On neotonality--the Wier version

I've identified as a "neotonalist" composer. That's a pretty broad term in itself, but in my case:

1. I use what could be called "tonal serialism", where all twelve notes of the chromatic scale (or more!, in case of microtonality) are given a tonal center. Often, this means the first note of a row is established as the tonal center, and this note is repeated at the end as a thirteenth note of the row. An example of mine is the tone row leitmotif in the Hamlin Pond score, first heard against a D minor key. The D pitch class keeps "asserting" itself.
2. Extended tonality: higher harmonics (for major keys) and lower subharmonics (for minor keys) than the classic triad (4:5:6) are incorporated into functional tonality. This requires the use of extra harmonics. For the seventh (sub)harmonic, the one-and-a-half flat (sharp) symbols are used; these would creat subminor and supermajor intervals. For the eleventh: neutral, semi-augmented and semi-diminished intervals, and thus half-sharps and half-flats.
3. The use of non-Western scales: Arabic-Turkish maqams, Iranian dastgahs, Indian ragas, the scales of Thai piphat and Javanese-Balinese gamelan, and so on. Many of these could be considered mixed major-minor, such as the maqam Hijaz, which contains a minor second followed by a major third. (Tonal serialism would automatically mix major and minor.)

However, even in the radical twelve-tone music of mid-to-late Schoenberg, I hear at least "accidental tonality", which can't really be avoided. If a major or minor triad shows up in the interaction of rows and their permutations, that would force tonality upon the listener, albeit a transient tonality.

Schoenberg the thesis, Partch the antithesis...

The 45th movement of Symphony No. 2 is now finished (except for future edits). It’s kind of short, but maybe it needs to be that way.




It took me a while to get started, but about a day to actually write.

One of my main goals as a composer has been to reconcile two opposite musical schools:

• the masters of the twelve-tone system of the common-practice period and thereafter, from Bach to Schoenberg
• the ideas of Partch (and Helmholtz through him), who wished to return music to a pre-equal temperament ideal of natural harmony using the overtone and undertone scales; also the extended-meantone work of Vicentino and (maybe) Gesualdo

I have advocated 72 equal temperament as a compromise between the two: it’s twelve-tone multiplied by six, and it approximates Partch’s 43-tone just scale very accurately. This piece, which is for string quartet though part of a large symphonic project, is another example of my works in that tuning.

The violin plays a 31-tone row, as an expression of microtonal serialism:

• B Bd D‡ Fd C‡ D Db C# C B‡ G# Bb A Ad Cd F‡ F# A‡ A# G‡ Eb Ab Dd D# Ed E E‡ Gb F Gd G
  (‘‡’  = half sharp; ‘d’ = half flat)

The cello repeats it later in retrograde inversion. Though originally intended to be atonal, it ended up being in a vague G major, even using the appropriate key signature.

The beginning is an example of tonality flux—it has a major third, G–B, transform into a minor third by small movements of the two violins in opposite directions (33.33 cents, precisely), to G‡–Bd. The rest of the movement is harmonization of the row using various otonal and utonal chords, with major and minor tonalities extended to twelve harmonics or subharmonics, instead of the traditional six.

I also originally wanted a fugue, but it does end in a “fuguing tune”-like canon. (There’s another idea I took from old shape-note hymnody, besides using the notehead shapes as a microtonal notation.) However, the still-unfinished 43rd movement (New Mexico) is to be a fugue using a 19-tone row, also in 72-tone.

The 31- and 19-tone scales are derived by simple formulas: n*(72/31) rounded to the nearest whole number, and n*(72/19) also rounded to the nearest whole number. I’ll also need to explain these systems further at a future time (I have written something about them before).

Subtitles of the 52 Movements of Symphony No. 2

I have now decided on subtitles for all movements of the Second Symphony (Subject to change.)

1. Maine: Light, Hope
2. New Hampshire: White Mountains
3. Vermont: Green Mountains
4. Massachusetts: Welcome to America
5. Rhode Island: Roger Williams and the Narragansetts
6. Connecticut: Crimson vs. Bulldogs
7. New York: Tesla vs. Edison
8. New Jersey: A Garden!
9. Pennsylvania: Franklin's Fugue
10. Delaware: Gallus gallus domesticus (The Blue Hen)
11. Maryland: Lord Baltimore's Banner
12. Washington, DC: American Pantheon
13. Virginia: Enter Appalachia
14. West Virginia: Ursus americanus
15. Kentucky: Poa pratensis
16. Tennessee: Oak Ridge Prophecy
17. North Carolina: Pinus palustris
18. South Carolina: Sabal palmetto
19. Georgia: A City Grows in the Forest
20. Alabama: The Original Mardi Gras
21. Florida: Eudocimus albus
22. Puerto Rico: El viejo castillo
23. Louisiana: Where the Sun Rises in the West
24. Mississippi: The Old Shape Note Baptist Church
25. Arkansas: In the Valley of Walmart
26. Missouri: Gateway Imperial March
27. Iowa: An Old Dilapidated Schoolhouse (with Quarter Tones)
28. Illinois: Polka Dragged through the Garden
29. Indiana: A Lonely Barn in Winter
30. Ohio: The Polka Strikes Back
31. Michigan: Maqâm Basta Nikâr (for Fuller)
32. Wisconsin: The Frozen Tundra (The Day the Cowboys Lost)
33. Minnesota: Rhapsodie norvégienne
34. North Dakota: Alces alces
35. South Dakota: American Pantheon II
36. Montana: Ursus arctos horribilis
37. Wyoming: 63 Minutes
38. Colorado: Tesla at Pikes Peak
39. Nebraska: Thunder over Zea mays
40. Kansas: Helianthus annuus (Is the Highest Point)
41. Oklahoma: Original Americans
42. Texas: A Russian Cosmonaut and a Mexican-American Astronaut on the International Space Station Communicate with Mission Control in Houston; All Contemplate the Earth's Place in the Universe and Ours, and Our Common Destiny. The Prognosis is Not Good.
43. New Mexico: Route 66 Fugue (with 19-Tone Rows)
44. Arizona: A Lonely Carnegiea gigantea in Summer
45. Nevada: The Road to No Man's Land (for Partch)
46. Utah: This Is the Place
47. Idaho: Almost There... (for Knievel)
48. Oregon: Another Portland
49. Washington: Emerald City
50. Alaska: Hell Freezes Over
51. Hawaii: The Gates of Hell are in Paradise
52. California: The Great Gate of San Francisco

Again, I include Puerto Rico, despite being a territory and not a state, because it has a greater population than 22 states, and because 52 (largest prime: 13) is a rounder number than 51 (largest prime: 17). Obviously, the District of Columbia needs its own movement as well.

The entire work is subtitled "52 American Preludes", and mostly inspired by Mussorgsky (as can be determined by several of the movements).

About shape notes again

There are two shape note systems (see also this NMB article) you may find in old hymnals (the Baptist church I attended as a child had them): the four-shape (mi-fa-sol-la) system used in Sacred Harp and Southern Harmony, and the seven-shape system of Jesse B. Aikin (1808-1900) and the Christian Minstrel. I use the latter for my microtonal notation, especially 72-tone.

These two systems were proposed to ease the sight reading of hymns, especially in Protestant churches. Each of the shapes can only be placed on particular notes; in the seven shape system, the 'do' shape is always a C, 're' a D, and so on. (The 'sol' shape in both systems is the regular oval used in standard staff notation.)

However, for my microtonal notation, the shapes can be used with any note. In 72-tone, the 'sol' shape indicates a note unchanged from conventional equal temperament. A 're' shape lowers the note by 50 cents; the 'do' shape raises it by the same, and the other shapes indicate deviations of 16.67 and 33.33 cents. From the lowest to the highest, the shapes in order are re, mi, fa, so (sol), la, ti (si) and do.

There has been another unconventional usage of shape notes--one that has nothing to do with pitch, but with rhythm. The American composer Henry Cowell (1897-1965; website) proposed shape notes as an alternative to tuplets. A graphic can be found on this page, but the image is of low resolution.

Huygens and 31-tone

A few days ago, we observed the 386th anniversary of the birth of the Dutch polymath Christiaan Huygens. Among his discoveries and inventions were Saturn’s moon Titan, the pendulum clock, and the division of the octave into 31 equal parts as an extension of quarter-comma meantone. Centuries later, another Dutchman, Adriaan Fokker, built an organ using this tuning.

The last of these is lesser known than the others, since we normally use twelve equally-tempered tones per octave today. Also, over a century before, Nicola Vicentino proposed his archicembalo, which was to be tuned in what was essentially 31 equal plus five additional notes in each octave.

Huygens found that 31-tone tuning approximated septimal intervals better than other tunings. The seventh harmonic is often thought of as a flat minor seventh, but he found that in his tuning, it really approximated the augmented sixth. Likewise, 7/6 would be an augmented second, and the 7/5 tritone the augmented fourth.

It is also an example of miracle temperament, discovered by George Secor in 1974. What Huygens missed is that 31-tone also well approximates the 11th harmonic, which would be a perfect fourth raised by a 38.71-cent diesis, or an augmented fourth lowered by the same. Other miracle equal temperaments are 41 and 72.

I have been using a 41-tone subset of 72-equal, using 31 pitch classes; these can be heard in the fourth movement of my First Symphony and one of my early microtonal compositions, "The Waterloo Rag". These are given below using my shape note notation, with alternates for some pitch classes beamed together (these are there to avoid ‘wolves’), with measurements in 72-tone and the just intonation ratios that they represent:

I’ve found that these could also be used for tuning Arabic maqams, with some modifications. For quarter-tone music, A double flat is equivalent to G half sharp, and G double sharp to A half flat, and so on. The fifths, 700 cents, are more in tune than the 696.77-cent fifths of 31-equal. Also, in augmented-second scales such as Hijaz, the minor second is slightly sharp and the major third slightly flat, unlike as in conventional 24-tone tuning.

(I started on G, instead of C or another note, because Yigah, the lowest pitch of the two-octave traditional maqam scale, is a low G, and Partch based his 43-tone just scale on G.)

Doktoro Esperanto: an opera project

A few months ago, the idea of writing an Esperanto-language opera about the language’s inventor came to me. I took it as a challenge, because not only have I never written (or even cared to write) an opera, and I know relatively little about the art form. But the first two operas I ever heard were Don Giovanni and The Magic Flute, so I’m at least familiar with the Mozartian version of the art form. (I’m listening to Wagner’s epic Ring cycle now.)

Now on to the plot. This is indeed based on the life of L.L. Zamenhof, but with some artistic license, as long as the known facts of the story are preserved.. It’s part history, part historical fantasy, part hysterical fantasy. (Most of the libretto will have to be written by someone else, but I’ll try to write a little.)

The opera will be dramatic while occasionally comic, a Bildungsroman of sorts for Zamenhof and for modern civilization. The music itself progresses, through the three acts, from Classical to Romantic to Modern--from Mozart to Wagner to Berg.

Act I: Białystok, Poland, then under Russian rule.

This act is in Russian, Polish and Yiddish (Esperanto had not yet been invented), with brief portions using four liturgical languages. Groups of singers from the Orthodox, Catholic, Jewish and Islamic faiths (the last being the Lipka Tatars) function as a "Greek chorus" throughout the opera.

Zamenhof was born in 1859. The January Uprising took place four years later. He witnesses the divisions between the ethnic and religious groups with which he becomes acquainted. He dreams of a future where there would be no war and no persecution, and all nations and peoples would be free. The idea comes to him that if people spoke a common language, then there would be world peace.

Act II: Warsaw, Moscow, Vienna and elsewhere. In this act, among the other languages, Esperanto is first introduced.

Now in secondary school, Zamenhof, a glossophile like his father, has studied numerous other languages, and also discovered one called Volapük, published in 1880 by a German Catholic priest named Johann Martin Schleyer (the language is briefly featured in the act). By now, Esperanto was already well into development.

His career, however, was to be in medicine, and he continued his medical studies in Russia and in Poland. He would begin his practice in Lithuania in 1885, then as an ophthamologist in Poland and Vienna. But while he healed people, he hoped to heal mankind through his universal language.

He would get married, and with the help of his father-in-law, publishes Международный язык. Предисловие и полный учебник (An International Language: Introduction and Complete Textbook). He also developed a religious humanistic philosophy based on the teachings of the rabbi Hillel the Elder, called Homaranismo, which taught the Golden Rule as the supreme moral principle which unites all religions.

Act III: Warsaw, April 1917. This act is entirely in Esperanto.

Zamenhof is now in the last days of his life. The Russian Revolution had begun the month before, and the First World War raged on. Great changes would sweep the Old and New Worlds.

He has a vision of the past and the future. He is told that a great evil will befall Europe, caused by a "German-speaking Haman" (all three of his children were murdered in the Holocaust). His language endures both success (however limited) and suppression. He sees that there may never be world peace, but it will always a noble effort to try to bring it about.

My microtonal notation systems, revisited

I've written about this before, but I've devised two systems of notating microtonal intervals, usually 72 equal divisions to the octave.

The first uses a combination of standard accidentals, quarter-tone symbols (called Tartini-Couper) and standard accidentals with attached arrows. Bartók wrote arrows above notes or accidentals to mark small inflections of time, what he vaguely called "quarter tones", in his own works and transcriptions of folk tunes. I call this system "Tartini-Couper-Bartók", or TCB for short.

The fractions underneath mark fractions of a whole tone (200 cents in 72-edo; 1/12 tone = 16.67 cents):

If I want to indicate exact degrees of 72-tone, I may use shape notes instead. These are normally used in old American hymnals. I took the seven-shape system and adapted them, in a very nonstandard way, where any shape can fall on any note. The regular oval, or 'so' shape, indicates a pitch that does not deviate from 12-tone equal temperament.

Remember, the shapes do not represent the notes of the diatonic "do-re-mi" scale. Also, the shape for 'fa' is turned 180° when the note stem points downward (or, in the case of whole notes, would be if it had a stem).

These are just two possible notation systems. I don't pretend to offer a be-all-end-all solution to anything.

Fundamental intervals

Many important intervals in the 72 tone matrix, organized into 31 pitch classes (two are in italics because they lie outside the 41-tone Miracle scale).

This method could be used to determine ideal tuning of notes in a chord, if microtonal intervals are available (as in vocals, violin-type instruments and trombones).

0. perfect prime (unison), 1/1
2. semiaugmented prime, 64/63
4. augmented prime, 27/26
5. augmented prime, 21/20
7. minor second, 16/15
9. neutral second, 12/11
10. neutral second, 11/10
11. major second (less than whole tone), 10/9
12. major second (whole tone), 9/8
14. diminished third, 8/7
16. augmented second ("blues" third), 7/6
18. minor third (Pytagorean), 32/27
19. minor third, 6/5
21. neutral second, 11/9
23. major third, 5/4
25. diminished fourth, 14/11
26. diminished fourth, 9/7
28. semidiminished fourth, 21/16
30. perfect fourth, 4/3
32. semiaugmented fourth, 15/11
33. semiaugmented fourth, 11/8
35. augmented fourth (tritone I), 7/5
37. diminished fifth (tritone II), 10/7
39. semidiminished fifth, 16/11
40. semidiminished fifth, 22/15
42. perfect fifth, 3/2
44. semiaugmented fifth, 32/21
46. augmented fifth, 14/9
47. augmented fifth, 11/7
49. minor sixth, 8/5
51. neutral sixth, 18/11
53. major sixth, 5/3
54. major sixth (Pythagorean), 27/16
56. septimal major sixth, 12/7
58. septimal minor "blues" seventh, 7/4
60. minor seventh (Pythagorean), 16/9
61. minor seventh, 9/5
62. neutral seventh, 20/11
63. neutral seventh, 11/6
65. major seventh, 15/8
67. diminished octave, 40/21
68. diminished octave, 27/16
70. semidiminished octave, 63/32
72. perfect octave, 2/1

On transliteration of Cyrillic into Latin

(Note: this does not cover scientific transliteration of the Cyrillic alphabet, which uses letters borrowed from Czech and other languages, and operates on a one-to-one character principle, e.g. Челябинск > Čeljabinsk. This has to do with transcribing Cyrillic-alphabet languages into the Anglo-Latin alphabet.)

The way Russian and other languages which use the Cyrillic alphabet are transliterated into Latin, the way the English language uses it, generally follows the standard set by the Library of Congress. It is simplified a bit, so it can be typed out using only the ASCII character set (all the characters US-English keyboards can write). This requires using multi-letter groups to replace certain Cyrillic letters.

The scheme is a bit outdated. Since the breakup of the USSR, the West has become more familiar with current Russian usage and pronunciation. To properly represent Russian names for laymen (not linguists), and I want Latinization to best represent how the language is pronounced, not necessarily written, I would make a few changes.

(This does not take into account two features of standard Russian pronunciation: the devoicing of voiced stops and fricatives at the end of words, and the neutralization of unstressed vowels, i.e. why до свидания is pronounced "da svidaniya".)

• Ее: At the beginning of words, after vowels and after Ьь, I would write this as "ye": Достоевский > Dostoyevskii. After consonants (where it usually, but not always, palatizes or softens the preceding consonant), I'd have it simply as 'e': Евгений Онегин > Yevgenii Onegin.
• Ёё: When pronounced as "yo" in "your" (and in that situation, always stressed), the two dots are optional in everyday writing. Still, I would always transliterate this as "yo", and simply 'o' after consonants that are always soft or hard: Пётр > Pyotr; Горбачёв > Gorbachov.
• Жж: The usual practice is to write this as "zh", and I would do the same. I have seen it as 'j', but I feel that would be too easily confused as the 'y' or "dzh" sound. Therefore, I would have, Нижинский > Nizhinskii.
• Йй: This is usually found after vowels, as part of a diphthong, and I would write it as 'i': Николай > Nikolai; Дмитрий > Dmitrii. Before vowels, I would write it as 'y'.
• Хх: Though this is sometimes recorded as 'h' (as in Serbo-Croatian), I would continue to write it as "kh" to avoid confusion: Чехов > Chekhov.
• Цц: This should be continued to be transliterated as "ts", not as "tz", "cz" or 'c' (before front vowels in the latter case): царь > tsar'.
• Щщ: Traditionally, this letter has been pronounced as "shch", and still is in Ukrainian. In current Russian, however, it is pronounced as a long, soft "sh" sound. I would transliterate this not as "shch", but as "sch", still distinct from the always-hard "sh", used for Шш: Хрущёв > Khruschov; Щедрин > Schedrin.
• Ъъ: Since the spelling reforms of 1918, this letter has become exceptionally rare. It has no pronunciation of its own, and only serves to block palatization (softening) of the final consonant of a couple of prefixes. Therefore, I would not transliterate the letter itself, but write the following vowel with a 'y': съезд > syezd.
• Ьь: On the other hand, this letter is frequently used--to palatize (soften) a preceding consonant. I would maintain the current practice of transliterating this as an apostrophe: цар > tsar'. However, after consonants that are either always hard or always soft, I would ignore it: ночь > noch. (Here, the letter serves to indicate that the noun is feminine, despite ending in a consonant.)
• Ээ: This letter is always pronounced 'e' and does not soften a preceding consonant. It is usually, but not always, found at the beginning of words, more often in foreign loans. I would write it simply as 'e': электроника > elektronika; поэт > poet.

About that Requiem... (enter the "3D fugue")

I just discovered a level of fugal polyphony that, as far as I know, not even Bach attempted. (But Mozart did, or was beginning to at least.)

This is a variation of the multiple fugue (double, triple etc.). However, instead of developing the first subject, than the second later, and so on, this is a fugue with the subjects being developed at the same time. In other words, a "three-dimensional fugue", with fugues layered on top of fugues. If the fugue is the musical equivalent of chess, than this is the music equivalent of 3D chess.

It just so happens that a two-by-two fugue has been written: the Kyrie from Mozart's Requiem. Here it is again; start at 4:46:

I'm writing a three-by-three example in the Second Symphony, fourth movement (Massachusetts), and the final reprise of the Ouverture in the First Symphony has a two-by-four-voice fugue.

(It's not quite the same as the twelve-part "group fugue" in the sixth movement of my First Symphony--that's a four-voice fugue, with each voice harmonized in three parts, but it is another example of a 3D fugue.)

Also, what I'm talking about would be an example of what could be called "hyperpolyphony". The best established example, from the Renaissance, would be Thomas Tallis' Spem in alium:

This isn't a fugue, but it's still 40-voice polyphony.

But does this mean that someday I have to write a 40-part fugue?

What if he had lived longer?

I am convinced that Mozart, had he not died so tragically young, could've done for music what Beethoven did later.

Anyway, one more post: his unfinished Requiem. Specifically, the Kyrie.

A couple things to notice:

• the use of basset horn (marked by the gold diamonds) in the introduction; the basset horn is an alto/tenor clarinet in F with a low range extended to C (in modern instruments)--and remember, Mozart was the first major composer to write extensively for clarinets in general
• the double fugue in the Kyrie--I told you he was getting good at this counterpoint thing

I am planning on writing a Mass someday, maybe a Requiem. So I'll be studying this tonight, another thing I wish I had done when I was much younger.

The "Jupiter" Symphony: when Mozart challenged the musical gods

Wolfgang Amadeus Mozart was born in Salzburg, Austria on this day in 1756.

I have had an obsession with the finale of his 41st and final Symphony for many years now.

This is when the composer triumphantly goes all out with all of his techniques. It's almost a fugue worthy of Bach himself, working five different themes together. He had used Baroque counterpoint in other works late in life: the Requiem, the Magic Flute (especially the Overture) and of course the Adagio and Fugue, K.546.

His music was also becoming heavily dramatic, especially in his unfinished Requiem. Had he not have died much too soon, he would've started the Romantic era of music decades before Beethoven did. (He already was part of the Sturm und Drang movement since the 25th Symphony, in G minor.)

Also, the chromatic stuff that began the development in the final movement of the 40th symphony? (See Leonard Berstein's commentary, beginning at 7:48.) He uses note of the chromatic scale except G, since it was a symphony in G minor and he had used the note enough. Almost a tone row à la fellow Austrians Arnold Schoenberg and company, but with a few repeated notes. And he had done it at other times.

I mean it; Mozart was really getting good near the end.

The Rite again, and what it did to me

The beginning of the still-unfinished First Symphony was 35 years ago, when I first heard The Rite of Spring on vinyl in my bedroom. It was part of a compilation (one of those boxed "best-of" classical compilations); I don't remember what orchestra recorded it, or what year.

I consider that event when and where I, the composer, was born. I had heard Bach, Mozart, Beethoven, Chopin, Tchaikovsky, the rest, but it was Stravinsky that made me decide I wanted to be like them all, and innovate in my own way. The first ideas of the First Symphony came to me around that time. The ideas came to me in little pieces, from movies, television, video games, everything I was exposed to.

I didn't start writing and recording any of it until about ten years ago. I wasn't ready until then. I would have rather been a bassist for a metal band somewhere. This was even long after I went to college to study musical composition and theory, but dropped out my freshman year due to illness. (I'm entirely self-taught after that.)