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Eight Pieces for Eight Guitars

Casey Mongoven

March 19, 2007

 

This set of pieces for eight guitars features the use of eight sequences related to the Fibonacci numbers and the golden ratio.

 

Although these eight pieces were written to be performed on the same concert, they can be performed separately as well. If all pieces are played on the same concert, they should be played in the following order:

 

1. Horizontal Para-Fibonacci Sequence no. 20

2. Min Fibbit Running no. 12

3. Vertical Para-Fibonacci Sequence no. 25

4. Zeck Reps Density no. 10

5. Max Binary Fibonacci Rep 0s Density no. 7

6. Fibonacci Stolarsky Vertical no. 21

7. Max Fibbit Running no. 9

8. Fibonacci Stolarsky Horizontal no. 19

 

instruments: All the guitars should have a similar sound; the more similar the sound, the more ideal. The strings should be of approximately the same age. Either metal or nylon strings may be used, but the performers should all have the same type of strings. A pick should be used to pluck each note, and all performers should use the same weight and type of pick. Capos should not be used.

 

arrangement of performers on stage: The integers in the sequences are represented not only by pitch and loudness, but also by the spatial location of the performers; there are two possible methods of arranging the performers on stage:

 

1) straight line: This arrangement can be used in almost any concert hall or room, provided there is space. There should be at least 2 meters separating the performers, measuring from the center of the body. The larger the audience and room, the greater the space between performers should be. This space should not exceed 15 meters:

 

2) a half-circle around the audience: The performers are arranged in a half circle around the audience, and are equally spaced from each other. A long piece of string and a compass can be used to create the half circle using the center of the audience as a center point. Not all concert-spaces will allow this arrangement:

 

 

In both the above arrangements the guitarists should be seated on stools in a uniform style.

 

In the line arrangement, the performers should direct their instruments’ sound holes all in the same exact direction: straight toward the audience. In the semi-circle arrangement all sound holes should be directed toward the center-point of the circle.

 

Guitar one is always on the right side from the perspective of the audience, guitar eight on the left.

 

tuning of the guitars: These works use scordatura tuning. They do not use the lowest or highest strings of the guitar. The open strings of the guitars are to be tuned as follows:

 

guitar 1 – A-string = 216.58 Hz, D-string = 276.32 Hz, G-string = 358.72 Hz, B-string = 478.90 Hz

guitar 2 – A-string = 217.35 Hz, D-string = 267.12 Hz, G-string = 351.24 Hz, B-string = 493.08 Hz

guitar 3 – A-string = 218.12 Hz, D-string = 273.58 Hz, G-string = 364.37 Hz, B-string = 479.19 Hz

guitar 4 – A-string = 218.89 Hz, D-string = 264.47 Hz, G-string = 356.78 Hz, B-string = 493.39 Hz

guitar 5 – A-string = 219.66 Hz, D-string = 270.87 Hz, G-string = 349.34 Hz, B-string = 479.49 Hz

guitar 6 – A-string = 208.07 Hz, D-string = 261.85 Hz, G-string = 362.40 Hz, B-string = 493.69 Hz

guitar 7 – A-string = 208.80 Hz, D-string = 268.18 Hz, G-string = 354.85 Hz, B-string = 479.79 Hz

guitar 8 – A-string = 209.54 Hz, D-string = 274.67 Hz, G-string = 368.11 Hz, B-string = 466.27 Hz

 

In each of the eight pieces, the player plays the same note for the entire length of the piece; each player’s note represents an integer in the sequence. Each player plays 8 different tones in this set of pieces. Here are the frequencies played by each performer in the pieces, and the frets and strings where the frequencies are found:

 

Horizontal Para-Fibonacci Sequence no. 20:

all guitars play on B-string:

guitar 1 – 904.04 Hz – 11th fret                  guitar 2 – 829.26 Hz – 9th fret

guitar 3 – 760.67 Hz – 8th fret                    guitar 4 – 697.76 Hz – 6th fret

guitar 5 – 640.04 Hz – 5th fret                    guitar 6 – 587.10 Hz – 3rd fret

guitar 7 – 538.54 Hz – 2nd fret                   guitar 8 – 494.00 Hz – 1st fret

 

Min Fibbit Running no. 12:

all guitars play on A-string:

guitar 1 – 324.50 Hz – 7th fret                    guitar 2 – 307.37 Hz – 6th fret

guitar 3 – 291.15 Hz – 5th fret                    guitar 4 – 275.78 Hz – 4th fret

guitar 5 – 261.22 Hz – 3rd fret                    guitar 6 – 247.43 Hz – 3rd fret

guitar 7 – 234.37 Hz – 2nd fret                   guitar 8 – 222.00 Hz – 1st fret

 

Vertical Para-Fibonacci Sequence no. 25:

all guitars play on D-string:

guitar 1 – 368.84 Hz – 5th fret                    guitar 2 – 356.56 Hz – 5th fret

guitar 3 – 344.69 Hz – 4th fret                    guitar 4 – 333.21 Hz – 4th fret

guitar 5 – 322.12 Hz – 3rd fret                   guitar 6 – 311.39 Hz – 3rd fret       

guitar 7 – 301.02 Hz – 2nd fret                   guitar 8 – 291.00 Hz – 1st fret

 

Zeck Reps Density no. 10:

all guitars play on G-string:

guitar 1 – 451.96 Hz – 4th fret                    guitar 2 – 442.54 Hz – 4th fret

guitar 3 – 433.31 Hz – 3rd fret                   guitar 4 – 424.28 Hz – 3rd fret

guitar 5 – 415.44 Hz – 3rd fret                   guitar 6 – 406.78 Hz – 2nd fret      

guitar 7 – 398.30 Hz – 2nd fret                   guitar 8 – 390.00 Hz – 1st fret

 

Max Binary Fibonacci Rep 0s Density no. 7:

all guitars play on G-string:

guitar 1 – 639.16 Hz – 10th fret                  guitar 2 – 625.84 Hz – 10th fret

guitar 3 – 612.80 Hz – 9th fret                    guitar 4 – 600.03 Hz – 9th fret

guitar 5 – 587.52 Hz – 9th fret                    guitar 6 – 575.27 Hz – 8th fret       

guitar 7 – 563.28 Hz – 8th fret                    guitar 8 – 551.54 Hz – 7th fret

 

Fibonacci Stolarsky Vertical no. 21:

all guitars play on D-string:

guitar 1 – 521.62 Hz – 11th fret                  guitar 2 – 504.25 Hz – 11th fret

guitar 3 – 487.46 Hz – 10th fret                  guitar 4 – 471.23 Hz – 10th fret

guitar 5 – 455.54 Hz – 9th fret                    guitar 6 – 440.37 Hz – 9th fret       

guitar 7 – 425.71 Hz – 8th fret                    guitar 8 – 411.54 Hz – 7th fret

 

Max Fibbit Running no. 9:

all guitars play on A-string:

guitar 1 – 458.92 Hz – 13th fret                 guitar 2 – 434.69 Hz – 12th fret

guitar 3 – 411.75 Hz – 11th fret                  guitar 4 – 390.01 Hz – 10th fret

guitar 5 – 369.42 Hz – 9th fret                    guitar 6 – 349.92 Hz – 9th fret

guitar 7 – 331.45 Hz – 8th fret                    guitar 8 – 313.96 Hz – 7th fret

 

Fibonacci Stolarsky Horizontal no. 19:

all guitars play on B-string:

guitar 1 – 1278.50 Hz – 17th fret               guitar 2 – 1172.75 Hz – 15th fret

guitar 3 – 1075.75 Hz – 14th fret               guitar 4 – 986.78 Hz – 12th fret

guitar 5 – 905.16 Hz – 11th fret                 guitar 6 – 830.29 Hz – 9th fret

guitar 7 – 761.62 Hz – 8th fret                    guitar 8 – 698.62 Hz – 7th fret

 

Tuning files and scores can be found in the catalogue entries for these works at www.caseymongoven.com/catalogue/b497.htm.1

 

synchronization: A click track or a conductor with a metronome should be used to aid synchronization. If a conductor is used in the semi-circle arrangement, he should stand or sit in the center of the audience. In the line arrangement, he should keep some distance from the performers, as to minimize blocking the path of the sound to the audience.

 

tempo: The tempo of the works is left up to the performers. However, the duration of one note should not be longer than .2 seconds. The tempo of the pieces should either be the same for every piece, or different for every piece so that no piece has the same tempo. The tempo should never change in the course of a piece.

 

notation: The notation of these works is a mixture of graph notation and standard notation. Rests have been omitted for the sake of a better and clearer visual representation of the sequence. Higher notes are always represented higher on the page, and player 1 is always at the top of the page, player 8 is on the bottom.

 

style of playing: The picking technique used should be uniform. Because the integers in the sequence are represented by loudness (in addition to pitch and location), it is very important that the performers play their note exactly the same each time at exactly the indicated dynamic level. There should be a perfectly gradual increase in loudness or softness from player 1 to 8 for each work.2 This can be seen in the scores.

 

The players should mute the string during every rest; no muting is necessary when a player has two or more notes in a row.

 

techniques and strategies: Performers can draw in barlines if they feel this aids them in performance, although this will most likely not be necessary for works 1 through 3 and 6 through 8.

 

Works 4 and 5 are the most difficult: Zeck Reps Density no. 10 and Max Binary Fibonacci Rep 0s Density no. 7 have 1597 notes each. The barlines should be drawn at points according to the rabbit sequence:3

 

 

The rabbit sequence is particularly useful for counting in these two works (and all of the works for that matter). The rabbit sequence is normally seen in the form: 1,0,1,1,0,…, but one can substitute the numbers (1,0) with any adjacent Fibonacci numbers, (34,21) or (21,13), for example. The notes above are grouped in a rabbit sequence according to the Fibonacci numbers 34 and 21; 34,21,34,34,21…. Using this technique of grouping can provide a useful structure for counting in the work; it provides some insight into the structure of the sequence as well.

 

A conductor can use the rabbit sequence as well. Each measure of 34 above could be conducted in groupings as such: 8,5,8,8,5. Each measure of 21 as such: 8,5,8. For example:

 

 

Another possibility is to create a click track using such groupings as beats.

It is difficult to overemphasize the importance of using the fractal nature of the work as a tool for performance. Repeated patterns such as the following should be recognized and utilized as an aid to performance:

 

 

Some performers may find it helpful to draw rests into the score. It may also be helpful for a performer to highlight his or her part.

 

Notes

 

1. The following diagrams illustrate the tuning of the guitars and show the pitches available on each fret:

 

 

Guitar 8

Guitar 7

Guitar 6

Guitar 5

Guitar 4

Guitar 3

Guitar 2

Guitar 1

Pieces I, VIII

 

 

 

 

 

 

 

 

open B-string

466.27

479.79

493.69

479.49

493.39

479.19

493.08

478.90

fret 1

494.00

508.32

523.05

508.00

522.73

507.68

522.40

507.37

2

523.37

538.54

554.15

538.21

553.81

537.87

553.46

537.54

3

554.49

570.57

587.10

570.21

586.74

569.86

586.37

569.51

4

587.46

604.50

622.01

604.12

621.63

603.74

621.24

603.37

5

622.40

640.44

659.00

640.04

658.60

639.64

658.18

639.25

6

659.41

678.52

698.19

678.10

697.76

677.68

697.32

677.26

7

698.62

718.87

739.70

718.42

739.25

717.97

738.79

717.54

8

740.16

761.62

783.69

761.14

783.21

760.67

782.72

760.20

9

784.17

806.90

830.29

806.40

829.78

805.90

829.26

805.41

10

830.80

854.89

879.66

854.35

879.12

853.82

878.57

853.30

11

880.20

905.72

931.97

905.16

931.40

904.59

930.81

904.04

12

932.54

959.58

987.38

958.98

986.78

958.38

986.16

957.80

13

987.99

1016.64

1046.10

1016.00

1045.46

1015.37

1044.80

1014.75

14

1046.74

1077.09

1108.30

1076.42

1107.62

1075.75

1106.93

1075.09

15

1108.98

1141.13

1174.20

1140.43

1173.49

1139.71

1172.75

1139.02

16

1174.93

1208.99

1244.03

1208.24

1243.26

1207.48

1242.48

1206.75

17

1244.79

1280.88

1318.00

1280.08

1317.19

1279.28

1316.37

1278.50

 

Pieces II, VII

 

 

 

 

 

 

 

 

open A-string

209.54

208.80

208.07

219.66

218.89

218.12

217.35

216.58

fret 1

222.00

221.22

220.44

232.72

231.90

231.08

230.27

229.46

2

235.20

234.37

233.55

246.56

245.69

244.83

243.96

243.10

3

249.19

248.31

247.43

261.22

260.30

259.38

258.47

257.56

4

264.00

263.07

262.15

276.75

275.78

274.81

273.84

272.87

5

279.70

278.71

277.73

293.21

292.18

291.15

290.12

289.10

6

296.33

295.29

294.25

310.65

309.55

308.46

307.37

306.29

7

313.96

312.85

311.75

329.12

327.96

326.80

325.65

324.50

8

332.62

331.45

330.28

348.69

347.46

346.24

345.01

343.80

9

352.40

351.16

349.92

369.42

368.12

366.82

365.53

364.24

10

373.36

372.04

370.73

391.39

390.01

388.64

387.26

385.90

11

395.56

394.16

392.77

414.66

413.20

411.75

410.29

408.85

12

419.08

417.60

416.13

439.32

437.77

436.23

434.69

433.16

13

444.00

442.43

440.87

465.44

463.80

462.17

460.54

458.92

 

Pieces III,VI

 

 

 

 

 

 

 

 

open D-string

274.67

268.18