Talk:Frequency

Hertz

@Boppennoppy: Hi, see it implicitly means 1 event, and 1 here is full of meaning, but (number of) event has no dimension, because it is a number. So Hertz not means "one per second", but it means "one event per second", i.e., one (or 1) here, is full of meaning. Hertz does not mean 1 "kilometer" per second or one "kilogram" per second or anything else, it means one "event" per second. Hooman Mallahzadeh (talk) 16:14, 30 December 2021 (UTC)

One "what" per second?

Many people, when they read "one per second", have a natural tendency to ask (themselves), "one what per second?" And the associated tendency is to insert some general term for a periodic event such as "cycle". But consider a digital readout on a clock showing the number of seconds passing since some starting time. The number on the display is increasing at a rate of ONE per second—which makes perfectly good sense. Since (temporal) frequency is the quotient of the number of periodic events and the corresponding elapsed time, the dimension is number/time, 1/T, where "1" is the symbol for the dimension number, to be shown in the same special font (e.g. Helvetica) as that used for other dimension symbols. [It is not the numeral 1.] The SI (implicit) "unit" for (the quantity) number is one, 1. For time, it is second, s. So the SI coherent unit for frequency is 1/s or "one per second", also written as s^–1. This refers to a periodic event, so one per second gets a special name: hertz, symbol Hz. One hertz is one per second: Hz = 1/s. Some authors have gone further and equated "cycle" to "revolution" (of some imaginary rotating body). Thus hertz becomes "cycles per second", which then becomes "revolutions per second", which then becomes "2π radians per second"—a unit for angular velocity. Since, in the SI, the hertz is one per second and rad/s is also one per second, the same authors call this the "2π problem". To "resolve" this dilemma, they have proposed officially redefining the hertz as 2π rad/s (!).

Any quantity that is of the nature of a time-varying number (for example, a varying strain in a tensile test, or a torque-ratio in a continuously varying transmission, or the phase of a time-varying sinusoid), has a time derivative (strain-rate, torque-ratio-rate, phase-rate) with the dimension of number/time. The appropriate coherent unit for this time rate of change is one per second, 1/s, usually spoken as "per second". [Of course, other time units can be used: per hour, per year, . . . .] There are two special cases, the hertz, Hz, and the becquerel, Bq.

By the way, a similar "problem" occurs with quantities like number density: number per volume. Some people will ask (themselves) "number of what per volume?" And will have an urge to insert some "downstream-from-the-SI" symbol such as mcl (molecule), pcl (particle), or some other descriptive symbol—analogous to inserting cyl (cycle) in the case of frequency. If we are dealing with number density, the correct coherent unit is 1/m³, "one per cubic metre", or m^–3 (per cubic metre). On the other hand, if we are dealing with amount density, the appropriate unit is "entity per cubic metre", ent/m³ or ent m^–3, where "ent" stands for one entity, the smallest amount of any substance—which should be recognised as the appropriate atomic-scale unit for amount, paralleling the dalton for mass. The unit ent m^–3, can be changed to mol m^–3, since one mole is exactly 6.02214076 x 10²³ ent. The Avogadro constant (not number) is exactly 1/ent, one per entity. The Avogadro number is (very very nearly) equal to the quotient g/Da, gram per dalton (to within an order of 10^–10). A mole is (very very nearly) an aggregate of g/Da entities: mol ≈ (g/Da) ent. So dalton per entity, Da/ent, the appropriate atomic-scale unit for molar mass, is (for all practical purposes) equal to g/mol = kg/kmol. Boppennoppy (talk) 14:56, 17 November 2024 (UTC)

Is there something about the article that you want to change? Constant314 (talk) 16:04, 17 November 2024 (UTC)

Good question. On my first reading, it seemed like a waste of time. So I tried again, which just reinforced that conclusion. I don't know if there is an actual frustration, or a constructive criticism, or testing the waters for a new article, or checking for agreement/disagreement, or something else. After reading the subject line, I was expecting a discomfort (which I would agree) with against dimensionless quantities in the SI coherent units convention. E.g. there is no distinction between cycles/sec and samples/sec... they are both just hertz. Maybe that is handy for abstract theorems and proofs, but in my world, that lack of detail makes it hard or at least error-prone to interpret those results back to the physical world and the reassurances of dimensional analysis.

--Bob K (talk) 20:39, 18 November 2024 (UTC)

Thank you Bob K and Constant314 for your careful reading of my note. The main point I was trying to make is that the coherent SI unit for frequency is one per second—not one "cycle" per second (or one "sample" per second)—sometimes written as 1/s or s^–1 (the latter called a reciprocal second)—and, since frequency refers to periodic events, one per second is given the special name hertz and symbol Hz. A "cycle" (or "sample") is not a unit. This is the main reason why the International Electrotechnical Commission introduced hertz (in the mid 1930s)—to get away from the colloquial term "cycles per second". And why the SI adopted it about a couple of decades later. Unfortunately, the older term still persists and is to be found in almost all dictionaries and online tutorials as the "definition" of hertz. It does not appear in the current (9th edition of the) BIPM Brochure. [There was quite a fuss about removing it from language in previous editions.] Fortunately, hertz is not called "cycle per second" in this Wikipedia article on frequency. My note is more of a warning to keep it that way.

There is, however, a problem associated with the term "angular frequency" for ω in the expression sin(ωt). This is not a frequency; nor does it have anything to do with angles. It is the time rate of change of the dimensionless phase of the sinusoid, ωt , with dimension 1/T and unit 1/s (or s^–1) but not Hz, and certainly not "rad/s". Its appropriate name is phase rate. Stating its unit as "radian per second" confuses it with the angular velocity of some (non-existent) rotating body. However, this is an ongoing problem with the SI's treatment of angle (and solid angle), which seems unlikely to be remedied in the near future. Boppennoppy (talk) 22:59, 18 November 2024 (UTC)

I share your frustration, but all we do here is paraphrase reliable sources. You may want to read this: WP:RGW. Constant314 (talk) 01:45, 19 November 2024 (UTC)

Thank you for clarifying your intention, as a warning. I've read a few things to try to understand why it's so important to you:

• International_System_of_Units
• Dimensionless quantity
• Dimensional analysis
• Quantity calculus
• Conversion of units
• https://pmc.ncbi.nlm.nih.gov/articles/PMC7727271/
• https://pubmed.ncbi.nlm.nih.gov/38868451/

But it hasn't helped. For one thing, all those documents seem to avoid DSP, multi-rate DSP, and Normalized frequency (which is commonly used in DSP). Cycles/sec and samples/sec are completely different concepts, but I've had people insist they are both just Hz. And what about their ratio, cycles/sample? People have told me that cycles and samples are both dimensionless. So the unit is simply "one". To quote your own post, "one what?" Sorry if I'm being obtuse. But SI seems to be very controversial, so I am not alone. Can you offer better arguments or examples to help us understand?
--Bob K (talk) 20:47, 19 November 2024 (UTC)

Again, thank you for continuing this important discussion. The subject is frequency, but similar questions arise in a number of related fields. Dictionaries report on definitions according to common usage. Presumably, encyclopaedias go further and sometimes discuss controversies surrounding the definitions. One of the most controversial subjects concerns so-called dimensionless quantities, sometimes called "quantities of dimension one" or "quantities of dimension number". [The latter—naming the dimension "number" rather than "one"—is due to Michael Krystek, who proposed the symbol Z for for this dimension (German: Zahl). Whereas, in fact, the symbol 1 (not to be confused with the numeral 1) is probably a better choice: the identity element for dimensional analysis. The dimension symbol 1 is pronounced "one" but refers to the dimension "number", just as the dimension symbol T is pronounced "tee" but refers to the dimension "time", for example.]

Of course, many quantities have the dimension 1, occurring in two categories: continuous and discrete. Examples of the former include any nondimensionalized continuous physical variables, ratios of two (possibly varying) continuous variables with the same dimension (such as strain, for example: L/L = 1), arguments of transcendental functions. Discrete quantities of dimension number include anything that can (in principle) be counted (including continuous variables that have been digitised). An interesting quantity that is (in fact) discrete is the number of entities of a substance in a chemical reaction (that is changing in time). This cannot (usually) be counted directly and is such a large number that it makes sense to treat it as a continuous differentiable variable. This is, after all, the basic assumption of continuum mechanics. [Of course, the same is true of the mass of the substance because of the discrete nature of matter—but it is (usually) automatically treated as continuous without further thought.]

The unit for any quantity of dimension 1 is (the number) 1. If we are considering the number density of the above substance, this would be the quotient of the number of entities (dimension 1) and the volume of the container (dimension L³): dim[number density] = 1/L³ = L^–3. [This is a good example of the use of 1 as the symbol for the dimension number.] The appropriate unit is 1/m³ = m^–3. [Not the similarity with the above dimensions relationship.] The most appropriate name for this unit is "per cubic metre". For example, if we have 2.468 × 10²⁵ entities in a container of volume 2 m³, the number density is 1.234 × 10²⁵ m^–3, read as 1.234 × 10²⁵ per cubic metre. It is NOT 1.234 × 10²⁵ "entities" per cubic metre (or 1.234 × 10²⁵ ent/m³)—that would be an amount density: the amount of the substance in the container is 2.468 × 10²⁵ ent, where ent is the appropriate atomic-scale unit for amount (the smallest amount of any substance). The number of entities in the substance is 2.468 × 10²⁵. [One mole is (exactly) 6.02214076 × 10²³ ent.]

Similarly, frequency is the quotient of the number of periodic events occurring in a given time interval and that time interval. Its dimension is (number)/(time) = 1/T = T^–1. Its unit is 1/s = s^–1, best read as "per second". So, if I clap my hands 20 times in 5 seconds, the clap frequency is 20/(5 s) = 20/5 s^–1 = 4 s^–1, four per second, NOT "four claps per second"—a "clap" is not a unit, it is a name of the periodic event. For the same reason, this is not "four cycles per second"—a cycle is not a unit, it is a generic name of a periodic event. And, finally, "per second" is replaced by "hertz". So we write the clap frequency as 4 Hz ("four hertz"). [And, by the way, it is certainly not four revolutions per second (referring to some imaginary rotating body), as some authors would claim (by having the SI redefine the hertz as 2π rad/s—a unit of angular velocity).]

There are an unlimited number of quantities that have the dimension 1/T and unit 1/s: the time rate of change of any quantity of dimension 1 that is changing with time. The unit for the time rate of change of periodic events gets the special name and symbol hertz, Hz. The unit for the time rate of occurrences (on average) of aperiodic "radioactive decay" events gets the special name and symbol becquerel, Bq. All the rest (continuous or discrete) get stuck with 1/s = s^–1(per second). All of these can use an appropriate SI prefix.

As I mentioned in my initial note, when we read "four per second", the natural tendency is to ask (ourselves) "four WHAT per second"? The correct answer is: the number 4 per second. For frequency, this is the number of (periodic) events occurring per second—not a "description" of the events themselves per second.

All this is probably too far "in the weeds" to try to include any hint of in an encyclopaedia. But I hope it helps clarify where some of the confusion about "dimensionless quantities" arises. Boppennoppy (talk) 17:50, 20 November 2024 (UTC)

It's not that simple. It seems like semantics are getting in the way of the actual topic. Four "numbers" per second is not what happens in sound wave frequency, for instance. Sound wave frequency tracks the number of cycles of rarefaction and compression, usually referring to molecules in air. The frequency is composed of these cycles, not of an abstract number. Binksternet (talk) 19:40, 20 November 2024 (UTC)

Actually, it is that simple. It doesn't matter whether it's cycles of compression-and-rarefaction of air or widgets coming off a production line, frequency is defined as the NUMBER of (periodic) events divided by the TIME interval between those events. It has the dimension of (number)/(time) = 1/T. And is expressed in the base "number unit" (= 1) divided by a time unit, where the numerator is "the number of ones". The coherent SI unit for frequency is therefore 1/s, one per second. The corresponding period is defined as the TIME interval between events divided by the NUMBER of events occurring in that time interval. The period has the dimension of (time)/(number) = T/1 = T, and is expressed in a time unit divided by THE number unit (one), so that this quotient becomes just the time unit. [Any quantity can be multiplied or divided by 1 without changing its value.]

One of the problems causing some confusion is that there has never been an accepted symbol for the dimension number. By using 1, we can see all kinds of important relationships among dimensions. For example, if D is the symbol for a general dimension, we have: 1D = D1 = D/1 = D; 1/D = D^–1; D/D = D⁰ = 1. It may take a while to get used to. But an event of 20 claps in 5 s has a frequency of (number of claps)/(time interval) = (20)/(5 s) = 4/s (read as "four per second"), or 4 Hz. NOT "four claps per second". NOT "four cycles per second". [And DEFINITELY NOT "four revolutions per second" or "eight-pi radians per second"—as some authors would have us believe.]

The word "per" is from the Latin, meaning "for each". So, in the above example, the answer to "how many (what number of) claps for each time interval?" is:

"four for each second" = four per second = 4 Hz.

An athletic resting heart-rate is 60 beats occurring in one minute. The frequency is the number of beats divided by the time interval: (60)/(1 min) = 60/min (sixty per minute). Colloquially, this is universally called sixty "beats per minute" and written 60 BPM. But a "beat" is not a unit (of anything). The number of beats occurring in one second divided by one second would give us the heart-beat frequency expressed in hertz, in this case: (60)/(60 s) = 1/s = one per second = 1 Hz. Boppennoppy (talk) 22:29, 20 November 2024 (UTC)

We paraphrase reliable sources. I don't have a problem with samples=cycles=radians=nepers=events=1. Constant314 (talk) 23:33, 20 November 2024 (UTC)

As I have already said, I disagree with samples=cycles, and nothing written here has changed my mind. My specific question about cycles/sample remains unanswered. I haven't seen any actual examples of the supposed superiority of SI vs quantity calculus (a new term for me). So far SI reminds me of the English language, whose arbitrary rules and pronunciations were simply settled on by the necessity of settling on something... anything. Boppennoppy's "warning" seems no different to me that someone claiming French is better than English, or vice versa. I do again apologize if I'm being obtuse. Just callin' it like I see it.
--Bob K (talk) 21:55, 21 November 2024 (UTC)