## 18 December 2012

### 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^4/2 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. :-)

## 16 December 2012

### 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 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 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.

## 10 July 2012

### 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.

## 09 July 2012

### 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.

## 04 July 2012

### 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.

## 25 February 2012

### 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!

## 24 February 2012

### 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...)

## 31 December 2010

### Changing Musical Time in the Renaissance and Today

The concepts of time, rhythm, and musical notation have changed dramatically from the Middle Ages to the Present. Music has slowed dramatically over the past millennium, and composers have repeatedly taken advantage of new resources. This pre-print of a short paper in honor of Joseph Connors documents this change and shows how it can affect how we think of nearly every piece written from 1100 to today. It concludes with an in-depth discussion of a piece from a particularly important moment in the history of notation, the lauda, O Regina from the manuscript Siena, Archivio di Stato, Fondo Vicariato di Gavorrano (1568-69), Ravi 3, giving the first transcription of this work with its unique notation. (.pdf 600k)

## 22 April 2010

### Managing with no idea about random fluctuations

Ah, Jerry Manuel, "manager" of the Mets, is up to it again. Now he's suggesting that he will bench or move Jason Bay or David Wright for batting .241 and .240 after 54 and 50 at bats, respectively. Of course batting .240 after an amazingly huge sample size of 50 at bats definitely proves that you're doing terribly, right? I mean it's utterly inconceivable that a .280 or .300 batter would go through a 50 at-bat period where they batted that low.

Okay, enough sarcasm, let's run the statistics for Jason Bay. Here's a little program in python than can do it for us.

------
`import randoml = 0for k in range(1000):    i = 0.0    for j in range(54):        if random.random() < .300:            i += 1.0    if (i/54) < .241:        l += 1print "%2.0f%%" % (l/10.0)`

------
What do we learn? It turns out that if Jason Bay were a pure .300 hitter -- that is he had a 30% chance of getting a hit on every at bat, there's a 22% chance that he'd have an average as low as .241 after 54 at bats. If he were a decent .275 hitter, he'd have 35% chance of having an average that low.

The numbers are about the same (or even a bit more disappointing) for Wright, which surprised me. I would've thought 4 extra at bats would far more than cancel out one point of BA, but it doesn't because the last digit isn't significant yet! With 50 at bats, it's impossible for Wright to be slightly better than Bay by batting, say .242, because each hit still represents a BA difference of .020! If Wright had one more hit, he'd be batting a respectable .260. If one or two fewer, he'd be at .220 or .200 and probably already traded by the kinds of short-sighted people who get paid millions to run things that they don't understand. (see: Management, Mets [or most non-winning teams]).

The point isn't that Bay and Wright are great hitters going through bad times. They could have morphed into terrible terrible hitters who are lucky to even be batting the .240 that they have. The point is that we just don't know yet. And given the number of millions you're paying them based on prior performance, maybe, just maybe, it makes sense to let them play long enough so we can find out what's actually happening?

Addendum, 3 hours later: I originally decided to leave out a criticism of the utter stupidity of messing with an approach that is causing Wright to be leading the league in walks (with a .457 OBP), but nah, they deserve all the criticism they can get. So there you are.

## 15 April 2010

### Changing Musical Time Over the Past 1000 Years

One of the fundamental changes in the West’s conception of music over the previous millennium was the idea that music’s rhythm and meter could be precisely measured not in terms of seconds or minutes, nor in borrowed notions such as poetic feet, but in purely musical terminology like quarter notes or dotted eighth notes.[2] This new way of thinking is intimately connected to the rise of musical notation, and in particular, the use of notation to show relationships among two or more coordinated parts. Pitch needed its own vocabulary even in the era of unwritten music—-organs needed to be built; lyres tuned—-but the necessity of specific terminology for musical rhythm arose only when it was written down. And while the notation of pitch became largely standardized within the first few centuries of the last millennium to the point where undergraduate music students can (more or less) read pitch from thirteenth-century manuscripts, rhythmic notation remained unsettled much longer.

Seemingly from the first moment when music became measured (musica mensurata), it began to slow down. The earliest measured music, that of Leonin and Perotin, the twelfth and early-thirteenth century composers of Notre Dame, used the figure of the “long” as a basic value, to be paired up and combined into a maxima or duplex long, or to be divided into three breves. By the late thirteenth century, music had slowed to such an extent that the breve became the basic note value, and the semibreve, worth either one-half or one-third the value of the breve, sprang into use.[2] This word, semibreve, is still the most common name in the British world for the note that in the States is called the whole note. Because the process of slowing continued unabated, this note is now the longest value in frequent use. By the fourteenth century, the tempo of music had slowed sufficiently that a note even faster than the semibreve needed to be introduced. This note came to be called the minima or minim--still the British word for half note—-signifying that this was to be the minimum, that is, the final, indivisible smallest possible note. (The debates of the music theorists of the Trecento are echoed in the discussions of modern physics that postulate that time itself may move in discrete and indivisible minima, each lasting about 1/200,000,000,­000,000,­000,000,­000,000,­000,000,­000,000,­000,000th of a second.) But the force of the slowing trend could not be stopped: it was less than fifty years later that the first “semiminim” would appear. Its name embodies a contradiction, “half of the shortest possible note,” yet though it was an extremely fast note for its time, it eventually became our most basic beat, the quarter note.

The process of slowing itself slowed slightly during the fifteenth, sixteenth, and seventeenth centuries. Faster notes continued to be introduced, but the fundamental tempos of music changed little. There are exceptions, and in some of these cases the slowness of the fundamental beat and the speed of the shortest notes produced dramatic effects: tuplet 128th notes appear in Beethoven’s third piano concerto, 256th notes in a concerto by Vivaldi (F. IV. n. 5), and most exceptionally, 1024th notes (incorrectly notated as 2048th notes) in the little known “Toccata Grande Cromatica” from The Sylviad by Anthony Phillip Heinrich (ca. 1825), a piece that in its own continual slowing of the fundamental beat is a microcosm of all of the notated music history of the West.[3] (See Example 1).

Example 1: Anthony Phillip Heinrich, “Toccata Grande Cromatica,” excerpt, with 512th and 1024th notes (incorrectly notated as 1024th and 2048th notes) at the end of the example.

Between Perotin and Heinrich, there exists a range of time that is almost incomprehensible to our sense of how music unfolds. Within a single maxima, there are 1024 128th notes, and, as we have seen, even longer and shorter notes have occasionally been used. Example 2 shows (on a logarithmic scale) the amount of time various notes would take if they were all played at the same adagio tempo of one quarter note per second. Obviously it is absurd to use the entire range of note values in a single piece at a single tempo mark. In fact the pieces using the shortest notes tend to have the slowest tempos, and the contrary is true for pieces with the longest notes. But Figure 1 demonstrates the enormous pull towards expanding the range of rhythmic resources that composers have felt over the centuries.

Figure 1: Lengths of different note values at M.M. qtr. = 60.
(with actual CD frequency and Ring lengths for reference)

In contemporary art music, composers have played with all extremes of lengths, but the most significant innovations in notation have come, as they did in the Middle Ages and Renaissance, in the notation of the most fleeting notes.[4] One of the most interesting case studies in the continuation of the medieval tradition of shorter notes comes in the American composer George Crumb’s string quartet Black Angels (1970), where, in the excerpt shown in Example 2, he uses a time signature of 7/128 in measure 2--written with the denominator as a note--along with the almost as equally unusual 7/64 with each note receiving an equal accent, thus providing the shortest non-compound meter ever published.

Example 2: Unusually short beat values in George Crumb’s Black Angels.

Other unusual and brief meters appear occasionally in modern music. Karlheinz Stockhausen’s 1956 composition Zeitmaße uses meters that are nearly as short as Crumb’s, such as 2/32. Composers allied with the New Complexity school, including Brian Ferneyhough and Thomas Adès, have used meters such as 5/6 and 2/10 that allow “tuplet” values (in these cases, triplets and quintuplets) to be fundamental and independent units. Finally, though multiple simultaneous meters were used in the works of both Bach and Mozart, the metrical experiments of Conlon Nancarrow use multiple meters in ways that particularly stretch the definition of meter. His “Transcendental” Etude contrasts two canonic lines whose rhythms are in the ratio of the transcendental irrational numbers e to π.

I am particularly interested in these unusual ways of specifying beat and meter in today’s music because the same spirit of experimentation was at work throughout the Italian ars nova. The feeling that the resources of the past were insufficient to express the creative metrical impulses of the present dominated musical thought in the late Trecento and early Quattrocento. Over the next few blog posts I hope to show how.

### Endnotes

[1] This post is adapted from part of an article "Changing Musical Time at
the Beginning of the Renaissance (and Today)" to be published by L. Olschki in an as-yet-unannounced surprise Festschrift in 2011.

[2] On attempts to measure the slowing of medieval musical time in terms of clock time, see Marco Gozzi, “New Light on Italian Trecento Notation, part 1”, Recercare 13 (2001), pp. 5–78.

[3] The source for these (and many other) extremes of musical notation is Donald Byrd’s excellent on-line resource. I thank him for many stimulating conversations on this topic. The score of Example 1 has been republished recently as Anthony Phillip Heinrich, The Sylviad, or, Minstrelsy of Nature in the Wilds of North America: opus 3, intr. J. Bunker Clark, Greenleaf, Wis. 1996. The term chromatic in the title of the piece refers not to notes out of the key, but to the amount of ink (or color) needed to express the short notes in the work.

[4] On recent uses of extremely long durations in recent music see Alex Rehding, “The Discovery of Slowness in Music,” forthcoming.

## 31 March 2010

### Cantus scriptus: Technologies of Medieval Song

 A two-day conference at University of Pennsylvania, November 19-20, 2010 will explore technology and medieval notation, both medieval notation as a technology in itself and new technologies used to study medieval musical manuscripts. I will be presenting on digital restorations, "bad" facsimiles (photos designed not to convey the look of the original, but instead facilitate further study), and how statistical models can help us estimate the number of French pieces of the late Middle Ages that once existed but now are lost.

## 26 December 2009

### Separating out Music Informatics topics

Hi all -- just wanted to mention that I've separated out the Music Informatics topics in a separate blog at http://music21-mit.blogspot.com/ . I suspect that this blog will mainly be about other musical and non-musical topics.

## 06 May 2009

### Put figures etc. in MS Word without the "Figure" tag

One of the annoying things (at least for me) about Microsoft Word is that it wants all your figure numbers to have "Figure" prepended to them. It's all great if you always say "Figure 1" or "Figure 2" but it doesn't work if you like to say things such as "As we see in Figures 1 & 2" and not "As we see in Figure 1 and Figure 2." Furthermore, Word doesn't work well with things such as "Figure 4.12" or, my favorite, having Figure 1 followed by Example 2 followed by Table 3 -- I'm big into continuous numbering.

So here's what I finally came up with as a solution. Type whatever preface you want to appear in the caption of the document ("Figure ") and then run this macro and give your figure a short name with no spaces ("voiceRangesDunstaple" for instance), then two numbers will appear. Leave the first alone; It'll put a number after "Figure" ("Figure 12" for instance). Copy and cut the second number and then paste it in your text somewhere -- it's the reference number to the figure ("See figure 12" -- I like lowercase figure references). You can paste this number multiple places if you need multiple references. And a quick "Ctrl-A, F9" will update all the tables.

You'll need to give this macro a keyboard shortcut if you use it often.

---
Sub MakeFieldCrossRef()
'
' Macro recorded 3/25/2006 by Myke
'
Dim Message, Title, Default, macroName
Message = "Enter a short, unique name for the Xref (no spaces!)"
Title = "Create Crossref"
Default = ""
macroName = InputBox(Message, Title, Default)
"AUTONUMLGL \e", PreserveFormatting:=False
Selection.MoveLeft Unit:=wdCharacter, Count:=1, Extend:=wdExtend
With ActiveDocument.Bookmarks
.DefaultSorting = wdSortByName
.ShowHidden = False
End With
Selection.MoveRight Unit:=wdCharacter, Count:=1
Selection.InsertCrossReference ReferenceType:="Bookmark", ReferenceKind:= _
IncludePosition:=False
Call UpdateAllDocFields
End Sub

## 29 December 2008

### Pachelbel lives on.

Great recent song by The Fratellis on their 2006 album Costello Music, "Ole Black 'N' Blue Eyes." The video seems not to have any connection to any plausible interpretation of the lyrics for me; I'd suggest ignoring it and just listening: (click here<)

The first thing that jumped out at me was how closely the progression of the song mirrored the Pachelbel Canon in D's ground bass (though transposed to B-flat):
(Not sure if the right hand part matches the actual guitar parts used in the piece, but they're close enough for illustrative purposes).

But better than seeing similarities, I thought a little remix of the opening minute or so would show just how compatible the two pieces were. Enjoy.

## 05 October 2008

### Overwrite IE Favorites with Firefox Bookmarks

I love Firefox, but on Windows IE has one big advantage over it: it's made by the same people who made the OS. This lets IE access its favorites from all sorts of places, not just from the browser. I'm a task-oriented person rather than application-oriented: I tend to think "Gotta pay the bills now" not "Gotta open up the browser and then do things that require a browser, such as bill pay." IE 4.0+ (and Windows98) was oriented towards people like me, by putting the Favorites option in the start menu, every folder, etc. It never really caught on, and in fact the Favorites option was quietly removed from the start menu in either XP or Vista, though it can be re-enabled from the taskbar options.

There are a couple of ways to get your Firefox bookmarks in the Favorites. There are lots of bookmark synchronization programs, though they tend to need to be run manually--which is messy for me. There's also the PlainOlFavorites plug-in for Firefox which lets you just use IE favorites instead of Bookmarks. I loved this plug in for a few years, but I like FF's native Bookmark system enough that I wanted to make my own synchronization system.

Here's a little Python script I came up with. It copies YESTERDAY's Firefox backup over your Favorites (copying the current version would just have been too messy). It requires Python 2.6 or you'll need to install the "simplejson" module and replace all references to "json" with "simplejson" in the code. Do not run this if you use IE favorites: it will DELETE THEM ALL and overwrite them with Firefox favorites. You can comment out the "nukeFavorites()" line if you just want to add to your existing IE favorites. I plop this script into my Startup folder and it works great!

`# bookmarks_json.py -- Michael Scott Cuthbertimport jsonimport copyimport sysimport osimport shutilclass BookmarkMaker(object):        startdir = os.environ["USERPROFILE"] + r'\Favorites'    internetShortcut = "[InternetShortcut]\nURL="    folderpath = os.environ["APPDATA"] + "\\Mozilla\\Firefox\\Profiles\\"    backuppath = r'\bookmarkbackups'        def nukeFavorites(self):        '''deletes everything in the Favorites dir -- do not use this if you update Favorites!'''        nukeem = os.listdir(self.startdir)        for thisfile in nukeem:            if thisfile != "desktop.ini":                thisFullFile = self.startdir + os.sep + thisfile                try:                    os.remove(thisFullFile)                except OSError:                    shutil.rmtree(thisFullFile)    def run(self):        stringFile = self.openRightFile()        self.parse(stringFile)        def openRightFile(self):        backupPath = self.backupPath()        theseFiles = os.listdir(backupPath);        theseFiles.sort(reverse=True)        for thisFile in theseFiles:            if thisFile.startswith('bookmarks-') and thisFile.endswith('.json'):                return self.openJson(backupPath + os.sep + thisFile)    def backupPath(self):        profile = self.mostRecentProfile(self.folderpath)        return profile + self.backuppath        def mostRecentProfile(self, profilesPath):        '''returns the most recently modified Profile in the profilesPath'''        newestName = ""        newestTime = 0                allDirs = os.listdir(profilesPath)        for thisDir in allDirs:            thisFullPath = profilesPath + os.sep + thisDir            try:                thisModTime = os.path.getmtime(thisFullPath + os.sep + "places.sqlite")                if thisModTime > newestTime:                    newestTime = thisModTime                    newestName = thisFullPath            except:                pass        return newestName    def openJson(self, jsonFile):        try:            myfile = open(jsonFile) #folderpath + '\\' + bookmarkfile)        except:            raise Exception("Could not open file " + jsonFile)        return myfile.read()      def parse(self, stringData):        self.jsonStuff = json.loads(stringData)        self.realBase = self.findRealBase(self.jsonStuff)        self.parseReally(self.realBase)            def findRealBase(self, searchIn):        return searchIn["children"][0]["children"]                def parseReally(self, startingPoint, currentDir = []):        for thing in startingPoint:            if thing.has_key("children"):  ## Bookmark folder                tempdir = copy.copy(currentDir)                tempdir.append(thing["title"])                self.parseReally(thing["children"], tempdir)            elif thing.has_key("uri"):  ## Bookmark                self.createFavorite(thing["uri"], os.sep.join(currentDir) + os.sep + thing["title"] + ".url")        def createFavorite(self, uri, filename):        shortcutOut = self.internetShortcut + uri        if not(filename.startswith(os.sep)):            filename = os.sep + filename        filename = self.startdir + filename        try:            self.recursiveMakeDir(filename)            filehandle = open(filename, "w")            filehandle.write(shortcutOut)        except:            pass    def recursiveMakeDir(self, filename):        if filename.count(os.sep) > 12:            raise Exception("Whoa, you sure you want to go so deep and make " + filename + "?")                    parentDir = os.path.dirname(filename)        if os.path.exists(parentDir):            return True        else:            self.recursiveMakeDir(parentDir)            os.mkdir(parentDir)if (__name__ == "__main__"):    bmm = BookmarkMaker()    bmm.nukeFavorites()    bmm.run()`

## 20 September 2008

### New mail script

I used to use this little program called "nfrm" (or frm -n) to tell me on log in to a unix system if I had new mail. Unfortunately what worked in the past no longer was keeping accurate lists in the world where I was also checking my mail via IMAP and the iPhone, where there's a lot of mail I've been informed (in some way) about, but have never actually read.

Here's a quick Perl script that I wrote which checks three folders (my inbox, a work-related box, and mailing lists) for new messages. It requires Mail::MboxParser (install via "cpan" then "install Mail::MboxParser" or have your sysadmin do it for you):
`#!/usr/bin/perluse Mail::MboxParser;use Term::ReadKey;use strict;my (\$wchar, \$hchar, \$wpixels, \$hpixels) = GetTerminalSize();my \$available_space = \$wchar - 13;my \$from_length    = int(\$available_space * .33);my \$subject_length = int(\$available_space * .67);my @mailboxes = @ARGV;if (scalar(@mailboxes) == 0) {   @mailboxes = (["**** ", \$ENV{'MAIL'}],                 ["NBER ", "/homes/nber/cuthbert/mail/nber"],                 ["Lists", "/homes/nber/cuthbert/mail/lists"]);}my \$parseropts = {        enable_cache    => 1,        enable_grep     => 1,        cache_file_name => '/tmp/msc-mbox-cache-file',      };foreach my \$box (@mailboxes) {  next unless -e \$box->[1];  my \$mb = Mail::MboxParser->new(\$box->[1],                                 decode     => 'ALL',                                 parseropts => \$parseropts);  while (my \$msg = \$mb->next_message) {    if (\$msg->header->{status} !~ /R/) {      my \$name = \$msg->from->{name};      if (!\$name) { \$name = \$msg->from->{email} }      \$name =~ s/\"//g;      \$name =~ s/^\s+//;      \$name = substr(\$name, 0, \$from_length);      my \$subject = \$msg->header->{subject};      \$subject = substr(\$subject, 0, \$subject_length);      printf("%-8s  %-\${from_length}s  %-\${subject_length}s\n", \$box->[0], \$name, \$subject);    }  }}`

## 14 September 2008

### Sexist ol' folks! (maybe not so much)

I was reading Matt Bai's most recent post on the presidential campaign; not as good as his usual missives, but still basically entertaining (and informative I didn't realize he was white; I guess I assumed Bai as an Asain name). But I focused mainly on a graphic that the New York Times placed in a sidebar showing the percentage of Americans who believe that a woman will be elected president during their lifetimes. Another statistical graphic without any exegesis; when will the Times learn? Based on the main gist of the article, I suppose we are to conclude that younger people are more open to the idea of a female president, and, therefore, more likely to conclude that a woman will be elected. Of the under-45 voters who expressed an opinion (95%), 17% believed that a woman would not be elected in their lifetimes. On the other hand, of the over-65 voters with an opinion (83%), 47% believed that no woman would be elected while they lived.

But the young have something more important than open-mindedness that accounts for the statistical difference: life expectancy. Simply put, the under 45 voting crowd will witness many more elections, and, therefore many more chances of electing a female president. Let's plot this out. Given a probability X of a woman being elected in any election, the probability that a woman would NOT be elected after N elections (or 4N years) is:
`(1-X)^N`
The following chart gives the probability that a woman would not be elected given various values for X (left column) and N (top row):

Let's say that an "Under 45 voter" is 35 (a mean of 18-45 weighed toward the fact that younger people vote in lower percentages) and let's call an "Over 65 voter" 70 years old. It's obvious that a woman will not be elected president this year, so we'll say that the 70 year olds have, on average 3.5 more elections in their lifetime where a woman could be elected (I'm giving a them more longevity than you'd expect since there is some probability that McCain will be elected and die in his first term, giving a Mrs. President Palin. Though this poll was taken in June, after Hillary had dropped out but before McCain's choice of a female VP seemed likely). The 35-year-old voter has, on average, 10 more elections (I'm giving them a few fewer years on average to live because a 35 year old hasn't proved they won't die before 70). And we'll define "likely to become president" the standard way: over 50% chance. That is, if the voter thinking that it was likely had to bet with even odds, he or she would bet for a woman president being elected.

What can we extrapolate from the chart and formula? If a (rational) 35-year-old voter said that he thought it likely for a woman to be elected, he would need to believe that a woman has over a 7% chance of being elected in any given election, while a 70-year-old voter needs to believe that there's an 18% chance of a woman being elected.

A quick-and-dirty way of extrapolating from the results of the poll to the table is to say if 25% of people believe X is likely, we should look at the point in the table where X is 25% likely to happen. (There are better ways depending on what distribution of beliefs you think voters have, but this should work well enough for now). So since 47% of over-65 voters don't believe a woman will be elected in the next 3.5 elections, we can suppose that they believe women have a 20% chance of being elected any given election.

What about the younger crowds? 17% of the under-45 crowd believe that after the next 10 elections we still won't have a woman president. That works out to belief in a 16% chance of a woman being elected president any given election. So if anything, the young are less optimistic of women's chances of breaking that ultimate glass ceiling than older voters are.

What CBS (the poll takers) should have done is removed a variable by asking, "Do you think it is likely that a woman will be elected president in the next 30 years?" so that all their respondents were thinking along the same time frame. The woman vs. men answer (an apparent +5% optimism gap toward women) is similarly skewed by age and probably underestimates women's optimism about the topic. Since women live longer, they probably disproportionately make up the (more optimistic) over-65 audience. In fact, we may be able to explain the entire age difference in this response by the gender differences and women's propensity to live longer.

(I realize now that I have used the word "optimism" throughout this article, assuming, dear readers, that like me you'd view the possibility of a female president as a good thing. If for some reason you disagree, please comment with your reasons. Oh and the graphic's title should have been Madam President. No one said that the female president needed to be married.)

## 06 August 2008

The most important stat for determining the quality a pitcher (at least over the course of a career) is ERA. There's really no argument that can be made against it. Wins are far too dependent on how the rest of the team (i.e., batters) do; strikeouts certainly show domination, and career strikeout numbers show longevity as well. But in terms of doing his job, preventing runs, career ERA is the single number that shows how well this was accomplished. (Okay, perhaps ERA over a 5-10 year peak period might be better, to not diminish the careers of those who choose to keep playing into their "worse than average but far better than replacement" years). The main problems with ERA are that it does not adjust for ballparks, leagues, mound heights, steroid usage of batters, or other factors that changed over the years.

Fortunately, the concept of adjusted ERA takes care of all this. And adjusted ERA+ makes reading adjusted ERA even easier. An aERA+ of 125 roughly means an ERA 25% lower than the league, adjusting for park effects etc. (okay, actually it's 20% lower; since an aERA+ of 200 means an ERA 50% of that of the league). An aERA+ of 125 also means you're likely to need to prepare a speech for Cooperstown. The vast majority of pitchers above 125 and almost all eligible starters above 130 are in the Hall.

For a long time, there was a big gap between aERA+ #1 and #2, with Pedro Martinez's 160 towering over Lefty Grove's 148. This disappointing season has lowered Pedro to a mere 157, but it would take more than a few years of (relative) mediocrity to drop him below Grove or Walter Johnson. (Next on the list, for those curious, is Dan Quisenberry which is probably the best Hall of Fame case for Quisenberry that could ever be made).

But why bring all this up now? Because despite Martinez's struggles, there's now (as of last week) a much bigger gap between #1 and #2:
`  1.  Mariano Rivera   197  2.  Pedro Martinez   157`

What happened? Rivera pitched his 1000th inning last week, making him eligible for career ERA awards. Perhaps the IP requirement should be raised so that the list isn't dominated by closers. But maybe that's not necessary. With the development of the minor-league (or even high school) relief specialist, fewer and fewer will reach this plateau with each passing year. Bochy and Bud Black's cautious use of Trevor Hoffman since he returned from surgery six years ago has kept him from breaking the 1000 IP mark. As of this evening, he's 22 IP short. Since he's been averaging .35 IP per game over the past four seasons, he's likely to end the season (and possibly his career? the fan in me hopes not) about five innings short. His aERA+ of 145 would place him 9th or 10th all time. Prior to this disastrous season, he would have been third.

(For the complete list, see Baseball-Reference.com)

The only other reliever on the ERA title horizon is Billy Wagner, whose aERA+ of 180 would place him comfortably in second, though still at a great distance below Rivera. However with only 819 IP, he will likely need three more seasons to qualify. And anything could happen by then. Troy Percival, with an aERA+ of 151 but only 687 IP, averaging fewer than 50IP/year at age 38, seems highly unlikely to qualify.

Of the other active high saves players, Roberto Hernandez already qualifies at 131; a Hall of Fame number for a starter, but unlikely to get a notice for a reliever with "only" 326 saves. Jose Mesa is what we always thought him to be, exactly an average pitcher (aERA+ of 100), and thus probably a below average reliever. Todd Jones is a little better, but nothing to write to Cooperstown about. Ditto Jason Isringhausen, 100IP short; though kudos to him for having such dreadful recent performances that his manager is bringing back the (great) idea of a closer by committee.

And as far as the brilliant young closers of today. Yes, very exciting. Yes, I know Papelbon has an aERA+ of 269(!); K-Rod at 186; Nathan at 152; Lidge at 147. But for an article on a stat that requires 1000 IP, please ring again when there's just a few hundred innings left to go. Why so pessimistic? Robb Nen, John Wettland, Randy Meyers, Tom Henke, Jeff Montgomery, Rod Beck, Ugueth Urbina, and (ah heck, throw in the newest member of the fraternity) Eric Gagne. If these names mean something to you, you'll understand.

## 04 August 2008

### Hate the stat, love the statter

I spent part of the last week of July reading the various, mostly extremely short, obits of Jerome Holtzman, sportswriter extraordinaire (and the Chicago cubs follower who did not incite the city of Sandy Eggo to riot by calling the selling of Sushi and the installation of baby-changing stations in Men's rooms at S.D.'s ballpark "the beginning of the decline of America." q.v., one M. Royko). Throughout all the obits, the discussion centered on Holtzman's invention of the "save" as baseball statistic.
(N.Y. Times obit.)

What puzzled me the most is, with the notable exception of The Hardball Times, how often commentators were blasting Holtzman for what a bad statistic the save is. To be sure, it is a bad statistic: it rewards (in the end, financially) players for participating in one facet of the game that has never been shown to be more significant than several others. It gives the same benefit to pitchers who bail their team out of the toughest of all situations as to closers who record a few outs with a pretty sizable lead. But consider again what the save replaced: a world where relief pitching had no worth on paper (and in a world where the "Win", another terrible stat, was even more important). Also consider what it meant in the early 60s to introduce a new stat into the world. Computing saves retroactively for every baseball player was not as simple as a few lines of Perl and a download from retrosheet. And, the tinkering that Holtzman and others applied to the save suggests that he did not believe he had discovered the perfect, never to be supplanted statistic. Admire what he did for the time in which he lived.

No, marshal your scorn for those who with the hindsight of time and better access to information still defend the save. Those who cling to a bad idea are far worse than those who throw the idea out to the marketplace in the first place.

P.S. I wish there were a stat with more sophistication than Ari Kaplan's "Fan Save Value", but with fewer lookup charts than his more accurate "Save Value." The former is not much of an improvement over the Save (and couldn't easily translate to a generalized "relief value") while the latter has too many "hidden" constants relating to expected runs (that aren't really constant, but actually change from year to year that I can't see it actually being adopted.

Here's a simple(r) formula for calculating expected runs, that you can carry around with you:
`ER = (5 + total_runners + 3 * (total bases occupied))             * outs_left / 30`

total bases occupied simply means to sum up the base numbers with runners. So runners on second and third is 2+3 = 5. Outs left is 3 - outs. So no outs = 3.

Here's how this version of expected runs compares to the standard expected runs chart:
`Outs 1B 2B 3B  My ERs   Table  Difference0    0  0  0    0.50    0.54   -0.040    0  0  1    1.50    1.46    0.040    0  1  0    1.20    1.17    0.030    0  1  1    2.20    2.14    0.060    1  0  0    0.90    0.93   -0.030    1  0  1    1.90    1.86    0.040    1  1  0    1.60    1.49    0.110    1  1  1    2.60    2.27    0.331    0  0  0    0.33    0.29    0.041    0  0  1    1.00    0.98    0.021    0  1  0    0.80    0.71    0.091    0  1  1    1.47    1.47    0.001    1  0  0    0.60    0.55    0.051    1  0  1    1.27    1.24    0.031    1  1  0    1.07    0.97    0.101    1  1  1    1.73    1.60    0.132    0  0  0    0.17    0.11    0.062    0  0  1    0.50    0.38    0.122    0  1  0    0.40    0.34    0.062    0  1  1    0.73    0.63    0.102    1  0  0    0.30    0.25    0.052    1  0  1    0.63    0.54    0.092    1  1  0    0.53    0.46    0.072    1  1  1    0.87    0.82    0.05  `

As you can see, the formula slightly over-predicts expected runs (though with the important exception of the most-common occurrence, no outs, no one on, which almost balances out the rest of the error). The only case where it's over 13 hundreths of a run off is the rare case of no outs, bases loaded, which it over-estimates by 1/3 of a run. If a formula is to overestimate any situation, I'm happy with it being this rare situation: an "oh shit!" moment for any incoming reliever. In any case, it's an easier formula to remember than 24 "random" numbers.

## 30 May 2008

### Are bullpens underused?

This post was mostly written during Spring Training, so 2007 figures are used throughout. Life prevented posting until now.

Back in the "good old days" of baseball, bullpens were nearly non-existent. Two-man rotations were common, 600 inning-pitched seasons were possible, and one man pitched every inning of an entire major league season (Wondering who? See below). Since then, we've created four-man rotations, dedicated bullpens, closers, five-man rotations, long relievers, setup men, and, coming soon to a ballpark near you, the seventh-inning specialist. And throughout all thus, grumpy old men--along with grumpy young men, grumpy old women, grumpy young women, and grumpy transexuals of all ages--have decried the changes as a weakening of the quality of starting pitching.

But is it possible that everyone is wrong? Could the problems with pitching be traced to an underuse of those 6 to 8 "guns" not considered durable enough to start a game? Let's consider some baseball axioms before we look further. I like math, so I'll use some weird symbols, but try to explain them.

Axiom 1: ∂(ERA)/∂(IP) > 0

That is just to say that we expect ERA to rise for every additional inning pitched. I'll admit that this axiom might not hold for a former AA-pitcher getting his first few innings in at the major league level, or for someone coming back just off the disabled list. But for the most part, it's hard to disagree with, whether we're dealing with innings pitched within a game or innings pitched over a season.

Axiom 2: ERA(closer) < ERA(ace) ; ERA(setup man) < ERA(#2 starter) ; etc...

The ERA of your relief squad tends to be lower than the starting pitchers. Or to put it in another way, if you only had to put your best pitcher in for one inning of work, he would almost certainly come from the bullpen. Compare the best reliever to the best starter on almost every club and you'll see that the best reliever comes out ahead. Usually even the best two or three relievers have better ERAs than aces--this holds true for great teams and miserable ones. In 2007, the Detroit tigers had three relievers log more than 40 innings with a lower ERA than their ace, Justin Verlander, and two more with a lower ERA than their second best pitcher to log at least 15 starts. The Padres (the team I know best): Even a triple-crown winning Cy Young pitcher (Peavy) had an ERA bested by a setup man, Heath Bell (with 93 2/3 IP, not a small sample); and four more relief pitchers (Hoffman, Brocail, K. Cameron, and Justin Hampson) bested Chris Young, their second best pitcher. We see the same patterns even for lousy teams: four Royals relievers outpitched Gil Meche's 3.67 ERA.

Compare the ERAs of bullpens as of mid-2007 to this analysis of starters at the end of the season. The average bullpen is about as good as the average no. 2 starter! In other words, it does not really matter if by the sixth inning your starter is still "feeling good." Unless he's your ace, or throwing a shutout or no hitter, or your bullpen is absolutely drained from a recent 18-inning game, it's time to call in some new arms. Do so and your expected chance of winning just went up.

All else being equal, every inning that you have someone on the mound with a higher ERA than someone else you could put out there is an inning of poor managing. All else being equal, bullpens should be used more until their ERAs rise to meet that of the starters.

Now here's the argument I'm ready to hear: all else is not equal. Not every inning is as important as every other. That's definitely true! The best relievers pitch in the most important situations: with the game close, and a win on the line if only he can not allow any runs. So it makes sense that you want some of your lowest ERA-men pitching then. But when do starters begin pitching? They begin with the game tied, where any run allowed or not makes a huge difference in the probability of winning or losing the game -- nearly as important a situation as what setup men and closers pitch in. But there are several members of the bullpen who tend to pitch in less important innings; that is, blowouts in either direction. So if anything, the ERAs of bullpens should be substantially higher than starters. That they are not, shows that the bullpens are being over-rested and under used.

Trivia answer: Jim Devlin of the 1877 Louisville Grays pitched every inning in a 61 game season, compiling 559 innings pitched and allowing a total of 4 HRs. Though his ERA was a very good 2.25 (146 ERA+), it says something about the way official scorers have changed over the years: though he allowed only 140 earned runs, he allowed a total of 288 runs, or 8 more unearned runs than earned.

(Before I'm accused of copying this trivia information from Wikipedia, be sure to check who added it there in the first place).