# Dissociate Dimensions to Improve Sound Metaphors Guiding the Hand Towards 3D Non-visible Targets

Coline Fons, Sylvain Huet, Denis Pellerin, Christian Graff

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Coline Fons, Sylvain Huet, Denis Pellerin, Christian Graff. Dissociate Dimensions to Improve Sound Metaphors Guiding the Hand Towards 3D Non-visible Targets. SN Computer Science, 2025, 6 (4), pp.387. 10.1007/s42979-025-03877-7. hal-05034794

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# Dissociate dimensions to improve sound metaphors guiding the hand towards 3D non-visible targets

Coline Fons<fnref n="1" />,<fnref n="2" />*, Sylvain Huet<fnref n="2" />, Denis Pellerin<fnref n="2" />, Christian Graff<fnref n="1" />

*Corresponding author(s). E-mail(s): coline.fons@univ-grenoble-alpes.fr;

## Abstract

This paper examines the improvement of sound metaphors guiding the hand of people deprived of sight towards virtual 3D targets. The quality of the metaphors is compared by target-reaching time and qualitative measures, but also by Fitts' law's throughput, a metric that can be generalized to other experimental parameters, improving the possibilities of comparison between studies. First, a unified metaphor that indicates deviation to the target with a single 3-D angle was compared with a metaphor that separates the horizontal angle (azimuth) and the vertical distance. This spatial dissociation of vertical and horizontal dimensions (DVH) enabled faster target acquisition. Based on these results, we hypothesized that providing one dimension deviation at a time (DVHseq, for sequentially) would show a similar advantage (both spatial and temporal dissociation) compared to providing them in parallel (DVHpar). Beforehand, we compared the most efficient sounds to express each dimension. The results showed better performance for a metaphor using binary vowels to indicate height and for a metaphor combining continuous coding of direction in pitch and binary stereo to indicate horizontal direction. Although DVHseq did not significantly reduce the target reach time compared to DVHpar, the majority of participants preferred the simplicity of the metaphor DVHseq, which guides one dimension at a time, a strategy largely adopted by participants. However, a minority managed to take advantage of the benefit of having all the information at the same time (DVHpar) to move directly to the target.

**Keywords:** Spatial Interaction, Fitts' Law, Sound Guidance, Sonification, Visual Impairment, User Performance Evaluation

<footnote n="1">¹*Univ. Grenoble Alpes, Univ. Savoie Mont-Blanc, CNRS, LPNC, Street, 38000, Grenoble, France.</footnote>

<footnote n="2">²*Univ. Grenoble Alpes, CNRS, Grenoble INP, GIPSA-lab, Street, 38000, Grenoble, France.</footnote>
# 1 Introduction

Visually impaired people (VIP) face significant challenges in spatial interactions, such as navigating unfamiliar environments or manually locating and retrieving objects. Among various assistive technologies, sensory substitution devices (SSDs) offer a solution by transmitting essential information collected by artificial sensors through an alternative sensory channel, such as auditory or tactile modalities.

In this study, we aim to identify the most effective sounds for guiding a user's hand to a target—an abstract task that is fundamental to many daily activities. A target object could be a door handle, an intercom button, or a product on a supermarket shelf. Our goal is to develop a highly accurate device while considering human cognitive limitations and the practical demands of the task.

We conduct tests using a hot-cold game in which participants must reach virtual targets in a virtual environment guided by sound. Rather than relying on spatialized sonification (such as stereo or Head-Related Transfer Function (HRTF)), we use an arbitrary mapping between spatial metrics and sound parameters. These mappings, known as metaphors, establish no direct connection between the sound properties and the spatial data they encode. For example, the deviation angle between a pointed direction and the target direction may be translated into pitch.

The effectiveness of a given metaphor depends on how easily users can grasp its encoding rules, either through innate perception or learning. Some mappings seem intuitive—such as associating upward movement with a higher pitch—but they may be uncomfortable or difficult to process alongside other auditory cues representing different dimensions simultaneously. A previous study [13] showed that using separate auditory parameters to convey vertical and horizontal deviations, rather than encoding a single 3D spatial metric into one auditory parameter, leads to faster target acquisition and is generally easier to use.

The effectiveness of sound guidance systems is typically assessed by measuring the time required to reach the target. However, this metric is highly dependent on experimental parameters, such as target size and the distance between the starting point and the target, making it difficult to compare studies in the literature. To address this limitation, we evaluate the performance of metaphors using both the most common quantitative metric—target-reaching time—and Fitts' law, which facilitates inter-study comparisons. Additionally, we analyze qualitative feedback from participants to assess the system's ergonomics.

The remainder of this section reviews the state of the art in sound-based SSDs for target guidance. We then explore the potential of Fitts' law as a generalizable metric for evaluating the performance of such systems. The first study (Section 2), as detailed in [14], validates the use of Fitts' law for assessing sound metaphors in a 3D virtual target-reaching task and highlights the benefits of explicitly separating vertical and horizontal dimensions. In a second series of experiments (Section 3), we apply Fitts' law to compare different sound metaphors for vertical and horizontal guidance separately, using a paradigm where metaphors are presented in random order. The findings from this second study are used to select the most suitable sound parameters for the third study (Section 4). The third study compares two approaches: one metaphor that presents vertical and horizontal dimensions in parallel and another that dissociates them but presents them sequentially.
# 1.1 Sensory Substitution Devices for guidance to a target

An Sensory Substitution Device (SSD) designed to guide a blind person to a target object using sound requires at least three key components:

a) a sensor that captures relevant spatial data;

b) a transcoder that converts this data into sound;

c) an actuator that delivers the sound to the user.

The sensor gathers the necessary data, while the transcoder and actuator work together to guide the user toward the target.

## 1.1.1 Frame of Reference

When designing an SSD for guidance, selecting an appropriate frame of reference is crucial. The spatial encoding must align with the body's motor and cognitive logic—providing excessive data, such as an entire pixel matrix, overwhelms the user, while encoding only the target's absolute position can lead to misinterpretation [15].

Spatial data is typically captured by cameras, whose placement determines the frame of reference. Some devices position the camera on the head [22, 25, 48], others on the hand or forearm [19, 30, 44], while some combine both [32, 56]—using head-mounted guidance for navigation and hand-mounted guidance for final reach.

A camera-based reference is allocentric, encoding data relative to the camera. In contrast, an egocentric frame encodes data relative to the user's body [43], which is generally more effective for Visually Impaired People (VIP), as aligning external coordinates to one's own body is challenging without visual feedback.

Given that reaching tasks center around the hand, an egocentric hand-based reference is likely the most intuitive. Here, we express deviations relative to the reaching hand rather than the sensor capturing the data.

From the chosen spatial reference frame, different types of deviations can be calculated to guide the user's hand toward the target. These deviations can be distance- or angle-based, depending on the coordinate system used (Cartesian, cylindrical, or spherical). Here, we use either a spherical or cylindrical system, depending on the metaphor. Distance deviations are measured from the pointing finger to the target, while angles are defined between two lines: one from the torso (O) to the pointing finger (P) and another from the torso (O) to the target (T).

## 1.1.2 Sound Guidance

Once spatial data is gathered, it can be transformed into sound using either verbal instructions or sonification. Verbal instructions, such as “turn right” or “turn left” [19, 22, 30, 44, 48, 49, 56], are often considered slow and cumbersome [32] as they disrupt the perception-action loop. If the user moves the sensor too quickly, the instruction may become outdated.

Sonification, which uses non-verbal sounds to convey spatial information [39], is more effective for tasks involving body movement due to the close connection between auditory and sensorimotor systems. It falls into two categories:

- Spatialized sonification exploits the auditory system's ability to locate sounds by virtually recreating the target's position (e.g., through stereo or head-related transfer functions
(HRTF)) [22, 25, 27, 32]. HRTF employs a mathematical transfer function to model how a listener's body alters sound waves. Combined with interaural time (ITD) and level (ILD) differences, this allows humans to determine a sound's location in both horizontal and vertical planes.

- Non-spatialized sonification encodes spatial data using properties like pitch, volume, and tempo [22, 25, 27, 30, 32, 49]. For instance, volume may indicate distance, increasing as the user approaches the target. A meta-analysis [8] identified pitch, intensity, spatialization, timbre, and rhythm as the most commonly used sonification parameters.

The chosen sonification strategy is constrained by human auditory capabilities, particularly the just-noticeable difference (JND) of the selected parameter. This limitation challenges the effectiveness of spatialized sonification: the highest localization accuracy for sound sources is $1° ± 3°$ on the horizontal plane, while distance estimation is limited to several decimeters in close range and worsens with increasing distance [53]. For a review, see [34].

In contrast, non-spatialized sonification, such as pitch-based encoding, offers greater precision, as the human ear can distinguish between 640 and 4000 pitches [57]. Studies indicate that pitch and tempo are more accurate and faster for target localization than intensity and brightness [39]. Additionally, [38] found that non-spatialized sonification improves distance perception compared to spatialized approaches.

Among non-spatialized sonification strategies, two key paradigms emerge: auditory icons and earcons. Auditory icons, as introduced by [16], use sounds that mimic real-world events or objects, leveraging the user's ability to associate sounds with their sources. For example, a creaking door sound could indicate an open or closed state in a virtual environment. These sounds are intuitive because they build on pre-existing knowledge of physical acoustics. In contrast, earcons, as defined by [3], are structured, abstract sounds that often rely on musical motifs and rhythmic patterns rather than real-world analogies. They can be systematically combined to encode complex interactions, making them particularly useful in hierarchical or layered auditory feedback systems. The relationship between the sounds and the information they represent is abstract, metaphorical.

Given the limitations of the auditory system in pinpointing the location of sound sources [34], we opted for non-spatialized sonification. This choice leverages the strong ability of the auditory system to perceive specific sound attributes such as pitch and intensity. ([53] for review).

Ziemer et al. [52–55] identified key conditions for an effective 3D guidance system based on non-spatialized sonification:

- All three spatial dimensions must be integrated into a single auditory stream.

- Dimensions must be perceptually orthogonal, meaning that simultaneous sonification of variables remains unambiguous.

- Dimensions must be continuous and linear.

- Dimensions must offer high perceptual resolution, measurable by the just-noticeable difference.

While these criteria enhance precision and efficiency, cognitive load must also be considered. Ziemer et al. developed a metaphor using a single auditory stream with five continuously varying psychoacoustic parameters. This approach enables precise guidance but risks cognitive overload. As [38] emphasized, sonification strategies should be task-dependent. Here,
<figure><img src="image_2.png" /><figcaption>**Fig. 1** Overview of Fitts' law parameters.</figcaption></figure>

the system must balance accuracy with ease of use, ensuring both effective guidance and quick learnability for novices. Our approach incorporates elements of both auditory icons and earcons: while we try to maintain intuitive mappings akin to auditory icons to facilitate learnability (by using vowels, for example), we also introduce structured auditory cues inspired by earcons to ensure scalability and precision in spatial guidance. The relationship between the sounds we used and the actions they encode being metaphorical, we will call these mappings “sound metaphors”.

## 1.2 Evaluation of sound guidance systems

The effectiveness of sound guidance systems can be assessed through target-reaching tasks conducted in real [19, 30, 44, 48, 49, 56] or virtual environments [25, 27, 32]. Quantitative performance evaluation typically involves measuring target-reaching times in multiple trials [25, 27, 30, 32, 44, 48, 49, 56], success rates [19, 27, 32, 44, 49], or hand trajectories [25, 27, 32]. These quantitative measures are often complemented by subjective qualitative assessments [30, 32, 56].

We will focus on a different evaluation method: Fitts' law, which is an empirical model that describes the trade-off between speed and accuracy in target-reaching tasks. This law is particularly significant in the HCI community as it is commonly used to assess mouse movements on computers.

The effectiveness of sound guidance systems is typically evaluated through target-reaching tasks in real [19, 30, 44, 48, 49, 56] or virtual environments [25, 27, 32]. Performance is measured using target-reaching times [25, 27, 30, 32, 44, 48, 49, 56], success rates [19, 27, 32, 44, 49], and hand trajectories [25, 27, 32], often complemented by qualitative assessments [30, 32, 56].

Here, we adopt a different approach: Fitts' law, an empirical model describing the speed-accuracy trade-off in target-reaching tasks. Widely used in HCI, it is particularly relevant for evaluating cursor movements in computer interfaces.

### 1.2.1 First Formulation of Fitts' Law

Fitts' law predicts the time required to point at a target based on its size and the distance between the user and the target. Figure 1 summarizes the key details of the law.

To test whether body movement speed is limited by the human nervous system's information processing capacity, Fitts created a reciprocal tapping task. Participants alternately touched two rectangular targets with a stylus [10], varying in width and distance. Fitts introduced the Index of Difficulty (ID) to quantify the task's information load:
$$ID = \log_2\left(\frac{2A}{W}\right) \qquad (1)$$

where A is the amplitude (distance from the pointer's starting position to the target center), and W is the target width. ID, measured in bits, represents the binary information needed to complete the task. Research shows a strong correlation between ID and target-reaching time, which is unaffected by the stylus's weight. Smaller and more distant targets take longer to reach, aligning with the concept of information processing.

Fitts' law can be formulated as:

$$MT = a + b * \log_2\left(\frac{2A}{W}\right) \qquad (2)$$

where MT is the movement time, a and b are empirically derived constants, with b representing the time required to process one bit of information (s/bit). The constant a represents reaction time, as defined by Fitts and Peterson [11], although its exact definition remains debated.

Initially applied to continuous movements [10], Fitts later showed that the law could be generalized to discrete movements [11].

Fitts' law is linked to a performance metric, initially called the Index of Performance (IP) by Fitts in 1954 [10], but later generalized as throughput (TP) in the HCI literature. TP quantifies the number of bits transferred per second. It was first defined as:

$$TP = \frac{ID}{MT} \qquad (3)$$

However, there is another version of TP, and the choice between them is debated in the literature, which we will discuss further.

## 1.2.2 Other Formulations of Fitts' Law

Following Fitts, researchers have attempted to enhance the predictive power of the original law by modifying it. One widely adopted version is by MacKenzie (1989), known as Shannon's version, which has become a standard in the HCI community. However, some authors reject MacKenzie's arguments for this modification [6, 7, 20]. According to [7], the correct formula is mostly a theoretical issue, as practical results show only minor differences of a few milliseconds. He also suggests the HCI community disregard alternative formulas and related research.

## 1.2.3 Fitts' Law Adapted to 3D Tasks

Fitts' law, initially developed for 1D translational movements, has been successfully applied to 2D movements and is commonly used to evaluate systems like computer mice. About 15 years ago, researchers adapted Fitts' law to analyze user movements in 3D virtual environments [47]. However, 3D tasks present challenges, including additional degrees of freedom, the absence of haptic feedback, and potential system instabilities, such as hand tremors or tracking noise. As a result, Fitts' law may not directly apply to 3D motion.

Several studies have proposed adaptations of Fitts' law for 3D movements [2, 4, 5, 36]. However, to maintain a standardized metric for evaluating and comparing sound guidance
systems, we choose to retain the original version of Fitts' law (equation 2), as none of the adaptations focus on user performance or define an appropriate formulation of TP.

### 1.2.4 Fitts' Law Adapted to Sightless Target-Reaching Tasks

Fitts' law, originally applied to sight-guided target-pointing tasks, can also be used to assess performance in sensory substitution devices for blind and visually impaired individuals. In sightless target-reaching tasks, such as those using sound or vibration, Fitts' law helps evaluate performance. For instance, in [26], participants pointed at targets with a smartphone camera, using spatialized sound where frequency varied with the angle of elevation. The study compared frequency ranges by calculating angular error and throughput (TP, equation 3), showing that higher frequencies reduced angular error but increased time. Similarly, [31] used spatialized sound (white noise beats) for target-reaching, comparing mobility and feedback conditions via TP. [21] evaluated four feedback types for 3D target-reaching (spatialized sound, verbal instructions, vibrations, and sonification), finding better TP with spatialized sound.

Thus, Fitts' law is useful for describing the performance of sound guidance systems in 3D target-reaching tasks, as it effectively explains the time required to reach targets.

### 1.2.5 Evaluating sound guidance system performance using Fitts' law parameters

In the HCI community, it's common practice to calculate the correlation between ID and MT to determine how well the data fit Fitts' law, assuming that a correlation close to 1 indicates a good fit. However, as pointed out by [6], a nonzero correlation merely indicates a linear relationship between the variables, and additional statistical tests are needed to assess its significance. Furthermore, many researchers average reach times for different conditions of A and W yielding the same ID, which renders the correlation meaningless. This assumes Fitts' law's validity before testing it, though different combinations of A and W producing the same ID should have identical reach times [17].

After validating the model against the data, Fitts' law can be used to evaluate system performance through the throughput (TP) metric, first introduced by Fitts in 1954 (equation 3). TP indicates speed, showing how many bits are transferred per second. However, in 1966, [12] proposed using the slope as a performance index. This led to an alternative version of TP:

$$TP = \left(\frac{1}{b}\right) \qquad (4)$$

While some consider equation 3 superior [28, 29, 45] because it accounts for the y-intercept (a-), counterarguments exist:

* Equation 3 simplifies a multivariate metric into a single one [37, 51];

* TP calculated with equation 3 is not invariant, contrary to [29]'s claims [37];

* TP using equation 3 depends on the ID range used, making it difficult to generalize across experimental conditions [51];

* The reciprocal of the slope (equation 4) is seen as a better measure, as it is intrinsic to the system under study and can be generalized beyond the experimental conditions [51].
In theory, both versions of throughput (TP) should yield the same value if the y-intercept (a-) were zero, which would be the case if the ID were 0, and MT = 0. However, in practice, this is never the case. The y-intercept (a-) can be attributed to reaction time [11], unavoidable delays in the psychomotor system, extra time to process feedback [12], or secondary tasks like pressing a button [29]. There is no consensus on this issue in the literature.

We align with [37] and [51] in using the reciprocal of the slope (equation 4) for our TP calculations, as this avoids conflating the two parameters, a- and b-, into a single metric. However, to present a complete view, we will report both a- and b-, along with the reaching times for each condition of size and distance.

The data suggest that TP, as derived from Fitts' law, is a suitable metric for evaluating and comparing sound guidance systems. In the next section, we describe an experiment to test Fitts' law's applicability in evaluating sound metaphors for guiding people to reach 3D targets with their hands [14]. The results of the first study confirm the applicability of Fitts' law and replicate previous work showing advantages in dissociating vertical and horizontal deviations from the target [13].

## 2 First study: spatial dissociation of vertical and horizontal dimensions

<figure><img src="image_3.png" /><figcaption>**Fig. 2** Illustration of the experimental setup (Reproduced with permission from [14]). Possible positions of the target are represented by circles. Targets can be positioned at 12 cm (red circles), 18 cm (green circles) or 27 cm (blue circles) from the starting point (the start trial button).</figcaption></figure>

We begin by revisiting the results of our previous study [14], which had two main objectives:

* Demonstrate that Fitts' law applies to reaching virtual 3D targets with the hand, guided by sound metaphors.
<figure><img src="image_4.png" /><figcaption>**Fig. 3** Spatial metrics used for metaphors UVH and DVH. The $\vec{OP}$ vector defines the participant's pointing direction. The distance to the target is calculated from P to the center of the target T. Possible positions of the target are represented by circles. Targets can be positioned at 12 cm (red circles), 18 cm (green circles) or 27 cm (blue circles) from the starting point (the start trial button). For UVH, direction deviation $\theta$ is converted into pitch, with a white noise added when OP intersects the target. For DVH, $\theta_h$ is converted into pitch, with a white noise added when $\Delta Z < r$ (i.e. the radius of the sphere).</figcaption></figure>

* Replicate a prior study [13] showing that a metaphor which dissociates the vertical and horizontal dimensions into separate auditory streams is more effective than a metaphor that does not dissociate these dimensions.

Previous studies have used Fitts' law to evaluate sound guidance systems, but they involved either spatialized sonification [26, 31] or non-spatialized sonification that failed to guide participants effectively [33], so the applicability of Fitts' law in such contexts was unclear. We aimed to fill this gap by testing Fitts' law with sound metaphors.

To balance guidance from the literature and users' cognitive load, we developed two sound metaphors and evaluated them in a virtual target-reaching task. Performance was assessed using target-reaching times and TP metrics [14]. Spatial metrics converted into sound for each metaphor are shown in Figure 3.

The first sound metaphor, called Dissociated Vertical-Horizontal (DVH), was designed as follows:

1. Horizontal position is conveyed by the angle deviation between the direction pointed by the hand and the target, transcoded into pitch on a continuous scale;

2. Vertical position is conveyed by mild binary noise added to the main auditory stream when the hand is at the same height as the target;

3. Depth is represented by sound intensity on a continuous scale.

We compared DVH with another metaphor, Unified Vertical-Horizontal (UVH), which was designed as follows:

1. Both horizontal and vertical positions are conveyed by the angle deviation between the direction pointed by the hand and the target, transcoded into pitch on a continuous scale;

2. Depth is represented by sound intensity on a continuous scale;
3. An additional mild white noise is activated when the direction pointed by the hand intersects the target.

A previous study demonstrated an advantage of the DVH metaphor, which separates the vertical and horizontal dimensions into distinct sound streams, in a 3D target-reaching task where participants walked and reached the target with their hand [13]. In the current study, the task was adapted to the Fitts' paradigm, with participants reaching virtual targets within their reach. These targets were spheres of varying sizes (5.33, 8 or 12 cm) and distances from a starting button (12, 18 or 27 cm) in a 3D virtual space (see Figure 2 [14]). Participants searched for the target with their index finger, using the sound metaphors (UVH and DVH) to estimate its position. Each participant performed 3 different test blocks for each target size, for a total of 9 test blocks. A test block consisted of a series of 16 consecutive targets to be reached, selected semi-randomly so that the participant reached half of all possible targets, with an equal number of targets at each distance, without repeating the same target twice for the same target size. This procedure was repeated twice, one for each sound metaphor. Each session lasted one hour, so the experiment lasted two hours in total. For more details on materials and methods, see [14] and Figures 2 [14] and 3.

Five sighted people took part in the experiment (19.2 ± 0.9 years old). On average, participants took 10.21 ± 8.26 seconds with the UVH metaphor, and 8.09 ± 5.09 seconds with the DVH metaphor. A Cox analysis showed participants were significantly faster with DVH (z = 6.14, p < .001).

Fitts' analysis revealed that, in both metaphor conditions, MT increased linearly with ID. The regression analysis results, including complete equations and TP values for each metaphor, are presented in Table 1. Table 2 includes further, with average target-reaching time for each condition of difficulty of UVH and DVH metaphores.

Key points from the results include:

- Fitts' law is a useful model for evaluating target reaching, even with non-spatialized sonification feedback (both DVH and UVH).

- The study replicated [14]'s finding of faster reaching times with DVH compared to UVH.

- Fitts' law analysis confirmed that performance (measured with TP) was better with the DVH metaphor than with UVH.

For a detailed discussion on the differences between the two metaphors, see [13, 14]. The strategy most naturally adopted by participants appears to be searching for the target one dimension at a time—first addressing the angular gap on the horizontal plane, then focusing on height and reducing the distance. With DVH, dimensions are orthogonal, allowing for easier and separate interpretation. With UVH, the same strategy is possible, but it comes at the cost of higher cognitive load. As a result, participants performed better in the target-reaching task with the DVH metaphor compared to the UVH metaphor.

<table><caption>**Table 1** Results of the linear regression analysis for DVH and UVH metaphors.</caption><thead><tr><td>Metaphor</td><td>Regression equation</td><td>R<sup>2</sup></td><td>F</td><td>p</td><td>TP (bits/s)</td></tr></thead><tbody><tr><td>DVH</td><td>2.24(±0.72) + 2.70(±0.32)ID</td><td>0.91</td><td>73.27</td><td>&lt;.001</td><td>0.37</td></tr><tr><td>UVH</td><td>-0.33(±1.87) + 4.88(±0.82)ID</td><td>0.83</td><td>35.30</td><td>&lt;.001</td><td>0.20</td></tr></tbody></table>
<table><caption>**Table 2** Average target-reaching time for each condition of UVH and DVH metaphors.</caption><thead><tr><th rowspan="2">A (cm)</th><th rowspan="2">W (cm)</th><th rowspan="2">ID (bits)</th><th colspan="2">MT (s)</th></tr><tr><th>UVH</th><th>DVH</th></tr></thead><tbody><tr><td>12</td><td>5.33</td><td>2.17</td><td>12.06</td><td>9.03</td></tr><tr><td>12</td><td>8</td><td>1.58</td><td>7.89</td><td>6.78</td></tr><tr><td>12</td><td>12</td><td>1.00</td><td>5.70</td><td>5.27</td></tr><tr><td>18</td><td>5.33</td><td>2.75</td><td>13.28</td><td>10.15</td></tr><tr><td>18</td><td>8</td><td>2.17</td><td>8.91</td><td>7.47</td></tr><tr><td>18</td><td>12</td><td>1.58</td><td>6.72</td><td>6.04</td></tr><tr><td>27</td><td>5.33</td><td>3.34</td><td>17.88</td><td>11.50</td></tr><tr><td>27</td><td>8</td><td>2.75</td><td>10.66</td><td>9.24</td></tr><tr><td>27</td><td>12</td><td>2.17</td><td>9.07</td><td>7.30</td></tr></tbody></table>

This demonstrates that the most effective sound metaphor is one that aligns with the user's cognitive limitations and the most intuitive strategy for reaching the target. Since participants tend to focus on one dimension at a time, performance and comfort could be improved by creating a sequential sound metaphor that presents the deviation to the target one dimension at a time. This will be explored in the third study (see Section 4). Before that, in Section 3, we compare different metaphors that encode one spatial dimension at a time.

# 3 Second Study: Comparison of metaphors in one dimension

## 3.1 Introduction

In the previous experiment, we used Fitts' law's Throughput (TP) as a metric to compare different sound metaphors guiding a person deprived of sight towards a vertical 3D target. The results showed that Fitts' law can effectively compare sound metaphors for guiding people to 3D targets, with performance being better when the sound metaphor explicitly dissociates the vertical and horizontal dimensions (DVH).

This dissociation aligns with the common strategy users adopt: searching for the target along one spatial dimension at a time. Since the dimensions are orthogonal, they can be interpreted separately, making this approach intuitive. To create a more effective device that better adapts to both the human user and the task, we now aim to compare DVH with a sound metaphor that dissociates the dimensions temporally. In this model, information about deviations from the target along each dimension will be presented sequentially, rather than all at once. This method tailors the amount of information transmitted to the real-time needs of the user, aiming to reduce cognitive load. It also aligns with the way voice guidance systems, commonly used in devices for the visually impaired, typically provide instructions one dimension at a time (e.g., left, right, forward, backward). By applying this principle to sound metaphors, we can combine the intuitive clarity of verbal instructions with the efficiency of sound.

To test the effectiveness of this sequential metaphor—referred to as *DVH<sub>seq</sub>*—we will compare it with our previous DVH metaphor, which dissociates dimensions spatially and presents them in parallel (*DVH<sub>par</sub>*). Both sound metaphors will use the same spatial metric conversions, with the only difference being whether the information is presented sequentially or in parallel. However, our previous DVH metaphor is not compatible with a sequential
mode. For example, in the sequential model, when information about the vertical dimension is given, there would be no sound unless the finger is at the correct height. This could create confusion, as silence could indicate either that the finger is in the wrong position or that the system is malfunctioning. Therefore, we need to select sound parameters that are compatible with both sequential and parallel metaphors.

To do this, we conducted two studies that focused on the vertical and horizontal dimensions separately.

For the sequential metaphor, it is essential that participants can easily transition from one dimension to the other. Therefore, the chosen sound parameters for each dimension must be intuitive and simple enough to be quickly interpreted. To assess this, we compared four sound metaphors that code the difference in height between the finger and the target for the vertical dimension:

* **Continuous**: This metaphor continuously converts the distance between the finger and the target on the vertical plane into pitch. As the distance decreases, the sound becomes higher-pitched. This is one of the most commonly used sound parameters in studies on non-spatialized sonification for sound guidance [8]. The pitch range used (110 Hz to 440 Hz) has been chosen to avoid excessively high-pitched sounds, which can be unpleasant [23].

* **Discrete**: A constant high-pitched sound is generated when the target is above the hand, and a low-pitched sound is generated when the target is below the hand. This metaphor relies on the natural tendency of humans to associate higher-pitched sounds with higher spatial positions and lower-pitched sounds with lower positions [9, 18, 35, 40–42].

* **Shepard**: This metaphor generates a Shepard-Risset auditory illusion, where the sound appears to rise or fall indefinitely in pitch. Several tones spaced an octave apart are played simultaneously. As the illusion rises, the lowest-pitched tones gradually decrease in volume while the highest-pitched tones increase in volume. This creates the effect of the sound continuously rising or falling. In our metaphor, a rising pitch is used when the target is above the hand, and a falling pitch when the target is below. This is based on the natural association between pitch changes and hand movement direction.

* **Formant**: In this metaphor, the vowel sound [a] is generated when the target is below the hand, and [o] when the target is above the hand. This metaphor takes advantage of the natural association between these vowels and corresponding vertical positions in French: the vowel [a] is linked with the French word *bas* (meaning low), and [o] is linked with *haut* (meaning high). This metaphor resembles verbal instructions, which are used in many guidance systems [1, 19, 21, 30, 48, 49, 56] and are more natural and intuitive, while still being faster to interpret than full words.

In total, we have four sound metaphors of varying complexity and precision. The *Continuous* metaphor has the advantage of accurately indicating the distance to the target, thanks to the continuous pitch and the high perceptual resolution of sound. The other metaphors offer binary information about the target's direction, with the binary coding potentially being easier to interpret than continuous sounds. Among these, the *Shepard* metaphor is the most complex, but could be very intuitive if participants can associate the direction of the pitch change with the direction of their hand movement towards the target. The *Discrete* and *Formant* metaphors, while simpler, should still be intuitive due to the natural spatial associations of the sounds used.
For the horizontal dimension, we compared three sound metaphors encoding the angular
difference between the direction pointed by the hand and the direction of the target:

* *Continuous*: Similar to the vertical dimension, this metaphor continuously encodes the angular deviation in pitch. The pitch increases or decreases depending on the angular distance between the hand and the target.

* Stereo: A constant-pitched sound is generated in the right ear when the target is to the right, and in the left ear when the target is to the left. This metaphor is a form of spatialized sonification, which is often considered more intuitive and requires less time to learn than non-spatialized sonification. However, since spatialized sonification reproduces acoustic cues to simulate the target's position, it faces the same limitations as auditory spatial perception. The perception of a sound source in the horizontal plane is known to be relatively imprecise [34]. Thus, this metaphor presents a simplified binary cue: a sound in one ear or the other.

* *Continuous + Stereo*: This metaphor combines the previous two. The pitch of the sound varies continuously according to the angular distance, and the sound is presented in either the right ear (if the target is to the right) or in the left ear (if the target is to the left).

The *Stereo* metaphor offers the advantage of directly indicating the direction of the tar-
get, unlike the *Continuous* metaphor. Moreover, being a binary signal, it may be easier to
interpret than the *Continuous* metaphor. However, the *Continuous* metaphor provides more
precise information by encoding the distance to the target in pitch. The *Continuous + Stereo*
combination attempts to blend the advantages of both, though there is a possibility that the
combination could create an excess of information, making it more complicated to interpret
than the individual metaphors.

For both the vertical and horizontal tasks, the different metaphors offer distinct advan-
tages and disadvantages, ranging from more precise to simpler, more intuitive metaphors. To
determine which metaphor is most effective and comfortable for participants, we employed
a paradigm where metaphors were presented randomly from trial to trial for both tasks. This
design allows us to assess which characteristics—simplicity, precision, binary or continuous
signals, direction, or distance—are most beneficial in the context of a sequential metaphor.

**3.2 Materials and method**

**3.2.1 Participants**

Two separate groups of participants tested the metaphors for the vertical and horizontal dimensions.

For the vertical dimension, 14 participants without visual impairments (mean age: 20.3 ± 1.8 years) took part in the experiment. All participants were psychology students and received a bonus point on a course of their choice in exchange for their participation.

For the horizontal dimension, 11 participants without visual impairments (3 men, 8 women, mean age: 23.3 ± 4.5 years) were recruited. Seven of them were psychology students, receiving the same bonus point incentive. The remaining participants were students from other disciplines, recruited by the laboratory on a voluntary basis.

All participants gave their informed consent before participating in the study.
### 3.2.2 Materials

The tests were conducted in the form of a virtual game, where participants were required to reach a specific height (vertical dimension) or direction (horizontal dimension) with their index finger.

For the horizontal task, participants wore a harness on their back and a pointer on their finger, while for the vertical task, only the pointer on the finger was used. These objects were equipped with reflectors tracked by a Vicon motion capture system. The coordinates of these objects were used to determine the position of two anatomical points: the middle of the torso (point O) and the tip of the finger (point P). The coordinates were transmitted from the Vicon Tracker acquisition computer to the pilot computer via the Virtual Reality Peripheral Network (VRPN) protocol.

In the pilot computer, our C++ control software:

1) Immersed the participant in the virtual environment with the target, utilizing the Open-Scene Graph (OSG) 3-D toolkit;

2) Calculated spatial metrics for sound conversions;

3) Transmitted these metrics to the PureData sound system running on the same computer, which synthesized the sounds and delivered them to the participant through a bone conduction headset (Aftershokz Sportz3) connected to a receiver box (Mipro MI909R);

4) Managed the experimental protocol.

A button was fixed on a microphone stand at a height of 1.25 m, serving as both the start button and a reference for calculating distance from the target.

For the vertical task, each target was represented as a horizontal plane. The width and distance from the starting point varied, resulting in 72 possible conditions (6 heights, 3 sizes, and 4 metaphors, see Table 4). Each participant completed 3 test blocks, each containing 24 targets (72 trials in total), one for each condition. Due to an error in creating the blocks for the horizontal task, participants did not complete the A = 55° / W = 3.75° condition for the Stereo metaphor. This condition was therefore excluded from the reaching-time analyses for the other metaphors.

For the horizontal task, each target represented a direction to be reached, with varying width and angle, resulting in 54 conditions (6 angles, 3 sizes, and 3 metaphors, see Table 6). Participants completed 3 test blocks, each containing 18 targets (54 trials in total), one per condition.

For both tasks, each block contained an equal number of trials for each target size, height or angle, and metaphor. The trials were presented in a semi-random order to ensure that, in two successive trials, the size, height, and metaphor varied.

### 3.2.3 Participants’ Equipment

For the vertical dimension, the participant was equipped with: 1) a real object attached to their index finger defining point P at the tip of the finger; 2) bone conduction headphones connected to 3) a receiver box, as in the previous experiment. A button used to trigger the trials was fixed to a microphone stand, 1.25 metres above the ground.
<figure><img src="image_5.png" /><figcaption>Fig. 4 Illustration of the spatial metrics used in the vertical and horizontal tasks. For both metaphors, S represents the button that participants had to press to start a trial. A and W represent the distance from the starting point S, and W the width of the target.</figcaption></figure>

For the horizontal dimension, the participant was equipped with the same equipment as for the vertical dimension with a real object mounted on a harness attached to their back, defining point O at the center of their torso.

### 3.2.4 Spatial Metrics

Spatial metrics for the two one-dimensional tasks are represented in Figure 4.

For the vertical dimension, a single spatial metric was used to generate the sound metaphors:

$$ \Delta Z = ZP-ZT $$

This corresponds to the projection of the distance between the target T and the finger P onto the vertical Z axis. A positive value of ΔZ indicates that the target is above the finger, while a negative value indicates that the target is below the finger.

For the horizontal dimension, a single spatial metric was also used to generate the sound metaphors: θh. This corresponds to the angular distance between the direction of the target and the direction OP (torso-index). A positive value of θh means that the target is to the left of the pointed direction, a negative value means that the target is to the right.

### 3.2.5 Sound metaphors

For the vertical dimension, we created four sound metaphors:

* **Continuous**: a sinusoid is generated, the frequency of which varies as a function of ΔZ on a continuum ranging from fmin = 110 Hz for |ΔZ| = 2 m, to fmax = 440 Hz for |ΔZ| = 0 m. In this way, f is within the audible spectrum while avoiding frequencies that are too high, which are unpleasant.

* **Discrete**: if ΔZ < 0, a constant high-pitched sound at 440 Hz is generated. If ΔZ > 0, a constant low-pitched sound at 110 Hz is generated.
* Shepard*: a Shepard-Risset illusion is generated. If ΔZ <0, then the illusion is rising: the sound frequency gives the impression of rising continuously. If ΔZ >0, then the illusion is descending: the frequency of the sound gives the impression of continuously descending.

*Formant*: if ΔZ <0, then the vowel [o] is generated continuously. If ΔZ >0, then the vowel [a] is generated continuously.

For the horizontal dimension, we compared 3 sound metaphors:

*Continuous*: a sinusoid is generated, whose frequency varies as a function of θh, on a continuum ranging from fmin = 110 Hz for |θh| = 180, to fmax = 440 Hz for |θh| = 0.

*Stereo*: a sound with a constant frequency of 220 Hz is generated in the right ear when θh <0, and in the left ear when θh >0.

*Continuous + Stereo*: combination of the two previous metaphors. A sinusoid is generated and varies in the way described above, either in the right ear or in the left ear as in the *Stereo* metaphor.

For *Continuous* metaphors and *Continuous + Stereo* metaphors, the conversion of θh or ΔZ (X in the following equations) into pitch follows Stevens' [46] psychophysical law, a power law function:

$$log(f) = Af * log(X) + Bf \quad (5)$$

$$Af = \frac{log(fmax) - log(fmin)}{log(Xmin) - log(Xmax)} = 0.44 \quad (6)$$

$$Bf = log(fmin) - Af * log(Xmax) = 1.30 \quad (7)$$

Additional audio signals were used and will be described in the following.

### 3.2.6 Protocol

The experiment took place in the motion capture space described earlier. Participants were tasked with finding several targets presented successively, guided by the metaphors described in the previous sections.

**Procedure overview:** The experiment began with a training phase, where participants went through a training block for each sound metaphor (four in the vertical experiment, three in the horizontal experiment). The order of the training blocks was counterbalanced. Each training block consisted of 18 targets (3 target sizes * 6 target heights or directions). During this phase, the experimenter could communicate with the participant to clarify the sound metaphors if necessary.

After completing the training, the participants proceeded to the 3 experimental blocks. Only the data from these experimental blocks were used in the analyses. The total duration of the experiment was approximately 30 minutes.

**Experimental Block Procedure:**

1. The participant, equipped as previously described, stood in front of the start button (S).

2. The test began when the participant pressed the start button with their index finger. A sound indicated the beginning of the trial.
3. From that point, the participant heard the sound feedback corresponding to the current trial condition. The sound feedback varied according to the position of their finger relative to the target.

4. The trial continued until the participant's finger remained in the target area for 400 ms (to ensure that the target wasn't accidentally reached). Once this occurred, a victory sound played, the sound feedback was deactivated, and the trial ended.

5. If the participant failed to reach the target within 10 seconds, a defeat bell sounded, and the trial was stopped.

6. After each trial, the participant pressed the start button to begin the next trial. This process continued until the end of the block, which was signaled by the experimenter.

**Post-Experiment Questionnaire:** At the end of the session, the participant completed a questionnaire to assess their experience. They were asked to:

* Share their general opinion of the experience.

* Describe any difficulties they encountered during the experiment.

* Indicate whether the similarity between different sound metaphors caused any confusion.

* Rate each metaphor on a scale from 1 to 7 for three criteria: efficiency, comfort and ease.

### 3.2.7 Analysis

The analyses were performed using RStudio (RStudio 2023.12.1+402) and Jamovi (Jamovi 2.3.29).

For each dimension, we evaluated and quantitatively compared the performance of the different metaphors in several ways.

As our response variable, target-reaching time, is a duration with an asymmetric distribution, we used a Cox analysis to compare reaching times between conditions. A Cox model allows us to analyze repeated measures without averaging the data for each participant, so it takes into account intra- and inter-participant variability [24]. To carry out this analysis, we used the coxph function in the Survival package of the RStudio software.

We also used Fitts' law. To assess whether Fitts' law applied in these conditions of sound metaphors, we performed a linear regression analysis for each metaphor on Jamovi, as well as an F-test to attest the significance of the linear relationship between MT and ID. The inverse of the slope was used to calculate the TP for each metaphor.

Finally, the metaphors were qualitatively compared, using scores given by the participants on three criteria, on a scale of 1 to 7. For each of the three criteria, we want to explain the form and intensity of the relationship between the score and the metaphor. We are studying the probability distribution of the values taken, and we therefore consider the score as an ordinal categorical variable. We ran an ordinal regression to compare the scores as a function of the metaphor, using the clm() function in the ordinal package of the RStudio software.

Additional analyses of the trajectory of the finger during trial were conducted and will be detailed in next sections.
## 3.3 Results for the vertical dimension

### 3.3.1 Reaching Times

Average target-reaching time was 3.68 ± 1.45 s for the *Formant* metaphor, 4.35 ± 2.17 s for the *Continuous* condition, 4.18 ± 2.13 s for the *Discrete* condition (M = 4.18, SD = 2.13), and 4.46 ± 2.33 s for the *Shepard* condition.

Target-reaching times were compared using a Cox model. The results show that reaching times were shorter for the *Formant* condition than for the *Continuous* condition (z = 4.12, p < .001), for the *Formant* condition than for the *Discrete* condition ( z = 3.28, p < .01) and for the *Formant* condition than for the *Shepard* condition (z = -4.85, p < .001). The other comparisons were not significant.

Here, nine IDs are calculated according to the original formulation of Fitts' law. To determine whether a Fitts-type response occurs in this experiment, we run a linear regression between time to target (MT) and ID for each metaphor. MT is averaged for each condition of size and distance to the target (see Table 4). The fit of the model to our data is confirmed, with significant correlations between ID and MT for each metaphor (see Table 3). The calculated TP for each sound metaphor is reported in Table 3.

<table><caption>**Table 3** Results of the linear regression analysis for each metaphor of the vertical task.</caption><thead><tr><th>Metaphor</th><th>Regression equation</th><th>R<sup>2</sup></th><th>F</th><th>p</th><th>TP (bits/s)</th></tr></thead><tbody><tr><td>Shepard</td><td>MT = 1.89(±0.84) + 0.90(±0.28) ID</td><td>0.60</td><td>10.54</td><td>0.014</td><td>1.11</td></tr><tr><td>Continuous</td><td>MT = 2.12(±0.40) + 0.79(±0.13) ID</td><td>0.84</td><td>35.67</td><td>&lt;.001</td><td>1.27</td></tr><tr><td>Formant</td><td>MT = 2.00(±0.56) + 0.59(±0.18) ID</td><td>0.60</td><td>10.46</td><td>0.014</td><td>1.69</td></tr><tr><td>Discrete</td><td>MT = 1.53(±0.54) + 0.90(±0.18) ID</td><td>0.82</td><td>25.25</td><td>0.002</td><td>1.11</td></tr></tbody></table>

### 3.3.2 Qualitative metric

We ran an ordinal regression to compare scores by metaphor, which revealed no significant differences between metaphors for any of the criteria.

<table><caption>**Table 4** Average target-reaching time for each condition for the four metaphors of the vertical task.</caption><thead><tr><th rowspan="2">A (cm)</th><th rowspan="2">W (cm)</th><th rowspan="2">ID (bits)</th><th colspan="4">MT (s)</th></tr><tr><th>Shepard</th><th>Continuous</th><th>Formant</th><th>Discrete</th></tr></thead><tbody><tr><td>15</td><td>3.75</td><td>3</td><td>6.06</td><td>4.51</td><td>4.75</td><td>5.17</td></tr><tr><td>15</td><td>7.5</td><td>2</td><td>4.56</td><td>3.54</td><td>3.35</td><td>3.69</td></tr><tr><td>15</td><td>15</td><td>1</td><td>2.29</td><td>3.13</td><td>2.86</td><td>2.48</td></tr><tr><td>30</td><td>3.75</td><td>4</td><td>5.61</td><td>5.58</td><td>4.74</td><td>4.87</td></tr><tr><td>30</td><td>7.5</td><td>3</td><td>4.71</td><td>5.18</td><td>3.48</td><td>3.78</td></tr><tr><td>30</td><td>15</td><td>2</td><td>2.93</td><td>3.06</td><td>2.30</td><td>2.78</td></tr><tr><td>45</td><td>3.75</td><td>4.58</td><td>5.83</td><td>5.38</td><td>4.56</td><td>5.90</td></tr><tr><td>45</td><td>7.5</td><td>3.58</td><td>3.90</td><td>4.80</td><td>3.62</td><td>4.45</td></tr><tr><td>45</td><td>15</td><td>2.58</td><td>4.26</td><td>3.99</td><td>3.39</td><td>4.53</td></tr></tbody></table>
### 3.3.3 Relative direction of the first movement

In addition to the main analyses, we wanted to assess whether the metaphors helped participants direct their hand immediately towards the target. A binomial test was performed to determine if the proportion of first moves towards or away from the target differed from 0.5 for each metaphor. Three metaphors (*Shepard*, *Formant*, *Discrete*) should allow participants to identify the target's direction immediately. Thus, we expected more first movements towards the target for these metaphors. A one-tailed test was used for the *Shepard*, *Formant*, and *Discrete* metaphors, and a two-tailed test for the *Continuous* metaphor. The proportion of first moves towards the target was significantly higher than 50% for the *Discrete* metaphor (p <.001), but not for the others.

### 3.3.4 Absolute direction of the first movement

We aimed to check for any bias in the direction of the first movement (whether participants tended to move upwards or downwards, regardless of the target position). A two-tailed binomial test was conducted to determine if the proportion of first upward or downward movements differed from 0.5 for each metaphor. The results showed p <.001 for *Continuous*, p = .006 for *Discrete*, p = .002 for *Formant*, and p <.001 for *Shepard*. The proportion of first downward movements was significantly greater than 50% for all metaphors.

### 3.3.5 Changes of direction during the trial

An ANOVA was conducted to compare the number of direction changes between metaphors. The results revealed a significant difference in the number of direction changes depending on the metaphor (F(3, 1002) = 5.02, p = 0.002). Post hoc analyses showed fewer direction changes with the *Continuous* metaphor compared to the *Discrete* metaphor (t(1003) = -3.84, p <.001). The other comparisons were not significant.

### 3.3.6 Number of times the target was overshot during the trial

An ANOVA was conducted to compare the number of overshoots between metaphors. The results showed a significant difference in overshoots depending on the metaphor (F(3, 1000) = 6.23, p <.001). Post hoc analyses revealed fewer overshoots with the *Continuous* metaphor compared to the *Shepard* (t(1003) = 3.54, p = 0.002), *Formant* (t(1003) = -3.24, p = 0.007), and *Discrete* (t(1003) = -3.73, p = 0.001) metaphors. Other comparisons were not significant.

## 3.4 Results for the horizontal dimension

### 3.4.1 Reaching Times

Average reaching times were 2.76 ± 0.84 s for the *Continuous* + *Stereo* metaphor, 3.72 ± 1.74 s for the *Continuous* metaphor, and 3.69 ± 1.56 s for the *Stereo* metaphor.

Target-reaching times were compared using a Cox model. The results show that the reach times were shorter for the *Continuous* + *Stereo* metaphor than for the *Continuous* metaphor (z = 7.22, p <.001), and shorter for the *Continuous* + *Stereo* metaphor than for the *Stereo* metaphor (z = 7.22, p <.001). The comparison between *Continuous* and *Stereo* was not significant.
To determine whether a Fitts-type response occurs in this experiment, we run a linear regression between time to target (MT) and ID for each metaphor. MT is averaged for each condition of size and distance to the target (see Table 6). The fit of the model to our data is confirmed, with significant correlations between ID and MT for each metaphor (see Table 5).

<table><caption>**Table 5** Results of the linear regression analysis for each metaphor of the horizontal task.</caption><thead><tr><th>Metaphor</th><th>Regression equation</th><th>R<sup>2</sup></th><th>F</th><th>p</th><th>TP (bits/s)</th></tr></thead><tbody><tr><td>Continuous</td><td>MT = 0.81(±0.70) + 0.78(±0.19) ID</td><td>0.74</td><td>17.35</td><td>.006</td><td>1.33</td></tr><tr><td>Stereo</td><td>MT = 0.58(±0.49) + 0.83(±0.13) ID</td><td>0.87</td><td>40.27</td><td>&lt; .001</td><td>1.20</td></tr><tr><td>Continuous + Stereo</td><td>MT = 0.83(±0.24) + 0.56(±0.06) ID</td><td>0.93</td><td>77.28</td><td>&lt; .001</td><td>1.79</td></tr></tbody></table>

<table><caption>**Table 6** Average target-reaching time for each condition for the four metaphors of the horizontal task.</caption><thead><tr><th rowspan="2">A (°)</th><th rowspan="2">W (°)</th><th rowspan="2">ID (bits)</th><th colspan="3">MT (s)</th></tr><tr><th>Continuous</th><th>Stereo</th><th>Continuous + Stereo</th></tr></thead><tbody><tr><td>30</td><td>3.75</td><td>4.00</td><td>3.75</td><td>3.95</td><td>2.99</td></tr><tr><td>30</td><td>7.5</td><td>3.00</td><td>3.07</td><td>3.75</td><td>2.41</td></tr><tr><td>30</td><td>15</td><td>2.00</td><td>2.61</td><td>2.15</td><td>1.99</td></tr><tr><td>55</td><td>7.5</td><td>3.87</td><td>3.33</td><td>3.58</td><td>2.89</td></tr><tr><td>55</td><td>15</td><td>2.87</td><td>2.77</td><td>2.51</td><td>2.34</td></tr><tr><td>80</td><td>3.75</td><td>5.42</td><td>5.54</td><td>5.16</td><td>3.95</td></tr><tr><td>80</td><td>7.5</td><td>4.42</td><td>3.75</td><td>4.05</td><td>3.13</td></tr><tr><td>80°</td><td>15</td><td>3.42</td><td>4.30</td><td>3.53</td><td>3.07</td></tr></tbody></table>

### 3.4.2 Qualitative metric

We ran an ordinal regression to compare scores by metaphor, which revealed a better score for *Continuous + Stereo* than for *Continuous* metaphor and *Stereo* metaphor, for the efficiency criterion (z = 2.74, p = .02). There was no significant difference between *Continuous* and *Stereo* for this criterion, and no difference between the three metaphors for the ease and comfort criteria.

### 3.4.3 Relative direction of the first movement

A binomial test was performed to assess whether the proportion of first moves towards or away from the target differed from 0.5 for each metaphor. The *Stereo* and *Continuous + Stereo* metaphors typically allow participants to determine the target direction immediately, so we expected more first movements towards the target. A one-tailed test was performed for *Stereo* and *Continuous + Stereo*, and a two-tailed test for *Continuous*. The binomial tests yielded p = .04, p < .001, and p < .001 for *Continuous*, *Continuous + Stereo*, and *Stereo*, respectively. The proportion of first movements towards the target was significantly higher than 50% for all three metaphors.
### 3.4.4 Absolute direction of the first movement

A two-tailed binomial test was performed to assess whether the proportion of first leftward or rightward movements differed from 0.5 for each metaphor. The binomial tests showed that the proportion of first leftward movements was not significantly different from 50% for any of the metaphors.

### 3.4.5 Number of changes of direction during the trial

An ANOVA was performed to compare the number of direction changes across metaphors. The results showed a significant difference in the number of direction changes during the test depending on the metaphor (F(2, 591) = 18, p <.001). Post hoc analyses revealed that the number of direction changes was lower for the *Continuous + Stereo metaphor* than for the *Continuous* metaphor (t(591) = 4.87, p <.001), and lower for the *Continuous + Stereo* than for the *Stereo* metaphor (t(591) = -5.48, p <.001). The comparison between *Continuous* and *Stereo* was not significant.

### 3.4.6 Number of times the target was overshot during the trial

An ANOVA was performed to compare the number of overshoots across metaphors. The results showed a significant difference in the number of overshoots depending on the metaphor (F(2, 591) = 20.6, p <.001). Post hoc analyses revealed that the number of overshoots was lower for the *Continuous + Stereo* metaphor than for the *Stereo* metaphor (t(591) = -5.89, p <.001), and lower for the *Continuous* metaphor than for the *Stereo* metaphor (t(591) = -5.17, p <.001). The comparison between *Continuous + Stereo* and *Continuous* was not significant.

## 3.5 Discussion for the vertical dimension

We compared four metaphors for encoding the vertical dimension in a 1D target-reaching task.

The results showed that the *Formant* metaphor was superior in terms of time to target and TP. It encodes the dimension in a binary form, making it easy to interpret and linguistically intuitive, which allows for quicker understanding than verbal instructions. However, participant feedback showed no significant differences in appreciation between the metaphors. Opinions were divided, possibly due to confusion between metaphors during the test phase. We presented the metaphors randomly to identify the most intuitive metaphor, which would be easiest to interpret despite distractions. The metaphors were designed to be distinct, but confusion between *Discrete* and *Formant* may have occurred. Although both vowels [o] and [ɑ] were generated with the same fundamental frequency, some participants misperceived [o] (higher target) as lower-pitched and [ɑ] (lower target) as higher-pitched, confusing it with the *Discrete* metaphor. This is because the perception of vowel pitch is not only linked to the fundamental frequency, but also to the position of the formants. A formant is a concentration of acoustic energy in certain specific frequency bands that results from resonances in the human vocal tract during vowel production. They play an essential role in vowel recognition. The first and second F1 and F2 formant of the vowel [o] are lower than the F1 and F2 formant of the vowel [ɑ].
To investigate this, we analyzed target-reaching times during the training phase. Results showed that reaching times were shorter for *Discrete* than for *Continuous*, *Shepard*, and *Formant* metaphors. A significant time difference between the training and test phases was found only for the *Discrete* metaphor, suggesting confusion with *Formant*. However, trajectory analysis revealed that participants were more likely to start in the right direction with *Discrete*, unlike *Formant*. Another explanation is a confusion *Discrete* with *Continuous*. The *Discrete* metaphor enables the participant to automatically go in the right direction. Participants confusing *Discrete* with *Continuous* would expect to hear the sound change continuously as a function of their movement and would thus hesitate when not hearing the change. This result is supported by an analysis showing that participants changed direction more often during the trial with the *Discrete* metaphor than with the *Continuous* metaphor.

The Continuous metaphor allowed participants to overshoot less frequently, as it provided distance to the target, aiding precise stopping. In contrast, metaphors that only provide the direction towards the target theoretically allow the movement to be directed directly towards the target (which didn't work here because of confusion between metaphors), but not to adapt the speed. As a result, participants have to pass the target to hear a change in sound and know its position. Despite this, the performance of the *Continuous* metaphor in terms of reaching the target is inferior to that of the *Formant* metaphor (and *Discrete* metaphor if we take into account the training phase). The precision of the *Continuous* metaphor did not surpass the simplicity of the binary metaphors.

Participants had difficulty with the *Shepard* metaphor, despite it providing the correct direction. Many struggled with the rising and falling frequencies, often starting in the wrong direction, and it took longer for them to correct their mistake.

A bias was also observed in the first movement direction, with most participants moving downward, possibly due to comfort or arm fatigue from pressing the start button. This bias may explain why participants did not move directly towards the target with *Formant* and *Shepard* metaphors.

In conclusion, simple, intuitive binary sounds are preferable to continuously varying sounds, which are more precise but harder to interpret, or complex binary sounds like the Shepard-Risset illusion. Careful control of metaphor similarity is crucial to avoid misinterpretation.

## 3.6 Discussion for the horizontal dimension

We compared three sound metaphors in a 1D horizontal target-reaching task. One metaphor used a continuous frequency change to represent the difference in angle between the direction pointed at and the target (*Continuous*). Another metaphor indicated the target's relative direction in a binary manner using stereo sound (*Stereo*), and the third combined these two metaphors (*Continuous* + *Stereo*).

The results showed better performance with the *Continuous + Stereo* metaphor, including faster target reach times, higher TP, and better participant ratings. This was reflected in the finger trajectory analysis, consistent with the properties of each metaphor.

For most trials, metaphors providing direction to the target (Stereo and *Continuous + Stereo*) allowed the finger to point directly toward the target. Metaphors providing distance (*Continuous* and *Continuous + Stereo*) reduced overshooting, resulting in fewer instances where participants passed through the target without validating it. The *Continuous + Stereo*
metaphor combined these advantages, reducing the need for direction changes compared to
Continuous or Stereo alone.

The Stereo metaphor is a simplified form of spatialized sound, offering intuitive and natu-
ral direction cues. Although spatialized sound is generally less accurate, the binary form used
here makes it easier to detect the target’s position by shifting sound from one ear to the other.

The Continuous metaphor, while more precise, requires more cognitive effort to inter-
pret. It provides distance to the target, allowing participants to adjust their movement speed
and avoid overshooting. However, it requires an initial movement in a random direction to
determine the target’s position.

Combining these metaphors did not overwhelm users. Instead, it created a helpful
redundancy, improving both efficiency and user experience.

4 Third study: temporal dissociation of vertical and
horizontal dimensions

**4.1 Introduction**

This paper explores the best sound metaphor for guiding a hand toward a target, aiming to
create a device that is both accurate and user-friendly, considering cognitive limitations and
task requirements.

The first study demonstrated the advantage of spatially dissociating vertical and horizontal
dimensions when guiding visually impaired people to virtual targets. This separation allows
participants to focus on one spatial dimension at a time, leading to faster and more intuitive
target reaching. This suggests that metaphors presenting dimensions separately in time may
be simpler and better suited to the most intuitive task strategy.

The second study compared various metaphors for guiding movement in the vertical and
horizontal dimensions. Results showed that using vowels for vertical positioning improved
speed, while a continuous pitch coding of direction deviation in stereo was faster than pitch
or stereo coding alone for the horizontal dimension.

Building on these results, we developed two metaphors: one dissociating vertical and
horizontal dimensions spatially and temporally (*DVH<sub>seq</sub>), and one dissociating them only
spatially (*DVH<sub>par</sub>). These metaphors are compared in a 3D target-reaching task similar to the
one in the first study [14].

**4.2 Materials and method**

**4.2.1 Participant**

Ten participants (six men and four women, M = 29.7 ± 10.8 years old) were recruited for this
study. All participants gave their informed consent prior to participating in the study.

**4.2.2 Materials**

The experiment was carried out in the same motion capture space as described in previous
studies. The target configuration, target size, and distance from the target are the same as in
the first study (see Figures 2 from [14] and see [14]). The participant equipment was a harness
and a pointer on the index finger as in the first study. The spatial metrics are identical to those used for the 1D tasks in the second (see Figure 4).

### 4.2.3 Sound Metaphors

* $DVH_{seq}$: when $|\theta_h| > 3°$, a sinusoid is generated, the frequency of which varies as a function of $\theta_h$ on a continuum ranging from $fmin = 110$ Hz for $|\theta_h| = 90°$, to $fmax = 330$ Hz for $|\theta_h| = 0°$. When $|\theta_h| < 3°$, the sinusoid stops and a vowel is generated, of a constant pitch of 330 Hz. If $\Delta Z < 0$, then the vowel [o] is generated. If $\Delta Z > 0$, then the vowel [a] is generated.

* $DVH_{par}$: a vowel is generated, the frequency of which varies as a function of $\theta_h$ on a continuum ranging from $fmin = 110$ Hz for $|\theta_h| = 90°$, to $fmax = 330$ Hz for $|\theta_h| = 0°$. If $\Delta Z < 0$, then the vowel [o] is generated. If $\Delta Z > 0$, then the vowel [a] is generated.

Unlike previous experiments, the maximum pitch was set at 330Hz rather than 440Hz. This limit was set to allow for vowel recognition with $DVH_{par}$.

For both metaphors, a mild white noise is generated and superimposed on the main auditory stream when $\theta_h < 3°$ and $\Delta Z < r$ (target radius). A buzzer sound replaces all other sounds if the finger (point P) overshoots the target.

Additional audible signals are used to indicate when a trial has started, when a target has been reached, and when the time limit has expired.

### 4.2.4 Protocol

The experiment lasted approximately 45 minutes. Participants completed 3 blocks of 12 trials for each metaphor, one block for each target size. Within a block, there were 4 trials for each distance to the starting point, in random order. The participants completed the three blocks first for one metaphor and then for the other metaphor. The order in which the metaphors were performed was counterbalanced between the participants. All participants started with the block with the biggest targets, then the medium targets, and then the smallest targets. The test phase was preceded by a training phase in which participants had to reach the targets in the same way as during the test phase. The training lasted as long as the participant needed to become comfortable with the metaphors.

For each block, the participant sat in a chair, with the start button placed on a stool in front of them. They pressed the button to start the trial. Audible feedback was triggered and continued until the participant reached the target, or the 60-second time limit for reaching the target had elapsed. The participant pressed the button again to start the next trial, until the end of the block of targets. At the end of the six tests blocks, the participant answered a questionnaire identical to that of the second study.

<table><caption>**Table 7** Results of the linear regression analysis for $DVH_{par}$ and $DVH_{seq}$ metaphors.</caption><thead><tr><th>Metaphor</th><th>Regression equation</th><th>R<sup>2</sup></th><th>F</th><th>p</th><th>TP (bits/s)</th></tr></thead><tbody><tr><td><i>DVH<sub>par</sub></i></td><td>1.64(±0.43) + 4.52(±0.97)ID</td><td>0.68</td><td>14.75</td><td>0.006</td><td>0.61</td></tr><tr><td><i>DVH<sub>seq</sub></i></td><td>1.45(±0.51) + 4.49(±1.16)ID</td><td>0.54</td><td>8.06</td><td>0.025</td><td>0.69</td></tr></tbody></table>
<table><caption>**Table 8** Average target-reaching time for each condition of *DVH*<sub>par</sub> and *DVH*<sub>seq</sub>.</caption><thead><tr><th rowspan="2">A (cm)</th><th rowspan="2">W (cm)</th><th rowspan="2">ID (bits)</th><th colspan="2">MT (s)</th></tr><tr><th>par</th><th>seq</th></tr></thead><tbody><tr><td>12</td><td>5.33</td><td>2.17</td><td>9.13</td><td>8.78</td></tr><tr><td>12</td><td>8</td><td>1.58</td><td>7.30</td><td>7.73</td></tr><tr><td>12</td><td>12</td><td>1.00</td><td>6.07</td><td>6.20</td></tr><tr><td>18</td><td>5.33</td><td>2.75</td><td>8.28</td><td>7.38</td></tr><tr><td>18</td><td>8</td><td>2.17</td><td>6.99</td><td>6.51</td></tr><tr><td>18</td><td>12</td><td>1.58</td><td>6.68</td><td>5.58</td></tr><tr><td>27</td><td>5.33</td><td>3.34</td><td>10.42</td><td>10.20</td></tr><tr><td>27</td><td>8</td><td>2.75</td><td>8.51</td><td>8.05</td></tr><tr><td>27</td><td>12</td><td>2.17</td><td>9.37</td><td>8.20</td></tr></tbody></table>

### 4.2.5 Analysis

The analyses were performed using RStudio (RStudio 2023.12.1+402) and Jamovi (Jamovi 2.3.29).

Outliers were removed: reaching times greater than 1.5 times the interquartile range were excluded from the analyses, for all participants.

As for the previous study, a Cox analysis was conducted to compare reaching times between metaphors, as well as regression analysis and an ordinal regression to compare the scores given to each metaphor.

## 4.3 Results

### 4.3.1 Reaching Times

The average reaching time was 8.09 ± 5.27 s for metaphor *DVH*<sub>par</sub> and 7.62 ± 4.98 s for metaphor *DVH*<sub>seq</sub>. A Cox analysis showed no significant differences between the two metaphors (z = 1.19, p = 0.23).

Five IDs were calculated, and a linear regression between reaching time (MT) and the index of difficulty (ID) was performed for each metaphor. MT was averaged for each condition of size and distance to the target, resulting in 9 conditions (see Table 8). The model fit was confirmed with significant correlations between ID and MT for each metaphor (see Table 7). The calculated TP for each sound metaphor is reported in Table 7.

### 4.3.2 Qualitative metric

We ran an ordinal regression to compare scores by metaphor, which revealed a better score for *DVH*<sub>seq</sub> for the efficiency criterion (z = -2.25, p = 0.025). There was no significant difference between the two metaphors for the ease and comfort criteria.

## 4.4 Discussion

We compared two metaphors: one dissociating vertical and horizontal dimensions both spatially and temporally (*DVH*<sub>seq</sub>), and one dissociating dimensions only spatially (*DVH*<sub>par</sub>). We expected the sequential metaphor to perform better, based on previous studies suggesting
that searching for the target on one dimension at a time is more efficient. This should reduce
cognitive load and improve target reach time.

The results showed a preference for DVH<sub>seq</sub>, which was considered more efficient. Participants felt the parallel metaphor required complicated “cognitive gymnastics” to interpret all the sounds at once. The sequential metaphor allowed participants to focus on one dimension at a time, making it easier to learn.

However, this preference did not emerge significantly in terms of reaching times. TP were also very close for both metaphors. Of the 10 participants, 2 clearly stated that they preferred the parallel metaphor because, by interpreting all the sounds at the same time, it allowed a direct trajectory towards the target, whereas the sequential metaphor forced the participant to work dimension by dimension.

These results highlight a divide: most participants found the parallel metaphor more cognitively demanding, requiring them to interpret more information at once, while a minority benefited from its precision for a more direct target reach.

These findings emphasize the balance between simplicity and precision in designing effi-
cient and comfortable devices. The parallel metaphor may be better suited for experienced
users, while the sequential metaphor may be more suitable for beginners.

4.5 Limitations and suggestions for improvement

In our last study, participants sometimes struggled to distinguish the two vowels indicating target height. This encoding required the longest learning time, as the vowels were synthesized by computer and sounded somewhat unnatural. Participants occasionally had to overshoot the target to differentiate them. However, they all succeeded in finding the correct height. In the 1D study, some confused the *Formant* condition with the *Discrete* condition, where two pitches indicated whether to move up or down. This was because they perceived [o] as lower-pitched than [a] and assumed it signaled downward movement. A potential improvement would be to merge the Discrete and Formant metaphors by adjusting vowel pitch—using a higher [o] for upward movement and a lower [a] for downward movement.

In our third study, we compared a parallel and a sequential metaphor for horizontal and
vertical dimensions, but the task was in 3D. Distance was encoded by intensity in both con-
ditions and was not sequenced. Future studies could explore sequencing all three dimensions,
for instance, using tempo to encode distance.

Finally, as pointed out by [50], when working on auditory displays, it is necessary to
carry out tests outside the controlled laboratory environment, in more realistic contexts. It is
important to design a coherent sound ecosystem that takes into account integration with envi-
ronmental noise. If the environment is noisy, the sound parameters need to be adapted to avoid
confusion or cognitive overload. This supports our argument for transmitting only the amount
of information needed for the task, when it is needed, rather than replacing the eye. While
our system was tested in a controlled virtual environment, our goal is real-world applica-
tion. These sound metaphors could be integrated into a smartphone app for blind and visually
impaired users, helping them locate objects (e.g., an intercom or a supermarket product) using
the phone's camera for guidance. We also foresee applications in gaming and sports, both in
virtual and outdoor settings—for example, in laser-run events combining running and laser
pistol shooting.
# 5 Conclusion

This paper investigated the effectiveness of different sound metaphors for guiding visually impaired individuals toward virtual 3D targets. A first study showed that spatially dissociating the vertical and horizontal dimensions (DVH metaphor) resulted in faster target acquisition compared to a unified metaphor (UVH). The second study refined auditory parameters using 1D comparisons, finding that binary vowels for height and a combination of continuous pitch and binary stereo for horizontal direction provided the best performance. These findings guided subsequent 3D comparisons.

In the final study, we compared sequential (DVH<sub>seq</sub>) and parallel (DVH<sub>par</sub>) spatial dissociation. While DVH<sub>seq</sub> did not significantly reduce reach time, participants preferred it due to its simplicity, demonstrating a trade-off between information overload and movement efficiency. Our results show a dichotomy: most participants preferred the sequential metaphor, while a minority benefited from the parallel metaphor's simultaneous information for more direct movements. This suggests that the ideal metaphor may depend on the user's experience.

Fitts' law and the performance metric TP confirmed our findings, improving comparability with other studies.

A well-designed auditory metaphor provides the necessary metrics to accomplish the task without overwhelming the user with unnecessary information. This explains the advantage of a sequential metaphor, which reduces cognitive load by allowing users to process one dimension at a time. This approach aligns with verbal instruction-based guidance strategies, which are often more intuitive and natural than sonification.

In conclusion, our research highlights the need for customizable SSDs that cater to diverse user preferences and cognitive strategies. Optimizing auditory metaphors should take advantage of both quantitative metrics and user feedback, leading to more effective and user-friendly assistive technologies.

# Declarations

## Competing Interests

On behalf of all authors, the corresponding author states that there is no conflict of interest.

# Funding

This work was supported by the Agence Nationale de la Recherche (ANR-21CE33-0011-01).

This work has been partially supported by ROBOTEX 2.0 (Grants ROBOTEX ANR-10-EQPX-44-01 and TIRREX ANR-21-ESRE-0015) funded by the French program Investissements d'avenir.

## Author Contribution

All authors contributed to the study conception and design. The first draft of the manuscript was written by Coline Fons and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.
**Research Involving Human**

This work was authorized by the ethical committee CER Grenoble Alpes (Avis-2018-06-19-1).

**Informed Consent**

Informed consent was obtained from all individual participants included in the study.

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