Addition and Multiplication of Intervals

I was emailing back and forth with a student research assistant about some algebraic properties of adding, subtracting, multiplying, and dividing intervals, and I thought that I might share the email (in slightly revised form) with the Net.  I would be very surprised if this hasn't been written up already by someone else, but the only things I could find quickly about interval multiplication was all in the 12-tone/serial usage, which is less interesting to me today.   
 

An interval is just a ratio (fraction), so the octave is 2:1.  That ratio could apply to frequency (a note an octave up has a frequency twice that of the lower note), or the reciprocal gives the string length (so a string half the length of another is an octave higher than the longer string) or length from mouthpiece to first open tone hole, etc. 

The octave is always 2:1.  The other interval ratios depend on what temperament we’re talking about.  In equal temperament, all other ascending intervals are integer exponents of L , where L is a the 12th root of 2 (I’m using L because it’s the 12th letter).  So a major third (4 semitones) is L^4 or the cube-root-of 2.  In other temperaments, the ratios are integer ratios, but they’re not standardized.  So a major third can be either 5:4 or 81:64 (I like the latter, also called a ditone, for reasons below).  You just have to memorize them all.  But once you have P5 as 3:2 and major second as 9:8, you can basically derive the rest.

When we add intervals, what we’re actually doing is multiplying ratios.  So P5 + P4 = 3/2 * 4/3 = 2/1 = P8. When we subtract intervals we’re dividing ratios, which is multiplying by the reciprocal, so P5 – P4 = 3/2 * 3/4 = 9/8 = M2.  And P8 – P5 = 2/1 * 2/3 = 4/3 = P4.  Since multiplication of ratios is commutative, then addition of intervals is also commutative.  The generic interval (the "5" in "P5") of the sum of ascending intervals is always the sum of the two generic intervals minus 1.  For descending intervals or a mix or if adding multiple intervals at once, take each generic interval and subtract 1 from the absolute value and then restore the sign.  In the end add one to the absolute value of the result and then restore the sign.  If it was 0 before, make it positive, since we only have P1 for a unison, never P-1.  (Though how we should actually designate the sign for d1 or dd1, is unclear, maybe it should be d-1? but then it's not analogous to dd2, such as between E# and Fb, where the generic interval is definitely ascending -- in programming this possibility is always one of those cases that bites you in the ass later later, where you have an ascending diatonic interval that is a descending chromatic interval, so if you mix ASCENDING tests for generic intervals with > 0 tests for chromatic intervals, you'll get inconsistent results depending on which form of the interval you're looking at.)

Getting back to the main subject.  So what does it mean to multiply an interval by an integer?  M2 * 2 = ?  Well, what’s M2 + M2 = 9/8 * 9/8 = 81/64 = M3.  So M2 * 3 = M2 + M2 + M2 = 9/8 * 9/8 * 9/8 = A4/Tritone (729/512), and so M2 * 3 = (9/8)^3, so when we multiply an interval by a number it’s like taking the ratio to a power.

Since exponentiation is not commutative, 2 * P5 would be different than P5 * 2;  2 * P5 = 2^(3/2) = 2 radical 2, while P5 * 2 = (3/2)^2 = 9/4 which is a M9 (P8 + M2 = 9/8 * 2/1).  However, we’re defining our own algebraic system, so we could define * as always placing the integer in the exponent and thus make this commutative.  HS and early college math doesn’t talk much about defining our own algebras, but we do it all the time.  (otherwise we couldn’t define 11:00 + 3hours = 2:00, etc.)

So, does it make any sense to multiply intervals?  What would M3 * P5 be?  Well, if we convert it to ratios, then it’d be (81/64)^(3/2), or (9/8)^3, or 729/512, which we defined as an Augmented 4th.  Most ratios * other ratios though will create irrational ratios, which we don’t like unless we’re in Equal Temperament (“irrational ratio” is an oxymoron if you think about it).  In equal temperament though we’d end up with irrational numbers raised to irrational exponents.  Your calculator will calculate these things, by substituting in the nearest rational number, and in fact to take a number to an irrational ratio, you need to find the limit of the ratio of the base to the closest smaller rational number exponent with the base of the closest larger rational number exponent.  (btw – did you ever notice that any negative number to an irrational power is undefined? because it depends on whether the irrational number can be expressed as a ratio with an even or odd denominator, and irrational numbers are not ratios.  Fortunately, we don’t need to deal with negative ratios in music).

A nice property of defining multiplication of intervals as a form of exponentiation is that descending intervals (whose ratios are positive but < 1) can also be used.  I like M3 * P-8, or major third times descending perfect octave; or (81/64)^(1/2) power, or 9/8, or M2.

Consider what multiplying by an interval by an interval might be used for.  M2 * P8 = (9/8)^(2/1) = 81/64 = M3.  So a M2 occupies the same proportion of the harmonic space of one octave as a M3 does for two octaves.  This process (multiplying an interval by P5) could be used to convert intervals in standard, octave repeating, space into Bohlen-Pierce space which is based on the P12.  Or it can translate the ratios produced by fingering patterns in the lower vs. upper register of the flute (based on P8) into the ratio you’d get on the clarinet (based on P12).

Also notice that multiplying any interval (ascending or descending) by a descending perfect infinity (P-∞) (or the limit as the number of descending octaves increases without bound) condenses the available interval space to nothing. So every interval becomes a unison.  E.g., P4 * P-∞ = (4/3)^(1/(2^∞)) = (4/3)^(1/∞) = (4/3)^(0) = 1 (since any non-zero number to the zeroth power = 1) and 1 = 1:1 = P1.

The question of what diatonic intervals result from any addition or multiplication isn’t something I’ve touched on here.  It’s easy to figure out what the generic interval under addition will be as I described above.  The specifier (major, minor, augmented, diminished, perfect, etc.) is harder to determine.  I’ll leave that as an exercise – it’s messy and I solved it a while back, but I can’t remember the exact solution right now.  Under multiplication of an interval with an integer, it’ll be easy to solve what the diatonic interval will be, without converting to ratios, once you’ve solved the previous problem.  But for multiplication of an interval by another interval the math becomes harder.  The first question to solve there is, is the answer dependent on the temperament system chosen, or can it be generalized for any temperament?

Btw, raising intervals to the power of other intervals is just silly.  So say I. :-)

Litany of Ars Nova (Trecento) Saints

Lord have mercy on us
Christ have mercy on us
Lord have mercy on us.

Accept holy Trinity
This joyful cry of peace
And remove the cloud
Of horrible schism.

Holy Hildegard
Mother of Musicians
Virgin Composer -- have mercy on us.

Holy Philippe de Vitry -- pray for us
Holy Marchetto of Padua -- pray for us
Holy Guillaume de Machaut -- pray for us
Holy Jacopo da Bologna -- pray for us
Holy Giovanni da Cascia -- pray for us
Holy Master Piero -- pray for us
Blessed Egidio and Guglielmo -- pray for us
Holy Francesco the Blind -- pray for us
Holy Lorenzo of Florence  -- pray for us
Holy Johannes Ciconia -- pray for us
Blessed Anthony, called Zachara of Teramo -- pray for us
Holy Matteo of Perugia -- pray for us
Holy Bartolino of Padua  -- pray for us
Blessed Solage  -- pray for us
Blessed Engardus -- pray for us
Holy Christine de Pizan -- pray for us
Blessed Alanus -- pray for us
Holy Baude Cordier -- pray for us
Blessed Oswald of Wolkenstein -- pray for us
Holy Prosdocimus of Beldemandis -- pray for us
All you holy composers, singers, and musicians -- pray for us
All you holy theorists and poets -- pray for us
All you scribes and compilers of manuscripts -- pray for us.

Blessed Françoise-Joseph Fétis -- pray for us
Blessed Johannes Wolf -- pray for us
Blessed Friedrich Ludwig  -- pray for us
Venerable Heinrich Besseler -- pray for us
Blessed Kurt von Fischer -- pray for us
Blessed Susanne Clercx -- pray for us
Holy Nino Pirrotta -- pray for us
Blessed Billy Jim Layton -- pray for us
Blessed Giuseppe Vecchi -- pray for us
Blessed Pierluigi Petrobelli -- pray for us
All you holy scholars -- pray for us
All you thinkers about medieval composers -- pray for us
All you translators of music theory -- pray for us
All you searchers of manuscripts and fragments -- pray for us.

Lord, be merciful,
From all dissonances -- Lord, save your people
From all scribal errors -- Lord, save your people
From your tritones -- Lord, save your people
From bad ficta choices -- Lord, save your people
From a sudden and unprovided hexachordal mutation -- Lord, save your people
From the scourge of lost manuscripts -- Lord, save your people
From incorrect prolation and mensuration -- Lord, save your people
From unexplained coloration -- Lord, save your people.

By the mystery of minim equivalence,
By your dragmas,
By your custodes, -- Lord, save your people
By your ligatures of perfection,
By your ligatures of propriety,
By your ligatures of opposite propriety, -- Lord, save your people
By your alteration and imperfection,
By your dots of division, and of addition,
By your chains of perfect semibreves under similis ante similis,
By your knowledge that what cannot be transcribed
    should not be transcribed, -- Lord, save your people
On the day of publication -- Lord save your people.

Be merciful to us scholars, -- Lord hear our prayer
That you will guide us,
That you will help us discern the alignment of voices,
Through the logic of perfect consonances on strong beats,
And not invent alternative explanations for simple transcriptions -- Lord hear our prayer
That you will grant us your Apel to discern your will, -- Lord hear our prayer
That it may please you to bring us to true transcription -- Lord hear our prayer
Guide and protect your holy universities,
Preserve in holy religion the editors at LIM, Brepols, AIM,
    and all those in holy publishing houses -- Lord hear our prayer
Humble the fifteenth-century scholars,
Who assert that only complete polyphonic Mass cycles are pleasing to you,
And those who transcribe fourteenth-century music
    without rhythmic reduction -- Lord hear our prayer
Bring back to the unity of performance those who sing without ficta,
    those who choose moribund tempos, and all those who play
    shawms without thought of intonation -- Lord hear our prayer
Strengthen and preserve us at Certaldo, and Dozza, and Novacella,
Raise our databases to the level of true understanding,
Reward all your servants with everlasting tenure -- Lord hear our prayer
Deliver our souls from indecipherable tropes, and the souls of those who transcribe ars subtillior,
    who search in archives, and read clerical shorthand -- Lord hear our prayer
Give and preserve the fragments not yet found,
Yield to us productivity in our sabbaticals,
Grant three beats of rest to all perfect semibreves pausae
Never causing our Finales or Sibeliuses to think of imperfecting them,
That it may please You to hear us and our editions,
    Jesus, Son of the Living God -- Lord hear our prayer

Lamb of God, who takes away the sins of transcription -- Spare us, O Lord!
Lamb of God, who takes away the sins of musicology -- Spare us, O Lord!
Lamb of God, who takes away the sins of scholarship -- Grant us thy peace.

Christ, hear us,
Lord Jesus, hear our prayer,
Lord, have mercy on us,
Christ, have mercy on us,
Lord, have mercy on us,

Amen.

In Deutschland

Gruß Gott! Am Donnerstag um 10.00 Uhr, werde ich einen Vortrag an der Ludwig-Maximilians-Universität zum Thema "Codieren von Musiknoten für analytische Abfragen". Am Montag, meine Studenten und ich werde über das Thema Service-orientierte Architekturen für musikalische Analyse bei einem Workshop an der Digital Humanities Konferenz sprechen. Wenn Sie in München oder Hamburg und sind an einer Teilnahme interessiert sind, kontaktieren Sie mich und ich kann Himmelsrichtungen schicken. Oder wenn Sie in Berlin vom 18-20 Juli sein und wollen uns treffen, lass es mich wissen.

Unsere Reise wird von einem großzügigen Zuschuss von der deutschen Regierung den kulturellen Austausch und durch die Deutschland Seed Fund von MIT gefördert.

Fuga Trium Tempora from the Strasbourg Codex

[This is a draft “Working Paper” of research in progress; comments are welcome, but it should not be considered published work and may be removed before this is submitted for publication and replaced with a link to the published version.]

The manuscript, Strasbourg, Bibliothèque Municipale (olim Bibliothèque de la Ville), MS 222. C.22, was an extraordinary collection of music theory and secular and sacred music (sometimes in contrafact) from the first half of the fifteenth century.  In 1870, during the siege of Strasbourg in the Franco-Prussian war, the manuscript was destroyed.  Fortunately, portions of the manuscript survive in two important testimonies from before 1870: a short publication by Auguste Lippmann, “Essai sur un manuscrit du quinzième siècle decouvert dans la Bibliothèque de la ville de Strasbourg” Bulletins de la Société pour la Conservation des Monuments Historiques d'Alsace Serie 2.7 (1870), pp. 73–76, which reproduces a single page of the manuscript in facsimile (see Figure 1 below) and a partial copy of the manuscript made by Edmond de Coussemaker.  Coussemaker copied the table of contents, made an index with incipits, and transcribed some, but not all, of the pieces in the manuscript.  Coussemaker’s copy is now Brussels, Bibliothèque du Conservatoire Royal de Musique, MS 56.286 and has been published in facsimile in Albert van der Linden, editor, Le manuscrit musical M.222 C.22 de la Bibliothèque de Strasbourg: XVe siècle (Brussels: Office international de librairie, 1977). Many of the works in the manuscript can be identified through concordances in other manuscripts, though the process of finding concordances through incipits with contrafacted sacred texts has not always been easy.  Important work on the manuscript was conducted by Charles van den Borren (Le manuscrit musical M.222 C.22 de la Bibliothèque de Strasbourg (XVe siècle) brulé en 1870, et reconstitué d’après une copie partielle d’Edmond de Coussemaker [Antwerp: E. Secelle, 1924]) and in an excellent Habilitationschrift by Lorenz Welker (“Musik im Oberrhein im späten Mittelalter: Die Handschrift Strasbourg, olim Bibliothèque de la Ville, C.22” [Habilitationsschrift: Basel, 1993]).  Though most of the music that Coussemaker transcribed has appeared in modern editions, a few pieces have never been published in modern notation.

 Figure 1: Color image of folio 78v from the Strasbourg codex

One such neglected work was found on folio 38r (or perhaps 37v–38r, see Welker, “Folio-Synopse” p. 12) and transcribed on pp. 32–33 in Coussemaker’s edition.  It is a “Fuga trium temporum” whose top voices are attributed to J. de Climen and whose tenor is attributed to J. Cornelius (“Tenor J. Cornelii”).  The double attribution is unusual but as Virginia Newes notes, the tenor is inessential to the canon and could have been added later. (“Fuga and related contrapuntal procedures in European polyphony ca. 1350–ca. 1420,” [Ph.D. dissertation: Brandeis University, 1987], p. 403).  The description “Fuga trium temporum” implies a canon at the unison separated by three breves.  That the title appears under the top voice suggests that it is the top two voices which are in canon.  Though tenor canons are not unusual in the period and for much of the piece the tenor works in canon with itself at the distance of three breves, this effect is largely accounted for by the tenor’s need to support the upper-voice canon, and several cases of bare perfect fourths and long passages in parallel unisons strongly argues against an intention of four-voice performance (which is what Ensemble Leones, the only group I have found that has performed the piece, did in their reconstruction of “[Quatour voces in] fuga trium temporum”) or of one upper voice plus two tenor voices in canon (which would not fit the idea that J. Cornelius added an additional voice to an existing fuga).  Reproductions of Coussemaker’s transcription of the work are in Figures 2 and 3.

Figure 2: Top voice of J. de Climen, Fuga trium tempora.
 

Figure 3: Tenor by J. Cornelius of Fuga trium tempora.

The two-part form of the piece suggests that the work may have originally been a rondeau that no longer has a text (similar to Baude Cordier’s Tout par compas).  Less likely, the piece could have been a virelai or even a ballata (like Andrea da Firenze’s Dal traditor in the Squarcialupi codex), though the close spacing between entrances is more characteristic of French than Italian superius canons.  In any case, Charles van den Borren’s suggestion (p. 88) that the work could be an Italian caccia seems unlikely.

The work’s neglect in modern scholarship may be due to its lack of text, but is more likely attributable to the unsatisfactory nature of the piece which results when transcribed from Coussemaker’s edition.  Whether read as two upper voices without tenor, two upper voices with tenor, or a single upper voice, several problems emerge. The top voices have an unusual phrygian cadence.  The piece ends with a major third (C–E) between the top voice(s) and tenor: highly unlikely for the period.  Several intense dissonances appear in the second section that are out of style with the first. Parallel octaves, fifths, and unisons the tenor and the upper voices appear six times (10 if closely syncopated parallels are counted) in the second half; they are absent in either counting in the first half. Finally, the top voices have no motivic repetitions between the first and second sections.  Example 1 transcribes the ending as written.

Example 1: Ending of Fuga trium tempora as transcribed.

A small emendation to the piece, not previously explored, relieves all five of these problems.  After the first note of the second section, the top voice should be read a third higher than it is written.  Either the scribe of the Strasbourg codex or Coussemaker either wrote this section a third too high or he neglected to notice a change of clef for the second half of the piece.  A proposed emendation of the top voice is given in Figure 4.

Figure 4: Proposed emendation to Coussemaker’s transcription.

With this emendation, the second half of the top voice echoes many elements of the first half, all parallels are removed, the ranges of the first and second half become identical, and the piece ends with the upper voices on G supported by the C a perfect fifth below in the added tenor part.  (If the tenor was essential and conceived with the upper voices, we might expect the top voices to end on a high G with the D a fifth below with the tenor singing the G a fifth below.  The change in modal flavor added by the tenor is further evidence of it being a later addition).  The whole piece as I have transcribed it is given in Example 2 and, for the sake of understanding the piece better, a MIDI rendition as an .mp3 file is given in Example 3.  After the middle cadence, the second upper voice may rest or continue the canon from the first section; examples supporting both types of continuation are found in other pieces in Newes’s dissertation.

Example 2: Fuga trium temporum, new transcription.

Example 3: MIDI rendition.

The new transcription does nothing to explain who J. de Climen or Johannes Cornelius might be.  David Fallows has proposed that he might be the same as Jacobus de Clibano known from several compositions in the Aosta codex (“Jacobus de Clibano,” s.v. in The New Grove Dictionary of Music and Musicians, 2nd edition) or, less likely, he may be the same as Clement Liebert known from the piece Comment porray in the Strasbourg codex.  But neither this short contribution nor the musical style of Comment porray (though also in 2/4 but in white notation) give any aid in confirming or refuting this connection.  But I hope that the addition of a new contribution to the small repertory of fugae and canons of the early quattrocento can give a renewed urgency in discovering more about the identity of the composers of this finely crafted little work.

New news, Old news, and quotes...

Some news stories old and recent that I've forgotten to post:

Two responses to DarwinTunes (more coming):
Discover Magazine
Michael Scott Cuthbert, who works on computer-aided musical analysis at MIT, is sceptical that the approach tells us anything about the evolution of music. “They have shown that people can sense a glimmer of the things they like about music even when most of it consists of sounds they hate,” he says.  “But it doesn’t give any information about why music sounded differently in the past, why people like different things today, or how music might evolve in the future.”
 
“Suppose you randomly threw car parts into piles and asked people to rate those they’d most like to buy,” he says. “Then you took parts from the highest-rated heaps, and rearranged them into new heaps.  People might hate all of them at first, but they’d probably rate the ones with four tires or a trunk in the back or a steering wheel in the drivers’ seat higher than the rest. Do that long enough and I wouldn’t be surprised that you’d eventually get something that looked like a 2011 Honda Civic.  But that doesn’t mean that that’s how a car is made.”
 
L.A. Times
The study shows that people "can discern the little things they like about music even in the context of a lot of extraneous sounds," said MIT computational musicologist Michael Scott Cuthbert, who wasn't involved in the research. "But what they don't prove is why music today has changed from the popular music of the past. It doesn't show how changing tastes result in changing music and it doesn't give us a hint of what the future holds for music."

Five things from MIT:
MIT News Office on the ELVIS grant
MIT Tech interview on the ELVIS grant (posted previously)
MIT SHASS Magazine article (Spring 2010) on my research
A little blurb about my work (might change to someone else's in the future)
A little piece on a completion of a piece by Zachara da Teramo

Many new papers posted at Academia.edu.

Silly (but fun) little thing

I'm "Sara Does Science"'s "Science Crush Friday" for the week. Includes the wonderful (if fanciful) line "Michael Scott Cuthbert is music’s Indiana Jones" and the description "Not a bad looking guy at all! And he kind of reminds me of Ross from Friends, but the smile makes him seem less neurotic." For the record, I am equally neurotic, just in different ways.

Thanks Sara!

MIT Tech Article on Michael Cuthbert

Derek Chang of the MIT Tech published an interview with me in today's issue. Read it here. The opening appears below:

$500,000 grant for music research at MIT

Michael Scott Cuthbert, associate professor of music, was recently awarded a $500,000 grant from the Digging into Data consortium. This grant will support his work in using computational techniques to study changes in Western musical style. He has received $175,000 specifically for his music21 project . On Thursday, Cuthbert sat with The Tech to discuss his work with music21 and his passion for combining computational techniques with music.

The Tech: Many of us with a musical background must be interested in your computational work and how it applies to music. What is the motivation behind your project?

Cuthbert: One of the main ways artists analyze art work or music is examining a piece very carefully, from all possible dimensions. But it’s really hard to put the work into the context of the time. How is the piece representative of its time period, or how does it break the mold? It takes us a very long time to look at one piece. In contrast, computers are good at getting an overview of a particular problem. For example, what patterns exist in how chords progress from one to another? Is the piece being looked at representative of the music grammar for the period? (...read more...)