Happenings

1. Mixing with Mike course: I am currently going through the Advanced Mixing Course. The current focus is on compression, what the different parameters do (knee, threshold, attack, release, etc….) and how compression controls adsr (attack, decay, sustain, release) and how to manipulate this primarily through the attack and release settings along with the compression ratio and knee. Gaining a better understanding of knee has been really helpful. In this, the current focus is on learning to hear the effects that compression has when it is applied, the emphasis being on how subtle it is. My understanding around the inner workings of compression has become much more clear and so I am confident that I will be able to apply compression more competently in my mixes going forward.
2. Reason Operation Manual: I am gearing up to start working on the new record sometime in February, and I will be producing it in Reason. I wanted to go through the whole manual up to the end of the discussion of the main daw, but I might need to return to finish. So far I’m 8 chapters in and have learned so much. My process involves opening the DAW and cueing up one of my projects so I can click around in the DAW while I’m reading. The current focus is ‘Audio editing in the sequencer’. A big takeaway from all of this is the way in which audio files can be edited in the comp editor to cobble together a master take from multiple loop recordings. The techniques are fairly sophisticated.
3. Online Music School: Work continues. I am adding a little bit everyday. The inspiration is there to try and actually create something viable in terms of a music education platform centered around fundamental Western theory. Plans to expand are simmering in the background.
4. Counterpoint: Working through fifth species in two voices over cantus firmi constructed using the six modes. Looking forward to the next steps, which might entail setting texts for three or more voices. I intend to absorb actual compositional technique from the Renaissance masters for use in my own music.

Monday January 29th, 2024

Intentions

1. Finishing the first half or so of the Reason manual. Looking to be done asap; ideally to start in on new record mid-February
2. Finishing up mixingwithmike advanced mixing course for starting work on the new record
3. Continuing to chip away at beta/template version of online theory course on website

Happenings

Working through the Reason manual, on the editing audio in the comp editor, Need to start plowing through more aggressively to get through to where I need to be.

The current videos I am watching in the mixingwithmike series are amazing. The inner workings of compression are becoming a lot more clear to me now.

Making good progress with fifth species counterpoint. Really happy to be coming to the end of species in two voices

Theory:

The Study Of Counterpoint(https://tinyurl.com/bp6jhnkc)

Chapter Three: Third Species Counterpoint

Continuinig on from the commencement of third species counterpoint with its 4:1 rhythmic texture, today I constructed two counterpoints on a Mixolydian cantus firmus. Here are the examples. To get the most out of the experience, play the uploaded mp3 while reading along:

Electronic Music Production:

Now I am getting back on track, and things are moving more quickly. Today I started layering tracks. I am currently focusing on tracks using VCV Rack2:

From here I started inputting the notes using the piano roll. I will be doing a fair amount of this VCV Rack2 with Piano roll work:

I have been trying to practice and learn the various parts of the score, but I have realized that with the need to start making progress, using the piano roll will be mandatory. The sacrifice of a more natural feel for expedience is the primary thing.

Theory:

The Study of Counterpoint (https://tinyurl.com/4nzasujx)

Third Species

What was going to be a few days off turned into a couple of weeks, but things are coming back online. Continuing with my theory studies I will be focusing on counterpoint. Continuing on with this, we are looking at third species counterpoint, which has a rhythmic relattionship of four quarter notes against each whole note, or 4:1. Whereas second species deals with half notes against whole notes (2:1 rhythmic ratio). After that introduction, here are the third species examples provided as samples. To get the most out of the experience, play the mp3 at the bottom of this post while reading through the examples:

Electronic Music Composition:

Getting back into it, I listened to the current track a few times, tried playing through the notation for the next instrument to record, and decided to use the piano roll:

I am thinking to use the piano roll more extensively, if only to help expedite finishing.

Electronic Music Production:

I recently completed a fairly major goal and decided to take a few days off. Getting back into music creation mode after a break is always a little bit jarring. Once I adjusted to things and started getting settled in, things started flowing. Not my best session but the first in this new cycle, so we should be getting into a groove soon. 

Theory:

The Complete Musician(https://tinyurl.com/5b6prtrx)

Chapter 32 of 32: Metrical and Serial Techniques

At long last we come to the last chapter of the text!!!!! We have come to the end of the book. It will be interesting to scroll back and find the very first blog post about this task of summarizing ‘The Complete Musician’ by Steven J. Laitz. We might have more to say after the chapter is finished and the book complete. Let's be on our way. To The End!!!

While tonality is skewered by symmetry, meter is actually clearer when it is performed symmetrically. The first part of this chapter will look at asymmetrical rhythmic techniques that forgo musical regularity.

After this we will pic up the thread of atonal pitch structure from Chapter 31, but gaze at batches of pitch classes - including the aggregate of al twelve pitch classes - including the aggregate of all twelve pitch classes - in a different way: as ordered series rather than as unordered sets. How can a composer manipulate a consistently ordered row of pitch classes, and how might we approach the resulting music analytically?

Metrical Irregularity

Changing Meters

This refers to time signatures changing regularly. This results in phrasing that never settles into a groove, that requires serious concentration and commitment to follow, but which can lead to some wild results.  Different measure lengths, different beat emphases, everything is much more heightened in this state. 

Meters that Accommodate Rhythms (as Opposed to Vice Versa)

Changing meters can rebalance the relationship between meter and local rhythms. Rather than having a rigidly defined time grid superimposed on the music, the relationship between the various musical cues dictates the measure length, allow for organic phrase development and interesting notions for full-scale formal development. With the flexibility to change meters rapidly, there is an increase in rhythmic fluidity in rhythmic design in period and sentence construction, up to full forms. Everything ends in full forms. Olivier Messiaen employed the technique of additive rhythm to add up to a full extra beat into an already existing time signature, which would result in an almost-regular time signature. 

Perceived Vs. Notated Meter

Another way to increase irregularity in meter is to build in a discrepancy between the notated meter vs. the perceived meter.

Polymeter

In polymeter, the pulses in two (or more) parts of a musical texture do not line up because a single span of time is divided into different numbers of pulses in the two (or more layers). We name polymeters according to the ratio between the two conflicting parts, 2:3 polymeter, 7:3 polymeter, etc….

When a polymeter features a conflict at the beat level or the beat grouping level, it is often more disruptve than a conflict at the division level.

Twelve-Tone Music

The atonal music studied in chapter 31 used pitch classes vertically to produce harmonies and horizontally to produce melodies. The nature of a set is that we group pitch classes together without focusing ont he specific order in which they appear. 

A SERIES is different from a set because, rather than lumping pitch classses together as a single unit, it arranges them in a particular order. Angle brackets indicate an ordered series, for example <27t> indicates D, then G, then Bb.

When the concern for the ordering of pitch classes encompasses all twelve pitch classes - the aggregate - the music is called twelve-tone music, and one specific ordering of the twelve pitch classes is called a twelve-tone row (or just a row). A piece's row is a source of both melody and harmony. A composer chooses whether the twelve pitch classes sound one at a time, simultaneously, or in some mix of ways. In order to find one's way doing analysis of atonal pieces, one tool to help with orienting onself in the atonal music being analyzed, look for any unharmonized, single note lines, that present notes one at a time. That will go a long way towards helping you determine the home row. 

When we arrive at a series of twelve pitch classes without any duplications, we have found the piece's row. Immediately we sign the order numbers 1 through 12 to the row's pitch classes and we also label the OPCI's between each adjacent pair of pitch classes. After that we can search the row for any type of scalar or harmonic structure we can conjure up using the ordered arrangement of our row as the raw material we are working with. 

Transforming a Row: From One Row to 48 Rows

A single 12-tone row can be transformed to create any member of a family of 48 related rows called a row class. This involves using what are called ‘twelve-tone operations’. Of interesting note, there are 12 PRIME rows that can be generated by transposing the main row. 

There are also 12 inverted rows that can be made by inverting the main row at each index number. The inversion of a row takes place in pitch-class space, so an inverted row need not actually descend in pitch space where the main row ascended, and vice versa.

There are twelve retrograde rows that simply present each of the twelve prime rows in reverse order. We label a retrograde row according to its last pitch class, which is the first pitch class of the prime form that it states in reverse. The OPCI's of a retrograde row form are the mod-12 inverses of a prime form in reverse order.

Finally, there are twelve retrograde-inversions  that simply present each of the 12 inverted rows in reverse order. We also label this row with it's last pitch class, which is the first pitch of the inverted form that it states in reverse. The OPCI's of a retrograde inversion row form can be found by reading the OPCI's of a prime form in reverse order. 

The Twelve Tone Matrix

This enables us to visualize all 48 rows all at once. The end result is a big table called the matrix, wherein we can:

• read from left to right to show the prime forms of the row, named by their loeftmost pitch classes
• read from top to bottom to reveal the inverted forms of the row, name by its uppermost pitch classes
• read from left to right to provide the retrograde from of the row, named by it's leftmost pitch class.
• read from bottom to top to get the retrograde inversion forms of the row named by their uppermost pitch class

With that, my summary study of chapters 5-32 of this text, The Complete Musician, by Steven J. Laitz. Things were grueling as we got closer to the end, but it's done. We have achieved the goal. I will continue to peruse the book to deepen what I have learned, but it will be in short spontaneous bursts, as needed as much as for continued interest. 

So now, onwards to working on composing music in a couple different mediums, strict composition and electronic music. I feel like I have come to the end of a musical journey. There is more to learn, but now I have re-established my base. From here I plan on being as prolific as I can be for a while, to build up my recorded output and my profile. Here's to it. Cheers.

Theory:

The Complete Musician(https://tinyurl.com/5b6prtrx)

Chapter 31 of 32: Analysis with Sets

Building on our brief study of Chapter 30, we looked at sets. This chapter builds on that to begin talking about analyzing non-tonal music that cannot be easily handled through other means.

Pitch Space and Pitch Class Space

Pitch space: each pitch's register is important.

pitch-class space: There are just the twelve pitch classes regardless of octave transpositiong.

Octave equivalence: the understanding that the basic pitch of a tone doesn't change regardless of octave

Enharmonic equivalence: the understanding that any single pitch can have several different names, i.e. C = Dbb = B#. All three spellings refer to the same sounding pitch. 

Combining all these concepts together enable us to represent the sum total musical space with the integers 0-11.

Intervals in Pitch Space

In this system, there is a concept known as an ‘ordered pitch interval’ or ‘OPI’. This measures both the distance and the direction from one pitch to another pitch. It also has an integer sign to show if the distance is in ascending motion or descending one, i.e. a plus sign for ascent and a minus sign for descent. So the OPI would look something like this: +4. This would mean ‘this note is four half steps higher than the other note.’

There is another concept known as the ‘unordered pitch interval', which measures just distance. This is helpful when pitches sound simultaneously, or when we're more interested in the distance between them and not the ordering. 

Intervals in Pitch-Class Space

Unlike pitch-space, pitch-class space has no ‘up’ or ‘down’, because pitch classes have no register. Distance is measured using something called a pitch-class clock.

Ordered pitch-class interval (or OPCE): restricted to clock wise motion. ‘How many hours (or rising semitones in pitch-class space) elapse from the first pitch class to the second pitch class?’. Because 12 hours (clock face) pitches ‘travel’ from 0 to 11.

Unordered pitch-c.ass interval (or UPCI): allows both clockwise and counterclockwise motion and take the ‘shortest path’. Allowing for both directions of travel, no two spots on the are more than 6 hours away, as UPCI's range from 0 to 6. Another name for an unordered pitch-class interval is an Interval class (IC).

Pitch-Class Sets

Pitch-Class Set: a group of two or more pitch classes. The cardinality of the set is the unique number of members it contains. Two-member set: dyad, three-member: trichord; four-member = tetrachord; five-member = pentachord; six-member = hexachord. Sets larger than this exist, but are less common. The creation of a pitch class is referred to as segmentation. From here these is the need to utilize theoretical tools to allow us to recognize these sets.

Arranging a Pitch-Class set in normal order

normal: the pitch class compressed into its most compact arrangement in order to facilitate comparisons between sets. To find the normal order, construct the pitch class on an interval clock with arrs that can encompass all of its pitch classes. Whichever is shortest shows the normal order. The pitchclass is then notated in clockwise order as they appear within that shortest arc. 

Sets Related by Transposition

There are different types of transposition techniques to create sets closely related to the original but not exact.

Inversions in Pitch Space

Transposed sets retain the same interval structure, every note in the set moving up or down the same number of half steps. Inversion does something different: it reflects the set around some imagined line. The imagined line is referred to as the axis of inversion. Of the line is a specific pitch class, so the inversions are built on the understanding of ‘how many half steps’ in either direction. When reflected in this way, the type of inversion is called ‘mirror reflection’. 

Inversion in Pitch-Class Space

 When inversion takes place in pitch-class spacer rather than in pitch space, the reflection takes place within the pitch-class clock. Referencing this, it will become apparent that each pair of pitch classes adds up to the same number mod-12. This sum is how the inversion is labelled. 

What is Invariance?

Using transposition and inversion, most pitch-class sets can be transformed into 23 different pitch-class sets without any exact duplicates resulting in 24 unique sets. Many sets have fewer than 12 distinct transpositions and/or fewer than 12 distinct inversions. This is where invariance plays a role. The meaning of invariance is that any given pitches in a set will become members of other sets.

When inverting a set produces no pitch-class sets beyond what can be produced by transposing the set, it is because the set is inversionally symmetrical. In other words, these sets look the same when read forward or backwards. Invariance is helpful to composers, since it permits varying degrees of pitch-class similarity when a set is transformed.

Interval-Class Vector

This is a way of determining how many pitches will be invariant. It summarizes how may times each interval class occurs among the dyads within a set. To construct an IC vector, locate each pair of pitch classes that can be found within the set. Trichords have 3 different dyads, tetrachords 6, pentachords 10, and hexachords 15 different dyads. We determine which interval class each dyad belongs to, then tally instances of each of the 6 interval classes in order, enclosed within angle brackets without any commas.

Combining Transposition and Inversion: Set Class and Prime Form

Unlike a normal order, a prime form will no longer captuer the specific pitch clases or our original set. Instead, it will represent a whole group of 24 pitch-class set. Before finding a set's prime form, we need it in normal form. The prime form is the form with the smallest second digit. The prime form is also called the set class.

Theory:

The Complete Musician(https://tinyurl.com/3an5zcen)

Chapter 30 of 32: Centricity, Extended and Non-Tertian Sonorities, and Collections

The discussion in this chapter is focused on post-tonal music. Basically, as the 19th century drew to a close, the main underpinnings of tonality had been wrenched loose from their moorings, and composers, in their constant search for new ways of organizing sound, started pushing past the strictures imposed by traditional tonal expression, and ended up creating the body of music that has come to be called post-tonal. For this chapter, the focus is on two basic techniques of post-tonal organization: centricism and atonal music. 

There are three questions posed at the beginning of this chapter that sets the stage for the rest of the chapter:

1. How can centric music emphasize a particular pitch class and create a sense of closure, even in the absence of tonality
2. What are some sonoroties, including ones constructed of dissonant intervals rather than just htirds, that play roles in post-tonal music?
3. What are some pitch-class collections other than major and minor scales that can serve as color palettes for the pitch content of post-tonal music?

What is centricity?

In centric music, a pich class - called the centric pitch class, or just the center - stands out as the primary one, and even the most stable one, but it is not a tonic, because there are not functional progressions or tonal cadences. Other factors come into play here, such as repetition, symmetry, register, and/or gesture serve to highlight the centric pitch class.

One important technique in centric music for emphasizing a specific pitch class is ostinato. Ostinato is basically continuous repetition. This approach is especially effective where there is no conclusive harmonic motion to give formal coherence to the unfolding-in-time.

Achieving Closure in Centric Music

Centric music does not feature the same cadential formulas of tonal music. There are other means for creating this sense of closure, other techniques that composers have at their disposal. Through artful manipulation of rhythm, pitch meter, and any other parameters that may be open to articulation, the sense of tonal resolution can be emulated. 

Adding Colors to Teritan Chords

It is a feature of some twentieth-century music that tertian chords extend beyond seventh chords now, to include ninths, elevenths, and thirteenths.

Added Sixths

These are not the same as thirteenths. This refers to a chord that is basically a triad with the sixth added, so CMAJ6 would be C-e-G-a, C Major Sixth. The sixth is always major, bu the way. 

Adding Sharp-Ninth Chords

The sharp nine is often represented enharmonically as a minor third. THinking of this as a chord with a ‘split’ third, meaning both major and minor thirds are available.

Non-Tertian Sonorities: Stacking by Fourths and Fifths

Quartal Harmony: Stacking chords by 4ths

Quintal Harmony: Stacking chords by 5ths

Collections:

A collection is a musical color palette, a palette of pitches which is a repository of pitch classes from which a composer can draw any number of melodic and harmonic shapes. They can be any size and contain any number of pitches. 

The largest collection is the collection of all twelve half steps which is called the chromatic collection or the aggregate.

Seven-Note Collections: Diatonic Modes

Major and natural minor scales are examples of diatonic collections. Diatonic collections:

• contains seven pitch classes
• It does not divide the octave evenly
• each pitch class has one letter name
• Contains five whole steps and two half steps when arranged as a scale
• are referred to by their key signatures i.e.: Eb/c = ‘3-flat collection’
• each diatonic collection contains seven diatonic modes

A Five-Note Collection: Pentatonic

The pentatonic collection contains five pitches. The one focused on in this chapter takes the diatonic collection and removes the two pitches that participate in the tritone. This collection contains no half-steps; adjacent ptiches are separated by either a major second or a  minor third. It is also asymmetrical. No two diatonic collections are identical

Six-Note Collections:

• whole-tone collection: partitions octave symmetrically; there are two whole-cone collections, each consisting of six tones; it cannot be rotated because it has not half-steps; it is not possible to construct an odd-numbered interval; the huge degree of symmetry in the whole-tone scale makes it difficult for any single pitch to stand out as a center
• the hexatonic collection: there are only four distinct augmented triads; are generally created by combining two augmented triads that are either a whole of half-step apart; there are four different pairs; this is called hexatonic; each collection contains three different major triads and three different minor triads

An Eight-Note Collection: Octatonic:

Now that we have looked at building two kinds of six-note collections by combining pairs of augmented triads. Now we can use the same concept to create octatonic (8 note) collections by combining pairs of diminished 7ths. There are only three distinct diminished seventh chords. Therefore there are only three ocatonic collections. The octatonic collection repeats itself every three half-steps. The octatonic collection has many interesting subsets, smaller collections found within larger ones. The harmonic possibilities are rich. 

Theory:

The Complete Musician(https://tinyurl.com/4mz2n4t8)

Chapter 29 of 32: Symmetry Stretches Tonality: Chormatic Sequences and Equal Divisions of The Octave

For tonal music to function properly, it's structure need to be asymmetrical. Asymmetrical chords and scales, the asymmetry being the thing that enables us to orient ourselves. 

Symmetry on the other hand skews tonality and introduces ambiguity. If you take a major or minor scale, choose any of its pitches other than the tonic, and build the same type of scale starting on that pitch, you will not end up with the same set of pitch slasses as in the original scale. The asymmetry built in to this system of chord and scale construction brings a sense of differentiation, because the asymmetry builds variegatedness into the tonal topography.

Symmetrical structures: whole tone scales, diminished half/whole scales, the chromatic scale, etc… have absolutely no variegatedness in the profile. They are endlessly smooth structures, no jagged edges or hanging resolutions. 

Symmetrical chords vs. asymmetrical chords

Major and minor triads, are chord that involve three intervals of different sizes, the major third (four semitones), the minor third (three semitones) and the perfect fourth (five semitones). Augmented triads are completely symmetrical, consisting of three major thirds.

Seventh chords: Despite its dissonant nature, the dominant seventh is crystal clear tonally, because it points towards one single tonic. It is asymmetrical, whereas the fully diminished seventh is symmetrical, being generated entirely by minor thirds. 

Symmetrical vs Asymmetrical Tonal Paths

The straightforward harmonic paths of tonal music are also asymmetrical. The tonic is flanked symmetrically by the dominant (a P5 above the tonic) and the subdominant (a P5 below the tonic) but those perfect 5ths are comprised of interval combinations of various sizes, in other words each P5 is ultimately asymmetrical.

How are diatonic and chromatic sequences different?

Overall motion:

• From model to copy (and from copy to copy), the diatonic sequence is transposed down by a diatonic interval. This is because the size of the interval will change based on where ever the transposition is moved to.
• In contrast, symmetrical structures are constructed by the exact interval type to the point that structures do not stay diatonic

Chord Qualities:

• The qualities of chords change from arrangement to arrangement (modes, etc…) in asymmetrical structures. This helps orient to the tonic, etc….
• In contrast, the chord qualities of a chromatic sequence that are highly consistent

Tonal Clarity

• The asymmetry of a diatonic sequence permits it to remain firmly within the key in which it begins. 
• In contrast, the symmetry of a chromatic sequence causes it to spiral outside of any one key.

Voice exchange can extend dissonant chords, especially dominant seventh chords

The Omnibus

The omnibus is a sizeable expansion that involves expanding the dominant 7th chord a full chromatic octave in every voice of the chord, and there will often be stable, root position harmonies that will be evenly spaced throught the octave. Also, at least two voices move in contrary motion. 

Equal Divisions of the Octave:

The octave can be partitioned into equal pieces. It's a matter of finding those intervals that divide up the octave into smaller pieces that are the same.

Theory:

The Complete Musician(https://tinyurl.com/4mz2n4t8)

Chapter 28 of 32:  Tonal Ambiguity and Symmetrically Constructed Harmonies:

This chapter is significant for a couple reasons, the chief being that it is the first chapter of the last section, section 7: ‘Near, At, and Past the edges of tonality’. So, this means that I have completed the review for what would be considered the standard view of tonal organization. So, one whole section of this treatise has been gone through from end to end. Also, after these five chapters are complete, the book is finished. I am very much looking forward to ending this theory review and getting on with making music. It is almost time.

This chapter looks at tonal ambiguity, the idea that a pitch can have a couple different, but equally valid, meanings. A seeming tonic is also a submediant, something along those lines. It is not vague, harmonically unstable, or sure of itself, it is defined, harmonically stable and self-confident in multiple interpretations.

Diminished seventh chords:

Fully diminished sevenths can act as leading tones to multiple different keys, each note of the chord capable of acting as leading tone to a couple different major and minor keys. Example:

1. chord: b-d-f-ab: root: b key: viid in C
2. chord: b-d-f-g#: root: g# key: viid in A
3. chord: b-d-e#-g#: root: e# key: viid in F#
4. chord: cb-d-f-ab: root: d key: viid in Eb

This chord can be enharmonically reinterpreted. In other words, by respelling the names of different notes, we can achieve different resolution goals. This is because the dminished chord is a symmetrically constructed harmony. In other words, the fully-diminished seventh chord divides the octave into four equal intervals, each three half-steps big. This can be represented by squares inside a circle made up of all twelve pitch-classes. Each of the four points of the square would touch four different note names in the circle. The diminished seventh is also capable of being an enharmonic pivot chord, because it is capable of enharmonic presentation. Enharmonic basically means that any given note has different names that can refer to the same physical relationship but change the theoretical one. 

The diminished seventh is capable of not just harmonic ambiguity, but it is also capable of tonal ambiguity. This can be useful in constructing off-tonic beginnings. This possibility can start to really stretch the concept of applied dominant, as you will see things like V of V of V. Wild! This ambiguity can also be employed at the end of a section of a piece, and includes the use of two familiar techniques: the use of modal mixture to color major mode pieces with the sounds of parallel minor, and the frequent use of plagal motions that approach the tonic not from the dominant but from a pre-dominant harmony such as ii or IV.

Common-tone diminished seventh chords:

This is what happens when the diminished seventh chord you're working with contains the root of the chord that follows. This changes the resolution, partly because of the inclusion of a common tone between the chords. Diminished sevenths serve in multiple musical contexts, and you have to evaluate a diminished seventh based on what follows it.  Sometimes voice exchanges occur in the process of progression.

Beware of: vii/V-V6/4-5/3. It is tricky to sort out.

Common-tone Augmented Sixth chords:

Common-tone Augmented sixth chords arise from contrapuntal motions, but it is not a diminished seventh chord. It is an augmented sixth, and the specific interval in question resolves outward between two inner voices over a stable bassline, so it is effective for coloring expansion of the tonic harmony. The chords in question share one or more tones, hence the name.

Altered Dominant Sevenths:

A dominant seventh chord already possesses two tendency tones that demand resolution by step, the leading tone up a half step to 1, and the chordal seventh down to 3, either by half or whole-step depending on mode. Similarly we can create a third tendency tone by raising the chordal fifth from 2 to 2#. This will transform 2 into a tendency tone seeking resolution by moving up a half-step to 3. The spelling is now V7/#5. This includes the highly dissonant augmented sixth between 4 and #2. Discharging alll of that tension requires a tonic chord with doubled third. V7/#5 is common in first inversion. We can also create an altered dominant by lowering 2 to b2. This creates a highly charged tendency tone seeking downward resolution to 1. Played in first inversion with b2 in the bass, the chord is identical to a French augmented sixth chord, except that it resolves not to the dominant but to the tonic,

The FrV4/3 chord can function in a reciprocal process: it's resolution can sound just like a half-cadence even when it's actually functioning as part of an authentic cadence (FR V4/3 → I)

Augmented Triads:

This is another symmetrical harmony like the fully diminished seventh. Whereas the fully diminished seventh divices the octave into four equally sized minor thirds, so the Augmented triad divides the octave into three equally sized major thirds. This can be represented as a triangle inscribed in the circle. Each of the vertices of the triangle will touch a different note, telling us what three notes are needed for the augmented triad to be constructed. There are four unique augmented triads. Sometimes augmented triads are generated by chormatic voiceleading, and sometimes it comes into its own as an indpendent sonority and can be used to intentionally cloud the tonal clarity of a work.