First, the wrong way: forward maximum summation using geometric mean. ¼c/(√(F1 * F2))

It works, but it’s not the most efficient method.

After a discussion with Merlijn van Veen I learned to space the gradient subwoofer array (commonly known as a cardioid array) using arithmetic mean to match the region of greatest summation to the operating range.

Now, the right way: forward maximum summation across the entire operating range.

1. Find the center of the operating range using arithmetic average. eg (F1 + F2) / 2
2. Find the wavelength. eg. λ = c / Fc
3. Find one quarter. eg. spacing = ¼λ

For more on different kinds of averaging, please see Know Your Audio Analyzer Averages.

geometric mean: the nth root of the product of n numbers.

Wikipedia

arithmetic mean: the average of a set of numerical values, calculated by adding them together and dividing by the number of terms in the set.

Oxford Languages

Example

Nexo RS15

“Hey Siri, what’s the average of 35 and 100?”

It’s 67.5.

“Hey Siri, what’s the wavelength of 67.5 Hz?”

It’s 4441 km.

Hmmm, that’s not very helpful.

“Hey Siri, what’s 345 divided by 67.5?”

It’s about 5.11.

“What’s a quarter of 5.11?”

It’s about 1.277.

Ok, let’s try using a spacing of 1.27m in Merlijn’s Subwoofer Array Designer to see if we can fit the operating range in between the preferred filters (yellow triangles), which designate 3dB of summation.

Bingo.

Here’s an excerpt from my recent workshop: Follow the Sound System Tuning Roadmap

More Questions

What about the aim and spacing between gradient arrays?

You could space two gradient arrays so that they meet in the middle of the audience at their off-axis (-6dB) points, but you risk introducing further asymmetry into the design my moving the arrays out from under their respective main arrays. Instead, I would prefer to leave the left and right subwoofer arrays underneath left and right main arrays, but aim the sub array out until off-axis left (OFFAXL) matches crossover left-right (XLR) in the center.

Here’s a prediction with effectively the same result, but achieved through spacing and then aiming. Only one sub array is on to make the result more clear.

Here’s a top view showing both arrays on together, demonstrating reduced interaction and power alley in the middle.

Alternatively, if real estate for rotating the array is not available, you could rotate the alignment position in the rear in towards the stage until the desired aim in the front is achieved.

How do you space an inverted gradient stack in landscape mode?

Don’t space them evenly across the front of the stage. You may be unnecessarily lengthening the line and narrowing the coverage.

Do leave 6 inches between enclosures to improve efficiency. For more, see this article.

Why do you polarity invert the rear sub in a cardioid array?

Do you have other questions about gradient arrays? Let me know!