Guitar Fingering For Music Performance (Page 2 of 4) in pdf

Figure 3. Chords may be considered as the third dimen-

sion of the graph: the ﬁrst layer represents alternative ﬁn-

gerings for the same chord shape, whereas in the second

and third layer, single ﬁngered positions (for the individ-

ual notes D and G) are displayed.

Figure 2. Graph generated for a three notes (F2-D4-E3)

culties are considered and how they are evaluated. Weights

fragment. Each vertex represents a ﬁngered position (e.g.,

estimating difﬁculties are based on the work in motor be-

<6,1,1>). Weights on the edges are omitted.

havior by Heijink & Meulenbroek [3], on the ergonomic

approach to keyboard ﬁngering by Parncutt et al. [5], and

on the intuitive introspection of the authors. The elements

struction procedure guarantees that the graph has a layered

designed for the model have been quantitatively tuned on

structure, in that the vertices can be grouped in layers,

an implemented prototype.

and all the edges connect vertices of adjacent layers. The

Underpinned by [3], we assume that moving hands hor-

number of layers is equal to the number of events, be them

izontally along the ﬁngerboard and vertically across

chords or one-note events, and any path from the leftmost

the ﬁngerboard

constitute two different sources of dif-

layer (the ﬁrst note) to the rightmost layer (the last note)

ﬁculty. Horizontal movements can span over a large dis-

represents a ﬁngering for the score. Thus, the problem of

tance, and are named hand repositionings, while vertical

ﬁnding a suitable ﬁngering for a piece corresponds to the

movements are considered less complex and are named

problem of ﬁnding a path in the graph.

ﬁnger displacements. The weight estimation proceeds by

Chords are represented as groups of vertices (collapsed

arranging the transitions between two positions into two

into one vertex), distributed along a third dimension (Fig-

classes, along the neck (henceforth ALONG) and across

ure 3); in facts, while layers correspond to the succession

the neck (ACROSS) respectively. The weight (WEIGHT)

of ﬁngered notes, these groups model the simultaneity of

between two ﬁngered positions p =

<string , fret ,

positions in chords. Such grouped representations result

and q =

is the

finger >

from a modeling of chord ﬁngering as a constraint satis-

linear sum of the two difﬁculties:

faction problem (CSP) [2].

W EIGHT

= ALON G

+ ACROSS

(1)

Since performers pursue an overall effort-saving be-

havior, we assign a difﬁculty score (a cost) to the tran-

In turn, ALONG and ACROSS are computed by means of

sitions between vertices, and seek for a path that ensures

the following expressions:

the lowest overall difﬁculty score. An effective traversal

of the graph is then obtained by means of the dynamic pro-

ALON G

= f ret stretch

+ locality

(2)

gramming technique, that can ﬁnd the shortest path with

ACROSS

= vertical stretch

(3)

linear, O(m), time, where m = Edges . The piece is

split into phrases, each one being ﬁngered separately. The

We assume the following value ranges for the factors men-

motivations and the salience of phrasing and musical ar-

tioned here: ALONG varies between [0, 13.25], with fret

ticulation in the ﬁngering process have been investigated

stretch varying from [0.5, 5] and locality varying from [0,8.5]

by Radicioni et al. [6]. Since evidence was provided that

(some combinations are not possible); ACROSS as well as

hand replacements are most likely to occur at the phrase

vertical stretch varies between [0.25, 0.5]. The rationale

boundaries than within the phrases, we adopt a local opti-

for these assignments will be clear after the description of

mization approach, where the model might choose a glob-

the individual factors.

ally harder ﬁngering, but simplifying the execution of the

ALONG difﬁculties. Two factors contribute to ALONG

individual phrases.

difﬁculties: Fret Stretch, which accounts for the geomet-

ric distance between two positions on the ﬁngerboard, and

Locality, which accounts for transitions around some area

3. ESTIMATING BIO-MECHANICAL

of the ﬁngerboard.

DIFFICULTY

Fret Stretch. Given two ﬁngered positions p and q, Fret

Here we show how weights are estimated to account for

Stretch is a measure of the difﬁculty due to moving hori-

performers’ bio-mechanical comfort, which kind of difﬁ-

zontally. We introduce a directed distance, deltaf ret =

Guitar Fingering For Music Performance Page 2

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