Chord Studies. 374 Chords, including: Triads Sixths Sevenths Ninths. Chord Adjustments in Just Intonation Triads Sixths Sevenths

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1 Chord Studies 374 Chords, including: Triads Sixths Sevenths Ninths Chord Adjustments in Just Intonation Triads Sixths Sevenths Intervals and their Derivations from Equal Temperament Edited y Nikk Pilato

2 Triads Triads are three-note chords that can e stacked in thirds (therefore, a chord such as C-F-G is not technically a triad, ut rather a trichord). The three components of a triad are the root, the third, and the fifth. There are four types of triads: (M3 + m3, ex: C-E-G) Can e expressed as C, CM, or C (m3 + M3, ex: C-E-G) Can e expressed as Cm, or C- Diminished (m3 + m3, ex: C-E-G) Can e expressed C " (M3 + M3m ex: C-E-G") Can e expressed C+ Sixth Chords A sixth chord can also e interpreted as a minor seventh chord in first inversion. Whether it should e regarded as a sixth or a minor seventh ill depend on its harmonic function and context in the music. In modern music, a sixth chord is any triad ith an added sixth aove the root. Sixth (major triad + a M6, ex: C-E-G-A) Sixth (minor Triad + a M6, ex: C-E-G-A) There are also special sixth chords, such as: Neapolitan Sixth: First inversion chord of a major triad uilt on the flat second (ex: In C, this ould e F-A-D, here D ould e the flat second, and F the first inversion). It is notated II 6 or N 6. Sixth: Instead of a M6, these chords have an augmented sixth present along ith a major third. There are three possile variants: e Italian (It +6 or It 6 ): "4 (ex: in c minor: A-C-C-F", the root is douled). e French (Fr +6 4 or Fr 3 ): "4. Similar to the Italian, ut ith an added second aove the root (ex: in c minor: A-C-D-F"). e German (Gr +6 6 or Ger 5 ): "4. Similar to the Italian, ut ith an added flat third aove the root (ex: in c minor: A-C-E-F"). Edited y Nikk Pilato

3 Triads and Sixths Diminished 6th 6th C n n C D D n D E E n F n F G G n G A A n A B B n Note: For easiest reading, some and notes are sustituted y their enharmonics.

4 Seventh Chords A seventh chord is a triad ith an added note, hich is either a major, a minor, or a diminished seventh aove the root. There are eight possile types of tertian 7th chords, seven of hich are commonly used (the 8 th ould essentially equal a Triad ith a douled root). Seventh (aka Dominant 7 th ): When not otherise specified, the name "seventh chord" may refer to a Triad + minor seventh. (ex: C-E-G-B). can e expressed: Cdom 7, C7, or C 7. Seventh: A Triad + a seventh. (ex: C-E-G-B). can e expressed: Cmaj 7, CM7, or C 7. - Seventh: A minor triad + a seventh. (ex: C-E-G-B). can e expressed: Cmin maj7, Cm 7, C- M7, or Cm M7. Seventh: A minor triad + a minor seventh. (ex: C-E-G-B). can e expressed: Cmin 7, Cm 7, C Seventh: An triad + a seventh. (ex: C-E-G"-B). can e expressed: Caug maj7, C+ M7, or CM7 +5 Half-Diminished Seventh: A diminished triad + a minor seventh. (ex: C-E-G-B). can e expressed: Cmin 7dim5, C Ø7, Cmin7 $%'( (Fully) Diminished Seventh: A diminished triad + a diminished seventh. (ex: C-E-G-B). can e expressed: Cdim 7, or C )7. The folloing seventh chords are not tertiary (they include the interval of a diminished or augmented third), ut are used often enough to learn: Seventh: triad + a minor seventh. (ex: C-E-G"-B). can e expressed: Caug 7, C+ 7, C7 5, or C7 +5. Seventh Flat Five " : Essentially a dominant seventh chord, ut ith a flatted fifth. (ex: C-E-G-B). can e expressed: C 7 5, Cdom 7dim5. Diminished Seventh " : A diminished triad + a seventh. (ex: C-E-G-B). can e expressed: Cdim Maj7, or C )7 (addmaj7) " These chords are not shon in the accompanying chord chart. Edited y Nikk Pilato

5 Seventh Chords Dominant Half Diminished Diminished C n C D D n D E E n F n F G G n G A A n A B B n

6 Ninth Chords Ninth chords are uilt y adding the interval of a ninth to a seventh chord. A common practice is to omit the fifth hen voicing ninth chords. The three most common forms of ninth chords are: Ninth (aka Dominant 9 th ): When not otherise specified, the name "ninth chord" may refer to a dominant seventh + an added ninth. (ex: C-E-G-B-D). can e expressed: Cdom 9, C9, or C 9 Ninth: A major seventh chord + an added major ninth. (ex: C-E-G-B-D) can e expressed: Cmaj 9, CM 9, or C 9 Ninth: A minor seventh chord + an added major ninth. (ex. C-E-G-B-D). can e expressed: Cmin 9, Cm 9, C- 9. Additionally, though much less frequent, you may find the folloing forms: - Ninth: a minor-major seventh chord + an added major ninth. (ex. C-E-G-B-D). can e expressed: Cmin maj9, C- M9, or Cm M9. - Ninth: major seventh + a major ninth. (ex. C-E-G"-B-D). can e expressed: Caug maj9, or C+ M9. Ninth: seventh + a major ninth. (ex. C-E-G"-B-D). can e expressed: Caug 9, C+ 9, C9 5, or C9 +5 Half-Diminished Ninth: A half-diminished seventh + minor ninth. (ex. C-E-G-B-D) can e expressed: C Ø 9 Diminished Ninth: A diminished seventh + a minor ninth. (ex. C-E-G-B-D). can e expressed: Cdim dim9, or C 9. Seventh Flat Nine " : A dominant seventh + a minor ninth. (ex: C-E-G-B$%' ' can e expressed C7 9 Seventh Sharp Nine " : A dominant seventh + an ninth. (ex: C-E-G-B$%"' ' can e expressed C7 " 9 " These chords are not shon in the accompanying chord chart. Edited y Nikk Pilato

7 Ninth Chords Dominant Half Diminished Diminished C n C D D n D E E n F n F G G n G A A n A B B n

8 Adjustments Tuning There are to main types of Temperament (Tuning) Systems likely to e encountered y performing musicians: Just Temperament (sometimes called pure tuning ecause it occurs naturally as a result of harmonic ratios), and Equal Temperament (the system used y keyoard instruments and some fretted string instruments to divide the octave into telve equal pitches). These are y no means the only temperaments availale, ut they are the most common. Just Temperament (JT) occurs naturally as a result of the harmonic series. Every note in the scale is related y rational numers. With JT, tuning is dependant on the scale you are using. In this tuning, notes are related to each other y ratios of small integers such as the octave (2:1), the perfect fifth (3:2), the perfect fourth (4:3), etc. The Just Scale is ased on the octave and an attempt to have as many nice intervals as possile. Semitones are not equally spaced. This sort of tuning is often used y ensemles (ands, choirs, orchestras) ith the ensemle memers listening to each other and matching their pitches. Equal Temperament (ET) as developed for keyoard instruments, ho could not make quick intonation adjustments hen traveling to different key areas. ET is a compromised tuning system. It uses constant frequency multiples eteen each note of the chromatic scale and divides the octave into equal semitones. This allos the instrument to sound equally good (and equally ad) in any key. Although the resulting notes in each of the to temperaments are close, there are enough discrepancies to result in severe intonation deficiencies. For this reason, instrumentalists (and vocalists) must constantly adjust their pitches dependant on their instrument s tendencies and their position in any given sustained chord. For example, all things eing equal (i.e., a perfectly in-tune instrument) a trumpet player ho is the fifth of a Triad in one measure and the third of the same Triad in the next measure ould have to raise his pitch very slightly (to cents*) for the fifth, then loer the pitch significantly (fourteen cents) for the third, in order to have all the surrounding chord pitches ring in true harmonic ratios. While this sounds like a complicated task, many professional musicians do this so quickly and efficiently that they rarely think aout hat they are doing anymore. With practice and a lot of listening, a performer s role in a chord can e ascertained more and more quickly, resulting in etter-sounding chords and intervals. The folloing chart shos the chord and interval adjustments necessary in order to make chords ring true in their correct harmonic ratios. For more information on the suject, it is recommended to consult the to folloing resources: Ho Equal Temperament Ruined Harmony y Ross Duffin. The Physics of Sound y Richard Berg and David Stork. *Cents are a logarithmic unit of measure used to divide an octave into 12 semitones/half-steps of 100 cents each. Edited y Nikk Pilato

9 Chord Adjustments in Just Intonation The folloing adjustments are ased on the root of the chord (in this case, "C") hich is an equal-tempered pitch. The adjustments (given in cents) indicate the difference necessary for that note to e in tune ith the chord (as opposed to the given equal tempered pitch). The adjustments are applicale to all chords, regardless of starting pitch* Triad (CM) Triad (Cm) Diminished Triad (C ) Triad (C+) -4 Sixth (C6) -16 Sixth (cm6) +19 Dominant Seventh (C7) -31 Seventh (CM7) Seventh (CmMaj7) +11 Seventh (Cm7) Seventh (CM7 5 or C+M7) 6-4 Seventh (C+7 or C7 5 ) -31 Half-Diminished Seventh ( CØ7 or Cm75 ) +18 (Fully) Diminished Seventh ( Cº7 or Cdim7 ) +19 Seventh Flat Five ( C75) Diminished Seventh ( CºM7) Intervals and their deviations from Equal Temperament Second +12 Second Third Third +4 Perfect Fourth Fourth Diminished Fifth Perfect Fifth Sixth Sixth Seventh Seventh *Note that some intervals have multiple adjustments, dependant on their function in a chord.

The diminished harmonic system produces three eight-note

Diminished System Diminished Scale Construction The diminished harmonic system produces three eight-note symmetric scales ased on the minor third interval (four equal divisions of the octave). Each scale