Sing a Song of Spectrum

Is 100 MHz of radio spectrum a lot? It depends on whether it’s at 200 MHz (say, 200-300 MHz) where it’s a significant chunk of the entire band, or at 20 GHz (say, 20.0-20.2 GHz) where it’s just a sliver. Musicians don’t encounter this puzzle because they express intervals in terms of frequency ratios. Thus, a soprano can sing the same song as a bass, though at a much higher pitch; the same notes but transposed upwards. 

When we think about spectrum allocations, we shouldn’t think about raw hertz, but rather about the “song” that a given number of hertz allows an operator “to sing.” (H/t Nathan Simington, Ofcom/William Webb, and SPT; see endnote [1].)

Expressing ratios in music and engineering

Musical intervals

Singers, from sopranos down to basses, all have approximately the same vocal range in terms of the number of notes they can cover. This means that a soprano can sing the same song as a bass but does so a higher pitch. [2]

Notes in a musical scale are constructed to reflect this. Musical intervals are expressed as frequency ratios; for example, an octave doubles the pitch.

An octave is divided into 12 semitones, where the frequency of each semitone is 2^(¹/₁₂) times that of the preceding semitone. [3]
(“2^(¹/₁₂)” means 2 to the one-twelfth power, aka the twelfth root of 2.) 

Let’s start with a note of some frequency f₀. If f₁ is the frequency of the note one semitone higher than f₀, followed by f₂, f₃, f₂, … up to the note f₁₂, the note an octave above f₀, we have:

f₁ = 2^(¹/₁₂) * f₀

f₂ = 2^(¹/₁₂) * f₁ = 2^(¹/₁₂) * 2^(¹/₁₂) * f₀ = 2^(²/₁₂) * f₀

…

f₁₂ = 2^(¹²/₁₂) * f₀ = 2 * f₀

So: the frequency of f₁₂ is double that of f₀, as intended.

Put another way, the frequency ratio between two notes k semitones apart, f_n and f_n+k, is given by:

f_n+k/f_n = 2^(^k/₁₂)

Taking logarithm base 2 on both sides gives log₂(f_n+k/f_n) = ^k/₁₂, or k = 12 * log₂(f_n+k/f_n). Generalizing, frequencies f_b and f_a are k semitones apart if 

k (semitones) = 12 * log₂(f_b/f_a)   .   .   .   .   .   eq. (1)

Engineering ratios

This should start looking familiar to spectrum engineers. Recall the definition of the unit "decibel" (symbol dB). A value in bels is the base-10 logarithm of the ratio between a measured quantity and a reference quantity; for example, one bel reflects a 10:1 ratio. A decibel is one-tenth of a bel, so a value in decibels is 10 times the base-10 logarithm of the ratio between two quantities. (https://en.wikipedia.org/wiki/Decibel)

For example, the power P in watts can be expressed in decibels relative to a power of 1 watt as:

P (dBW) = 10 * log₁₀( P (watt) )

By analogy, let’s define dBHz (decibel hertz) as a unit of frequency used to indicate that a frequency is expressed in decibels (dB) with reference to one hertz (Hz). For example, we write frequency f in Hz in dBHz as 

f (dBHz) = 10 * log₁₀( f (Hz) )

By analogy to music, we could describe radio frequency intervals in terms of “dBtones,” a synonum for dBHz

k (dBtones) = 10 * log₁₀(f_b/f_a)   .   .   .   .   .   eq. (2)

To paraphrase

• Eq. (1) says that increasing a musical pitch by 12 semitones results in a frequency increasing by 2x, i.e. doubling (an octave).
• Eq. (2) says that increasing a frequency by 10 dBtones results in the frequency increasing by 10x (a decade).

In music theory, a semitone is defined to be 100 “cents”, so that an interval in cents is given by

k (cents) = 1200 * log₂(f_b/f_a)   .   .   .   .   .   eq. (3)

Similarly, we could write a radio frequency interval in terms of dBcents as 

k (dBcents) = 1000 * log₁₀ (f_b/f_a)   .   .   .   .   .   eq. (4)

Since a decibel is 100 millibels, dBcents intervals are the same as mBHz (millibel.hertz) intervals.

To sum up, music (to be precise, Western music in equal temperament) and spectrum engineering both use logarithmic scales.

Musical Range and Spectrum Bandwidth

Frequency fraction for a range of notes

Let’s define an interval δf between frequencies f_a and f_b as

δf := f_b – f_a   .   .   .   .   .   eq. (5)

Consider a range of N notes, starting with frequency f0 and ending with frequency f_N. By definition, 

f_N = f₀ * (2^(¹/₁₂))^N   .   .   .   .   .   eq. (6)

For this interval, then 

δf /f = (f_N – f₀) / f₀= (2^(¹/₁₂))^N – 1   .   .   .   .   .   eq. (7)

So: no matter what frequency you start with, N notes always have the same δf /f. 

Bandwidth in dBHz

Eq. (7) shows that Ofcom’s spectrum weighting is equivalent to the logarithmic intervals in a musical scale, and that it is essentially a ^δf/_f (or equivalently, a logarithmic) metric. [4] Let’s explore expressing spectrum intervals logarithmically a little further.

The interval δf defined in eq. (5) could be called the (band)width of a spectrum band with start and end frequencies of f_a and f_b. It’s measured in a frequency unit like Hz or, more typically in spectrum, MHz or GHz. If “BW” denotes bandwith, then the bandwidth in Hertz is 

BW(Hz) = δf 

But we can also write it as a fraction of the characteristic frequency of the band (the center frequency, say) that we’ll call f. 

BW(fraction) = ^δf/_f, where f is roughly equal to f_a and f_b.

Let’s define 

BW(dBHz) := dBHz(f_b) – dBHz(f_a) 

BW(dBHz) = 10 * log₁₀( f_b / f_a ) = 10 * log₁₀ ( (f_a + δf)/f_a ) = 10 * log₁₀( 1 + ^δf/f_a )   

But since f ≈ f_a we can write this

BW(dBHz) = 10 * log₁₀( 1 + ^δf/_f )   .   .   .   .   .   eq. (7)

Now recall the Taylor expansion of the natural logarithm of 1 + x, where x << 1, and the rule for converting logarithm bases, respectively:

ln(1+x) = x - x2/2 + x3/3 - …, so that ln(1+x) ≈ x for x << 1

log_a(b) = log_c(b) / log_c(a)

Setting x = ^δf/_f and c = 10 and assuming ^δf/_f << 1 gives a useful approximation:

BW(dBHz) ≈ (10 / ln(10)) * (^δf/_f) ≈ 4 * (^δf/_f)

So What?

The same songs in different spectrum bands

As an example of very different MHz bandwidths at different frequencies that amount to the same “song” being sung, consider some bands auctioned in FCC auctions. 

Here is a selection of broadband spectrum license auctions since 1995 that illustrate how the bandwidth of spectrum licenses increased. 

Auction	Description	nominal f (MHz)	Sub-block BW (MHz)	BW (dBHz)	BW (mBHz)	BWs offered
73	700 MHz	700	6	0.04	3.7	6, 10, 12, 22
35	Bbd PCS, C & F block	1,950	15	0.03	3.3	10, 15, 30
107	3.7 GHz	3,700	20	0.02	2.3	20
103	Upper 37 GHz	38,000	100	0.01	1.1	100
103	47 GHz	48,000	100	0.01	0.9	100

Note:

• “BW” is the bandwidth of license sub-blocks auctioned; since auctions often offered a variety of license bandwidths, we note the rest in the “BWs offered” column.
• I’ve given the logarithmic bandwidth, equivalent to the musical interval, in both dBHz and mBHz; they simply differ by a factor of 100 – just like a musical semitone is 100 cents – but mBHz is a little more intuitive since the significant figures are before the decimal point.
• I have chosen the auctions and bandwidths to make the point. Some of the lower bands offered quite wide license bandwidths, and overall, bandwidth hasn’t increased as quickly as band frequency.

Observe that even though the bandwidths vary dramatically, from 6 MHz to 100 MHz, the logarithmic measures (dBHz and mBHz) don’t increase as much and are all the same order of magnitude. Thus, the “songs being sung” in all these bands use roughly the same number of “notes.”

A logarithmic alternative to $ per dBHz-POP

Using a logarithmic bandwidth metric might also make tracking auction proceeds across widely separated bands more sensible.

The customary metric, of course, is $ per MHz-POP. Everything else being equal, this number goes down as the frequency of a bands go up, since the MHz on offer increases at higher frequencies. Thus, it’s hard to compare auction outcomes in very different bands. (It’s already hard to compare outcomes, given that every band has different encumbrances, but still.)

If our intuition is correct that a logarithmic bandwidth measure is better than a linear one, a better metric would be $ per dBHz-POP. It would change less as one went to higher frequency bands on the assumption that the “number of notes in the song” remained roughly constant.

I’m currently trying to find numbers to check if this hypothesis holds up. There are many factors that go into auction prices; frequency is one of them, but it can be swamped by other considerations, as in the recent 3.45 and 3.7 GHz auctions. (H/t Coleman Bazelon for helping me reduce my ignorance a little in this area.)

Coda

After a reviewing a draft, SPT asked an excellent question: “Why should one expect a spectrum band to be like a song?”

I don’t have a good answer. Saying that it feels intuitively right is insufficient in these rationalist times. Part of a reasoned answer might be that both music and radio communicate using vibrating waves that cover many orders of magnitude in frequency (octaves or decades, respectively). Since musical notation uses a logarithmic scale to ensure that songs in different ranges use the same number of notes, one might expect something like that to be useful in radio, too. Another part of the answer is empirical: we see service bands getting wider as radio frequencies increase, and a logarithmic scale is an economical way to represent that. (And an intuitive one to spectrum geeks, since they already use dB all over the place.)

I’ve pointed out that music uses octaves (2x increases) and engineering uses decades (10x increases). Equal temperament uses base-2 logarithms, and the decibel (dB) scale uses base-10 logarithms. Current FCC allocations range from 3.8 kHz to 275 GHz (table download, revised June 28, 2021); that’s seven decimal orders of magnitude. Most pianos have 88 keys and span just over seven octaves. Coincidence? [5]

Endnotes

[1] This thinking was prompted by Nathan Simington asking how one explains that a MHz in one band isn’t equal to a MHz in another band. I was reminded that Ofcom, the UK spectrum regulator, talked about “weighted use of the spectrum” back in 2005; it divided the width of a band by the middle frequency of a band (section 2.4 of the 2005 Spectrum Framework Review, see Figure 2.1). I wasn’t surprised to hear that this was another of William Webb’s genius ideas. (I’m still hoping to see Spectrum Usage Rights, aka SURs, implemented; they were one of the inspirations for harm claim thresholds.)

Simington suggested a musical analogy, noting that the perceived change in pitch of a fixed number of Hz differs immensely depending on the frequency of the reference tone. For example, a deviation of 10 Hz from A2 (the bottom A on the staff in bass clef, frequency 110 Hz) is almost exactly 1.5 equal-tempered semitones, a very obvious pitch change even to someone with no musical training. However, for A6 (two octaves above the orchestra tuning pitch, frequency 1760 Hz) a deviation of 10 Hz upward is only ¹/₁₅th of a semitone and is imperceptible as a musical interval to even a trained ear. Thanks also to SPT for reviewing and commenting on a draft.

[2] For example, here are the vocal ranges according to The New Harvard Dictionary of Music; note names are given in Scientific Pitch Notation (SPN)

Voice	Note range	Range (semitones)	Start (Hz)	End (Hz)	Range (Hz)
Soprano	C4 to A5	21	261.63	880.00	618.4
Mezzo soprano	A3 to F5	20	220.00	698.46	478.5
Alto	F3 to D5	21	174.61	587.33	412.7
Tenor	B2 to G4	20	123.47	392.00	268.5
Baritone	G2 to E4	21	98.00	329.63	231.6
Bass	E2 to C4	20	82.41	261.63	179.2

Note that a soprano’s range is more than three times that of a bass’s when measured in Hz, but both are 21 semitones. [See endnote 3]

[3] Technically, these are equal-tempered semitones, but I won’t get into the delightful weeds of tuning systems.)

[4] Cf. Ofcom’s requirement that a 1 MHz allocation at 100 MHz be given equal weighting to a 10 MHz allocation at 1 GHz in band (section 2.4 of the 2005 Spectrum Framework Review). Since the number of cents (n) in an interval (f_a, f_b) is given by

n = 1200 * log₂(f_b/f_a)

one can easily check that both frequency intervals (100, 101) MHz and (1000, 1010) MHz are 17.2 cents wide, in musical notation. In dBHz we get:

fa (MHz)	fb (MHz)	fa (dBHz)	fb (dBHz)	Diff (MHz)	Diff (dBHz)
100	101	50.00	50.04	1	0.04
1,000	1,010	60.00	60.04	10	0.04

[5] Coincidence? Yes and no. It is a coincidence since the frequency range of the allocation table keeps growing as higher frequencies become accessible due to technology advance; five or 10 years from now there will probably be allocations well beyond 275 GHz (the current unallocated tranche is 275 – 3000 GHz), so that it’ll be more than seven orders of magnitude. On the other hand, I suspect that the “focus frequency range” for radio services – those bands that generate, say, 80% of the revenues (I wish I knew where to find this data) – are a sliding window that keeps moving up. In the early 20th century, radio spectrum was considered (Wikipedia) to consist of longwave, medium-wave, and short-wave radio bands, that is 30 kHz to 30 MHz (three orders of magnitude). Today I’d guess the sweet spot is 30 MHz to 30 GHz (three orders of magnitude again, though I suspect 300 MHz would be a more realistic better low end; and there is some activity in the 70/80/90 GHz band).