Added twelve-tone resources

contents-hidden.md

@@ -72,63 +72,73 @@ title: Table of Contents
 
 ## Form ##
 
-Thematic Structure in the Classical Style
+**Thematic Structure in the Classical Style**
 
-- [Introduction to Thematic Structure in the Classical Style.](thematicStructureInTheClassicalStyle.html) 
-- [The sentence.](sentence.html)  
-- [The period.](period.html)  
-- [Hybrid themes.](hybridThemes.html)  
-- [Compound themes.](compoundThemes.html)
+[Introduction to Thematic Structure in the Classical Style.](thematicStructureInTheClassicalStyle.html) 
+[The sentence.](sentence.html)  
+[The period.](period.html)  
+[Hybrid themes.](hybridThemes.html)  
+[Compound themes.](compoundThemes.html)
+[External Expansions.](externalExpansions.html)
+[Internal Expansions.](internalExpansions.html)
 
-- [External Expansions.](externalExpansions.html)
-- [Internal Expansions.](internalExpansions.html)
-
-[Minuet form.](minuet.html)
+[**Minuet form**.](minuet.html)
 
-Sonata Form
+**Sonata Form**
 
-- [Introduction to Sonata Theory.][SonataIntro]  
-- [Sonata form: exposition types.][SonataExpo]  
-	[Structural Points of Arrival.](sonataStructuralPointsOfArrival.html)
-	[Thematic Zones.](sonataThematicZones.html)
-- [Sonata form: the recapitulation.][SonataRecap]
-- [Sonata form: the development.](sonataDevelopment.html)
+[Introduction to Sonata Theory.][SonataIntro]  
+[Sonata form: exposition types.][SonataExpo]  
+- [Structural Points of Arrival.](sonataStructuralPointsOfArrival.html)
+- [Thematic Zones.](sonataThematicZones.html)
+[Sonata form: the recapitulation.][SonataRecap]
+[Sonata form: the development.](sonataDevelopment.html)
 
-Rondo Form
+**Rondo Form**
 
-- [Introduction to Rondo form.](rondo.html)
-- [Thematic Function in Rondo Form.](thematicFunctionInRondo.html)
-- [Five-Part Rondo.](fivePartRondo.html)
-- [Sonata Rondo.](sonataRondo.html)
+[Introduction to Rondo form.](rondo.html)
+[Thematic Function in Rondo Form.](thematicFunctionInRondo.html)
+[Five-Part Rondo.](fivePartRondo.html)
+[Sonata Rondo.](sonataRondo.html)
 
-[Form in pop/rock music.][popRockForm]
+[**Form in pop/rock music.**][popRockForm]
 
 [Classical theme types.][classicalThemes] 
 [Introduction to musical functions.](functions.html)
 [Galant schemata.][Schemata]  
 
 ## Post-tonal music ##
 
-Basics
+**Basics**
+
+[Pitch (class).](pitch(Class).html)
+[Interval (class).](interval(Class).html)
+[Modular 12 arithmetic.](mod12.html)
+
+**Organizing Forces**
 
-[Pitch (class).](pitch(Class))
-[Interval (class).](interval(Class))
-[Modular 12 arithmetic](mod12)
+[Collections and Scales.](scales.html)
+[Symmetry and Centricity.](symmetryAndCentricity.html)
 
-Organizing forces
+**Set Theory**
 
-[Collections and Scales](scales)
-[Symmetry and Centricity](symmetryAndCentricity)
+Pitch-Class Sets.
+[Normal Order.](normalOrder.html)
+[Tranposition.](transposition.html)
+[Inversion.](inversion.html)
+[Set Class and Prime Form (1).](setClassAndPrimeForm1.html)
+[Set Class and Prime Form (2).](setClassAndPrimeForm2.html)
+[Complements.](complements.html)
+[Common Tones under Transposition.](commonTonesUnderTransposition.html)
+Common Tones under Inversion.
+Tranpositional Symmetry.
 
-Set Theory
+**Twelve-Tone Theory**
+[Basics.](twelveToneBasics.html)
+[Operations.](twelveToneOperations.html)
+[Intervallic Structure.](twelveToneIntervallicStructure.html)
+[Derivation.](twelveToneMusicDerivation.html)
+[Invariance.](twelveToneMusicInvariance.html)
 
-[Normal Order](normalOrder)
-[Tranposition](transposition)
-[Inversion](inversion)
-[Set Class and Prime Form (1)](setClassAndPrimeForm1)
-[Set Class and Prime Form (2)](setClassAndPrimeForm2)
-[Complements](complements)
-[Common Tones under Transposition](commonTonesUnderTransposition)
 
 [Analyzing atonal music.][atonal]  
 [Analyzing 12-tone music.][twelveTone]  

internalExpansions.md

@@ -3,9 +3,6 @@ layout: post
 title: Internal Expansions
 ---
 
-{{ page.title }}
-================
-
 For historical reasons, and perhaps biological ones as well, the length of sub-phrases and phrases are generally duple. But plenty of classical music displays irregular lengths. This topic has been the subject of a great deal of musical theoretical exploration, for more than two centuries. Those interested will find a great deal more about *expansion* and *compression* in writings by William Rothstein, William Caplin, Janet Schmalfeldt, and Dinuta Mirka, among many others. 
 
 Here, we will concentrate on a simple, common type of expansion. These expansions occur when a sub-phrase or phrase is made larger through simple repetition *within itself.* Notably, irregular sub-phrase and phrase lenghts are not always the result of expansion. When in doubt, try to recompose the passage in question without destroying its melodic, harmonic, and motivic structure. If you can, there is a case to be made that an *expansion* is present.

twelveToneBasics.md

@@ -0,0 +1,30 @@
+---
+layout: post
+title: Twelve-Tone Theory — Basics
+---
+
+Twelve-tone music is most often associated with a compositional technique, or style, called *serialism*. The terms are not equivalent, however. *Serialism* a broad designator referring to the *ordering* of things, whether they are pitches, durations, dynamics, and so on. Twelve-tone composition refers more specifically to music based on orderings of the *twelve pitch classes*. 
+
+This style of composition is most associated with a group of composers whose figurehead was Arnold Schoenberg and which also included the influential composers Anton Webern and Alban Berg. But twelve-tone compositional techniques and ideas associated with such techniques were very influential for many great composers, and serial and twelve-tone music is still being written today. Much of this music shares similar axioms, outlined below, but composers have used these basic ideas to cultivate entirely original approaches.
+
+Twelve-tone music is based on _series_ (sometimes called a _row_) that contains all twelve pitch classes in a particular order. There is no one series used for all twelve-tone music; most composers write a unique row for each piece. (There 12!—that is, 12 factorial—twelve-tone series, which is equal to 479,001,600 unique row forms. Quite a lot of possibilities!) Here's an example, the row for Webern's Piano Variations, Op. 27:
+
+[![](Graphics/postTonal/basicRow.png)](Graphics/form/basicRow.png)
+
+There are some general rules for using a twelve-tone row, though as I said, individual approaches are always different:
+
+	1. Pitch classes are played in order; 
+	2. Once a pitch class has been played, it isn't repeated until the next row. 
+
+A twelve-tone row might be used as a theme or as a source for motives. Chords might be derived from the row, or the row may be used for both thematic and harmonic purposes. We call the basic ordering, shown above for Op. 27, the *prime form* (P). And because it begins on B (pitch class 11), we label it **P11**.
+
+Rows can be transposed, inverted, retrograded, or any combination of those operations. Inverting the prime form results in an "I-form." Like P-forms I-forms are labeled by their first pitch-class. Hence, the row below, an inversion of the one above, is called **I0**. Note that it starts on C (0).
+
+
+[![](Graphics/postTonal/inversion.png)](Graphics/form/inversion.png)
+
+Prime forms and inversion forms can be also be played backwards, also called retrograde. In the example below notice how this work in relation to the P11 and I0 rows from above. When a P-form is retograded, we call it a "R-form." When an I-form is retrograded, it's called an "RI-form." As the example shows, R- and RI-forms are labeled according to their _last pitch class_.
+
+[![](Graphics/postTonal/family.png)](Graphics/form/family.png)
+
+That graphic shows only four row forms, but each of those forms has twelve transpositions. Thus, a single row breeds a total of 48 rows: 12 *4.  That collection of rows is called a *row class*, and it is the *row class* that the composer draws from when writing his or her music. 

twelveToneIntervallicStructure.md

@@ -0,0 +1,16 @@
+---
+layout: post
+title: Twelve-Tone Music — Intervallic Structure
+---
+
+Pitch-class orderings are not the only things ordered by twelve-tone rows. Because pitch classes are always in relationships with one another, a twelve-tone row is also an ordered collection of *intervals.* Understanding the intervallic structure of a row class is the best way to get a sense of what it will sound like.    
+
+Below, you'll see the figure from resource on [operations](twelveToneOperations.html). Below each of the row forms in that example, I have shown the series of [ordered pitch-class intervals](interval(Class).html).
+
+Rows that are **transpositionally-related** (as **P11** and **P10** are) have _the same_ series of ordered pitch-class intervals.
+
+Rows that are **inversionally-related** (as **P10** and **I0** are) have *complementary* ordered pitch-class intervals. That is, intervals in corresponding locations in the row forms "sum to 12."
+
+Rows that are **retrograde-related** have ordered pitch-class intervals that are _reverse complements_. Compare **P10** and **R10**. Reading **R10** backwards, the *final* three intervals (for example) are 4 1 8. Those are the complements of **P10**′s *first* three intervals: 8 11 4.
+
+Rows that are **retrograde-inversion related** have ordered pitch-class intervals that are _reverses_ of one another. Compare **P10** and **RI0**. Reading **RI0**′s intervals backwards, you'll notice that they are the _same_ as **P10**′s read forwards.

twelveToneMusicDerivation.md

@@ -0,0 +1,12 @@
+---
+layout: post
+title: Twelve-Tone Music — Derived Rows
+---
+
+_Derived rows_ are rows whose non-overlapping segments (*discrete* segments) belong to the same set class. Because they must not overlap, discrete subsets divide the row into either six 2-note segments, four 3-note segments, three 4-note segments, or two 6-note segments. (In general, we are mostly concerned with rows that are trichordally- or tetrachordally-derived.)
+
+An example is offered below. This row (from Webern's String Quartet, Op. 28) has been divided into discrete tetrachords and discrete dyads. All of the tetrachords belong to the set class (0123). Thus, we say that the row is tetrachordally derived, and that it is *generated* by (0123). Further, the discrete dyads indicated an interesting dyadic derivation, by (01).
+
+[![](Graphics/postTonal/derivedRow.png)](Graphics/postTonal/derivedRow.png)
+
+From a compositional and listener-oriented perspective, derived rows are very suggestive. Because the set-class content of a row doesn't change when it's transposed, inverted, etc., these set classes will circulate constantly throughout a piece, even if different row forms are used. Therefore, a derived row guarantees the regular recurrence of a very small selection of set class—thus ensuring a particular type of unity throughout a piece.

twelveToneMusicInvariance.md

@@ -0,0 +1,26 @@
+---
+layout: post
+title: Twelve-Tone Music — Invariance
+---
+
+*Invariance* refers to the preservation of something: intervals, dynamics, rhythms, pitches, and so on. In elementary twelve-tone theory, we are mostly concerned with *intervallic* invariance and *pitch class segmental* invariance.
+
+**Intervallic Invariance**
+
+Any time a row is [transposed](twelveToneIntervallicStructure.html), the ordered intervallic content of the row is unchanged. Thus, transposition always results in intervallic invariance. [Retrograde inversion ](twelveToneIntervallicStructure.html)creates retrograde intervallic invariance.
+
+**Segmental Invariance**
+
+When a pitch-class segment of a row is unchanged when that row is transformed, we say that the segment is "held invariant." Consider the following example, from Webern's String Quartet, Op. 28:
+
+[![](Graphics/postTonal/invariance.png)](Graphics/postTonal/invariance.png)
+
+The brackets show the [discrete tetrachords](twelveToneMusicDerivation.html) of the row. Notice that these tetrachords are the *same* amongst the to different rows. That is, the tetrachords are *invariant segments.* These segments are held invariant because of they share the same *relationship* with one another that the rows share. Because the tetrachords are related by _T8_, when the row _as a whole_ is transposed by _T8_, those tetrachords are "held invariant." (Think of the process like this: when the first tetrachord [6789] is transposed by _T8_, it becomes the last tetrachord [2345]. And therefore, when the whole row is transposed by _T8, _the last tetrachord _becomes_ the first tetrachord.)
+
+To determine when and if a pitch-class segment of a row will be held invariant:
+
+(1) Find an equivalent set-class elsewhere in the row. This may be a dyad, trichord, tetrachord, etc.
+
+(2) Determine the transpositional or inversional relationship between them.
+
+(3) When the row is transposed or inverted by that *same* relationship a segment will be held invariant.

twelveToneOperations.md

@@ -0,0 +1,21 @@
+---
+layout: post
+title: Twelve-Tone Music — Operations
+---
+
+Like pitch-class sets, twelve-tone rows can be transposed (*Tn*),  inverted (*I*), or transposed and inverted (*TnI*). Like [transposing a pitch-class set](transposition.html), transposing a row is accomplished by *adding* a constant value to all of the pitch-classes of the row *while maintaining the order*. In the example below, I have transposed **P11** by *T11* by adding 11 to each of the pitch classes of **P11**. The new row is called **P10** because it begins with pitch class 10. 
+
+Inversion occurs when we *subtract* each pitch class of the row from a constant value. Again referring to the example below, when I do *T10I* of **P11** is accomplished by subtracting every pitch class of **P11** from 10.
+
+Twelve-tone rows can be _retrograded_ as well, symbolized as *R*. To retrograde a row we read it backwards. Reading **P11** backwards results in the row form shown below **P11** in the example: R11. (Remember that retrograde rows are labeld according to their *final* pitch class.) Reading **I0** backwards results in **RI0**—just below it in the example.
+
+[![](Graphics/postTonal/operations.png)](Graphics/postTonal/operations.png)
+
+It's important that you know the difference between *operations* and row forms, because they are often labeled similarly. In these resources *operations* like transposition and inversion will always be italicized. ***Row forms*** will be bolded.
+
+You need to memorize the effect of transposing, inverting, or retrograding any particular type of row. That is, you should know what kind of ***row form*** results when you perform any *operation* on it. For example, if you *transpose* a **P-form,** what kind of row form results? What about when you retrograde an **RI-form**? The flowchart below will be helpful:
+
+[![](Graphics/postTonal/abstractedRowClass.png)](Graphics/postTonal/abstractedRowClass.png)
+
+
+