We know then for the most part speakers push air. They radiate on a manner that is perpendicular to the surface. Cone speakers generally describe a three dimensional 180 degree arc. Flat panels radiate a flat 180 degree surface. They both propagate waves.
It is generally accepted that wide dispersion is better. At least one person disagrees
If you listen in stereo with both speakers pointed into the room you are listening as mu h as 45 degrees off axis. You can of course tilt the speaker directly at your ear.
How do we describe a sweet spot?
How dramatic is the fall pff as we move away from the point where the speaker sounds best?
Is there a trade off between the ultimate quality of the sweet spot and its width?
.Can we automatically assume the sweet spot is achieved with the tweeter pointed directly at our ear at ear level?
I'm not so sure there's as much of a direct correlation between a speaker's dispersion and size of the sweet spot or even much relevance to speaker dispersion as some may think. Especially since the speaker is not a music instrument but rather an electrical to mechanical sound converter.
That said, I'd like to offer two thoughts:
1.
You overlook far and away the most important aspect of speaker dispersion and that's the percentage of music info retrieved from a given recording that remains audible at the speaker due to a much lowered noise floor vs that which remains inaudible below a much raised noise floor. Accumulated distortions of course determining the noise floor level.
If we take into consideration a potentially high percentage of quantity of music info remaining inaudible at the speaker, then every discussion, speaker design, philosophy, theory, and research project and findings about speaker dispersion, including those of Floyd Toole, is potentially severely compromised, inconclusive, and incomplete.
It's probably not much different than bantering or measuring a Formula 1 car's performance using 84 octane fuel when far superior fuels are readily available that will dramatically change the F1's overall performance around the track. IOW, dramatically increased percentage of music info remaining audible at the speaker doesn't just dramatically impact the dispersion of the sound but also every aspect of what we hear while also greatly impacting the significance of other aspects like room acoustic anomalies.
And the more limited the percentage of music info remaining audible at the speaker the more limited its dispersion ability. The more limited a speaker's ability to disperse sound due to lack of quantity of music info, the more beamy, flat, lifeless, and fatiguing the sound at the ear.
2.
Theoretically, I suspect a second reason a speaker's ability to disperse sound may be overrated. To one degree or another all instruments on a soundstage are omni-directional but certainly some more than others. Take the strike of a snare drum at a live performance which disperses sound 360 degrees. A small percentage of this dispersion travels straight toward the recording mic's (or audience) i.e. direct sound. Simultaneously, much of the remainder is directed toward and interacting with soundstage boundaries while much of the sound is still traveling throughout the air within the soundstage interacting with other instruments' notes are generating a collective whole within the soundstage i.e. indirect sound and ambient info. Well-positioned recording mic's seem to easily capture the bulk of the direct and indirect sounds and are transferred to the recording medium.
If (and it is so) much of the direct and indirect sound of the live performance is captured at the mic's and to the recording, then why might a speaker's sound dispersion seem so important a topic to some?
Hopefully few would argue that the lowest of low-level detail is the volumes and volumes of a live performance's ambient info. This is what makes a live performance so exhilarating and believable and the same goes for playback performance. However, when distortions enter into the playback chain, (think everybody to one degree or another), the playback system's noise floor is raised and the very first thing to become inaudible at the speaker is the volumes of ambient info. And of course other low-level indirect detail will follow suit depending on the severity of a playback system's noise floor.
That said, if (and it is so) the vast majority of direct and indirect sounds of a live performance are indeed embedded within a given recording but due to playback equipment-induced distortions much of the indirect sound remains inaudible at the speaker what are we left listening to? Seems we're listening only to great percentage of the direct sound very little of the volumes of indirect low-level sound. In such a case, do words like fatiguing, flat, lifeless come to mind? Or maybe beamy, distorted, half-baked notes, making a bee-line for the ear? Or maybe phrases like me-too, mainstream, or hi-fi sound come to mind.
In other words, if well-placed mic's at a live performance are indeed capturing the vast majority of direct and volumes of indirect sound and it's embedded in the recording and for argument's sake setting aside all playback distortions (lowest possible noise floor), shouldn't a speaker's dispersion of sound becomes less important? In such a hypothetical, won't we hear every direct and indirect sound captured at the mic and embedded in the recording?
But the reality is that we're all dealing with distortions, much raised noise floors, and me-too hi-fi sounds to one degree or another and we're all left wanting.
The point being, with very very little indirect / ambient info remaining audible at the speaker we look to introduce substitutions for missing music info i.e. sound that we know should be there. For example. We to look at room acoustics and room reflections because there's little ambient audible at the speakers. Others entertain multi-channel while others entertain omni-directional speakers (perhaps a clever means of incorporating multi-channel into 2 speakers?) to somehow make up for this lack of ambient info that defines the soundstage and live music. All of which fail because the same indirect / ambient info missing before is still missing now becuase the technology to replace the indirect / ambient info missing at the speaker has not yet been invented.
Anyway, not sure if I explained it very well but those are my suspicions why the importance of a speaker's dispersion ability is overrated as well as room acoustic anomalies, multi-channel, and perhaps radial speakers too. Because in every one of these cases we're dealing with the effects rather than the cause. And as usual, dealing with the effects always results in paltry benefits at far greater frustration and costs than dealing with the cause. But the main point is, when far more of the music info embedded in a recording remains audible at the speaker, dispersion becomes essentially a non-issue. Then again so does room acoustic anomalies, multi-channel, omni-directional speakers, etc.