Baroque music. Why that specific number? Again, a practical compromise: more or less in the ballpark of what was (German) Baroque practice, and exactly an even-tempered semitone down from A=440 Hz. This means that modern fixed-pitch instruments can still perform in an ensemble with period instruments: all that is required is transposing the former’s part down by a half-step.

 

Writers, which classical composer are you?

Some time ago, Sarah Hoyt wrote a long post at MadGeniusClub on making oneself “write like the wind”.  Somewhat tangentially related, Christopher Nuttall blogged in the same venue about the three types of writer: the “wannabe”, the “amateur”, and the “professional”.

The wannabe was memorably depicted by Albert Camus (in “The Plague”) as a civil servant named Joseph Grand who wants to write a number but has spent ten years looking for the perfect opening sentence. The amateur does get things written and published, but for him, the writing is just a hobby (perhaps even with a therapeutic aspect) and (s)he treats it as such: if he is too tired to write or the creative juices aren’t flowing, then so be it. For the professional, on the other hand, writing is a day job, and bills don’t get paid unless books get written. [A commenter would add a fourth category: the “moonlighting” writer who has a day job to pay the bills, but treats the writing as a second job.]

In a nutshell, the recommendations of both writers boil down to: if you want to make a living from writing, (a) treat it like a job, and make yourself write whether you want to or not, whether you have inspiration or not; (b) make yourself write fast, because the more you write, the better you get at it, and the more there is for people to buy. Especially (b) is a far cry from the romantic fantasy of the writer as an “artiste”-demiurge.

But is there only one path to being a writer? As I see it — if I may use a musical metaphor — it boils down to which composer one wants to be.

At one extreme are the Mozarts and Vivaldis. A snide joke among some classical musicians has it that Vivaldi didn’t write 400+ concertos, but 400+ times the same concerto. Of course this is a cheap shot, but even people who generally love classical music would be hard-pressed to name any Vivaldi composition other than the Four Seasons.

Disparaging comments aside, “The Ginger Priest” did help establish the concerto as a classical music form, which in itself is no mean musical legacy. None other than J. S. Bach  studied his work diligently and transcribed several of his concerti for keyboard instruments. You might say Vivaldi was a kind of E. E. “Doc” Smith among composers.

Mozart was every bit as prolific and crowd-pleasing as Vivaldi—which elicited from the always sharp-tongued Glenn Gould the notorious quip that “Mozart died too late rather than too early”. The main reason so many of Mozart’s compositions did endure (while the likes of Ditters von Dittersdorf have been forgotten) is very simple: Mozart was a transcendent genius. He simply preferred to write lots and in a very accessible idiom, even as he occasionally ventured into ‘learned’ composition for his own pleasure and was very skilled at it. He was perennially short of money, as he enjoyed the good life and had no steady patron—he might have nodded in agreement at Heinlein’s statement that his favorite five words in the English language were ‘pay to the order of’.

At the other extreme stood Maurice Ravel. Nicknamed ‘the Swiss watchmaker’ by colleagues, he was a total perfectionist with a very modest output (after WW I, he averaged about one composition per year). Financially secure as a tenured professor of composition, he had no need to “compose to pay the bills”. In fact, he was as well known as an orchestrator of other people’s music as for his own compositions — Mussorgsky’s Pictures At An Exhibition was originally solo piano music, which Ravel orchestrated. Impervious to others’ opinions but relentlessly self-critical, he quipped to fellow composer Arthur Honegger: “I’ve written only one masterpiece – Boléro. Unfortunately there’s no music in it.”

 

Anton Bruckner was eternally plagued by self-doubts. In response to suggestions and criticism from colleagues, he would endlessly revise his symphonic works  — for example, there are at least five extant versions of the First Symphony — in response to suggestions and criticism from colleagues.

 

The creative struggles of Beethoven in his major works — the symphonies and the 32 piano sonatas, which have been named “the New Testament of solo piano music” — are legend, as evident from the minefields of corrections and erasures in his manuscripts , which drove copyists and score engravers alike crazy.  The resulting financial woes he addressed in part through patronage, but also through a parallel “pay the bills” compositional output written to publisher’s order: folk song arrangements, variation cycles, and “Bagatelles” — short, inventive piano pieces accessible to a somewhat skilled amateur. Yes, even this archetype of the “artiste-demiurge” wrote to pay the bills sometimes! The writer who churns out popular romances or mysteries under one pen name in order to subsidize a more literary output under his/her own name took a page, so to speak, from Beethoven’s playbook.

Liszt is a special case that has no real parallel among fiction writers: a wildly popular concert virtuoso who wrote piano showpieces and arrangements of symphonic works for his own use in performance (making the most of his unique technical skills), then retired to focus on more profound compositions.

And then there was Johann Sebastian Bach. As Mauricio Kagel memorably put it, “Perhaps not all musicians believe in G-d, but all believe in Bach”. A virtuoso on multiple instruments (not just keyboards), with an intellect that probably exceeded even Mozart’s, Bach was a scholar as much as an artist, and had an active interest in the practical side of instrument design. His work ethos struck an interesting balance.

In Bach’s days, one couldn’t live strictly off composing: Bach’s principal income streams were as a church organist (legendary in his day), a court musician, and eventually as the musical director and assistant principal of the St.-Thomas high school in Leipzig. (He generated secondary income streams from music all his married life, as an organ building consultant, as a reseller of Silbermann’s early pianofortes, and the like.) As part of his duties in Leipzig, for several years he wrote a fresh cantata every week [!], many of which have sadly been lost (though over 200 have been preserved). He was thus every bit capable of churning out works on demand on a tight production schedule. (To be sure,he routinely recycled material, and levels of thematic borrowing that nowadays would be considered plagiarism were accepted in his day.)

At the same time, at the works that mattered most to him, he worked slowly and painstakingly — the first autograph MS for The Art Of The Fugue, for instance, dates from 1741, while the final fugue was still unfinished at Bach’s death nine years later. Most unlike Beethoven, few erasures are evident in Bach’s manuscripts until his eyes started faltering near the end of his life: he would write the whole composition in his head and at the keyboard, then put it down on paper when it met his exacting standards of quality. Fairly little of his music was published during his lifetime, primarily cycles of keyboard works. Bach studied music of other composers with great attention and mined it for forms as well as themes (to which he applied his never-surpassed mastery of counterpoint), but was essentially indifferent to public acclaim or fashion: while he took his musical duties seriously, the very notion of writing to fit the popular fashion of the day was alien to him. At the end of the day, this devout Lutheran with an impressive collection of theological books wrote his major works S. D. G. (soli Dei gloria, for the glory of G-d alone), as he was wont to write on his manuscripts.

Coming back to Sarah’s “Write Like The Wind” post. She mentions that endless revising and agonizing about language will not be noticed by one’s readers. Like with so many things in life, the ‘law of diminishing returns’  — just like a $20m Stradivarius obviously will not sound 1000x better than a violin custom-built by a luthier for, say, $20K. Sure, there is a reader segment (yours truly included) that enjoys exquisitely crafted prose for its own sake. However, beyond a certain point one is better off finishing up and moving on to the next work. One would like to believe that there is a golden mean between assembly-line hack work on the one hand, and toxic perfectionism on the other hand.

 

Closing the circle and ending the spiral: of well-temperament and equal temperament

I’ve been struggling for a while with how to present this in an accessible way. Stumbling upon this video below made me want to try: let’s have a go at this.

“Circle of fifths” or “spiral of fifths”?

It’s been well-known since the days of the ancient Greeks that the musical intervals we perceive as “purest” are the ones where the frequencies are simple fractions.
• octave: 2:1
• perfect fifth: 3:2   (and its inversion the perfect fourth, 4:3)
• pure major third: 5:4 (and its inversion the minor sixth, 8:5)
Less consonant are smaller ratios such as:
• pure minor third: 6:5 (and its inversion the major sixth, 5:3)
• greater whole tone, 9:8 (and its inversion the minor seventh, 16:9)
Now if you have a continuous-pitch instrument such as the violin or the human voice, you can just sing/play the intervals perfect (in “just intonation“) and be done with it. But what about fixed-pitch instruments such as guitars/lutes or pianos/harpsichords?
Big deal, you say: just stack up pure fifths, thus build the circle of fifths, and we can span a whole scale. Twelve fifths up and seven octaves down, and you should be where you started.
Sounds good in theory, except: the circle of pure fifths isn’t — it’s really a spiral of fifths. (3/2)^12 divided by 2^7 works out to 1.013643265…, almost a quarter of a semitone too wide. This small interval 531,441/524,288 is known as the Pythagorean comma.
Big deal, you say, if it doesn’t close: at least our fifths are pure. But what about the thirds?
Okay, let’s stack up four fifths, C-G-D-A-E, and drop two octaves down. We get… ((3/2)^4)/4=81/64, a Pythagorean third which is considerably wider than the pure major third. The ratio between them, (81:64)/(5:4)=81:80, which is known as the syntonic comma (or just plain “comma”). It’s just slightly (2% of a semitone) flat of the Pythagorean comma.
Even to an untrained listener, Pythagorean thirds sound unpleasantly sour. There is no way to fix this without either detuning the fifths or adding microtones to the scale.
So what if instead we start stacking up major thirds? Well, let’s see: C-E-G#-B#=C. That’s (5:4)x(5:4)x(5:4)=125:64, or 3:64 short of an octave!
In short: pure octaves, fifths, thirds: pick two.
The octave is the one interval nobody wants to mess with. (Well, there are “xenharmonic” scales, but nobody outside academia has even heard of them.)

Pythagorean intonation: pure fifths at all cost

You may effectively say: I must have the fifths (and hence fourths) pure, and if that means the thirds are sour, I’ll treat them as dissonant.
This is exactly what happened in Western music until the Renaissance.
Since all intervals in Pythagorean tuning are rational fractions that have no prime factors larger than 3, Pythagorean is also known as “3-limit rational tuning”.  [According to the same classification, “just intonation” as practiced by a cappella vocal ensembles is also known as “5-limit rational tuning”, since the largest prime involved is five (e.g., in the major third 5:4, the minor third 6:5).]
String instruments naturally lend themselves to Pythagorean tuning: anybody with musical hearing can tune pure fifths (or their inverses, fourths) by ear, just by tweaking until the “beats” stop. Violins are tuned in fifths; bass guitars, and the lowest four strings of a guitar in standard tuning, are tuned in fourths.

Quarter-comma meantone: pure major thirds at all cost

Alternatively, we can sacrifice the pure fifth in such a way as to restore the pure major third. The simplest way to do this is to narrow all fifths down by one-quarter of a Pythagorean comma, such that four narrowed fifths minus two octaves come out exactly a pure major third, 5:4. Such a fifth would be 5^(1/4)=1.495348781… (This quarter-comma meantone temperament was first proposed by a Spanish monk of Jewish origin named Pietro Aaron.)
Fifths in 1/4CM do “beat” (they are flat by about 1/20th of a semitone), but one can get used to them. The trouble: twelve fifths now stack up to three perfect major thirds (5:4)^3,  which we’ve seen above work out to 125:64, or 3:64 short of an octave. If you like: where Pythagorean tuning creates an expanding spiral of fifths, quarter-comma meantone gives rise to a contracting spiral of fifths.
Thus, you end up somewhere with one last fifth that is really bad, a so-called “wolf fifth” wide by two-fifths of a semitone. Since there are twelve possible places to start tuning, you can pick one such that the wolf fifth does not appear in the most frequently used keys. (Typically, it is put on G#—D#.) There are also several “wolf thirds” in the most remote keys.
Quarter-comma meantone was the prevalent tuning for much of the Renaissance. If one stays in the “safe” keys (with no more than two sharps or flats, say) and does not modulate to the more “remote” ones, it is quite tolerable. But don’t even think of playing in F# or Db on a keyboard tuned in quarter-comma meantone.

A first compromise: sixth-comma meantone

Musicians soon started experimenting with different meantone tunings.
In sixth-comma meantone [Ed.: a.k.a. Silbermann temperament], the wolf fifth can be reduced to about one-sixth of a semitone, at the expense of making the thirds just a little bit wide. This tuning still enjoys some popularity among “authentic Baroque practice” performers. Twelve fifths are now 1/5 semitone short of seven octaves.
Eleventh-comma or twelfth-comma meantone are nearly impossible to tune by ear, but are actually as close as makes no difference to equal temperament (see below).

Well-temperaments: closing the circle

As the Renaissance morphed into the Baroque era, composers started becoming ever more adventurous with modulations, and solutions that retained playability (to a greater or lesser extent) for all twelve major and all twelve minor keys were sought.
This led to the family of so-called “well-temperaments”, in which the Pythagorean comma is spread out over all twelve fifths, (at least) initially in an unequal fashion. Such temperaments are also called “circular”, in that twelve fifths now stack up to exactly seven octaves.
The term “well-tempered” (wohltemperiert) was originally coined in 1691 by the German organist and music theorist Andreas Werckmeister. He himself proposed several well-temperaments, one of which (Werckmeister III) is still in some use today among the HIP (historically informed performance) community.
In Werckmeister III, six of the fifths are tuned a quarter-comma flat (F-C, D-A-E, F#-C#-G#) while the remote G#–D#, to compensate, is made sharp by a quarter-comma and the remainder are tuned pure.
Another example is Vallotti temperament, in which the six diatonic fifths F-C-G-D-A-E-B are all tuned 1/6 of a comma flat and the rest are tuned pure. Young temperament, developed by the physicist and polymath Thomas Young, is based on the same pattern except cycled by one fifth to C-G-D-A-E-B-F#. There are many others: Kirnberger (advocated by a pupil of J. S. Bach), “tempérament ordinaire”,…
Common characteristics of all these well-temperaments include the following:
• all keys are at worst tolerable
• “nearer” keys approach just intonation
• “remote” keys approach Pythagorean intonation with its sharp thirds
Some of these are easier to realize by ear (i.e., without a digital tuner or other assistive device) than others.

Equal temperament: nothing perfect, everything equally imperfect

Mathematicians like Simon Stevin and Marin Mersenne in the West (and independently, Zhu Zaiyu in China) had proposed a more radical solution: to simply divide the octave into twelve equal parts, 2^(1/12)=1.059463094…, which is equivalent to narrowing all twelve fifths by one-twelfth of a Pythagorean comma to 2^(7/2)=1.498307077…
This is known as “equal temperament”, specifically “12-tone equal temperament” (12-TET). It is a special case of well-temperaments, and arguably the only “unbiased” or “universal” one. It is actually equivalent to tempering all fifths by 1/12th of a Pythagorean comma, and as close as makes no effective difference to tempering all fifths by 1/12 of the (slightly smaller) syntonic comma. So 1/11 comma meantone is functionally equivalent, and 1/12 Pythagorean comma meantone exactly so.
The luthier Vincenzo Galilei (father of Galileo) was perhaps the first to actually apply an approximate ET12 in instrument building, when he calculated fret spacings based on the ratio 18/17=1.058823529…, a fairly decent rational approximation to 2^(1/12)=1.059463094.
Ears used to the clean major thirds of quarter-comma meantone balked at first: also, 12-TET is not so easy to tune correctly with the naked ear. Despite the common misconception that everybody since Bach used equal temperament, other forms of well-temperament did not leave common practice until well into the 19th Century, but eventually 12-TET did become the Western standard for fixed-pitch instruments. Other well-temperaments have seen a modest revival in the HIP (“historically informed performance”) movement, particularly for harpsichord and organ tunings.
People with relative pitch may claim that in 12-TET, keys lose their “character”. To people with absolute pitch, they still have distinct sounds — though I have often asked myself a “cicken or egg” question here. For example, do I think of D minor as a “pensive, cerebral key” because it sounds like that (to someone with absolute pitch), or because I’ll forever associate it with Bach’s Art of Fugue BWV1080?

How ‘Well-Tempered’ was Bach’s Clavier?

Many people mistakenly assume J. S. Bach invented 12-TET. Of course he did not, nor was he even the first to write a composition exploiting it — that would have been Johann Caspar Fischer https://en.wikipedia.org/wiki/Ariadne_musica . Bach was however the first to write a major cyclical work, of transcendent musical value no less, that absolutely requires some form of well-temperament — and in doing so certainly hastened its adoption.
There is a scholarly consensus nowadays that Bach used not 12-TET but one or more well-temperaments, though it is not clear which. Bradley Lehman, in an article in Early Music, claimed that the ornament of the title page of the Well-Tempered Clavier actually encoded Bach’s own favored well-temperament [http://dx.doi.org/10.1093/em/cah067 and http://www.larips.com ], while a harpsichordist has recently argued [https://dominiceckersley.files.wordpress.com/2013/04/rosetta_revisited.pdf] that the temperament was in fact just the tempérament ordinaire described in Diderot’s Encyclopédie.

Equal temperament: blessing or curse?

Paraphrasing Winston Churchill about democracy: 12-TET is the worst possible solution for tuning fixed-pitch instruments…. except for all the others that have been tried.
On modern electronic instruments, when performing tonal music that also goes easy on modulation, one could in principle play in bespoke temperaments for each key. However, 12-TET is at this point so ingrained that people with fine musical hearing may actually consider just intonation or a favorably located well-temperament as ‘off’, even though it is objectively more in tune! Yet, unequal temperaments pop up in the strangest places — such as some guitarists slightly tuning down their B string in order to get just-intoned major thirds.
Allow me to end this post with one of my favorite Bach preludes played in two different temperaments on the same piano: the first time in Young temperament, the second time in modern 12-TET. Enjoy!

Of scales and tetrachords

Yesterday I discussed the Byzantine scales and their construction from two equal tetrachords. This actually inspired another post.

What happens if we

  • do construct tetrachords that span a perfect fourth, but
  • only allow whole tones and semitones, and
  • we allow two different tetrachords in the scale?

Well, if we only allow whole tones and semitones, and the beginning and ending notes are fixed at a perfect fourth, leaving only the two middle notes movable, then you basically only have the following choices. I will notate their interval sequences in semitones (half-steps):

  • major: 2-2-1 (i.e., whole-whole-half — major second, major third, perfect fourth)
  • minor: 2-1-2 (whole-half-whole — major second, minor third, perfect fourth)
  • upper minor: 1-2-2 (half-whole-whole — minor second, minor third, perfect fourth)
  • harmonic: 1-3-1 (half-third-half—minor second, major third, perfect fourth)
  • [an odd-duck fifth member is the augmented tetrachord 2-2-2, which ends on an augmented rather than a perfect fourth]

From mixing and matching pairs of those four tetrachords (plus one odd duck), we can assemble the following:

  • major+major: Ionian mode or classical major scale. In medieval church music, this was actually called the “lascivious mode” for its association with sprightly dances.
  • major+minor: mixolydian mode (major with a flattened seventh). Very common in Anglo folk tunes and in rock and pop music derived from it. Example: “Get Back” by The Beatles. (BTW, it also contains all five notes of the major pentatonic.) [On the piano, playing a scale on the white keys but starting from G rather than C gives you G mixolydian  — G major would have had F# rather than F natural.]
  • minor+minor: dorian mode (minor with a raised sixth). Very common in Anglo folk tunes and in rock and pop music derived from it. Has a more ‘minor’ feel than the mixolydian. Examples: “What shall we do with the drunken sailor”,  “Scarborough fair”,… (BTW, it also contains all five notes of the minor pentatonic, which is the backbone of the blues “scale” — which is really more like an Indian raga rather than a scale.) [On the piano, playing a scale on the white keys, but starting from D rather than C, gives you D dorian — D minor would have had Bb rather than B natural.]
  • minor+upper minor: aeolian mode, or natural minor scale.
  • minor+augmented: harmonic minor scale. In Western classical music that follows common-practice harmony, the need for major chords on the dominant (fifth) step automatically requires accidentals to temporarily raise the seventh. Hence a classical piece “in X minor” actually will pop in and out of harmonic (and melodic) minor rather than stay in natural minor. Using minor chords on the dominant instead automatically will give the piece a “modal” (“churchy” or “folky”) feel.
  • minor+major: melodic minor scale (ascending). Unlike the harmonic minor scale, the “un-flattened” sixth eliminates the minor third. Generally, classical melodies in a minor scale follow melodic minor when ascending, and natural minor when descending, although the locally prevailing harmony may dictate variances from this.
  • upper minor+upper minor: Phrygian mode. This “more minor than minor” or “martial” scale has occasionally been used in Western classical music (e.g., “Mars” from Holst’s The Planets), and is fairly common in darker heavy metal tunes. (E.g., “Harvester of Sorrow” by Metallica, the opening of “Sails of Charon” by The Scorpions, the fast middle section of “Seventh Son” by Iron Maiden) Among film composers, Phrygian for battle scenes is something of a cliché.
  • harmonic+upper minor: Phrygian major, a.k.a. Flamenco scale, a.k.a. “Jewish scale”. This Phrygian mode with a raised third is indeed a staple of Flamenco music, but can also be heard in the synagogue (the ahava rabba mode), in klezmer music, and indeed in some heavy metal tunes — for example: “Forty-six and two” by Tool, or the opening of “Killing in the name of…” by Rage Against The Machine.
  • harmonic+harmonic: double-harmonic or “Byzantine” or “Arabic” scale. It can be derived from the previous mode by flattening the sixth. Used for an exotic feel by classical composers (Debussy was rather fond of it), and sometimes in hard rock and metal. Dick Dale‘s instrumental “surf music” classic “Misirlou” is a very nice illustration.

An odd duck in the above list is the Lydian mode, which is built from augmented+major tetrachords. In a sense “more major than major”, it’s rarely used for a whole piece (the theme from The Simpsons is one example of a popular tune in the Lydian mode), but episodically can be used to set a mood of hope or anticipation. An example is the verse of “Freewill” by Rush.

OK, I guess this is an excuse for posting two of my favorite tunes: