Sacsayhuamán

¹ A word or phrase, the letters (or components) of which, when taken in reverse order read the same. Collins English Dictionary, 1982.

² A space in any finite number of dimensions, in which points are designated by coordinates (one for each dimension), the distance between two points is given by distance formula.

Pitch Palindromes:

Apart from the single note (unison), the interval (the space between two notes) is the smallest pitch palindrome. There are six interval-classes and when used in multiple, they all complete an equidistant palindromic cycle.

Class 1: minor 2^nd and Major 7^th.

Class 2: Major 2^nd and minor 7^th.

Class 3: minor 3^rd and Major 6^th.

Class 4: Major 3^rd and minor 6^th.

Class 5: Perfect 4^th and Perfect 5^th.

Class 6: tritone/diminished 5^th.

Class 1 and 5 are notable in the fact they cycle through the total chromatic before they repeat any notes. Class 1 will result in an equidistant row of total dissonance (chromatic octave) and Class 5 results in equidistant twelve-tone consonance (cycle of fourths/fifths).

Class 2 produces a six-note equidistant whole-tone scale, Class 3 results in a four-note equidistant diminished 7^th chord, Class 4 results in a three-note equidistant augmented triad and Class 6 simply produces the two-note interval of a tritone. Apart from the single note, the interval (or micro-interval) is the smallest pitch palindrome.

The most succinct way of referring to all intervals and interval-classes was to compose with the two All-Interval Tetrachords (AIT) 4-z15 (0146/0256) and 4-z29 (0137/0467).

The AIT are subsumed within the two All-Triad Hexachords (ATH) 6-z17 (012478) and 6-z17b (014678).

The ATH in turn can be found in prime form within the Symmetrically Inverted All-Interval Twelve-Note (SI AITN) Chords: (Elliott Carter numbers) SI AITN 1, 3, 58 and 60.

In addition (and perhaps incidentally), Dorian Mode (Heptachord 7-35) is the only mode in the tonal system that inverts to itself: a tonal, or ‘natural’ palindrome. The Dorian Mode subsumes a transposition of its own complement; the Major Pentatonic Scale (Pentachord 5-35): another scalic palindrome.

The Cycle of Fourths:

The cycle of fourths can be described as a twelve-tone equidistant chromatic scale. Like the six notes of the equidistant whole-tone scale, there is no inherent hierarchical reference to a home-tone unless one is established through the act of composition. The cycle of fourths could also be described as a Mirror Inverted All-Interval Twelve-Note Chord; a palindromic chromatic cycle with multiple equidistant palindromes within its self-contained Matryoshka Form.

The All-Interval Tetrachord (AIT):

4-z15 (0146/0256) and 4-z29 (0137/0467)

The two All-Interval Tetrachords (AIT) get their name due to the fact they contain all twelve interval-types without replication:

0146/0256 (4-z15/4-z15B) <111111> All-Interval Tetrachord 1

0137/0467 (4-z29/4-z29B) <111111> All-Interval Tetrachord 2

The all-interval tetrachords (AIT) are self-contained palindromes:

0256 (4-z15b) is a retrograde inversion of 0146 (4-z15)

0467 (4-z29b) is a retrograde inversion of 0137 (4-z29)

AIT Pitch Matrices:

4-z15:

(0146/0256) <111111> All-Interval Tetrachord 1 (Inverts to 4-z15)

4-z29:

(0137/0467) <111111> All-Interval Tetrachord 2 (Inverts to 4-z29)

The All-Triad Hexachord (ATH):

(012478/014678) <322332> Carter Hexachord 35, Forte 6-z17 [6-z17b]

A hexachord is any sequence of six notes laid out in a scalic pattern. The All-Triad Hexachord (ATH) takes its name from the fact it contains all twelve prime-form trichords (without replication) within the span of an octave:

ATH list of prime-form trichords:

i. 3-1* (012) <210000> BACH/Chromatic Tri-mirror: C - C^# - D

ii. 3-2 (013) <111000> Phrygian Trichord: C^# - D - E

iii. 3-3 (014) <101100> Major-minor Trichord: C - C^# - E

iv. 3-4 (015) <100110> Incomplete Major Seventh Chord: G - G^# - C

v. 3-5 (016) <100011> Rite Chord/Tritone Fourth: C^# - D - G

vi. 3-6* (024) <020100> Whole-tone Trichord: C - D - E

vii. 3-7 (025) <011010> Incomplete minor seventh Chord: D - E - G

viii. 3-8 (026) <010101> Incomplete Dominant Seventh Chord – Italian Sixth: D - E - G^#

ix. 3-9* (027) <010020> Quartal Trichord: C - D - G

x. 3-10* (036) <002001> Diminished Chord: C^# - E - G

xi. 3-11 (037) <001110> Major/minor Chord (047): C^# - E - G^#

xii. 3-12* [4] (048) <000300> Augmented Chord: C - E - G^#

ATH Pitch Matrices:

The ATH is a self-contained (combinatorial) palindrome: 014678 is a retrograde inversion of 012478:

The ATH subsume the AIT 0146/0256 and 0137/0467 in prime form:

The All-Triad Hexachords (ATH) are themselves subsumed in prime form within the Symmetrically Inverted All- Interval Twelve-Note (SI AITN) chords: 1, 3, 58 & 60.

Symmetrically Inverted All-Interval

Twelve-Note (SI AITN) Chords:

Elliott Carter’s Symmetrically Inverted All Interval Twelve Note (SI AITN) chords offer a flexible way to utilise the largest compositional units (12-tone scales) as pitch palindromes on the vertical and linear axis. There are many SI AITN to choose from³, but only four contain the All-Triad Hexachord (ATH) in prime form:

Intervallic Spelling:

SI AITN 1:    27431 6 E985T/27431 6 E985T

SI AITN 3:   27491 6 E385T/27491 6 E385T

SI AITN 58: 21497 6 538ET/21497 6 538ET

SI AITN 60: 21437 6 598ET/21437 6 598ET

SI AITN 1: 27431 6 E985T – 27431 6 E985T subsumes 6-Z17 twice

Literal Pitch Spelling:

029145 ET7386 = C – D – A – C^# – E – F – B – A^# – G – D^# – G^# – F^#

029145 ET7386 = C – D – A – C^# – E – F – B – A^# – G – D^# – G^# – F^#

³ Carter lists 88 but does not include inversions. Researchers have identified further alternates [Link Chords]

Sacsayhuamán

The Composition

Sacsayhuamán is composed from SI AITN 1. The tempo is marked crotchet equals 40 throughout. The work could be described as harmonically static, with the near omnipresence of the two framing pitches of the chords: C¹ (contrabasses) and F^#6 (violins).

The Main Movement is in a four-bar, 18-beat sequence: Prime, Retrograde, Inverted and Retrograde Inverted, which accumulates in dynamic and texture consistently towards the central extended and sustained crescendo, adding one more note of the relevant chord on each cycle. The process is then reversed after the crescendo to dissipate dynamically and harmonically in mirror-form. Harmonic motion moves between the chords: Prime occupies the first two bars and Inverted the second two bars (working from the bass up), with [R] and [I] overlapping in juxtaposition on the second and fourth bars of each cycle (working from the treble down). [P] is paired with [R] and [I] is paired with [RI] throughout. Most of the changes in chord type and density are marked in the score.

Sacsayhuamán

Main Movement Basic Form & Harmony

Whilst the vertical unfolding of texture and harmonic progression in the Main Movement was fairly systematic, there were many anomalies to negotiate as the overlapping chords grew more dense (not to mention choices of orchestration and timbral balance). A basic hierarchy was adopted so that the Prime and Inverted spellings would always be preferred in times of necessary choice. Therefore, some notes of the overlapping [R] and [RI] chords were omitted from three cycles either side of the crescendo, where the continuous expansion of register (and the subsequent mirror diminution) of the two main [P] and [I] chords needed to be emphasised.

SI AITN 1 Prime:

029145 ET7386 = C – D – A – C^# – E – F – B – B^b – G – D^# – G^# – F^#

SI AITN 1 Inverted:

0T3E87 125946 = C – B^b – E^b – B – G^# – G – C^# – D – F – A – E – F^#

Linear Writing:

Combined 20-note scale

Many of the linear gestures used the [P] and [I] chord-scales as fixed (or unordered) rows and subsets. Care was taken to avoid semi-tone clashes with the vertical harmony. The vibraphone concentrates upon varying elements from the combined 20-note version of the row:

C – D – A – B^b – D^b – E^b – E – F – B – G^# – A^# – G – C^# – D – Eb – F – A^b – A – E – F^#

Vertical Spaces:

Vertical space in the harmony for linear melodic and motivic materials …

Most of the linear gestures, chromatic statements and melodic material happen within the vacant vertical spaces of the combined chords, which were essentially precomposed and are present throughout. Although horizontal material always refers to the twelve and twenty note rows, composition was far more instinctive within these ‘vertical’ spaces, often using subsets and micro units of the rows as unordered cells.

Four intervals of vertical space:

C³ – G³, B³ – F^#4, A^b4 – C⁵, B^b5 – E^b6

A Natural Anomaly:

There are four shared notes between the combined chords: C¹ – B² – G⁴ – F^#6. The C¹ and F^#6 are omnipresent in the score by design and intent and quite literally frame the music. The B² was used as the catalyst to one of the motif on the bass clarinet and therefore is diluted. This left the G⁴. During earlier sketches (before the vertical harmony had reached G⁴), the trumpets and horns (and to a lesser extent viola) had started using the note of A^b4 in an important motif/phrase. One always tries to introduce a composed anomaly into any pre-compositional system, this one presented itself naturally through the organic tendency of linear phrases, so the rogue A^b4 was embraced within the linear statements and all occurring incidents where G⁴ and A^b4 wanted to occupy the same acoustic space were resolved on an individual basis. Mostly deferring to the linear A^b4 to heighten the recurring blurring effect of this anomaly.

SI AITN 1 [P] and [R]: C¹ - D¹ - A¹ - C^#2 - E² - F² - B² - B^b3 - G⁴ - E^b5 - A^b5 - G^b6

SI AITN 1 [I] and [RI]: C¹ - B^b1 - E^b2 - B² - A^b3 - G⁴ - C^#5 - D⁵ - F⁵ - A⁵ - E⁶ - F^#6

Introduction and Conclusion:

diū noctūque (by day and by night)

The Introduction and Conclusion share the same fundamental properties and four-bar cycles as the Main Movement. However, they both start in crescendo with a repeated complete SI AITN 1 Inverted chord and then essentially begin to dissipate the harmony (rotating through the chord-types) towards an ever-thinning texture and quieter dynamic. In contrast to the Main Movement, the vertical textures of the Introduction and Conclusion are far less reliant upon the harmonic method and are generally more instinctive in the approach to, and use of, passing harmony and linear statements.

In Summation:

Although this score is the result of an extended period in the composer’s life, took almost two years to simply sketch the initial framework and plot the harmonic changes into a musical form, and then numerous multiple month monastic ‘sittings’ spanning over a decade to finally reach the closing bars, it is pleasing to note that the finished music sounds alive, full of energy and ready to explore the world. It bears no scars from the battles it has put it’s composer through and actually sounds like it was always meant to be! Sometimes, things just work …

For Robin